title
listlengths
0
18
author
listlengths
0
4.41k
authoraffiliation
listlengths
0
6.45k
venue
listlengths
0
9
abstract
stringlengths
1
37.6k
doi
stringlengths
10
114
pdfurls
listlengths
1
3
corpusid
int64
158
259M
arxivid
stringlengths
9
16
pdfsha
stringlengths
40
40
text
stringlengths
66
715k
github_urls
listlengths
0
36
[ "Equivalence of the Generalized Zhang-Zhang Polynomial and the Generalized Cube Poly- nomial", "Equivalence of the Generalized Zhang-Zhang Polynomial and the Generalized Cube Poly- nomial" ]
[ "Petražigert Pleteršek " ]
[]
[]
In this paper we study the resonance graphs of benzenoid systems, tubulenes, and fullerenes. The resonance graph reflects the interactions between the Kekulé structures of a molecule. The equivalence of the Zhang-Zhang polynomial (which counts Clar covers) of the molecular graph and the cube polynomial (which counts hypercubes) of its resonance graph is known for all three families of molecular graphs.Instead of considering only interactions between 6-cycles (Clar covers), we also consider 10-cycles, which contribute to the resonance energy of a molecule as well. Therefore, we generalize the concepts of the Zhang-Zhang polynomial and the cube polynomial and prove the equality of these two polynomials.Mathematics Subject Classification (2010). Primary 05C31; Secondary 92E10.
null
[ "https://arxiv.org/pdf/1612.02986v1.pdf" ]
119,179,327
1612.02986
fac4a695034fbf35c81fd0b022cdbc22482cc972
Equivalence of the Generalized Zhang-Zhang Polynomial and the Generalized Cube Poly- nomial 9 Dec 2016 Petražigert Pleteršek Equivalence of the Generalized Zhang-Zhang Polynomial and the Generalized Cube Poly- nomial 9 Dec 2016Zhang-Zhang polynomialcube polynomialbenzenoid sys- temtubulenefullereneClar cover In this paper we study the resonance graphs of benzenoid systems, tubulenes, and fullerenes. The resonance graph reflects the interactions between the Kekulé structures of a molecule. The equivalence of the Zhang-Zhang polynomial (which counts Clar covers) of the molecular graph and the cube polynomial (which counts hypercubes) of its resonance graph is known for all three families of molecular graphs.Instead of considering only interactions between 6-cycles (Clar covers), we also consider 10-cycles, which contribute to the resonance energy of a molecule as well. Therefore, we generalize the concepts of the Zhang-Zhang polynomial and the cube polynomial and prove the equality of these two polynomials.Mathematics Subject Classification (2010). Primary 05C31; Secondary 92E10. Introduction A benzenoid system is determined with all the hexagons lying inside cycle C of the hexagonal lattice. They represent molecules called benzenoid hydrocarbons. These graphs are also known as the hexagonal systems and form one of the most extensively studied family of chemical graphs. For fundamental properties of benzenoid systems see [8], while some recent results can be found in [10,11,18,21]. If we embed benzenoid systems on a surface of a cylinder and join some edges we obtain structures called open-ended singlewalled carbon nanotubes also called tubulenes. Carbon nanotubes are carbon compounds with a cylindrical structure and they were first observed in 1991 [9]. If we close a carbon nanotube with two caps composed of pentagons and hexagons, we obtain a fullerene. More exactly, a fullerene is a molecule of carbon in the form of a hollow sphere, ellipsoid, tube, or many other shapes. The first fullerene molecule was discovered 30 years ago. In graph theory, a fullerene is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. The overview of some results on fullerene graphs can be found in [1]. Papers [6,17] present a sample of recent investigations. The concept of the resonance graph appears quite naturally in the study of perfect matchings of molecular graphs of hydrocabons that represent Kekulé structures of corresponding hydrocarbon molecules. Therefore, it is not surprising that it has been independently introduced in the chemical [5,7] as well as in the mathematical literature [20] (under the name Z-transformation graph) and then later rediscovered in [15,14]. The equivalence of the Zhang-Zhang polynomial of the molecular graph and the cube polynomial of its resonance graph was established for benzenoid systems [22], tubulenes [2], and fullerenes [19]. The Zhang-Zhang polynomial counts Clar covers with given number of hexagons, i.e. conjugated 6-cycles. For some recent research on the Zhang-Zhang polynomial see [3,4] The resonance energy is a theoretical quantity which is used for predicting the aromatic stability of conjugated systems. In the conjugated-circuit model, the resonance energy is determined with conjugated cycles of different lengths (see [13]), not only with 6-cycles. Among them, only 6-cycles and 10-cycles have uniquely determined structure. Therefore, we introduce the concept of the generalized Zhang-Zhang polynomial, which considers both of them. In this paper we prove the equivalence of the generalized Zhang-Zhang polynomial of a molecular graph and the generalized cube polynomial of the corresponding resonance graph. Preliminaries A benzenoid system consists of a cycle C of the infinite hexagonal lattice together with all hexagons inside C. A benzenoid graph is the underlying graph of a benzenoid system. Next we formally define open-ended carbon nanotubes, also called tubulenes [16]. Choose any lattice point in the hexagonal lattice as the origin O. Let − → a 1 and − → a 2 be the two basic lattice vectors. Choose a vector −→ OA = n − → a 1 +m − → a 2 such that n and m are two integers and |n|+|m| > 1, nm = −1. Draw two straight lines L 1 and L 2 passing through O and A perpendicular to OA, respectively. By rolling up the hexagonal strip between L 1 and L 2 and gluing L 1 and L 2 such that A and O superimpose, we can obtain a hexagonal tessellation HT of the cylinder. L 1 and L 2 indicate the direction of the axis of the cylinder. Using the terminology of graph theory, a tubulene T is defined to be the finite graph induced by all the hexagons of HT that lie between c 1 and c 2 , where c 1 and c 2 are two vertex-disjoint cycles of HT encircling the axis of the cylinder. For any tubulene T , if its chiral vector is n − → a 1 + m − → a 2 , T will be called an (n, m)-type tubulene, see Figure 1. A fullerene G is a 3-connected 3-regular plane graph such that every face is bounded by either a pentagon or a hexagon. By Euler's formula, it follows that the number of pentagonal faces of a fullerene is exactly 12. A 1-factor of a graph G is a spanning subgraph of G such that every vertex has degree one. The edge set of a 1-factor is called a perfect matching of G, which is a set of independent edges covering all vertices of G. In chemical literature, perfect matchings are known as Kekulé structures (see [8] for more details). Petersen's theorem states that every bridgeless 3-regular graph always has a perfect matching [12]. Therefore, a fullerene always has at least one perfect matching. A hexagon of G with exactly 3 edges in a perfect matching M of G is called a sextet. Let G be a benzenoid system, a tubulene or a fullerene with a perfect matching. The resonance graph R(G) is the graph whose vertices are the perfect matchings of G, and two perfect matchings are adjacent whenever their symmetric difference forms a hexagon of G. The hypercube Q n of dimension n is defined in the following way: all vertices of Q n are presented as n-tuples (x 1 , x 2 , . . . , x n ) where x i ∈ {0, 1} for each 1 ≤ i ≤ n and two vertices of Q n are adjacent if the corresponding n-tuples differ in precisely one coordinate. A convex subgraph H of a graph G is a subgraph of G such that every shortest path between two vertices of H is contained in H. The generalized polynomials Let G be a benzenoid system, a tubulene or a fullerene. A Clar cover is a spanning subgraph of G such that every component of it is either C 6 or K 2 . The Zhang-Zhang polynomial of G is defined in the following way: ZZ(G, x) = k≥0 z(G, k)x k , where z(G, k) is the number of Clar covers of G with k hexagons. A generalized Clar cover is a spanning subgraph of G such that every component of it is either C 6 , C 10 or K 2 . See Figure 2 for an example. The generalized Zhang-Zhang polynomial of G is defined in the following way: GZZ(G, x, y) = k≥0,l≥0 gz(G, k, l)x k y l , where gz(G, k, l) is the number of generalized Clar covers of G with k cycles C 6 and l cycles C 10 . Note that for a graph G number gz(G, 0, 0) equals the number of vertices of R(G) and gz(G, 1, 0) equals the number of edges of R(G). Furthermore, number gz(G, k, 0) represents the number of Clar covers with k hexagons. Let H be a graph. The Cube polynomial of H is defined as follows: C(H, x) = k≥0 α k (H)x k , where α k (H) denotes the number of induced subgraphs of H that are isomorphic to the k-dimensional hypercube. Let G be a graph and i ≥ 1 an integer. Then by G i we denote the Cartesian product of i copies of G, i.e. G i = G · · · G. Also, G 0 = K 1 . Furthermore, for any k, l ≥ 0 we define Q k,l = P k 2 P l 3 , where P 2 and P 3 are paths on 2 and 3 vertices, respectively. Obviously, Q k,0 is the k-dimensional hypercube. Moreover, if k + l > 0, vertices of the graph Q k,l can be presented as (k + l)-tuples (b 1 , . . . , b k , b k+1 , . . . , b k+l ), where b i ∈ {0, 1} if i ∈ {1, . . . , k} and b i ∈ {0, 1, 2} if i ∈ {k + 1, . . . , k + l}. In such representation two vertices (b 1 , . . . , b k , b k+1 , . . . , b k+l ) and (b ′ 1 , . . . , b ′ k , b ′ k+1 , . . . , b ′ k+l ) are adjacent if and only if there is i ∈ {1, . . . , k + l} such that |b i − b ′ i | = 1 and b j = b ′ j for any j = i. Let H be a graph. The generalized Cube polynomial of H is defined as follows: GC(H, x, y) = k≥0,l≥0 α k,l (H)x k y l , where α k,l (H) denotes the number of induced convex subgraphs of H that are isomorphic to the graph Q k,l . The main result In this section we prove that the generalized Zhang-Zhang polynomial of every benzenoid system, tubulene or fullerene equals the generalized cube polynomial of its resonance graph. Proof. Let k and l be nonnegative integers. For a graph G we denote by GZ(G, k, l) the set of all generalized Clar covers of G with exactly k cycles C 6 and l cycles C 10 . On the other hand, consider a graph H; the set of induced convex subgraphs of H that are isomorphic to a graph Q k,l is denoted by GQ k,l (H). Let us define a mapping f k,l from the set of generalized Clar covers of G with k cycles C 6 and l cycles C 10 to the set of induced convex subgraphs of the resonance graph R(G) isomorphic to the graph Q k,l f k,l : GZ(G, k, l) −→ GQ k,l (R(G)) in the following way: for a generalized Clar cover C ∈ GZ(G, k, l) consider all perfect matchings M 1 , M 2 , . . ., M i of G such that: • if cycle C 6 in C, then |M j ∩ E(C 6 )| = 3 for all j = 1, 2, . . . , i, • if cycle C 10 of C is composed of two hexagons, h 1 and h 2 , then |M j ∩ (E(h 1 ) ∪ E(h 2 ))| = 5 for all j = 1, 2, . . . , i, • each isolated edge of C is in M j for all j = 1, 2, . . . , i. Finally, assign f k,l (C) as an induced subgraph of R(G) with vertices M 1 , M 2 ,. . ., M i . Note first that in case when k = 0 and l = 0 generalized Clar covers are the perfect matchings of G and if C is such a generalized Clar cover then f k,l (C) is a vertex of the resonance graph and the mapping is obviously bijective. So from now on at least one of k and l will be positive. We first show that f k,l is a well-defined function. Lemma 4.2. For each generalized Clar cover C ∈ GZ(G, k, l) it follows that f k,l (C) ∈ Q k,l (R(G)). Proof. First we show that f k,l (C) is isomorphic to the graph Q k,l . Let c 1 , c 2 ,. . .,c k be the hexagons of C and let c k+1 , . . . , c k+l be cycles C 10 that are in C. Obviously, every hexagon of C has two possible perfect matchings. Let us call these "possibility 0" and "possibility 1". Moreover, for every cycle C 10 in C we obtain tree possible perfect matchings of graph V (C 10 ) , which will be denoted as "possibility 0", "possibility 1", and "possibility 2". Also, if cycle C 10 is composed of hexagons h 1 and h 2 , "possibility 1" denotes the perfect matching containing the common edge of h 1 and h 2 . For any vertex M of f k,l (C) let b(M ) = (b 1 , b 2 , . . . , b k , b k+1 , . . . , b k+l ), where b j = i if on c j possibility i is selected. It is obvious that b : V (f k,l (C)) → V (Q k,l ) is a bijection. Let b(M ′ ) = (b ′ 1 , b ′ 2 , . . . , b ′ k , b ′ k+1 , . . . , b ′ k+l ) for M ′ ∈ V (f k,l (C)). If M and M ′ are adjacent in f k,l (C), then M ⊕ M ′ = E(h) for a hexagon h of some c i , where 1 ≤ i ≤ k + l. Therefore, b j = b ′ j for each j = i and |b i − b ′ i | = 1, which implies that b(M ) and b(M ′ ) are adjacent in Q k,l . Conversely, if (b 1 , b 2 , . . . , b k , b k+1 , . . . , b k+l ) and (b ′ 1 , b ′ 2 , . . . , b ′ k , b ′ k+1 , . . . , b ′ k+l ) are adjacent in Q k,l , it follows that M and M ′ are adjacent in f k,l (C). Hence b is an isomorphism between f k,l (C) and Q k,l . To complete the proof we have to show that f k,l (C) is a convex subgraph of R(G). Therefore, let M and M ′ be two vertices of f k,l (C). Obviously, perfect matchings M and M ′ can differ only in the edges of hexagons that belong to cycles of C. Therefore, any shortest path between M and M ′ in R(G) contains perfect matchings that are vertices of f k,l (C). It follows that f k,l (C) is convex in R(G). The following lemma shows that f k,l is injective. Proof. Let C and C ′ be distinct generalized Clar covers in GZ(G, k, l). If C and C ′ contain the same set of cycles, then the isolated edges of C and C ′ are distinct. Therefore, f k,l (C) and f k,l (C ′ ) are disjoint induced subgraphs of R(G) and thus f k,l (C) = f k,l (C ′ ). Therefore, suppose that C and C ′ contain different sets of cycles. Without loss of generality we can assume that there is hexagon h such that h has at least five edges in C and h has at most three edges in C ′ . Hence at least one edge e of h does not belong to C ′ . From the definition of the function f k,l , e is thus unsaturated by those perfect matchings that correspond to the vertices in f k,l (C ′ ). However, there obviously exists perfect matching M ∈ V (f k,l (C)) such that e ∈ M . As a result, M / ∈ V (f k,l (C ′ )) and f (C) = f (C ′ ). The next lemma was proved in [22] for benzenoid systems. The same proof can be applied in the case of tubulenes or fullerenes. The following lemma shows that f k,l is surjective. 1, 0, . . . , 0), . . . , N k+l = (0, 0, 0, . . . , 1) be the vertices of Q. It is obvious that M N i is an edge of R(G) for every i, 1 ≤ i ≤ k + l. By definition of R(G), the symmetric difference of perfect matchings M and N i is the edge set of a hexagon of G. We denote this hexagon by h i and we obtain the set of hexagons {h 1 , . . . , h k+l } of graph G. If two of these hexagons were the same, for example if h i = h j for i, j ∈ {1, . . . , k + l} and i = j, then N i = N j -a contradiction. Hence, we have the set of k + l distinct hexagons. In the next claim we show that these hexagons are pairwise disjoint. Proof. It is easy to see that h i = h ′ i (otherwise M = O i ). Therefore, suppose that h i and h ′ i are disjoint. Since they are both sextets in the perfect matching N i , there is a vertex X of R(G), X = N i , which is adjacent to M and O i . If X ∈ V (Q), the string of X must differ from M for 1 in exactly one position and must differ from O i for 1 in exactly one position, which means X = N i -a contradiction. Therefore, X is not in Q. Since M XO i is a shortest path between M and O i , Q is not convex subgraph of R(G), which is a contradiction. Hence, h i and h ′ i have exactly one common edge. Proof. Define the following vertices in Q: h ′ i = h j (otherwise O i = Y , • Using Lemma 4.4 we can easily see that hexagon h ′ j corresponds to the edge X 1 X 2 and hexagon h ′ i corresponds to the edge X 1 X 3 . Since X 1 X 2 X 4 X 3 is a 4-cycle in the resonance graph, Lemma 4.4 again implies that h ′ i and h ′ j are disjoint and the proof is complete. Let C i , i ∈ {k+1, . . . , k+l} be a 10-cycle formed by h i and h ′ i . Moreover, let C be a spanning subgraph of G such that E(C) = M ∪ E(h 1 ) ∪ . . . ∪ E(h k ) ∪ E(C k+1 ) ∪ . . . E(C k+l ). Therefore, C is a generalized Clar cover with k hexagons and l 10-cycles. It is obvious that every edge in Q corresponds to some hexagon h i , i ∈ {1, . . . , k + l} or h ′ i , i ∈ {k + 1, . . . , k + l}. Therefore, V (f k,l (C)) = V (Q). Since both Q and f k,l (C) are induced subgraphs of the resonance graph, it follows f k,l (C) = Q. We have proved that f k,l is bijective function and hence, |GZ(G, k, l)| = |GQ k,l (R(G))|. Therefore, the proof is complete. An example In this final section we give an example of a benzenoid system G and calculate the generalized Zhang-Zhang polynomial of G, i.e. the generalized cube polynomial of the resonance graph of G. See Figures 3 and 4. The polynomials are GZZ(G, x, y) = GC(R(G), x, y) = = 34 + 53x + 35x 2 + 12x 3 + x 4 + 48y + 7y 2 + 37xy + xy 2 + 3x 2 y . For example, the coefficient in front of x 2 y is 3, since there are 3 generalized Clar covers in G with two C 6 and one C 10 . On the other hand, this coefficient counts the number of induced convex subgraphs of R(G) isomorphic to the graph P 2 2 P 3 . The vector −→ OA is called the chiral vector of T and the cycles c 1 and c 2 are the two open-ends of T . Figure 1 . 1Illustration of a (4, −3)-type tubulene. Figure 2 . 2A generalized Clar cover of a benzenoid system G. Theorem 4. 1 . 1Let G be a benzenoid system, a tubulene or a fullerene with a perfect matching. Then the generalized Zhang-Zhang polynomial of G equals the generalized cube polynomial of its resonance graph R(G), i.e. GZZ(G, x, y) = GC(R(G), x, y) . Lemma 4. 3 . 3The mapping f k,l : GZ(G, k, l) −→ Q k,l (R(G)) is injective for any integers k, l. Lemma 4. 4 . [ 22 ] 422Let G be a benzenoid systems, a tubulene, or a fullerene with a perfect matching. If the resonance graph R(G) contains a 4-cycle M 1 M 2 M 3 M 4 , then h = M 1 ⊕ M 2 and h ′ = M 1 ⊕ M 4 are disjoint hexagons. Also, we have h = M 3 ⊕ M 4 and h ′ = M 2 ⊕ M 3 . Lemma 4. 5 . 5The mapping f k,l : GZ(G, k, l) −→ Q k,l (R(G)) is surjective for any integers k, l.Proof. Let k, l be integers and Q ∈ Q k,l (R(G)). Then the vertices of Q can be identified with strings(b 1 , . . . , b k , b k+1 , . . . , b k+l ), where b i ∈ {0, 1} if i ∈ {1, . . . , k} or b i ∈ {0, 1, 2} if i ∈ {k + 1, . . . , k + l},so that two vertices of Q are adjacent in Q if and only if their strings b and b ′ differ in precisely one position i, such that |b i − b ′ i | = 1. Let M = (0, 0, 0, . . . , 0), N 1 = (1, 0, 0, . . . , 0), N 2 = (0, Claim 4 . 6 . 46The hexagons h i , 1 ≤ i ≤ k + l, are pairwise disjoint. Proof. Let i, j ∈ {1, . . . , k + l} and i = j. Let W be a vertex of Q having exactly two 1's (and these are in the ith and jth position) and 0 at every other position. Obviously, M N i W N j is a 4-cycle and therefore, by Lemma 4.4, h i and h j are disjoint hexagons. Next, we consider the vertices O i , i ∈ {k + 1, . . . , k + l}, such that O i has 2 in the ith position and 0 in every other position. Obviously, N i O i is the edge of R(G) for any i ∈ {k + 1, . . . , k + l}. Let h ′ i be the hexagon of G corresponding to the edge N i O i . Claim 4 . 7 . 47If i ∈ {k + 1, . . . , k + l}, the hexagon h ′ i has exactly one common edge with h i . Claim 4. 8 . 8Let i ∈ {k + 1, . . . , k + l}. Then the hexagon h ′ i is disjoint with every h j , j ∈ {1, . . . , k + l} \ {i}.Proof. Let X be a vertex in Q with 2 in the ith position, 1 in the jth position and 0 in every other position. Furthermore, let Y be a vertex in Q with 1 in the ith position, 1 in the jth position and 0 in every other position. Obviously, X 1 has 1 in the ith position, 1 in the jth position and 0 in every other position, • X 2 has 1 in the ith position, 2 in the jth position and 0 in every other position, • X 3 has 2 in the ith position, 1 in the jth position and 0 in every other position, • X 4 has 2 in the ith position, 2 in the jth position and 0 in every other position. Figure 3 . 3Benzenoid system G. Figure 4 . 4Resonance graph R(G). which is a contradiction). Since N i O i XY is a 4-cycle such that h ′ i corresponds to the edge N i O i and h j corresponds to the edge N i Y , it follows from Lemma 4.4 that hexagons h ′ i and h j are disjoint. Claim 4.9. Let i ∈ {k + 1, . . . , k + l}. Then the hexagon h ′ i is disjoint with every h ′ j , j ∈ {k + 1, . . . , k + l} \ {i}. AcknowledgmentSupported in part by the Ministry of Science of Slovenia under grant P 1 − 0297. Mathematical aspects of fullerenes. V Andova, F Kardoš, R Škrekovski, Ars Math. Contemp. 11V. Andova, F. Kardoš, R.Škrekovski, Mathematical aspects of fullerenes. Ars Math. Contemp. 11 (2016), 353-379. Equivalence of Zhang-Zhang polynomial and cube polynomial for spherical benzenoid systems. M Berlič, N Tratnik, P Žigert Pleteršek, MATCH Commun. Math. Comput. Chem. 73M. Berlič, N. Tratnik, P.Žigert Pleteršek, Equivalence of Zhang-Zhang polyno- mial and cube polynomial for spherical benzenoid systems. MATCH Commun. Math. Comput. Chem. 73 (2015), 443-456. Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations. C.-P Chou, J.-S Kang, H A Witek, Discrete Appl. Math. 198C.-P. Chou, J.-S. Kang, H. A. Witek, Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations. Discrete Appl. Math. 198 (2016), 101-108. ZZDecomposer: a graphical toolkit for analyzing the Zhang-Zhang polynomials of benzenoid structures. C.-P Chou, H A Witek, MATCH Commun. Math. Comput. Chem. 71C.-P. Chou, H. A. Witek, ZZDecomposer: a graphical toolkit for analyzing the Zhang-Zhang polynomials of benzenoid structures. MATCH Commun. Math. Comput. Chem. 71 (2014), 741-764. Generation of lattice graphs. An equivalence relation on Kekulé counts of catacondensed benzenoid hydrocarbons. S El-Basil, J. Mol. Struct. (Theochem). 288S. El-Basil, Generation of lattice graphs. An equivalence relation on Kekulé counts of catacondensed benzenoid hydrocarbons. J. Mol. Struct. (Theochem) 288 (1993), 67-84. On two Graffiti conjectures about fullerene graphs. M Faghani, T Došlić, MATCH Commun. Math. Comput. Chem. 76M. Faghani, T. Došlić, On two Graffiti conjectures about fullerene graphs. MATCH Commun. Math. Comput. Chem. 76 (2016), 723-730. Signifikante Elektronenstrukturen fur benzenoide Kohlenwasserstoffe. W Gründler, Wiss. Z. Univ. Halle. 31W. Gründler, Signifikante Elektronenstrukturen fur benzenoide Kohlenwasser- stoffe. Wiss. Z. Univ. Halle 31 (1982), 97-116. Introduction to the Theory of Benzenoid Hydrocarbons. I Gutman, S J Cyvin, Springer-VerlagBerlinI. Gutman, S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-Verlag, Berlin, 1989. Helical microtubules of graphitic carbon. S Iijima, Nature. 354S. Iijima, Helical microtubules of graphitic carbon. Nature 354 (1991), 56-58. The edge-Wiener index of benzenoid systems in linear time. A Kelenc, S Klavžar, N Tratnik, MATCH Commun. Math. Comput. Chem. 74A. Kelenc, S. Klavžar, N. Tratnik, The edge-Wiener index of benzenoid systems in linear time. MATCH Commun. Math. Comput. Chem. 74 (2015), 521-532. On symmetries of benzenoid systems. J Kovič, T Pisanski, A T Balaban, P W Fowler, MATCH Commun. Math. Comput. Chem. 72J. Kovič, T. Pisanski, A. T. Balaban, P. W. Fowler, On symmetries of benzenoid systems. MATCH Commun. Math. Comput. Chem. 72 (2014), 3-26. Die Theorie der regulären Graphen. J Petersen, Acta Math. 15J. Petersen, Die Theorie der regulären Graphen. Acta Math. 15 (1891), 193- 220. The conjugated-circuit model: application to nonalternant hydrocarbons and a comparison with some other theoretical models of aromaticity. D Plavić, S Nikolić, N Trinajstić, J. Mol. Struct. (Theochem). D. Plavić, S. Nikolić, N. Trinajstić, The conjugated-circuit model: application to nonalternant hydrocarbons and a comparison with some other theoretical models of aromaticity. J. Mol. Struct. (Theochem) 277 (1992), 213-237. Resonance in catacondensed benzenoid hydrocarbons. M Randić, Int. J. Quantum Chem. 63M. Randić, Resonance in catacondensed benzenoid hydrocarbons. Int. J. Quan- tum Chem. 63 (1997), 585-600. Resonance in large benzenoid hydrocarbons. M Randić, D J Klein, S El-Basil, P Calkins, Croat. Chem. Acta. 69M. Randić, D. J. Klein, S. El-Basil, P. Calkins, Resonance in large benzenoid hydrocarbons. Croat. Chem. Acta 69 (1996), 1639-1660. Kekulé count in tubular hydrocarbons. H Sachs, P Hansen, M Zheng, MATCH Commun. Math. Comput. Chem. 33H. Sachs, P. Hansen, M. Zheng, Kekulé count in tubular hydrocarbons. MATCH Commun. Math. Comput. Chem. 33 (1996), 169-241. Forcing and anti-forcing numbers of (3, 6)-fullerenes. L Shi, H Zhang, MATCH Commun. Math. Comput. Chem. 76L. Shi, H. Zhang, Forcing and anti-forcing numbers of (3, 6)-fullerenes. MATCH Commun. Math. Comput. Chem. 76 (2016), 597-614. The edge-Szeged index and the PI index of benzenoid systems in linear time. N Tratnik, MATCH Commun. Math. Comput. Chem. 77N. Tratnik, The edge-Szeged index and the PI index of benzenoid systems in linear time. MATCH Commun. Math. Comput. Chem. 77 (2017), 393-406. Resonance graphs of fullerenes. N Tratnik, P Žigert Pleteršek, Ars Math. Contemp. 11N. Tratnik, P.Žigert Pleteršek, Resonance graphs of fullerenes. Ars Math. Contemp. 11 (2016), 425-435. Z-transformation graphs of perfect matchings of hexagonal systems. F Zhang, X Guo, R Chen, Discrete Math. 72F. Zhang, X. Guo, R. Chen, Z-transformation graphs of perfect matchings of hexagonal systems. Discrete Math. 72 (1988), 405-415. On the global forcing number of hexagonal systems. H Zhang, J Cai, Discrete Appl. Math. 162H. Zhang, J. Cai, On the global forcing number of hexagonal systems. Discrete Appl. Math. 162 (2014), 334-347. A relation between Clar covering polynomial and cube polynomial. H Zhang, W C Shiu, P K Sun, MATCH Commun. Math. Comput. Chem. 70H. Zhang, W. C. Shiu, P. K. Sun, A relation between Clar covering polynomial and cube polynomial. MATCH Commun. Math. Comput. Chem. 70 (2013), 477- 492. Maribor Faculty of Natural Sciences and Mathematics. Maribor Slovenia17PetraŽigert Pleteršek Faculty of Chemistry and Chemical Engineering, University of Maribor Smetanova ulica ; University of Maribor Koroška cestaPetraŽigert Pleteršek Faculty of Chemistry and Chemical Engineering, University of Maribor Smetanova ulica 17, 2000 Maribor Faculty of Natural Sciences and Mathematics, University of Maribor Koroška cesta 160, 2000 Maribor Slovenia e-mail: [email protected]
[]
[ "Two sample inference for the second-order property of temporally dependent functional data", "Two sample inference for the second-order property of temporally dependent functional data" ]
[ "Xianyang Zhang [email protected] \nDepartment of Statistics\nUniversity of Missouri-Columbia\n65211ColumbiaMOUSA\n", "Xiaofeng Shao [email protected] \nDepartment of Statistics\nUniversity of Illinois at Urbana-Champaign\n61820ChampaignILUSA\n" ]
[ "Department of Statistics\nUniversity of Missouri-Columbia\n65211ColumbiaMOUSA", "Department of Statistics\nUniversity of Illinois at Urbana-Champaign\n61820ChampaignILUSA" ]
[ "Bernoulli" ]
Motivated by the need to statistically quantify the difference between two spatio-temporal datasets that arise in climate downscaling studies, we propose new tests to detect the differences of the covariance operators and their associated characteristics of two functional time series. Our two sample tests are constructed on the basis of functional principal component analysis and self-normalization, the latter of which is a new studentization technique recently developed for the inference of a univariate time series. Compared to the existing tests, our SN-based tests allow for weak dependence within each sample and it is robust to the dependence between the two samples in the case of equal sample sizes. Asymptotic properties of the SN-based test statistics are derived under both the null and local alternatives. Through extensive simulations, our SN-based tests are shown to outperform existing alternatives in size and their powers are found to be respectable. The tests are then applied to the gridded climate model outputs and interpolated observations to detect the difference in their spatial dynamics.
10.3150/13-bej592
[ "https://arxiv.org/pdf/1506.00847v1.pdf" ]
18,133,887
1506.00847
312bf25456cff62a3c1c6fb4ef28632088f80d1c
Two sample inference for the second-order property of temporally dependent functional data 2015 Xianyang Zhang [email protected] Department of Statistics University of Missouri-Columbia 65211ColumbiaMOUSA Xiaofeng Shao [email protected] Department of Statistics University of Illinois at Urbana-Champaign 61820ChampaignILUSA Two sample inference for the second-order property of temporally dependent functional data Bernoulli 212201510.3150/13-BEJ592climate downscalingfunctional data analysislong run variance matrixself-normalizationtime seriestwo sample problem Motivated by the need to statistically quantify the difference between two spatio-temporal datasets that arise in climate downscaling studies, we propose new tests to detect the differences of the covariance operators and their associated characteristics of two functional time series. Our two sample tests are constructed on the basis of functional principal component analysis and self-normalization, the latter of which is a new studentization technique recently developed for the inference of a univariate time series. Compared to the existing tests, our SN-based tests allow for weak dependence within each sample and it is robust to the dependence between the two samples in the case of equal sample sizes. Asymptotic properties of the SN-based test statistics are derived under both the null and local alternatives. Through extensive simulations, our SN-based tests are shown to outperform existing alternatives in size and their powers are found to be respectable. The tests are then applied to the gridded climate model outputs and interpolated observations to detect the difference in their spatial dynamics. Introduction Functional data analysis (FDA) which deals with the analysis of curves and surfaces has received considerable attention in the statistical literature during the last decade (Ramsay and Silverman [19,20] and Ferraty and Vieu [6]). This paper falls into a sub-field of functional data analysis: inference for temporally dependent functional data. Specifically, we focus on testing the equality of the second-order structures (e.g., the covariance operators and their associated eigenvalues and eigenfunctions) of two temporally dependent functional sequences. Our work is partially motivated by our ongoing collaboration with atmospheric scientists on the development and assessment of high-resolution climate projections through statistical downscaling. Climate change is one of the most urgent problems facing the world this century. To study climate change, scientists have relied primarily on climate projections from global/regional climate models, which are numerical models that involve systems of differential equations and produce outputs at a prespecified grid. As numerical model outputs are widely used in situations where real observations are not available, it is an important but still open question whether the numerical model outputs are able to mimic/capture the spatial and temporal dynamics of the real observations. To partly answer this question, we view the spatio-temporal model outputs and real observations as realizations from two temporally dependent functional time series defined on the two-dimensional space and test the equality of their secondorder structures which reflects their spatial dynamics/dependence. Two sample inference for functional data has been investigated by a few researchers. Fan and Lin [5], Cuevas et al. [4] and Horváth et al. [10] developed the tests for the equality of mean functions. Benko et al. [1], Panaretos et al. [16], Fremdt et al. [7], and Kraus and Panaretos [12] proposed tests for the equality of the second-order structures. All the above-mentioned works assumed the independence between the two samples and/or independence within each sample. However, the assumption of independence within the sample is often too strong to be realistic in many applications, especially if data are collected sequentially over time. For example, the independence assumption is questionable for the climate projection data considered in this paper, as the model outputs and real station observations are simulated or collected over time and temporal dependence is expected. Furthermore the dependence between numerical model outputs and station observations is likely because the numerical models are designed to mimic the dynamics of real observations. See Section 5 for empirical evidence of their dependence. In this paper, we develop new tests that are able to accommodate weak dependence between and within two samples. Our tests are constructed on the basis of functional principal component analysis (FPCA) and the recently developed self-normalization (SN) method (Shao [21]), the latter of which is a new studentization technique for the inference of a univariate time series. FPCA attempts to find the dominant modes of variation around an overall trend function and has been proved a key technique in the context of FDA. The use of FPCA in the inference of temporally dependent functional data can be found in Gabrys and Kokoszka [8], Hörmann and Kokoszka [9], Horváth et al. [10] among others. To account for the dependence, the existing inference procedure requires a consistent estimator of the long run variance (LRV) matrix of the principal component scores or consistent estimator of the LRV operator. However, there is a bandwidth parameter involved in the LRV estimation and its selection has not been addressed in the functional setting. The same issue appears when one considers the block bootstrapping and subsampling schemes (Lahiri [13] and Politis et al. [18]), since these techniques also require the selection of a smoothing parameter, such as the block length in the moving block bootstrap, and the window width in the subsampling method (see, e.g., Politis and Romano [17] and McMurry and Politis [15]). Since the finite sample performance can be sensitive to the choice of these tuning parameters and the bandwidth choice can involve certain degree of arbitrariness, it is desirable to use inference methods that are free of bandwidth parameters. To this end, we build on the bandwidth-free SN method (Shao [21]) recently developed in the univariate time series setup, and propose SN-based tests in the functional setting by using recursive estimates obtained from functional data samples. In time series analysis, the inference of a parameter using normal approximation typically requires consistent estimation of its asymptotic variance. The main difficulty with this approach (and other block-based resampling methods) is the sensitivity of the finite sample performance with respect to the bandwidth parameter, which is often difficult to choose in practice without any parametric assumptions. As a useful alternative, the self-normalized approach uses an inconsistent bandwidth-free estimate of asymptotic variance, which is proportional to asymptotic variance, so the studentized quantity (statistic) is asymptotically pivotal. Extending the early idea of Lobato [14], Shao [21] proposed a very general kind of self-normalizers that are functions of recursive estimates and showed the theoretical validity for a wide class of parameters of interest. The settings in the latter two papers are however limited to univariate time series. The generalization of the SN method from univariate to functional time series was first done in Zhang et al. [23] where the focus was on testing the structure stability of temporally dependent functional data. Here we extend the SN method to test the equality of the second-order properties of two functional time series, which is rather different and new techniques and results are needed. To study the asymptotic properties of the proposed test statistics, we establish functional central limit theorems for the recursive estimates of quantities associated with the second-order properties of the functional time series which seems unexplored in the literature and are thus of independent interest. Based on the functional central limit theorem, we show that the SN-based test statistics have pivotal limiting distributions under the null and are consistent under the local alternatives. From a methodological viewpoint, this seems to be the first time that the SN method is extended to the two sample problem. Compared to most of the existing methods which assumed the independence between the two samples and/or independence within each sample, the SN method not only allows for unknown dependence within each sample but also allows for unknown dependence between the two samples when the sample sizes of the two sequences are equal. Methodology We shall consider temporally dependent functional processes {(X i (t), Y i (t)), t ∈ I} +∞ i=1 defined on some compact set I of the Euclidian space, where I can be one-dimensional (e.g., a curve) or multidimensional (e.g., a surface or manifold). For simplicity, we consider the Hilbert space H of square integrable functions with I = [0, 1] (or I = [0, 1] 2 ). For any functions f, g ∈ H, the inner product between f and g is defined as I f (t)g(t) dt and · denotes the inner product induced norm. Assume the random elements all come from the same probability space (Ω, A, P). Let L p be the space of real valued random variables with finite L p norm, that is, (E|X| p ) 1/p < ∞. Further, we denote L p H the space of H valued random variables X such that (E X p ) 1/p < ∞. Given two sequences of temporally dependent functional observations, {X i (t)} N1 i=1 and {Y i (t)} N2 i=1 defined on a common region I, we are interested in comparing their secondorder properties. Suppose that the functional time series are second-order stationary. We assume that E[X i (t)] = E[Y i (t)] = 0. The result can be easily extended to the situation with nonzero mean functions. Define C X = E[ X i , · X i ] and C Y = E[ Y i , · Y i ] as the covariance operators of the two sequences respectively. For the convenience of presentation, we shall use the same notation for the covariance operator and the associated covariance function. Denote by {φ j X } ∞ j=1 and {λ j X } ∞ j=1 the eigenfunctions and eigenvalues of C X . Analogous quantities are {φ j Y } ∞ j=1 and {λ j Y } ∞ j=1 for the second sample. Denote by |v| the Euclidean norm of a vector v ∈ R p . Let vech(·) be the operator that stacks the columns below the diagonal of a symmetric m × m matrix as a vector with m(m + 1)/2 components. Let D[0, 1] be the space of functions on [0, 1] which are right-continuous and have left limits, endowed with the Skorokhod topology (see Billingsley [2]). Weak convergence in D[0, 1] or more generally in the R m -valued function space D m [0, 1] is denoted by "⇒", where m ∈ N and convergence in distribution is denoted by "→ d ". Define ⌊a⌋ the integer part of a ∈ R, and δ ij = 1 if i = j and δ ij = 0 if i = j. In what follows, we shall discuss the tests for comparing the three quantities C X , φ j X and λ j X with C Y , φ j Y and λ j Y , respectively. Covariance operator Consider the problem of testing the hypothesis H 1,0 : C X = C Y versus the alternative H 1,a : C X = C Y (in the operator norm sense) for two mean zero stationary functional time series {X i (t)} N1 i=1 and {Y i (t)} N2 i=1 . Let N = N 1 + N 2 . Throughout the paper, we assume that N 1 /N → γ 1 , N 2 /N → γ 2 , as min(N 1 , N 2 ) → +∞, where γ 1 , γ 2 ∈ (0, 1) and γ 1 +γ 2 = 1. Define the one-dimensional operator X i = X i , · X i = X i ⊗ X i and Y j = Y j , · Y j = Y j ⊗ Y j . LetĈ XY be the empirical covariance operator based on the pooled samples, that is, C XY = 1 N 1 + N 2 N1 i=1 X i + N2 i=1 Y i . (2.1) Denote by {λ j XY } and {φ j XY } the corresponding eigenvalues and eigenfunctions. The population counterpart ofĈ XY is then given byC XY = γ 1 C X + γ 2 C Y whose eigenvalues and eigenfunctions are denoted by {λ j } and {φ j } respectively. Further letĈ X,m = 1 m m i=1 X i be the sample covariance operator based on the subsample {X i (t)} m i=1 with 2 ≤ m ≤ N 1 . Define {φ j X,and Iφ i X,m (t)φ j X,m (t) dt = δ ij . Similarly, quantitiesĈ Y,m ′ , {φ j Y,m ′ } N2 j=1 and {λ j Y,m ′ } N2 j=1 are defined for the second sample with 2 ≤ m ′ ≤ N 2 . To introduce the SN-based test, we define the recursive estimates c i,j k = (Ĉ X,⌊kN1/N ⌋ −Ĉ Y,⌊kN2/N ⌋ )φ i XY ,φ j XY , 2 ≤ k ≤ N, 1 ≤ i, j ≤ K, which estimate the difference of the covariance operators on the space spanned by {φ j } K j=1 . Here K is a user-chosen number, which is held fixed in the asymptotics. Denote byα k = vech(C k ) with C k = (c i,j k ) K i,j=1 . In the independent and Gaussian case, Panaretos et al. [16] proposed the following test (hereafter, the PKM test), T N1,N2 = N 1 N 2 2N K i=1 K j=1 (c i,j N ) 2 ̺ i̺j ,̺ j = 1 N N1 i=1 ( X i ,φ j XY ) 2 + N2 i=1 ( Y i ,φ j XY ) 2 , which converges to χ 2 (K+1)K/2 under the null. To take the dependence into account, we introduce the SN matrix V (1) SN,N (d) = 1 N 2 N k=1 k 2 (α k −α N )(α k −α N ) ′ ,(2.3) with d = (K + 1)K/2. The SN-based test statistic is then defined as, G (1) SN,N (d) = Nα ′ N (V (1) SN,N (d)) −1α N . (2.4) Notice that the PKM test statistic can also be written as a quadratic form ofα N but with a different normalization matrix that is only applicable to the independent and Gaussian case. The special form of the SN-based test statistic makes it robust to the dependence within each sample and also the dependence between the two samples when their sample sizes are equal. We shall study the asymptotic behavior of G (1) SN,N (d) under the weak dependence assumption in Section 3. Eigenvalues and eigenfunctions In practice, it is also interesting to infer how far the marginal distributions of two sequences of stationary functional time series coincide/differ and quantify the difference. By the Karhunen-Loève expansion (Bosq [3], page 26), we have X i (t) = +∞ j=1 λ j X β Xi,j φ j X (t), Y i (t) = +∞ j=1 λ j Y β Yi,j φ j Y (t), where β Xi,j = I X i (t)φ j X (t) dt and β Yi,j = I Y i (t)φ j Y (t) dt are the principal components (scores), which satisfy that E[β Xi,j β Xi,j ′ ] = δ jj ′ and E[β Yi,j β Yi,j ′ ] = δ jj ′ . The problem is then translated into testing the equality of the functional principal components (FPC's) namely the eigenvalues and eigenfunctions. For a prespecified positive integer M , we denote the vector of the first M eigenvalues by λ 1:M X = (λ 1 X , . . . , λ M X ) and λ 1:M Y = (λ 1 Y , . . . , λ M Y ). Further define φ 1:M X = (φ 1 X , . . . , φ M X ) and φ 1:M Y = (φ 1 Y , . . . , φ M Y ) the first M eigenfunctions of the covariance operators C X and C Y , respectively. Since the eigenfunctions are determined up to a sign, we assume that φ j X , φ j Y ≥ 0 in order for the comparison to be meaningful. We aim to test the null hypothesis [1], where the authors proposed an i.i.d. bootstrap method which seems not applicable to the dependent case. The block bootstrap based method is expected to be valid in the weakly dependent case but the choice of the block size seems to be a difficult task in the current setting. To accommodate the dependence and avoid the bandwidth choice, we adopt the SN idea. Recall the recursive estimates of the eigenvaluesλ j X,m andλ j Y,m ′ which are calculated based on the subsamples {X i (t)} m i=1 and {Y i (t)} m ′ i=1 . Letθ j k =λ j X,⌊kN1/N ⌋ −λ j Y,⌊kN2/N ⌋ andθ k = (θ 1 k , . . . ,θ M k ) ′ with ⌊N ǫ⌋ ≤ k ≤ N for some ǫ ∈ (0, 1] , which is held fixed in the asymptotics. We consider the trimmed SN-based test statistic G (2) SN,N (M ) = N 3θ′ N N k=⌊N ǫ⌋ k 2 (θ k −θ N )(θ k −θ N ) ′ −1θ N . (2.5) The trimmed version of the SN-based test statistic is proposed out of technical consideration when the functional observations lie on an infinite dimensional space. It can be seen from the proof in the supplemental material [22] that the trimming is not required when functional data lie on a finite-dimensional space; see Remark 0.1 in the supplemental material [22]. Remark 2.1. To compare the difference between the eigenvalues, one may also consider their ratios. Defineζ k = (λ 1 X,⌊kN1/N ⌋ /λ 1 Y,⌊kN2/N ⌋ , . . . ,λ M X,⌊kN1/N ⌋ /λ M Y,⌊kN2/N ⌋ ) ′ for k = ⌊N ǫ⌋, . . . , N . An alternative SN-based test statistic is given bỹ G (2) SN,N (M ) = N (ζ N − l M ) ′ 1 N 2 N k=⌊N ǫ⌋ k 2 (ζ k −ζ N )(ζ k −ζ N ) ′ −1 (ζ N − l M ), (2.6) where l M is a M -dimensional vector of all ones. Since the finite sample improvement by usingG (2) SN,N (M ) is not apparent, we do not further investigate the properties of G We now turn to the problem of testing the equality of the eigenfunctions. To proceed, we letν j = (φ j+1 XY ,φ j+2 XY , . . . ,φ p XY ) (2.7) be a vector of p− j orthonormal basis functions for j = 1, 2, . . . , M with M ≤ p and p being a user chosen number. Recall thatφ j X,m (t) andφ j Y,m ′ (t) are the jth eigenfunctions of the empirical covariance operatorsĈ X,m andĈ Y,m ′ which are computed based on the first m (and m ′ ) samples. Here we require that φ j X,m ,φ j X,N1 ≥ 0 and φj Y,m ′ ,φ j X,N1 ≥ 0 for 2 ≤ m ≤ N 1 and 2 ≤ m ′ ≤ N 2 . As the eigenfunctions are defined on an infinite-dimensional space, we project the difference between the jth eigenfunctions onto the space spanned byν j . Formally, we define the projection vectorŝ η j k = ( φ j X,⌊kN1/N ⌋ −φ j Y,⌊kN2/N ⌋ ,φ j+1 XY , . . . , φ j X,⌊kN1/N ⌋ −φ j Y,⌊kN2/N ⌋ ,φ p XY ), where 1 ≤ j ≤ M and k = ⌊N ǫ⌋, . . . , N . Further letη k = (η 1 k ,η 2 k , . . . ,η M k ) ′ ∈ R M0 with M 0 = M(2p−M−1) 2 . The trimmed SN-based test statistic is then defined as G (3) SN,N (M 0 ) = Nη ′ N 1 N 2 N k=⌊N ǫ⌋ k 2 (η k −η N )(η k −η N ) ′ −1η N ,(2.8) for some 0 < ǫ < 1. Remark 2.2. It is worth noting that G (3) SN,N (M 0 ) is designed for testing the equality of the first M eigenfunctions. Suppose we are interested in testing the hypothesis for a particular eigenfunction, that is, the null φ j X = φ j Y versus the alternative φ j X = φ j Y . We can consider the basis functions ν j = (φ 1 XY . . . ,φ j−1 XY ,φ j+1 XY , . . . ,φ p XY ), and the projection vectorη j k = ( φ j X,⌊kN1/N ⌋ −φ j Y,⌊kN2/N ⌋ ,φ 1 XY , . . . , φ j X,⌊kN1/N ⌋ − φ j Y,⌊kN2/N ⌋ ,φ j−1 XY , φ j X,⌊kN1/N ⌋ −φ j Y,⌊kN2/N ⌋ ,φ j+1 XY , . . . , φ j X,⌊kN1/N ⌋ −φ j Y,⌊kN2/N ⌋ ,φ p XY ) ′ . The SN-based test statistic can then be constructed in a similar manner. We also note that when φ j X = φ j Y and φ i X = φ i Y for i = j, the choice ofν j may result in trivial power because φ j X − φ j Y ,φ i for i = j can be close to 0. In this case, one remedy is to consider alternative basis functions, for example, (4.5) and (4.6) as suggested in the simulation. Remark 2.3. The choice of the basis functionsν j is motivated by the Bahadur representation of the recursive estimates in the supplemental material [22]. Under suitable assumptions as given in the next section, it can be shown that φ a X,k , φ = φ a X , φ − 1 k k i=1 s =a β Xi,s β Xi,a λ s X − λ a X φ s X , φ + R a X,k , (2.9) with R a X,k being the remainder term and φ ∈ L 2 (I). The second term on the RHS of (2.9) plays a key role in determining the limiting distribution of the SN-based test statistic. When φ = φ j X with j = a, the linear term reduces to − 1 k k i=1 βX i ,j βX i ,a λ j X −λ a X , which satisfies the functional central limit theorem under suitable weak dependence assumption. Notice that the linear term vanishes when φ = φ a X and the asymptotic distribution of the projection vector is degenerate. It is also worth noting that the linear terms in the Bahadur representations of φa X,k , φ j X and φ j X,k , φ a X are opposite of each other which suggests that when testing the eigenfunctions jointly, the basis functions should be chosen in a proper way so that the asymptotic covariance matrix of the projection vector, that is,η k is nondegenerate. Theoretical results To study the asymptotic properties of the proposed statistics, we adopt the dependence measure proposed in Hörmann and Kokoszka [9], which is applicable to the temporally dependent functional process. There are also other weak dependence measures (e.g., mixing) or specific processes (e.g., functional linear processes) suitable for the asymptotic analysis of functional time series (see Bosq [3]), we decide to use Hörmann and Kokoszka's L p -m-approximating dependence measure for its broad applicability to linear and nonlinear functional processes as well as its theoretical convenience and elegance. Definition 3.1. Assume that {X i } ∈ L p H with p > 0 admits the following representation X i = f (ε i , ε i−1 , . . .), i = 1, 2, . . . ,(3.{ε j } j∈Z . The sequence {X i } is said to be L p -m-approximable if ∞ m=1 (E X m − X (m) m p ) 1/p < ∞, (3.2) where X (m) i = f (ε i , ε i−1 , . . . , ε i−m+1 , ε (i) i−m , ε (i) i−m−1 , . . .). Define B q (r) as a q-dimensional vector of independent Brownian motions. For ǫ ∈ [0, 1), we let W q (ǫ) = B q (1) ′ J q (ǫ) −1 B q (1), where J q (ǫ) = 1 ǫ (B q (r)− rB q (1))(B q (r)− rB q (1)) ′ dr. The critical values of W q := W q (0) have been tabulated by Lobato [14]. In general, the quantiles of W q (ǫ) can be obtained via simulation. To derive the asymptotic properties of the proposed tests, we make the following assumptions. Assumption 3.1. Assume {X i (t)} +∞ i=1 ⊆ L 2 H and {Y i (t)} +∞ i=1 ⊆ L 2 H are both L 4 -mapproximable and they are mutually independent. Assumption 3.2. Assume {(X i (t), Y i (t))} +∞ i=1 ⊆ L 4 H×H is an L 4 -m-approximable se- quence. Assumption 3.3. Assume λ 1 X > λ 2 X > · · · > λ m0+1 X and λ 1 Y > λ 2 Y > · · · > λ m0+1 Y , for some positive integer m 0 ≥ 2. Note that Assumption 3.2 allows dependence between {X i (t)} and {Y i (t)}, which is weaker than Assumption 3.1. To investigate the asymptotic properties of G It is seen from Theorem 3.1 that G (1) SN,N (d) has pivotal limiting distributions under the null and they are consistent under the local alternatives as L → +∞. It is worth noting that in our asymptotic framework, d (or K) is assumed to be fixed as n → ∞. Since K is usually chosen to make the first K principle components explain a certain percentage of variation (say 85%), the magnitude of K critically depends on the prespecified threshold and the decay rate of the eigenvalues. In some cases, d = (K + 1)K/2 can be quite large relative to sample size so it may be more meaningful to use the asymptotic results established under the framework that d → ∞ but d/n → 0 as n → ∞. This motivates the question that whether the following convergence result sup x∈R |P (G (1) SN,N (d) ≤ x) − P (W d ≤ x)| → 0 as n → ∞ holds when d diverges to ∞ but at a slower rate than n. This would be an interesting future research topic but is beyond the scope of this paper. To study the asymptotics of G SN,N (M ) and G SN,N (M 0 ), we introduce some notation. Let ω jk Xi = β Xi,j β Xi,k and r jk, j ′ k ′ X (h) = E[(ω jk Xi − δ jk λ j )(ω j ′ k ′ X i+h − δ j ′ k ′ λ j ′ )] be the crosscovariance function between ω jk Xi and ω j ′ k ′ Xi at lag h. Set r jk X (h) := r jk,jk X (h). Define v jk Xi = ω jk Xi − E[ω jk Xi ] = ω jk Xi − δ jk λ j . Analogous quantities r jk,j ′ k ′ Y (h) and v jk Yi can be defined for the second sample. We make the following assumption to facilitate our derivation. Assumption 3.4. Suppose that j,k j ′ ,k ′ +∞ h=−∞ |r jk,j ′ k ′ X (h)| 2 < +∞, j,k +∞ h=−∞ |r jk X (h)| < +∞ (3.3) and j,k j ′ ,k ′ i1,i2,i3∈Z |cum(v jk X0 , v jk Xi 1 , v j ′ k ′ Xi 2 , v j ′ k ′ Xi 3 )| < ∞. (3.4) The summability conditions also hold for the second sample {Y i (t)}. Assumption 3.4 is parallel to the summability condition considered in Benko et al. [1] (see Assumption 1 therein) for i.i.d. functional data. It is not hard to verify the above assumption for Gaussian linear functional process (see, e.g., Bosq [3]), as demonstrated in the following proposition. In order to study the asymptotic properties of G Proposition 3.1. Consider the linear process X i (t) = ∞ j=0 b j ε i−j (t), where ε j (t) = ∞ i=1 √ λ i z i,j φ i (t) with {z i,j } being a sequence of independent standard normal random variables across both index i and j. Let π(h) = i b i b i+h . Assume that ∞ j=1 λ j < ∞ and h |π(h)| < ∞. Then Assumption 3.4 holds for {X i (t)}.Xi,j = X (m) i (t)φ j X (t) dt, where X (m) i is the m-dependent approximation of X i (t) (see Definition 3.1). Suppose one of the following conditions holds: ∞ m=1 ∞ j=1 {E(β X1,j − β (m) X1,j ) 4 } 1/4 < ∞, ∞ j=1 (Eβ 4 X1,j ) 1/4 < ∞, (3.5) or +∞ s=1 | φ s X ,φ j | < +∞, 2 ≤ j ≤ p. (3.6) The same condition holds for the second sample {Y i (t)}. It is worth noting that the conclusions in Theorem 3.2, Theorem 3.3 and Proposition 3.2 also hold with Assumption 3.1 replaced by Assumption 3.2 and γ 1 = γ 2 . Finally, we point out that condition (3.5) can be verified for Gaussian linear functional process as shown in the following proposition. Numerical studies We conduct a number of simulation experiments to assess the performance of the proposed SN-based tests in comparison with the alternative methods in the literature. We generate functional data on a grid of 10 3 equispaced points in [0, 1], and then convert discrete observations into functional objects by using B-splines with 20 basis functions. We also tried 40 and 100 basis functions and found that the number of basis functions does not affect our results much. Throughout the simulations, we set the number of Monte Carlo replications to be 1000 except for the i.i.d. bootstrap method in Benko et al. [1], where the number of replications is only 250 because of high computational cost. Comparison of covariance operators To investigate the finite sample properties of G (1) SN,N (d) for dependent functional data, we modify the simulation setting considered in Panaretos et al. [16]. Formally, we consider the model, 3 j=1 {ξ i j,1 √ 2 sin(2πjt) + ξ i j,2 √ 2 cos(2πjt)}, i = 1, 2, . . . , t ∈ [0, 1], (4.1) where the coefficients ξ i = (ξ i 1,1 , ξ i 2,1 , ξ i 3,1 , ξ i 1,2 , ξ i 2,2 , ξ i 3,2 ) ′ are generated from a VAR process, ξ i = ρξ i−1 + 1 − ρ 2 e i ,(4.2) with e i ∈ R 6 being a sequence of i.i.d. normal random variables with mean zero and covariance matrix Σ e = 1 1+µ 2 diag(v) + µ 2 1+µ 2 1 6 1 ′ 6 . We generate two independent functional time series {X i (t)} and {Y i (t)} from (4.1) with ρ = 0.5 and µ = 1. We compare the SNbased test with the PKM test which is designed for independent Gaussian process, and the traditional test which is constructed based on a consistent LRV estimator (denoted by CLRV), that is, G CL,N (d) = Nα NΣ −1 ααN , whereΣ α is a lag window LRV estimator with Bartlett kernel and data dependent bandwidth (see Andrews (1991)). We report the simulation results for N 1 = N 2 = 100, 200, K = 1, 2, 3, 4, 5 (d = 1, 3, 6, 10, 15) and various values of v in Table 1. Results in scenario A show that the size distortion of all the three tests increases as K gets larger. The SN-based test has the best size compared to the other two tests. The PKM test is severely oversized due to the fact that it does not take the dependence into account. It is seen from the table that the CLRV test also has severe size distortion especially for large K, which is presumably due to the poor estimation of the LRV matrix ofα N when the dimension is high. Under the alternatives, we report the size-adjusted power which is computed using finite sample critical values based on the simulation under the null model where we assume that both {X i (t)} and {Y i (t)} are generated from (4.1) with ρ = 0.5, µ = 1 and v = v X . From scenarios B-D in Table 1, we observe that the PKM is most powerful which is largely due to its severe upward size distortion. The SN-based test is less powerful compared to the other two tests but the power loss is generally moderate in most cases. Furthermore, we present the results when choosing K by K * j = arg min 1 ≤ J ≤ 20 : J i=1λ i XY 20 i=1λ i XY > α * j , j = 1, 2, (4.3) where α * 1 = 85% and α * 2 = 95%. An alternative way of choosing K is to consider the penalized fit criteria (see Panaretos et al. [16] for the details). We notice that the performance of all the three tests based on automatic choice K * j is fairly close to the performance when K = 4 or 5 in most cases. To sum up, the SN-based test provides the best size under the null and has reasonable power under different alternatives considered here, which is consistent with the "better size but less power" phenomenon seen in the univariate setup (Lobato [14] and Shao [21]). Comparison of eigenvalues and eigenfunctions In this subsection, we study the finite sample performance of the SN-based test for testing the equality of the eigenvalues and eigenfunctions. We consider the data generating process, where ξ * i = (ξ i 1,1 , ξ i 2,1 , ξ i 1,2 , ξ i 2,2 ) ′ is a 4-variate VAR process (4.2) with e i ∈ R 4 being a sequence of i.i.d. normal random variables with mean zero and covariance matrix Σ e = 1 1+µ 2 diag(v) + µ 2 1+µ 2 1 4 1 ′ 4 . We set ρ = 0.5 and µ = 0. Under H 2,0 (or H 2,a ), {X i (t)} and {Y i (t)} are generated independently from (4.4) with δ 1 = δ 2 = 0 and v X = v Y (or v X = v Y ). Notice that the eigenvalues of {X i (t)} and {Y i (t)} are given respectively, by v X and v Y when δ 1 = δ 2 = 0. Under H 3,0 and H 3,a , we generate {X i (t)} and {Y i (t)} independently from (4.4) with v X = v Y , δ X,1 − δ Y,1 = δ, and δ X,2 = δ Y,2 = 0, where δ = 0 under the null and δ = 0 under the alternatives. We aim to test the equality of the first four eigenvalues and eigenfunctions separately and jointly. Because functional data are finite dimensional, we implement the untrimmed version of the SN-based tests, that is, ǫ = 0. To further assess the performance of the SN-based test, we compare our method with the subsampling approach with several choices of subsampling widths and the i.i.d. bootstrap method in Benko et al. [1]. Suppose N 1 = N 2 = N 0 . Let l be the subsampling width and λ j sub,i = λ j sub,X,i − λ j sub,Y,i , i = 1, 2, . . . , s N0 (l) = ⌈N 0 /l⌉, where λ j sub,X,i and λ j sub,Y,i are estimates of the jth eigenvalues based on the ith nonoverlapping subsamples {X k (t)} il k=(i−1)l+1 and {Y k (t)} il k=(i−1)l+1 , respectively. The subsampling variance estimate is given by σ 2 sub,j = l sN 0 (l) sN 0 (l) i=1 (λ j sub,i − 1 sN 0 (l) sN 0 (l) i=1λ j sub,i ) 2 , and the test statistic based on the subsampling variance estimate for testing the equality of the jth eigenvalue is defined as G sub,N = N 0 (λ j X,N0 −λ j Y,N0 ) 2 /σ 2 sub,j . Since the data-dependent rule for choosing the subsampling width is not available in the current setting, we tried l = 8, 12, 16 for N 0 = 48, 96. For testing the equality of eigenvalues jointly and equality of the eigenfunctions, we shall consider a multivariate version of the subsampling-based test statistic which can be defined in a similar fashion. Table 2 summarizes some selective simulation results for testing the eigenvalues with various values of v. From scenario A, we see that performance of the SN-based test under the null is satisfactory while the size distortion of the subsampling-based method is quite severe and is sensitive to the choice of block size l. It is also not surprising to see that the i.i.d. bootstrap method has obvious size distortion as it does not take the dependence into account. Under the alternatives (scenarios B-D), we report the size-adjusted power by using the simulated critical values as described in previous subsection. When the sample size is 48, the SN-based method delivers the highest power among the tests and it tends to have some moderate power loss when the sample size increases to 96. On the other hand, the subsampling method is sensitive to the choice of subsampling width and its power tends to decrease when a larger subsampling width is chosen. To test the equality of the first four eigenfunctions, we implement the SN-based test and the subsampling-based test with the basis functions, ν * j = (φ 1 XY +φ j XY , . . . ,φ j−1 XY +φ j XY , (4.5) φ j+1 XY +φ j XY , . . . ,φ p XY +φ j XY ), 1 ≤ j ≤ 4, p = 4, for testing individual eigenfunction and ν * * j = (φ j+1 XY +φ j XY ,φ j+2 XY +φ j XY , . . . ,φ p XY +φ j XY ), 1 ≤ j ≤ j * , p = 4,(4.6) with j * = 2, 3, 4, for testing the first j * eigenfunctions jointly (correspondingly M 0 = 3, 5, 6). The tests with the above basis functions tend to provide similar sizes but higher powers as compared to the tests with the basis functionsν i in our simulation study. The basis functionsν * j is constructed by adding the same estimated eigenfunctionφ j XY to each component ofν j , and the associated SN-based test is expected to be asymptotically valid in view of the Bahadur representation (2.9). Selective simulation results are summarized in Table 3 and Figure 1 which present the sizes of the SN-based test, the subsampling-based test and the i.i.d. bootstrap method, and the size adjusted powers of the former two respectively. It is seen from Table 3 that the sizes of the SN-based test are accurate while the subsampling-based test is apparently size-distorted. It is somewhat surprising to see that the i.i.d. bootstrap provides better sizes compared to the Table 2. Empirical sizes and size-adjusted powers of (i) the SN-based test, the subsamplingbased test with (ii) l = 8, (iii) l = 12 and (iv) l = 16, and (v) Benko et al.'s i.i.d. bootstrap based method for testing the equality of the first two eigenvalues separately (the columns with M = 1, 2) and jointly (the column with M = (1, 2)), and the equality of the first four eigenvalues jointly (the column with M = (1, 2, 3, 4)). subsampling-based approach which is designed for dependent data. Figure 1 plots the (size-adjusted) power functions of the SN-based test and the subsampling-based test which are monotonically increasing on δ. When N 1 = N 2 = 48, the SN-based test delivers the highest power in most cases. The subsampling-based test with a small subsampling width becomes most powerful when sample size increases to 96. Overall, the SN-based test is preferable as it provides quite accurate size under the null and has respectable power under the alternatives. Climate projections analysis We apply the SN-based test to a gridded spatio-temporal temperature dataset covering a subregion of North America. The dataset comes from two separate sources: gridded observations generated from interpolation of station records (HadCRU), and gridded simu- lations generated by an AOGCM (NOAA GFDL CM2.1). Both datasets provide monthly average temperature for the same 19-year period, 1980-1998. Each surface is viewed as a two-dimensional functional datum. The yearly average data have been recently analyzed in Zhang et al. [23], where the goal is to detect a possible change point of the bias between the station observations and model outputs. In this paper, we analyze the monthly data from 1980 to 1998, which includes 228 functional images for each sequence. We focus on the second-order properties and aim to test the equality of the eigenvalues and eigenfunctions of the station observations and model outputs. To perform the analysis, we first remove the seasonal mean functions from the two functional sequences. At each location, we have two time series from the demeaned functional sequences. We apply the SN-based test developed in Shao [21] to test whether their cross-correlation at lag zero is equal to zero. The p-values of these tests are plotted in Figure 2. The result tends to suggest that the dependence between the station observations and model outputs may not be negligible at certain regions as the corresponding p-values are extremely small. The two sample tests introduced in this paper are useful in this case because they are robust to such dependence. We perform FPCA on the demeaned sequences. Figure 3 plots the first three PC's of the station observations and model outputs. We then apply the SN-based tests G Table 4. It is seen from the table that the first two eigenvalues of the station observations and model outputs may be the same, at least statistical significance is below the 10% level, while there is a significant difference between their third eigenvalue. The SN-based tests also suggest that there are significant differences of the first and second PCs between the station observations and model outputs as the corresponding p-values are less than 5% while the difference between the third PCs is not significant at the 10% level; compare Figure 3. We also tried the basis functions ν * j and ν * * j for G SN,N (M 0 ) (see (4.5) and (4.6)), which leads to the same conclusion. To sum up, our results suggest that the second-order properties of the station observations and model outputs may not be the same. In climate projection studies, the use of numerical models outputs has become quite common nowadays because of advances in computing power and efficient numerical algorithms. As mentioned in Jun et al. [11], "Climate models are evaluated on how well they simulate the current mean climate state, how they can reproduce the observed climate change over the last century, how well they simulate specific processes, and how well they agree with proxy data for very different time periods in the past." Furthermore, different institutions produce different model outputs based on different choices of parametrizations, model components, as well as initial and boundary conditions. Thus there is a critical need to assess the discrepancy/similarity between numerical model outputs and real observations, as well as among various model outputs. The two sample tests proposed here can be used towards this assessment at a preliminary stage to get a quantitative idea of the difference, followed by a detailed statistical characterization using sophisticated spatio-temporal modeling techniques (see, e.g., Jun et al. [11]). In particular, the observed significance level for each test can be used as a similarity index that measures the similarity between numerical model outputs and real observations, and may be used to rank model outputs. A detailed study along this line would be interesting, but is beyond the scope of this article. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2015, Vol. 21, No. 2, 909-929. This reprint differs from the original in pagination and typographic detail. the L 2 norm sense). The problem of comparing the FPC's of two independent and identically distributed (i.i.d.) functional sequences has been considered in Benko et al. ( 1 ) 1SN,N (d) under the local alternatives, we consider the local alternative H 1,a :C X − C Y = LC/ √ N withC being a Hilbert-Schmidt operator, where L is a nonzero constant. Define ∆ = ( Cφ i ,φ j ) Ki,j=1 ∈ R K×K as the projection ofC onto the space spanned by {φ 1 ,φ 2 , . . . ,φ K } and assume that vech(∆) = 0 ∈ R d . The following theorem states the asymptotic behaviors of G(1) SN,N (d) under the null and the local alternatives. Theorem 3. 1 . 1Suppose Assumptions 3.1, 3.3 hold with m 0 ≥ K. Further assume that the asymptotic covariance matrices Λ * d (Λ * d ) ′ given in Lemma 0.3 is positive definite. Then under H 1,0 , G (1) SN,N (d) → d W d and under H 1,a , lim |L|→+∞ lim N →+∞ G (1) SN,N (d) = +∞. Furthermore, if γ 1 = γ 2 , then the conclusion also holds with Assumption 3.1 replaced by Assumption 3.2. Theorem 3 . 2 . 32Suppose Assumptions 3.1, 3.3, 3.4 hold with m 0 ≥ M and the asymptotic covariance matrixΛ MΛ ′ M given in Lemma 0.5 is positive definite. Then under H 2,0 , we have G (2) SN,N (M ) → d W M (ǫ). Under the local alternative H 2,a : λ = 0 ∈ R M , we have lim |L|→∞ lim N →+∞ G (2) SN,N (M ) = +∞. ( 3 ) 3SN,N (M 0 ) under the null and local alternative, we further make the following assumption. Theorem 3. 3 . 3Suppose Assumptions 3.1, 3.3, 3.4 and 3.5 hold with m 0 ≥ M and the asymptotic covariance matrixΛ M0Λ ′ M0 given in Lemma 0.7 is positive definite. Then under H 3,0 , we have G (3) SN,N (M 0 ) → d W M0 (ǫ). Proposition 3 . 2 . 32Define∆ by replacingφ j X,N1 ,φ j Y,N2 andφ j XY with φ j X , φ j Y andφ j inthe definition ofη N . Consider the local alternative H 3,a :∆ = Lψ/ √ N withψ = 0 ∈ R M0 . Suppose Assumptions 3.1, 3.3, 3.4 and 3.5 hold with m 0 ≥ M and the asymptotic covariance matrixΛ M0Λ ′ M0 given in Lemma 0.7 is positive definite. Then we have Proposition 3 . 3 . 33Consider the Gaussian linear process in Proposition 3.1. Assume that∞ j=1 λ j < ∞ and ∞ m=1 ( ∞ j=m b 2 j ) 1/2 < ∞.Then Assumption 3.4 and condition(3.5) are satisfied for {X i (t)}. Figure 1 . 1Size-adjusted powers of the SN-based test and the subsampling-based tests for testing the equality of the first two eigenfunctions separately and jointly, and the equality of the first four eigenfunctions jointly. Figure 2 . 2p-values for testing the nullity of lag zero cross-correlation between the station observations and model outputs at each location. The numbers 0-5 denote the ranges of the N (M 0 ) (with p = 3) to the demeaned sequences, which yields the results summarized in Figure 3 . 3The first three PCs of the station observations (left panels) and model outputs (right panels), and the associated eigenvalues and percentage of variations explained. 1 ) 1where the ε i 's are i.i.d. elements taking values in a measurable space S and f is a measurable function f : S ∞ → H. For each i ∈ N, let {ε(i) j } j∈Z be an independent copy of Table 1 . 1Empirical sizes and size-adjusted powers of (i) the SN-based test, (ii) the PKM test and (iii) the CLRV test for testing the equality of the covariance operators. The nominal level is 5%K The nominal level is 5% and the number of replications for i.i.d. bootstrap method is 250Table 3. Empirical sizes of (i) the SN-based test, the subsampling-based test with (ii) l = 8, (iii) l = 12 and (iv) l = 16, and (v) Benko et al.'s i.i.d. bootstrap based method for testing the equality of the first two eigenfunctions separately (the columns with M = 1, 2) and jointly (the column with M =(1, 2)), and the equality of the first four eigenfunctions jointly (the column with M =(1, 2, 3, 4)). The nominal level is 5% and the number of replications for i.i.d. bootstrap is 250M Parameter N1 = N2 1 2 (1, 2) (1, 2, 3, 4) A vX = (10, 0.5, 5, 0.3) 48 (i) 5.4 5.1 4.6 3.8 (ii) 24.2 38.5 52.4 90.8 (iii) 21.9 28.8 51.3 68.8 (iv) 21.8 28.1 57.9 44.7 (v) 11.2 9.2 11.6 11.6 vY = (10, 0.5, 5, 0.3) 96 (i) 5.2 5.6 4.8 5.1 (ii) 19.0 40.4 46.4 84.8 (iii) 16.3 29.6 38.2 77.0 (iv) 16.0 25.3 36.5 78.5 (v) 14.4 8.4 15.2 15.2 B vX = (20, 0.5, 5, 0.3) 48 (i) 25.1 4.3 21.8 15.5 (ii) 24.2 5.4 13.3 7.1 (iii) 19.8 6.8 8.8 8.8 (iv) 14.1 6.8 8.0 9.1 vY = (10, 0.5, 5, 0.3) 96 (i) 48.4 4.8 35.6 25.0 (ii) 58.4 6.9 29.4 11.4 (iii) 50.9 6.1 29.7 13.3 (iv) 53.8 6.0 29.3 11.4 C vX = (10, 0.5, 5, 0.3) 48 (i) 6.2 70.6 58.9 44.1 (ii) 5.5 68.1 54.6 13.9 (iii) 4.8 49.3 23.0 16.9 (iv) 6.1 34.2 15.4 21.2 vY = (10, 0.5, 1, 0.3) 96 (i) 4.7 91.4 84.6 77.6 (ii) 4.7 98.7 96.3 69.7 (iii) 5.5 97.9 92.5 51.6 (iv) 5.4 96.5 83.0 29.4 D vX = (20, 0.5, 5, 0.3) 48 (i) 27.0 70.1 68.4 55.3 (ii) 25.8 65.7 51.1 14.0 (iii) 20.9 58.4 23.9 19.9 (iv) 14.9 40.1 11.8 17.2 vY = (10, 0.5, 1, 0.3) 96 (i) 55.3 87.9 88.4 83.3 (ii) 54.5 98.3 96.9 62.3 (iii) 48.8 97.6 95.2 53.2 (iv) 50.1 95.2 88.0 27.7 M Parameter N1 = N2 1 2 (1, 2) (1, 2, 3, 4) A vX = (10, 0.5, 5, 0.3) 48 (i) 6.4 3.2 4.9 4.8 (ii) 39.5 36.0 78.7 67.0 (iii) 62.9 62.3 24.8 26.4 (iv) 32.6 25.9 9.6 7.7 (v) 2.4 11.2 2.4 2.4 vY = (8, 0.5, 4, 0.3) 96 (i) 4.5 3.3 4.3 4.7 (ii) 18.2 16.8 27.2 45.8 (iii) 24.9 21.0 49.0 71.2 (iv) 32.6 30.6 75.9 61.0 (v) 3.2 12.4 4.8 6.8 B vX = (4, 0.5, 2, 0.3) 48 (i) 8.2 3.8 7.0 6.1 (ii) 43.3 45.8 83.4 71.4 (iii) 66.6 65.2 23.8 18.1 (iv) 26.5 22.5 5.8 3.7 (v) 2.4 6.0 3.6 1.6 vY = (2, 0.5, 1, 0.3) 96 (i) 5.3 4.5 5.2 4.9 (ii) 20.3 24.2 36.7 54.6 (iii) 25.3 27.9 53.0 75.7 (iv) 33.1 32.6 78.1 60.1 (v) 2.8 8.4 6.8 3.6 p-values, that is, 0 denotes [0.1, 1]; 1 denotes [0.05, 0.1]; 2 denotes [0.025, 0.05]; 3 denotes [0.01, 0.025]; 4 denotes [0.005, 0.01] and 5 denotes [0, 0.005]. Table 4 . 4Comparison of the eigenvalues and eigenfunctions of the covariance operators of the station observations and model outputs Note: The first three rows show the results for testing individual eigencomponent, and the last row shows the results for testing the first three eigenvalues and eigenfunctions jointly.K G (2) SN,N (M ) p-value G (3) SN,N (M0) p-value 1 10.8 (0.1, 1) 126.4 (0.025, 0.05) 2 5.4 (0.1, 1) 295.4 (0, 0.005) 3 119.9 (0.005, 0.01) 34.2 (0.1, 1) - 326.2 (0.005, 0.01) 318.0 (0.005, 0.01) Note: Under the alternatives, we simulate the size-adjusted critical values by assuming that both {X i } and {Y i } are generated from (4.1) with ρ = 0.5, µ = 1 and v = v X . Note: Under the alternatives, we simulate the size-adjusted critical values by assuming that both {X i } and {Y i } are generated from (4.1) with ρ = 0.5, µ = 0 and v = v X . AcknowledgementsWe are grateful to an associate editor and a referee for helpful comments, which led to substantial improvements. Shao's research is supported in part by National Science Foundation Grant DMS-11-04545.Two sample inferenceSupplementary MaterialSupplement to "Two sample inference for the second-order property of temporally dependent functional data" (DOI: 10.3150/13-BEJ592SUPP; .pdf). This supplement contains proofs of the results in Section 3. Common functional principal components. M Benko, W Härdle, A Kneip, Ann. Statist. 372488343Benko, M., Härdle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1-34. MR2488343 P Billingsley, Series in Probability and Statistics: Probability and Statistics. New YorkWiley1700749Convergence of Probability MeasuresBillingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley. MR1700749 Linear Processes in Function Spaces. D Bosq, Lecture Notes in Statistics. 1491783138SpringerBosq, D. (2000). Linear Processes in Function Spaces. Lecture Notes in Statistics 149. New York: Springer. Theory and applications. MR1783138 An ANOVA test for functional data. A Cuevas, M Febrero, R Fraiman, Comput. Statist. Data Anal. 472087932Cuevas, A., Febrero, M. and Fraiman, R. (2004). An ANOVA test for functional data. Comput. Statist. Data Anal. 47 111-122. MR2087932 Test of significance when data are curves. J Fan, S.-K Lin, J. Amer. Statist. Assoc. 93Fan, J. and Lin, S.-K. (1998). Test of significance when data are curves. J. Amer. Statist. Assoc. 93 1007-1021. MR1649196 Nonparametric Functional Data Analysis. Theory and Practice. F Ferraty, P Vieu, Springer Series in Statistics. 2229687SpringerFerraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis. Theory and Practice. Springer Series in Statistics. New York: Springer. MR2229687 Testing the equality of covariance operators in functional samples. S Fremdt, J G Steinebach, L Horváth, P Kokoszka, Scand. J. Stat. 40Fremdt, S., Steinebach, J.G., Horváth, L. and Kokoszka, P. (2013). Testing the equality of covariance operators in functional samples. Scand. J. Stat. 40 138-152. MR3024036 Portmanteau test of independence for functional observations. R Gabrys, P Kokoszka, J. Amer. Statist. Assoc. 102Gabrys, R. and Kokoszka, P. (2007). Portmanteau test of independence for functional observations. J. Amer. Statist. Assoc. 102 1338-1348. MR2412554 Weakly dependent functional data. S Hörmann, P Kokoszka, Ann. Statist. 382662361Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845-1884. MR2662361 Estimation of the mean of functional time series and a two-sample problem. L Horváth, P Kokoszka, R Reeder, J. R. Stat. Soc. Ser. B. Stat. Methodol. 75Horváth, L., Kokoszka, P. and Reeder, R. (2013). Estimation of the mean of functional time series and a two-sample problem. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 103- 122. MR3008273 Spatial analysis to quantify numerical model bias and dependence: How many climate models are there?. M Jun, R Knutti, D W Nychka, J. Amer. Statist. Assoc. 103Jun, M., Knutti, R. and Nychka, D.W. (2008). Spatial analysis to quantify numerical model bias and dependence: How many climate models are there? J. Amer. Statist. Assoc. 103 934-947. MR2528820 Dispersion operators and resistant second-order functional data analysis. D Kraus, V M Panaretos, Biometrika. 99Kraus, D. and Panaretos, V.M. (2012). Dispersion operators and resistant second-order functional data analysis. Biometrika 99 813-832. Resampling Methods for Dependent Data. S N Lahiri, Springer Series in Statistics. 2001447SpringerLahiri, S.N. (2003). Resampling Methods for Dependent Data. Springer Series in Statistics. New York: Springer. MR2001447 Testing that a dependent process is uncorrelated. I N Lobato, J. Amer. Statist. Assoc. 96Lobato, I.N. (2001). Testing that a dependent process is uncorrelated. J. Amer. Statist. Assoc. 96 1066-1076. MR1947254 Resampling methods for functional data. T Mcmurry, D Politis, The Oxford Handbook of Functional Data Analysis. F. Ferraty and Y. RomainOxfordOxford Univ. Press2908023McMurry, T. and Politis, D. (2011). Resampling methods for functional data. In The Oxford Handbook of Functional Data Analysis (F. Ferraty and Y. Romain, eds.) 189-209. Oxford Univ. Press, Oxford. MR2908023 Second-order comparison of Gaussian random functions and the geometry of DNA minicircles. V M Panaretos, D Kraus, J H Maddocks, J. Amer. Statist. Assoc. 1052724851Panaretos, V.M., Kraus, D. and Maddocks, J.H. (2010). Second-order comparison of Gaussian random functions and the geometry of DNA minicircles. J. Amer. Statist. Assoc. 105 670-682. MR2724851 K-sample subsampling in general spaces: The case of independent time series. D N Politis, J P Romano, J. Multivariate Anal. 1012564342Politis, D.N. and Romano, J.P. (2010). K-sample subsampling in general spaces: The case of independent time series. J. Multivariate Anal. 101 316-326. MR2564342 . D N Politis, J P Romano, M Wolf, Subsampling. Springer Series in Statistics. 1707286SpringerPolitis, D.N., Romano, J.P. and Wolf, M. (1999). Subsampling. Springer Series in Statistics. New York: Springer. MR1707286 Applied Functional Data Analysis. J O Ramsay, B W Silverman, Springer Series in Statistics. 1910407SpringerRamsay, J.O. and Silverman, B.W. (2002). Applied Functional Data Analysis. Springer Series in Statistics. New York: Springer. Methods and case studies. MR1910407 Functional Data Analysis. J O Ramsay, B W Silverman, Springer2168993New York2nd ed. Springer Series in StatisticsRamsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. New York: Springer. MR2168993 A self-normalized approach to confidence interval construction in time series. X Shao, J. R. Stat. Soc. Ser. B Stat. Methodol. 72Shao, X. (2010). A self-normalized approach to confidence interval construction in time series. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 343-366. MR2758116 Supplement to "Two sample inference for the second-order property of temporally dependent functional data. X Zhang, X Shao, 10.3150/13-BEJ592SUPPZhang, X. and Shao, X. (2014). Supplement to "Two sample inference for the second-order property of temporally dependent functional data." DOI:10.3150/13-BEJ592SUPP. Testing the structural stability of temporally dependent functional observations and application to climate projections. X Zhang, X Shao, K Hayhoe, D J Wuebbles, Electron. J. Stat. 52870150Zhang, X., Shao, X., Hayhoe, K. and Wuebbles, D.J. (2011). Testing the structural stability of temporally dependent functional observations and application to climate projections. Electron. J. Stat. 5 1765-1796. MR2870150
[]
[ "Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results", "Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results" ]
[ "Carlos Alvarado ", "Paolo Amore ", "\nFacultad de Ciencias\nFacultad de Ciencias, CUICBAS\nUniversidad de Colima\nBernal Díaz del Castillo 340Colima, ColimaMexico\n", "\nUniversidad de Colima\nBernal Díaz del Castillo 340Colima, ColimaMexico\n" ]
[ "Facultad de Ciencias\nFacultad de Ciencias, CUICBAS\nUniversidad de Colima\nBernal Díaz del Castillo 340Colima, ColimaMexico", "Universidad de Colima\nBernal Díaz del Castillo 340Colima, ColimaMexico" ]
[]
We obtain systematic approximations to the states (energies and wave functions) of quantum rings (annular drums) of arbitrary shape by conformally mapping the annular domain to a simply connected domain. Extending the general results of Ref.[1] we obtain an analytical formula for the spectrum of quantum ring of arbirtrary shape: for the cases of a circular annulus and of a Robnik ring considered here this formula is remarkably simple and precise. We also obtain precise variational bounds for the ground state of different quantum rings. Finally we extend the Conformal Collocation Method of[1,2]to the class of problems considered here and calculate precise numerical solutions for a large number of states (≈ 2000).
10.1063/1.3601445
[ "https://arxiv.org/pdf/1004.2407v1.pdf" ]
115,171,372
1004.2407
5d5bd9874ffa24af526ac86e43cdb07cb92fe870
Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results Carlos Alvarado Paolo Amore Facultad de Ciencias Facultad de Ciencias, CUICBAS Universidad de Colima Bernal Díaz del Castillo 340Colima, ColimaMexico Universidad de Colima Bernal Díaz del Castillo 340Colima, ColimaMexico Spectroscopy of annular drums and quantum rings: perturbative and nonperturbative results numbers: 0230Mv0270Jn0365Ge We obtain systematic approximations to the states (energies and wave functions) of quantum rings (annular drums) of arbitrary shape by conformally mapping the annular domain to a simply connected domain. Extending the general results of Ref.[1] we obtain an analytical formula for the spectrum of quantum ring of arbirtrary shape: for the cases of a circular annulus and of a Robnik ring considered here this formula is remarkably simple and precise. We also obtain precise variational bounds for the ground state of different quantum rings. Finally we extend the Conformal Collocation Method of[1,2]to the class of problems considered here and calculate precise numerical solutions for a large number of states (≈ 2000). I. INTRODUCTION This paper extends the general results obtained in a recent paper by one of us, ref. [1], for simply connected drums and quantum billiards to domains with a hole, i.e. annular drums or quantum rings. Quantum rings are two dimensional regions of annular shape where an electron is confined and possibly subject to an external field (magnetic or electric); as for the simply connected case studied in ref. [1], only few special cases may be solved exactly, such as for the circular annulus, where the solutions may be expressed in terms of Bessel functions of first and second kind. However, quantum rings of general shape, for which exact solutions are not known, present interesting physical behaviors, thus justifying the effort of finding analytical or numerical approximations. For instance, it is known that the bendings of an infinite wire cause the appearance of bound states below the continuum threshold [3] and of localized states in quantum rings in correspondence of regions of maximum curvature [4]. In particular, Gridin and collaborators have formulated in Ref. [4] an asymptotic approximation to the modes of a quantum ring, based on the assumption that the ratio of the ring half-width to the radius of curvature is small. Their approach extends to ring like domain the classical Keller-Rubinow method [5,6] In this paper we adopt a completely different strategy, which is both systematic and simple, and which can be used to obtain precise analytical and/or numerical approximations for the energies and eigenfunctions of a given quantum ring. The approach that we propose is * Electronic address: [email protected] (correspond-ing˙author) based on a generalization of the methods described in Ref. [1] and allows one to solve Helmholtz equation for the quantum ring of arbitrary shape by conformally mapping the ring to a simply connected domain. The results obtained in this way are very precise and prove to be useful even for the exactly solvable circular ring: in this case we have found an extremely precise formula for the energies of the rings, which avoids the use of Bessel functions and their zeroes. It is important to underline that the method that we propose is systematic and that it can be applied to rings of arbitrary width and curvature. The paper is organized as follows: in Section II we describe the numerical implementation of our method, using the Conformal Collocation Method of [1,2]; in Section III we discuss the extension of the analytical techniques of [1] to the case of annular domains; in Section IV we apply the methods, both analytical and numerical, to the solvable problem of a circular annulus and to the less tractable problem of a "Robnik's ring", i.e. a ring whose external border corresponds to the family of quantum billiards studied by Robnik in [7] and known as "Robnik's billiards". Finally in Section V we summarize our results and draw our conclusions. II. CONFORMAL COLLOCATION METHOD In this section we describe the application of the Conformal Collocation Method (CCM) of Ref. [1,2] to the solution of Schrödinger equation in a quantum ring. Although this approach has been already described in detail in those papers, we briefly review it here to make the discussion self-contained and to highlight the modifications which are needed to implement the specific problem at hand. Our starting point is the homogeneous Helmholtz equation on a "ring-like" domain D, which can be conformally mapped to an inhomogeneous Helmholtz equation on a "simpler" domain Ω, which is assumed here to be a rectangle. Let w = f (z) where w = u + iv and z = x + iy ((u, v) ∈ D and (x, y) ∈ Ω). As a result one is left to work with the inhomogeneous Helmholtz equation on Ω: − 1 Σ(x, y) ∆ψ(x, y) = Eψ(x, y) .(1) where Σ ≡ df dz 2 . An explicit example of conformal map with the desired properties is clearly the exponential map f (z) = e z−Lx ,(2) which maps a rectangle of sides 2L x and 2π centered in the origin into an annulus of radiuses e −2Lx and 1 respectively. Fig. 1 displays the annulus obtained mapping a rectangle of sides L x = 2 and L y = 2π centered in the origin (clearly using a smaller L y one would obtain an arc instead of the full ring). Notice that the function of eq. (2) maps the horizontal sides into the horizontal segment which cuts the ring on the negative axis. This is clearly a different situation from those considered in Ref. [1]: as a matter of fact in this case the direct approach of Ref. [1] would describe an annulus with a straight cut on which Dirichlet boundary conditions are obeyed. Although this is also an interesting problem in itself, we want here to treat a ring with no cuts. The solution to this problem is obtained by imposing periodic (Dirichlet) boundary conditions on the horizontal (vertical) sides of the rectangle. Under the conformal map of eq. (2) the rectangle is then mapped into an annulus fulfilling Dirichlet boundary conditions on the smaller and larger circles. To implement the new boundary conditions in the problem we thus need to introduce a proper set of Little Sinc Functions (LSF): this set corresponds to LSF 1 of Ref. [8]. We report here the explicit form of the LSF 1 : s (I) k (N, L, x) = (−1) k (N + 1) sin (N +1)πx 2L sin πx 2L − πk N +1(3) with k = −N/2, . . . , N/2. Notice that we use here a different convention for N with respect to Ref. [8]. These functions define the homogeneous grid x k = 2Lk N + 1 and satisfy the orthogonality relation +L −L s (I) k (N, L, x)s (I) j (N, L, x)dx = 2L N + 1 δ kj .(4) Notice that h (I) ≡ 2L N +1 is the grid spacing of the LSF 1 . The LSF 2 fulfilling Dirichlet boundary conditions may be cast in the form s (II) k (N, L, x) = (−1) k N cos πk N sin N πx 2L sin πx 2L − sin πk N .(5) with k = −N/2+1, . . . , N/2−1 1 . In this case these function define the homogeneous grid x k = 2Lk N and obey the orthogonality relation +L −L s (II) k (N, L, x)s (II) j (N, L, x)dx = 2L N δ kj .(6) Notice that h (II) ≡ 2L N is the grid spacing of the LSF 2 . For a given N (even integer) there are N + 1 (N − 1) LSF functions obeying periodic (Dirichlet) boundary conditions, each peaked (with value 1) at a point x k and vanishing at the remaining grid points x j , j = k. A function f (x) obeying periodic bc may be interpolated using the s (I) k (h, N, x) as f (x) ≈ N/2 k=−N/2 f (x k )s (I) k (h, N, x) .(7) Similarly we may derive twice this expression to obtain d 2 f (x) dx 2 ≈ N/2 k=−N/2 f (x k ) d 2 s (I) k (x) dx 2 ≈ N/2 k=−N/2 N/2 j=−N/2 f (x k ) d 2 s (I) k (x) dx 2 xj s (I) j (h, N, x) ≡ N/2 k=−N/2 N/2 j=−N/2 f (x k ) c (I2) kj s (I) j (h, N, x),(8) where in the last line we have introduced the matrix c (I2) kj ≡ d 2 s (I) k (x) dx 2 xj , which provides a representation for the second derivative operator on the grid. The case of Dirichlet bc has already been discussed in Ref. [1], although it can be obtained straightforwardly repeating the same steps done here. To take into account the presence of different sets of LSF we modify the notation of Ref. [1] and call c (II2) kj the matrix elements of the second derivative obtained with Dirichlet bc. Notice that the latin indices k, j span different values in the two cases and that the grid points also differ in the two cases. Omitting some trivial steps (see Ref. [1] for more detail) we may now may easily discretize eqn. (1) using Dirichlet bc in the x direction and periodic bc in the y direction. A suitable "basis" for this discretization is obtained with the direct product of the LSF in each direction: − 1 Σ(x, y) ∆s (II) k (h, N, x)s (I) k (h, N, y) = − jj 1 Σ(x j , y j ) × c (II2) kj δ k j + δ kj c (I2) k j s (II) j (h, N, x)s (I) j (h, N, y) (9) where it is understood that the latin indices span different ranges for the two LSF. To obtain the matrix element of the operator on the grid we need to associate a single integer to any pair of indices which define an element of the two dimensional grid. We may write: k = K − (N − 1) K N − 1 + − N 2 (10) k = − N 2 + K N − 1 + ,(11) where → 0 + and [a] means integer part of a. In this way we are able to identify a point of the grid in terms of a single integer K, which takes values from 1 to N 2 − 1. Using these relations we may read off the matrix element ofÔ as O KK = − 1 Σ(x j , y j ) c (II2) kj δ k j + δ kj c (I2) k j .(12) Although the procedure described above uses grid with the same N for the x and y direction, a more appropriate choice is to use meshes with the same grid size on each orthogonal direction, i.e. h (I) ≈ h (II) . In this way we may establish the relation between the number of grid points on each direction N x ≈ L x L y (N y + 1) . We may easily understand the physics contained in this relation: for thin rings, L x L y , the number of grid points in the x-direction is much less than the number of grid points in the y-direction, since the excitation of trasverse modes requires much higher energy than the excitation of longitudinal modes. In this case one can extend the previous relations for the grid to: k = K − (N x − 1) K N x − 1 + − N x 2 k = − N y 2 + K N x − 1 + . The implementation of these considerations allows us to represent the differential operator as a ( N x − 1)(N y + 1) × (N x − 1)(N y + 1) hermitean matrix. It is useful to summarize the differences with the results of Ref. [1]: • The region Ω is a rectangle of sides 2L x and 2L y , where L y = π to allow a closed ring • The operatorÔ is represented on the grid by a (N x − 1)(N y + 1) × (N x − 1)(N y + 1) hermitean matrix • The collocation points, corresponding to the nodes of the LSF functions, differ for the x and y directions: on the x direction they are distributed following the zeroes of the LSF with Dirichlet boundary conditions, while on the y direction they are distributed following the zeroes of the LSF with periodic boundary conditions. Their numbers (N x and N y ) are also different, although the grid size is (approximately) the same. Apart from these small differences, the remaining features of the collocation method are unchanged. In particular, the matrix representingÔ on the grid is obtained from the product of a diagonal matrix, representing 1/Σ on the grid, with a non-diagonal sparse matrix, representing the Laplacian with mixed bc on the grid. While the first matrix is specific to the problem considered and therefore it needs to be calculated each time that a different shape is chosen, the second matrix is universal and therefore it can be calculated and stored once and for all. As before the calculation of the diagonal matrix is not computationally demanding since it just requires the evaluation of the function 1/Σ at the (N x − 1)(N y + 1) points forming the grid. We will illustrate later specific applications of the CCM. III. ANALYTICAL METHODS In this section we generalize the analytical approach of Ref. [1] to describe quantum rings. We may consider two different approaches: in the first one the quantum ring is obtained performing a conformal map of a rectangle centered in the origin (as described in the previous section); in the second one the quantum ring is obtained by applying a conformal map directly to a circular annulus. While the formulas of Ref. [1] hold both cases, one needs to work with different orthonormal basis in the two cases. In the first case where Ω is a rectangle of sides 2L x and 2π, the basis is obtained by the direct product of functions obeying Dirichlet and periodic boundary conditions, i.e. ψ nx (x) = 1 √ L x sin n x π 2L x (x + L x )(13) for the Dirichlet bc and χ 0 (y) = 1 √ 2π ,(14)χ ny (y) = 1 √ π cos (n y y) ,(15)φ ny (y) = 1 √ π sin (n y y) ,(16) for the periodic bc. Notice that n x,y = 1, 2, . . .. The basis on Ω may then be written as Ψ nx,ny,s (x, y) = ψ nx (x) × χ ny (y) , s = 1 φ ny (y) , s = 2 , where the value n y = 0 can only be reached for the states with s = 1. In the second case, the wave function of a circular annulus with a < r < b is (see for example Ref. [9]) Φ m,n,s (r, θ) = N mns Y m (k mn ) J m k mn r a − J m (k mn ) Y m k mn r a × cos nθ , s = 1 sin nθ , s = 2 , where J m and Y m are Bessel functions of first and second kind and k is the n th root of the equation Y m (k) J m kb a − J m (k) Y m kb a = 0 .(17) The energies of the annulus are then given by E mn = k mn a 2 , m = 0, 1, 2, . . . , n = 1, 2 . . . (18) Although one may work equally well with each of the two basis, from the point of view of an analytical calculation the first one offers the advantage of simplicity, since it involves only elementary functions. We will therefore focus on this basis. We consider the most general conformal transformation which maps the rectangle Ω onto a ring of arbitrary shape: g(z) = C ∞ k=0 η k e z−Lx k+1 ≡ Cḡ(z) ,(19) where η 0 = 1 and C > 0 is a constant factor, representing a dilation. The energies of the ring obtained using the mapping g(z) (E n ) are related to those of the ring obtained usingḡ(z) (Ē n ) by the simple relation 2 E n =Ē n C 2 . We may therefore work withḡ(z) and then simple rescale the energies obtained. Notice also that with the choice C = 1 and η k>0 = 0 we obtain the map to the circular annulus considered earlier. With simple algebra we obtain the conformal density Σ(x, y) ≡ dḡ dz 2 = ∞ k=0 ∞ j=0 η k η j (k + 1)(j + 1) · e x−Lx (k+j)+2 cos (y(k − j))(20) and σ(x, y) = Σ(x, y) − 1. As discussed in Ref. [1] we need to calculate the matrix elements of σ between the states of Ω. We find: n x , n y , s|σ(x, y)|n x , n y , s = −δ nx,n x δ ny,n y δ ss + ∞ k=0 ∞ j=0 η k η j (k + 1)(j + 1)∆ nx,ny,s,n x ,n y ,s (k, j) (21) where ∆ nx,ny,s,n x ,n y ,s (k, j) ≡ +Lx −Lx dx +π −π dy Ψ nx,ny,s (x, y) e x−Lx (k+j)+2 cos (y(k − j)) Ψ n x ,n y ,s (x, y) = α ny α n y δ ss W nx,n x (k, j)I (s) ny,n y ,k,j ,(22) and α 0 = 1/ √ 2πδ s,1 and α ny>0 = 1/ √ π. We have also introduced the definitions W nx,n x ,k,j ≡ +Lx −Lx ψ nx (x) e x−Lx (k+j)+2 ψ n x (x)dx = 8π 2 L x n x n x (j + k + 2)e −2Lx(j+k+2) (−1) nx+n x e 2Lx(j+k+2) − 1 (4L 2 x (j + k + 2) 2 + π 2 (n x − n x ) 2 ) (4L 2 x (j + k + 2) 2 + π 2 (n x + n x ) 2 )(23) and I (1) ny,n y ,k,j ≡ +π −π cos(n y y) cos(n y y) cos((k − j)y)dy = π 2 δ ny−n y +(k−j) + δ ny+n y +(k−j) + δ ny−n y −(k−j) + δ ny+n y −(k−j) I (2) ny,n y ,k,j ≡ +π −π sin(n y y) sin(n y y) cos((k − j)y)dy = π 2 δ ny−n y +(k−j) − δ ny+n y +(k−j) + δ ny−n y −(k−j) − δ ny+n y −(k−j) .(24) Notice that the I (s) ny,n y ,k,j have been already defined in Ref. [1]; the reader should also observe that there are no terms mixing states with different values of s and that I (1) ny,n y ,k,j = I (2) ny,n y ,k,j unless n y + n y = ±(k − j). We may understand the physical consequences of these properties by considering the approximate expression for the energy: E nx,ny,s ≈ nx,ny,s n x , n y , s|Σ(x, y)|n x , n y , s , where nx,ny,s ≡ n 2 x π 2 4L 2 x + n 2 y(26) are the "unperturbed" energies (i.e. the energy on the rectangle Ω). This expression has been derived in Ref. [1] resumming specific terms in the perturbative expansion corresponding to a geometric series: it has later been applied in Ref. [10] to obtain a Weyl-like law for inhomogeneous drums. With simple algebra we obtain n x , n y , s|Σ(x, y)|n x , n y , s = ∞ k=0 ∞ j=0 η k η j (k + 1)(j + 1) W nx,nx,k,k δ kj + (−1) s+1 W nx,nx,k,j δ 2ny−|k−j| = ∞ k=0 η 2 k (k + 1) 2 W nx,nx,k,k + (−1) s+1 ∞ k=2ny η k η k−2ny (k + 1)(k + 1 − 2n y )W nx,nx,k,k−2ny + (−1) s+1 ∞ k=0 η k η k+2ny (k + 1)(k + 1 + 2n y )W nx,nx,k,k+2ny ,(27) where the terms depending explicitly on s will be respon-sible of the breaking of the degeneration of levels. IV. APPLICATIONS We will now consider few applications of the numerical and analytical techniques developed here and in Ref. [1] to quantum rings of different shape. A. Circular annulus Our first application is to the calculation of the energies and wave functions of the circular annulus using the basis of the rectangle. We may use eq. (25) and obtain an explicit formula for the energies of a circular annulus: E nx,ny,s ≈ 2 log 2 (a) + π 2 n x 2 n y 2 log 2 (a) + π 2 n x 2 π 2 (a 2 − 1) n x 2 log(a) ,(28) where a = e −2Lx is the inner radius of the ring. Notice that the states corresponding to n y = 0 are nondegenerate, whereas the states corresponding to n y > 0 are doubly degenerate: in other words the basis that we are using reproduces the exact pattern of degeneration of the circular annulus. In Fig. 2 we compare the exact numerical results obtained solving eq. (18) (solid curve) with the results obtained using the analytical formula (28) (dashed curve) for the first 2000 states. The dotted line corresponds to the numerical results obtained using the CCM with a grid with N x = 14 and N y = 400 (remember that the ratio N x /(N y + 1) should approximately be L x /L y ). Finally the dot-dashed line are the results obtained using Weyls'law supplemented by Weyl's conjecture E n ≈ 4πn A + L A 4πn n + . . . ,(29) where A and L are area and perimeter of the ring respectively (Dirichlet boundary conditions are chosen). Amazingly the first three curves are practically indistinguishable. In Fig. 3 we display the the error of the analytical formula, which is about 0.1 % for all the states considered. We may also obtain precise results for the ground state of the ring using a variational approach. Following Ref. [1] we pick a trial state |χ = k c (0) k |k ,(30) where |k ≡ |k x , k y are eigenstates of the rectangle with mixed bc. Using the inverse operatorÔ −1 we may generate a new state |Ψ (1) 0 =Ô −1 |χ ,(31) which has a larger overlap with the true ground state of the system. The expectation value ofÔ in this state is E (1) 0 = Ψ (1) 0 |Ô|Ψ (1) 0 Ψ (1) 0 |Ψ (1) 0 ,(32) where Ψ By minimizing E (1) 0 with respect to the coefficients c (0) k one can now find precise estimates for the ground state energy. Let us now pick a specific trial state. We will work in the limit of a annulus with (b − a) 1, which corresponds to using L x 1. We use the trial state |χ = c (0) 0 ψ 1 (x)χ 0 (y) + ψ 1 (x) N ny=1 c (0) 2ny−1 φ ny (y) + c (0) 2ny χ ny (y) (35) neglecting in first approximation the excited states in the x direction. We need to calculate the matrix elements m|Σ 1/2 |l = 1, m y |Σ 1/2 |1, l y = +Lx −Lx ψ 2 1 (x)e x−Lx dx δ myly = π 2 e −Lx sinh(L x ) L 2 x + π 2 δ myly (36) m|Σ|l = 1, m y |Σ|1, l y = +Lx −Lx ψ 2 1 (x)e 2x−2Lx dx δ myly = π 2 1 − e −4Lx 4 (4L 2 x + π 2 ) δ myly (37) Therefore E (1) 0 = k c (0) k 2 k π 2 e −Lx sinh(Lx) L 2 x +π 2 2 k c (0) k 2 2 k π 2 e −Lx sinh(Lx) L 2 x +π 2 2 π 2 (1−e −4Lx ) 4(4L 2 x +π 2 ) = 1 π 2 (1−e −4Lx ) 4(4L 2 x +π 2 ) k c (0) k 2 k k c (0) k 2 2 k (38) where k = π 2 4L 2 x + k + 1 2 2(39) where k 2 is the integer part of k/2. Since one of the coefficients c k (for k ≥ 1) break the rotational invariance of the true ground state of the annulus and therefore they must raise the energy. This explains the very good precision of eq. (28) for annuli with b − a 1. To improve this formula for states with smaller inner radius we may form a variational trial state with a superposition of wave functions in the x direction: |χ = φ 0 (y) N nx=1 c (0) nx ψ nx (x)(40) Therefore we may write the matrix elements: m|Σ 1/2 |l = m x , 0|Σ 1/2 |l x , 0 = +Lx −Lx ψ mx (x)ψ lx (x)e x−Lx dx =    π 2 e −Lx m 2 x sinh(Lx) L 3 x +π 2 Lxm 2 x , m x = l x 8π 2 e −2Lx Lxmxlx(e 2Lx −(−1) mx +lx ) cos(π(mx+lx)) 16L 4 x +8π 2 L 2 x (m 2 x +l 2 x )+π 4 (m 2 x −l 2 x ) 2 , m x = l x m|Σ|l = m x , 0|Σ|l x , 0 = +Lx −Lx ψ mx (x)ψ lx (x)e 2x−2Lx dx =    π 2 e −4Lx (e 4Lx −1)m 2 x 4(4L 3 x +π 2 Lxm 2 x ) , m x = l x 16π 2 e −4Lx Lxlxmx(e 4Lx −(−1) lx +mx ) cos(π(lx+mx)) 256L 4 x +32π 2 L 2 x (lx 2 +mx 2 )+π 4 (mx 2 −lx 2 ) 2 , m x = l x(41) Notice that in this case n = π 2 n 2 x 4L 2 x . In Table I we report the ground state energy of the annulus calculated using the variational principle, i.e using the variational formula (38) together with eq. (40), with different numbers of variational parameters. The case N = 1 corresponds to the simple analytical formula (28), which is seen to work very well for larger values of a. It is easy to understand why more and more terms are needed as a gets smaller and smaller: since L gets larger, the conformal map strongly deforms the radial coordinate in the annulus. Looking at Fig. 1, which corresponds to a = 1/e 2 , we observe that the radial grid spacing is finer for smaller values of r. Therefore one needs more terms in eq. (38) to obtain good estimates of the energy. Notice that this argument also holds for the Conformal Collocation Method, since also the collocation points are also distributed non-uniformly in the radial direction, as the central hole is made smaller. Notice that in this case n = a N = 1 N = 2 N = 3 N = 4 N = 5 N =π 2 n 2 x 4L 2 x . In Fig. 4 we have plotted the exact energy of the ground state of the annulus as a function of the inner radius a and we have compared it with the different approximations considered earlier. The simple analytical formula of eq. (28) is seen to work quite well for thin annuli, while the most accurate variational calculation of Table I is seen to reproduce the exact results even for very small inner radiuses. We also report the CCM results corresponding to a grid with N = 50. B. Robnik's rings We consider the conformal map f (z) = e −Lx+z + αe −2Lx+2z(42) which maps the rectangle of sides 2L x and 2π into a deformed ring, which for α = 0 has the shape of a circular ring and for α = 1/2 has the shape of a cardiod ring. We have called the family of these annular billiards "Robnik's rings", in analogy with the simply connected billiards known as "Robnik's billiards" having the same external contour [7]. In this case we have: Σ(x, y) = 4α 2 e 4x−4Lx + 4αe 3x−3Lx cos(y) + e 2x−2Lx (43) Notice that in this case the width of the ring is not uniform: we call r ∓ the smallest and largest width of the ring respectively, which read r ± = ±α 1 − e −4Lx − e −2Lx + 1 .(44) Their average is insensitive to α: r ≡ r + + r − 2 = 1 − e −2Lx .(45) Notice also that 1 −r = e −2Lx = a (in this case a is the average inner radius of the quantum ring). Using the general results obtained earlier we may write the analytical formula: E nx,ny,s ≈ 2 log 2 (a) + π 2 n x 2 n y 2 log 2 (a) + π 2 n x 2 π 2 (a 2 − 1) n x 2 log(a) + α 2 R nx (a) ,(46) where R nx (a) ≡ 2π 2 a 4 − 1 n x 2 log(a) log 2 (a) + π 2 n x 2 . This formula reduces to eq. (28) for α = 0. In Fig.6 we display the energies of the first 2000 states of an annulus obtained conformally mapping a rectangle of sides 2L x = 1/5 and 2L y = 2π with the map f (z) = z + αz 2 using α = 1/10. The solid line represents the precise numerical values obtained using CCM with a grid with N x = 14 and N y = 400; the dashed line corresponds to the results of the analytical formula of eq. (46); finally, the dotted line corresponds to Weyl's formula eq. (29). In Fig. 6 we display a detail of the previous figure, for highly excited states around n = 2000. Looking at the figure we see that our analytical formula describes more accurately than Weyl's equation, eq. (29), the low lying part of the spectrum, although the latter describes better the high part of the spectrum. We also notice that the analytical formula displays a behaviour already observed for the circular annulus: around specific n, corresponding to the opening of a new trasversal mode, the curve displays a small change of slope. While this behaviour is correct for the circular annulus, where it is indeed observed, neither the numerical results nor (of course) Weyl's expression display such a behaviour. However we may easily understand the origin of it: the analytical formula eq. (46) is obtained using a resummation of the terms in the Rayleigh-Schrödinger series which contains only the expectation values of the conformal density in a given state. For the circular annulus, this approximation is quite good because there is no mixing between transversal and longitudinal modes as a result of the rotational symmetry of the domain: this is reflected in the amazing accuracy of the analytical formula in this case. For a general annulus, the domain is not invariant under rotation and transverse and longitudinal modes actually mix. Already to second order in perturbation theory, there are terms which allow this mixing, although these are not present in the resummed formula. As a check of the quality of the numerical results obtained with the CCM we have applied a method devised by Berry, ref. [11], which allows one to extract the geometrical feature of a domain from a limited sequence of eigenvalues, using improved eigenvalues sums. For the area, for example, one obtains a family of approximating functions whose first few elements read [11] A 0 (t) = 4πt N n=0 e −Ent (47) A 1 (t) = 4πt N n=0 e −Ent (2E n t − 1) (48) A 2 (t) = 4πt N n=0 e −Ent E n t(2E n t − 3) (49) A 3 (t) = 4πt N n=0 e −Ent E n t(4E 2 n t 2 − 12E n t + 3)/3 (50) . . . = . . . Analogous expressions are given in Ref. [11] for the length and constant terms in the asymptotic expasion of the function counting the number of states up to a given energy [12] n (E) ≈ AE/4π − L √ E/4π + C + . . .(52) The constant C should vanish for domains with one hole. In figs.8,9 and 10 we have displayed the approximants for the area, perimeter and for the constant term built using the first 2000 numerical eigenvalues obtained with the CCM with a grid 14 × 400. The horizontal lines are the exact values. To estimate the optimal value of t where the approximant should be evaluated we minimize the sum of the squares of the derivatives of each approximant. For example, for the area we minimize the quantity A 1 (t) 2 + A 2 (t) 2 + A 3 (t) 2 (we leave out A 0 (t) because it is clearly less precise). For the length and constant terms we use the approximants displayed in the figures. As shown in Table II the method of Berry allows us to extract the area, perimeter and constant terms of the annulus with a truly amazing precision. Notice that the exact values of the area and perimeter are obtained directly using the conformal density: A = Ω dxdy Σ(x, y) (53) L = ∂Ω ds Σ(x, y) ,(54) where Ω is the rectangle of sides 2L x = 1/5 and 2L y = 2π and ∂Ω is the border of Ω where Dirichlet bc are obeyed 3 . As we did before, we may also resort to a variational calculation and use the trial state: |χ = c (0) 0 ψ 1 (x)χ 0 (y) + ψ 1 (x) N ny=1 c (0) 2ny−1 φ ny (y) + c (0) 2ny χ ny (y) (55) neglecting in first approximation the excited states in the x direction. In the case of the circular ring we have seen that only the first term contributes, due to the rotational invariance of the ground state; this limit corresponds to the simple formula (25), which was seen to work remarkably well for thin circular rings. In the present case we do not expect that the lowest order formula work well even for thin rings, due to the varying width of the ring, which favors the breaking of the rotational invariance. We may apply this variational calculation to the Robnik annulus that we have considered before, which corresponds to α = 1/10 and L x = 1/10, shown in Fig. 5. In In Fig. 11 we show the variational estimates for the ground state energy of this ring obtained using a varying number of variational parameters N var (the case N var = 0 corresponds to the simple analytical formula (25)); the horizontal line is the precise value obtained with the Conformal Collocation method with a grid with N = 100. As anticipated, the eq. (25) in this case provides a poor approximation to the exact energy, leading to a substantial overestimation of it. On the other hand, the inclusion of few longitudinal modes leads to quite drastic improvement, which can be already appreciated for N var = 2. We should point out that, even letting N var ⇒ ∞ (assuming that we can perform the calculation), we do not expect that the variational energy converge to the exact energy: the reason of this is that the variational ansatz that we are using includes only longitudinal modes (in the y direction) while completely neglecting the trasversal ones (in the x direction). For thin rings, the latter are less important, and therefore we are able to obtain good (but not arbitrarily good) approximations; for thick rings, on the other hand, only the inclusion of both modes will lead to acceptable approximations. Finally in Fig. 12 we have plotted the wave function of the ground state of this annulus: as anticipated, this wave function is not invariant under arbitrary rotations and it is peaked in the region where the ring is wider, as expected. The wave function is multiplied by a factor 10 to better appreciate the localization in the region where the width of the annulus is larger. V. CONCLUSIONS In this paper we have extended the powerful techniques of Ref. [1,10] to the study of quantum rings; working on specific examples we have obtained both numerical and analytical results of remarkable precision. In fact our approach provides a systematic tool for the study of the spectrum of quantum rings of arbitrary shape, not relying on specific assumptions concerning the shape or the width of the ring. Clearly, the problem of finding the particular conformal map which sends the rectangle to a specific ring may be attacked numerically, when the explicit expression is unknown [13,14]. The results that we obtain in the paper maybe roughly classified into: • numerical results obtained using the Conformal Collocation Method (CCM) for a large number of levels (≈ 2000); the accuracy of these results has been verified using Berry's approximants to estimate the geometrical features of the rings with a selected number of levels, calculated numerically; • variational estimates for the ground state of a quantum ring; • explicit analytical formulas for the whole spectrum of a quantum ring; the relation of these formulas with Weyl's law is also obtained in light of the results of Ref. [10]; We believe that the importance of our methods thus relies in the possibility of describing with precision both the low and high end of the spectrum of a quantum ring and on the possibility of improving this precision in a simple and sistematic fashion. To the best of our knowledge there are no other approaches in the literature which share these features. Finally, we wish to mention that the approach that we have described here can be applied directly also to annular drums of variable density (for simply connected drums this has been done in Ref. [10]). FIG. 1 : 1(color online) Annulus obtained conformally mapping a rectangle of sides 2Lx = 2 and 2Ly = 2π centered in the origin. FIG. 2 : 2(color online) Energy of the first 2000 states of an annulus with a = 9/10 and b = 1. The solid line are the exact results of eq. (18); the dashed line corresponds to eq. (28); the dotted line are the numerical results obtained with CCM; the dot-dashed line are the results obtained with Weyl's law supplemented by Weyl's conjecture. FIG. 3: (color online) Relative error of the analytical formula (28) with respect to the exact results for k ≥ 1, which is equivalent to working with eq. (25). This fact should not be surprising, since the states corresponding to nonvanishing c I: Energy of the ground state of the annulus with varying inner radius calculated using the variational formula (38) together with eq. (40), with different numbers of variational parameters (N − 1 is the number of variational parameters) . FIG. 4 : 4(color online) Ground state energy of the annulus as a function of the inner radius keeping the outer radius fixed (b = 1). The dashed line is the exact result, while the dashed line is the analytical formula of eq. (28). The dotted line is the result obtained using CCM with N = 50; the pluses are the variational results of Table I corresponding to N = 10. FIG. 5: (color online) Asymmetric annulus obtained conformally mapping a rectangle of sides 2Lx = 1/5 and 2Ly = 2π centered in the origin. α = 1/10. FIG. 6 : 6(color online) Energies of the first 2000 states of an annulus obtained conformally mapping a rectangle of sides 2Lx = 1/5 and 2Ly = 2π with the map f (z) = z + αz 2 using α = 1/10. The solild line are the numerical results obtained using CCM; the dashed line corresponds to the analytical formula, the dotted line is Weyl's law. FIG. 7 : 7(color online) Detail of the previous figure. FIG. 9 : 9(color online) Length approximant functions Lm(t) for the first 2000 eigenvalues obtained with CCM with a grid 14 × 400. FIG. 10: (color online) Constant approximant functions Cm(t) for the first 2000 eigenvalues obtained with CCM with a grid 14 × 400. FIG. 11 : 11(color online) Energy of the ground state of the asymmetric annulus with Lx = 1/10 and α = 1/10. The horizontal line is the precise numerical value obtained with the CCM with a grid with 14 × 400; the pluses are the variational results obtained with a varying number of variational parameters. this case we have r − = 0.148 , r + = 0.214 . FIG. 12 : 12(color online) Wave function (multiplied by a factor of 10) of the ground state of the Robnik's annulus with Lx = 1/10 and α = 1/10. TABLE TABLE II : IIEstimates for the geometrical features of the annulus obtained conformally mapping a rectangle of sides 2Lx = 1/5 and 2Ly = 2π with the map f (z) = z + αz 2 using α = 1/10 using the method of Berry,[11] on the numerical results obtained with the CCM. The last row contains the exact results.FIG. 8: (color online) Area approximant functions Am(t) for the first 2000 eigenvalues obtained with CCM with a grid 14 × 400. The reader may check that the expression given here for these function is equivalent to the one used before in Ref.[1] In ref.[10] these relations are used to derive Weyl's law from perturbation theory. AcknowledgmentsP.Amore ackowledges support of Conacyt through the SNI fellowship. P Amore, arXiv:0910.4798v1Spectroscopy of drums and quantum billiards: perturbative and non perturbative results. quant-phP. Amore, Spectroscopy of drums and quantum billiards: perturbative and non perturbative results, accepted on the Journal of Mathematical Physics, arXiv:0910.4798v1 [quant-ph] (2009) . P Amore, Journal of Physics A. 41265206P. Amore, Journal of Physics A 41, 265206 (2008) . J Goldstone, R L Jaffe, Phys.Rev.B. 4514100J.Goldstone and R.L. Jaffe, Phys.Rev.B 45, 14100 (1992) . D Gridin, A T I Adamou, R V Craster, Phys. Rev. B. 69155317D. Gridin, A. T. I. Adamou, and R. V. Craster, Phys. Rev. B 69, 155317 (2004) . J B Keller, S I Rubinow, Ann. Phys. Leipzig. 924J.B. Keller and S.I. Rubinow, Ann. Phys. Leipzig 9, 24 (1960) . J B Keller, Rev, 27485J.B. Keller, SIAM Rev. 27, 485 (1985) . M Robnik, J. Phys.A. 17M. Robnik, J. Phys.A 17, 1049-1074 (1984) . P Amore, F M Fernández, K Salvo, R A Saénz, J. Phys.A. 115302P. Amore, F.M.Fernández, K. Salvo and R.A. Saénz, J. Phys.A, 115302 (2009) . J R Kuttler, V G Sigillito, SIAM Review. 26J.R. Kuttler and V.G. Sigillito, SIAM Review 26 (1984) 163-193 . P Amore, arXiv:0912.1402v1math-phP. Amore, arXiv:0912.1402v1 [math-ph] (2009) . M V Berry, J. Phys.A. 20M.V.Berry, J. Phys.A 20, 2389-2403 (1987) H P Baltes, E R Hilf, Spectra of Finite Systems. Mannheim: B-I WissenschaftsverlagBaltes H P and Hilf E R 1976 Spectra of Finite Systems (Mannheim: B-I Wissenschaftsverlag) A MATLAB Toolbox for Schwarz-Christoffel mapping. T A Driscoll, ACM Trans. Math. Soft. 22T. A. Driscoll. A MATLAB Toolbox for Schwarz- Christoffel mapping, ACM Trans. Math. Soft. 22 (1996), pp. 168-186. Algorithm 843: Improvements to the Schwarz-Christoffel Toolbox for MATLAB. T A Driscoll, ACM Trans. Math. Soft. 31T. A. Driscoll. Algorithm 843: Improvements to the Schwarz-Christoffel Toolbox for MATLAB, ACM Trans. Math. Soft. 31 (2005), 239-251.
[]
[ "EXTENSIONS AND RENORMALIZED TRACES", "EXTENSIONS AND RENORMALIZED TRACES" ]
[ "Denis Perrot [email protected] \nUMR 5208\nUniversité de Lyon\nUniversité Lyon 1\nCNRS\nInstitut Camille Jordan\n43, bd du 11 novembre 191869622Villeurbanne CedexFrance\n" ]
[ "UMR 5208\nUniversité de Lyon\nUniversité Lyon 1\nCNRS\nInstitut Camille Jordan\n43, bd du 11 novembre 191869622Villeurbanne CedexFrance" ]
[]
It has been shown by Nistor [8] that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms[11,12], leading to "local" index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue.
10.1007/s00208-011-0665-0
[ "https://arxiv.org/pdf/0908.1757v2.pdf" ]
115,171,742
0908.1757
d54e7190b50d4f19563a7697039c40ef899d4ca7
EXTENSIONS AND RENORMALIZED TRACES 1 Nov 2009 November 1, 2009 Denis Perrot [email protected] UMR 5208 Université de Lyon Université Lyon 1 CNRS Institut Camille Jordan 43, bd du 11 novembre 191869622Villeurbanne CedexFrance EXTENSIONS AND RENORMALIZED TRACES 1 Nov 2009 November 1, 2009extensionsK-theorycyclic cohomology MSC 2000: 19D5519K5646L8046L87 It has been shown by Nistor [8] that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms[11,12], leading to "local" index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue. Introduction Some years ago Cuntz and Quillen were able to show that excision holds in complete generality for periodic cyclic (co)homology of associative algebras [6]. That is, given any extension (short exact sequence) of algebras over C, (E) : 0 → B → E → A → 0(1) there exists an associated six-term exact sequence relating the periodic cyclic homology of B, E , A , and similarly for cohomology. Using the abstract properties of the theory, Nistor [8] then proved that the connecting morphism HP 1 (A ) → HP 0 (B) of the cyclic homology exact sequence is compatible, via the Chern-Connes character, with the index map induced by the extension (E) on algebraic K-theory in low degrees [7]: Ind E : K 1 (A ) → K 0 (B) .(2) In principle this allows to state a general "higher index theorem", in the sense that the pairing of any periodic cyclic cohomology class [τ ] ∈ HP 0 (B) with the image of (2) can be computed as the pairing of its boundary E * ([τ ]) with K 1 (A ). Here E * : HP 0 (B) → HP 1 (A ) denotes the connecting morphism in cohomology. Nistor proves this theorem first in the case of a universal extension, for which the periodic cyclic cohomology is simply represented by traces over B, then the general case follows from the naturality of the index morphism in K-theory and the naturality of the boundary map in periodic cyclic cohomology. Although very elegant and general, this proof does not provide explicit formulas for the cocycle E * ([τ ]). In principle the proof of excision in [6] should lead to explicit formulas, but they turn out to be extremely complicated in general, and moreover are not local in contrast with, for instance, the residue index formula of Connes and Moscovici [3]. The goal of the present article is to present an explicit construction of the connecting morphism E * avoiding as much as possible the use of excision, and giving an alternative (direct) proof of Nistor's index theorem. One knows from the work of Cuntz and Quillen [5] that any cyclic cohomology class [τ ] ∈ HP 0 (B) can be represented by a trace over an adequate extension 0 → J → R → B → 0 of B, or equivalently by a trace over some power of this extension (think for example about the operator trace on a Schatten ideal). Our basic observation is the following: if the extensions 0 → J → R → B → 0 and 0 → B → E → A → 0 fit together in a commutative diagram (see (9)), then E * ([τ ]) is explicitely given by a fairly simple formula based on a "renormalization" procedure explained in section 2. The proof that actually any cyclic cohomology class over B can be represented in this way requires the knowledge of excision. Fortunately many cyclic cohomology classes appear naturally equipped with the required diagram, so we are able to circumvent excision completely in this situation. Let us mention that the term "renormalization" is inspired by our previous work on the bivariant Chern character for quasihomomorphisms [11,12,13], where it was argued that this procedure yields local index formulas automatically. This is related to the well-known anomalies of quantum field theory [10]. In fact we show in section 3 that when the extension (E) is invertible in a specific sense (Definition 3.1), the map E * coincides with the bivariant Chern character of the odd quasihomomorphism associated to the extension. This allows to give an alternative proof of Nistor's index theorem in section 4: for any [τ ] ∈ HP 0 (B) and [g] ∈ K 1 (A ), one has the equality of pairings [τ ], Ind E ([g]) = √ 2πi E * ([τ ]), [g](3) that is, the index map is adjoint to the connecting morphism in periodic cyclic cohomology. The overall factor √ 2πi comes from our particular choice of normalization for the pairings between cyclic cohomology and K-theory: just note that this choice is the only one compatible with the bivariant Chern character and Bott periodicity for topological algebras. Since we will consider only algebras without additional structure in this article, this factor is irrelevant. The index theorem is shown in two steps: first we reduce to the case of an invertible extension, and then (3) is the consequence of an explicit computation. Thus in contrast with [8], excision is not directly used in the proof. In section 5 we show on the example of the family index theorem that our construction of E * ([τ ]) effectively leads to local formulas. Thus we consider a proper submersion of smooth manifolds without boundary M → B. A canonical extension (E) is obtained by taking E as the algebra of smooth families of (fiberwise) classical pseudodifferential operators of order zero, B as the ideal of order −1 pseudodifferential operators, and A as the commutative algebra of smooth functions over the cotangent sphere bundle of the fibers. The projection E → A thus carries a family of pseudodifferential operators to its family of leading symbols. Then any de Rham cycle in the base manifold B gives rise to a cyclic cocycle τ over the algebra B; notice however that this requires to choose a connection on the submersion. Using zeta-function renormalization, we find that the cyclic cohomology class E * ([τ ]) ∈ HP 0 (A ) is given explicitely in terms of a fiberwise Wodzicki residue applied to some families of pseudodifferential operators, involving the connection and its curvature. Interestingly, the formula is a higher analogue of the famous Radul cocycle [14]. Finally if Q is a family of elliptic pseudodifferential operators with symbol class [g] ∈ K 1 (A ), the pairing between [τ ] and the "index bundle" Ind E ([g]) ∈ K 0 (B) is the evaluation of this higher Radul cocycle on certain polynomials built from Q, its parametrix P , and the connection. Connecting morphism Let us recall the Cuntz-Quillen formalism of cyclic cohomology [5], since it is particularly well-adapted to extensions. The basic fact is that any cyclic cohomology class of even degree over an associative algebra B can be represented by a trace over some extension R 0 → J → R → B → 0(4) vanishing on the large powers of the ideal J . A cyclic cohomology class of odd degree over B can be represented by a cyclic one-cocycle on R with similar vanishing properties. This motivates the definition of the X-complex of any algebra R. It is the Z 2 -graded complex X(R) : R ♮d ⇄ b Ω 1 R ♮ ,(5) where Ω 1 R ♮ = Ω 1 R/[R, Ω 1 R] is the quotient of the R-bimodule of universal one-forms by its commutator subspace. The class of a generic element (x 0 dx 1 mod [, ]) ∈ Ω 1 R ♮ is usually denoted by ♮x 0 dx 1 . The map ♮d : R → Ω 1 R ♮ thus sends x ∈ R to ♮dx. Also, the Hochschild boundary map b : Ω 1 R → R vanishes on the commutator subspace, hence passes to a well-defined map b : Ω 1 R ♮ → R. Explicitly the image of ♮x 0 dx 1 by b is the commutator [x 0 , x 1 ]. These maps satisfy ♮d • b = 0 and b • ♮d = 0, so that X(R) endowed with the boundary operator ∂ = ♮d ⊕ b indeed defines a Z 2 -graded complex. If J ⊂ R is a two-sided ideal, Cuntz and Quillen define a decreasing filtration of X(R) by the following subcomplexes indexed by integers n ∈ Z F 2n J X(R) : J n+1 + [J n , R] ⇄ ♮J n (+) d R (6) F 2n+1 J X(R) : J n+1 ⇄ ♮(J n+1 (+) d R + J n (+) d J ) , where J n (+) is equal to the power J n for n > 0 and equal to the unitalized algebra R + = R ⊕ C for n ≤ 0. The J -adic completions of the algebra R and of the complex X(R) are defined as projective limits R = lim ← − n R/J n , X( R) = lim ← − n X(R)/F n J X(R)(7) where R is viewed as a pro-algebra and X( R) as a pro-complex [6]. It follows that any cocycle τ : X( R) → C represents a cyclic cohomology class over B (here τ is viewed as a linear map between pro-complexes, that is, a linear map on X(R) vanishing on F n J X(R) for some n ≫ 0). In particular let T B = B ⊕ B ⊗ B ⊕ . . . be the non-unital tensor algebra and denote by JB the kernel of the multiplication homomorphism T B → B. Then the extension 0 → JB → T B → B → 0 is universal among all extensions 0 → J → R → B → 0 in the sense that one has a classifying homomorphism T B → R defined up to homotopy, which restricts to a homomorphism JB → J . Thus any cocycle over X( R) can be pulled back to a cocycle over X( T B). In particular the cohomology group H * (X( T B)) is isomorphic to the periodic cyclic cohomology HP * (B), see [5]. It follows from the proof of excision in periodic cyclic cohomology [6], that any extension (E) : 0 → B → E → A → 0(8) gives rise to a connecting morphism HP i (B) → HP i+1 (A ), i ∈ Z 2 . Here we shall present a way to calculate the connecting morphism, assuming that the cyclic cohomology classes of B are put into a suitable form. They will be represented not only by traces over some extension R, but more generally by traces over some power R n : indeed cyclic cohomology classes often arise as traces over finitely summable operator ideals [1]. Since one has to choose extensions of both algebras B and A to represent their cyclic cohomology, we first define the notion of a lifting for the extension (E): Definition 2.1 We say that an extension 0 → R → M → P → 0 is a lifting of 0 → B → E → A → 0, if both fit into a commutative diagram 0 0 0 0 / / J / / N / / Q / / 0 0 / / R / / M / / P / / 0 0 / / B / / E / / A / / 0 0 0 0 (9) where all rows and columns are extensions. Morally the columns of (9) will be used to represent cyclic cohomology classes of B, E , A respectively. Because the central algebra M has two distinguished ideals R and N , there are several ways to filter the complex X(M ). First we focus on the middle column, i.e. the extension 0 → N → M → E → 0. We denote the N -adic completions with a hat: M = lim ← − n M /N n , X( M ) = lim ← − n X(M )/F n N X(M ) .(10) There is a second extension 0 → R + N → M → A → 0 associated to the diagonal of (9). The corresponding (R + N )-adic completions will be denoted with a tilde. Of course this completion of X(M ) could be defined via the filtration by the subcomplexes F n R+N X(M ), but there is an equivalent construction starting from the above pro-complex X( M ) filtered by the subcomplexes F n R X( M ) = lim ← −k F n R X(M )/(F n R X(M ) ∩ F k N X(M )): M = lim ← − n M /(R + N ) n , X( M ) = lim ← − n X( M )/F n R X( M ) .(11) The cocycles over F n R X( M ) are chain maps τ : F n R X( M ) → C. We will show below that they represent cyclic cohomology classes over B. Let us describe now how to compute the connecting morphism associated to the initial extension (E) : 0 → B → E → A → 0. Definition 2.2 Let τ be a cocycle over the subcomplex F n R X( M ) for some n ≥ 1. A renormalization of τ is a linear map τ R : X( M ) → C extending τ . Of course τ R is usually not a cocycle over X( M ). Its coboundary τ R ∂ is however a cocycle vanishing on F n R X( M ) by construction. Here ∂ = ♮d ⊕ b is the X-complex boundary map. Since F n+k R X( M ) ⊂ F n R X( M ) for any k ≥ 0, one sees that τ R ∂ descends to a unique cocycle over X( M ). It remains to pull it back to X( T A ) in order to get a periodic cyclic cohomology class over A . Choose a linear splitting σ : A → M of the diagonal homomorphism M → A . The universal property of the tensor algebra T A allows to extend σ to a homomorphism σ * : T A → M by setting σ * (a 1 ⊗. . .⊗a k ) = σ(a 1 ) . . . σ(a k ). Then σ * sends the ideal JA = Ker(T A → A ) to the ideal R +N = Ker(M → A ). This may be depicted through the following commutative diagram where all arrows except the dashed one are homomorphisms of algebras: 0 / / JA / / σ * T A / / σ * A / / σ } } 0 0 / / R + N / / M / / A / / 0(12) Consequently σ * extends to a homomorphism of pro-algebras T A → M . This in turn induces a chain map σ * : X( T A ) → X( M ). (9). The map sending a cocycle τ over F n R X( M ) to the cocycle τ R ∂ • σ * over X( T A ), for an arbitrary choice of renormalization τ R , descends to a morphism in cohomology Proposition 2.3 Consider an extension (E) : 0 → B → E → A → 0 with lifting 0 → R → M → P → 0 as in diagramE n : H i F n R X( M ) → HP i+1 (A ) , i ∈ Z 2 .(13) The latter does not depend on the choice of renormalization, nor on the linear splitting σ : A → M . If the exact sequence (E) is split by a homomorphism A → E , then E n vanishes. Finally, E n+1 composed with the natural pullback H * F n R X( M ) → H * F n+1 R X( M ) coincides with E n . Proof: If τ ′ R is another choice of linear extension for τ , the difference τ ′ R − τ R vanishes on F n R X( M ), hence descends to a cochain over X( M ). The difference τ ′ R ∂ − τ R ∂ = (τ ′ R − τ R )∂ is therefore a coboundary over X( M ). The cyclic cohomology class of τ R ∂ • σ * is thus renormalization-independent. If τ = ϕ∂ is the coboundary of a cochain ϕ over F n R X( M ), then one can extend ϕ to a cochain ϕ R over X( M ) and take τ R = ϕ R ∂. Then τ R ∂ = 0 and E n is well-defined in cohomology. As observed by Cuntz and Quillen [5], two different choices of linear splittings σ : A → M induce homotopic homomorphisms σ * : T A → M , and the resulting chain maps X( T A ) → X( M ) are homotopy equivalent. Thus the cyclic cohomology class of τ R ∂ • σ * is independent of σ. Suppose that (E) is split by a homomorphism ρ : A → E . Then choose any linear splitting ℓ : E → M of the projection homomorphism M → E and put σ = ℓ • ρ. By the universality of T A one gets a commutative diagram 0 / / JA / / σ * T A / / σ * A / / ρ σ | | 0 0 / / N / / M / / E / / ℓ c c 0 Since JA lands in N , the homomorphism σ * : T A → M actually factors through M . Hence the composite map τ R • σ * is a well-defined cochain over X( T A ), the cocycle τ R ∂ • σ * = (τ R • σ * )∂ is a coboundary, and E n vanishes. The last assertion is obvious. Remark 2.4 If N is nilpotent then the projective limit M reduces to M . This has the following important consequence concerning the cocycles of even degree τ : F 2n+1 R X(M ) → C. Indeed, such a cocycle is a linear map τ : R n+1 → C vanishing on the commutator subspace [R n+1 , M ] + [R n , R]. In particular for n ≥ 1 the inclusion [R n+1 , M ] ⊂ [R n , R] holds, hence any reference to M disappears. We can conclude that a cohomology class in H 0 (F 2n+1 R X(M )) is simply represented by a trace over the (n+1)-th power of the nilpotent extension 0 → J → R → B → 0, that is, a linear map τ : R n+1 → C vanishing on [R n , R] for n ≫ 0. In the general case N is not nilpotent, and an even cocycle τ has to verify the additional condition that it vanishes on N k for some k ≫ 0. However one recovers the nilpotent situation after replacing the second row of (9) by the new extension 0 → R/(R ∩ N k ) → M /N k → P/Q k → 0, and the first row by 0 → J /(J ∩ N k ) → N /N k → Q/Q k → 0. Then τ still defines a cocycle for this diagram and the new ideal N /N k is nilpotent. We nevertheless prefer to stay in the general context since important examples of universal extensions are not nilpotent. Given any extension (E) : 0 → B → E → A → 0 there always exists a lifting in the sense of Definition 2.1. Indeed one can consider the following universal lifting 0 0 0 0 / / J(B : E ) / / / / A / / 0 0 0 0 (14) where the ideal T (B : E ) (resp. J(B : E )) denotes the kernel of the homomorphism T E → T A (resp. JE → JA ). The universal property of the tensor algebras T E and T A induce classifying maps from the second and third column of (14) to the second and third column of (9) respectively, and this in turn implies a classifying map for the first column also. The central classifying homomorphism T E → M and all other ones are defined up to homotopy, which ensures a canonical pullback morphism H * F n R X( M ) → H * F n T (B:E ) X( T E ) for all n. In fact the excision property of periodic cyclic cohomology [6] shows the following Lemma 2.5 Let (E) : 0 → B → E → A → 0 be an extension. Then for any n ≥ 1 one has an isomorphism H * F n T (B:E ) X( T E ) ∼ = HP * (B) .(15) Proof: According to the terminology of Cuntz and Quillen the pro-algebra T E is a quasi-free extension of T A . Moreover T A is also quasi-free, hence its homological dimension is ≤ 1. The results of [5] imply that the quotient complex X( T E )/F n T (B:E ) X( T E ) computes the periodic cyclic homology of A provided that n ≥ 1. Consequently all the complexes F n T (B:E ) X( T E ) are homotopy equivalent for n ≥ 1. In particular taking n = 1 one computes easily that the complex X( T E )/F 1 T (B:E ) X( T E ) is isomorphic to X( T A ), whence a short exact sequence of Z 2 -graded complexes 0 → F 1 T (B:E ) X( T E ) → X( T E ) → X( T A ) → 0 The associated six-term cohomology exact sequence relates H * F 1 T (B:E ) X( T E ) to HP * (E ) and HP * (A ). Now excision ( [6]) precisely says that the natural in- clusion X( T B) → F 1 T (B:E ) X( T E ) is a homotopy equivalence, which yields an isomorphism HP * (B) ∼ = H * F n T (B:E ) X( T E ) for any n ≥ 1. The above lemma remains unchanged if the tensor algebras T E and T A are replaced by any quasi-free extensions of E and A respectively in Diagram (14). Thus the cohomology groups H * F n R X( M ) provide an alternative way to represent the periodic cyclic cohomology of B. We will show in section 4 how to recover the pairing between HP 0 (B) and the K-theory group K 0 (B) in this context. For the moment observe that the morphism E n : H i F n R X( M ) → HP i+1 (A ) factors through the universal group H i F n T (B:E ) X( T E ) ∼ = HP i (B) . We summarize these results in a corollary. E * : HP i (B) → HP i+1 (A ) , i ∈ Z 2 (16) which coincides with the connecting morphism of the extension (E) given by excision. Proof: We have only to show that the map HP i (B) → HP i+1 (A ) is the connecting morphism of the extension. Set T E = lim ← −n T E /(T (B : E ) + JE ) n and denote by ι : T E → T E the natural homomorphism. Also let π * : T E → T A andπ * : T E → T A be the homomorphisms induced by the projection π : E → A . Then one hasπ * • ι = π * , whence a commutative diagram of Z 2 -graded complexes and chain maps 0 / / F 1 T (B:E ) X( T E ) / / X( T E ) π * / / ι X( T A ) / / 0 X( T E )π * : : u u u u u u u u u where the row is an exact sequence. Consider a linear splitting σ : A → T E in Diagram (14) as follows: first choose a linear splitting A → E , and then map E into the subspace of one-tensors in T E . The induced homomorphism σ * : T A → T E provides a right inverse forπ * : it is indeed sufficient to check the identityπ * •σ * = Id b T A on the subspace A which generates the whole tensor algebra T A . Then by excision, we know that any class [τ ] ∈ HP * (B) can be represented by a cocycle τ over F 1 T (B:E ) X( T E ). The connecting morphism of the extension (E) is nothing else but the boundary map associated to the above exact sequence of complexes: first extend τ to a linear map τ R over X( T E ). Then its coboundary τ R ∂ descends to a unique cocycle ϕ over X( T A ) such that ϕ • π * = τ R ∂. By definition the cyclic cohomology class of ϕ is the image of [τ ]. But observe that ϕ •π * is a cocycle over X( T E ), whose pullback via the map ι is precisely τ R ∂. Hence ϕ •π * is the (unique) descent of τ R ∂ over X( T E ). The composite map τ R ∂ • σ * = ϕ •π * • σ * = ϕ, which represents E * ([τ ]) , therefore coincides with the image of [τ ] under the connecting morphism. Remark 2.7 Excision has been used only to show that any class in HP * (B) can be represented as a cocycle over F n R X( M ) for an adequate diagram (9) and some n. Once this is known, the connecting morphism E * is given by the straightforward computation τ → τ R ∂ • σ * . Quasihomomorphisms The previous description of the connecting morphism is intimately related to our construction of a bivariant Chern character for quasihomomorphisms [11,12], if we restrict to a particular class of extensions: Definition 3.1 An extension 0 → B → E → A → 0 is invertible if there exists an algebra homomorphism ρ : A → M 2 (E ) , such that the off-diagonal entries of the matrix ρ are linear maps from A to the ideal B ⊂ E , and the upper left corner of the matrix is a linear splitting of the projection homomorphism E → A . From an invertible extension we construct a quasihomomorphism of odd degree as follows [11]. Let C 1 = C ⊕ εC be the first Clifford algebra: it is the Z 2 -graded algebra generated by the unit 1 in degree zero and the element ε in degree one, with ε 2 = 1. Define the algebra E s = C 1 ⊗E s + , where E s + is the (trivially graded) matrix algebra E s + = E B B E ⊂ M 2 (E ) .(17) E s is therefore Z 2 -graded and E s + can be identified with its subalgebra of even degree. The invertibility of the extension 0 → B → E → A → 0 is thus equivalent to the existence of a homomorphism ρ : A → E s + . Finally let I s be the Z 2 -graded algebra C 1 ⊗ M 2 (C). Then I s ⊗ B = C 1 ⊗ M 2 (B) is a two-sided ideal in E s . This situation is depicted through a quasihomomorphism of odd degree from A to B: ρ : A → E s ⊲ I s ⊗ B .(18) The Chern character of this quasihomomorphism lives in the bivariant cyclic cohomology of A and B. The construction of [11] uses the formalism of section 2. Thus consider a lifting 0 → R → M → P → 0 of 0 → B → E → A → 0, that is (Definition 2.1) a diagram of extensions 0 0 0 0 / / J / / N / / Q / / 0 0 / / R / / M / / P / / 0 0 / / B / / E / / A / / 0 0 0 0(19) Notice that we do not require the extension 0 → R → M → P → 0 be invertible. J and R are ideals respectively in N and M . Moreover J = R ∩ N . We introduce as above the Z 2 -graded algebras N s = C 1 ⊗ N s + and M s = C 1 ⊗ M s + , with N s + = N J J N ⊂ M 2 (N ) , M s + = M R R M ⊂ M 2 (M ) .(20) By construction N s + is a two-sided ideal in M s + and coincides with the kernel of the projection homomorphism M s + → E s + . Choose a linear lifting ℓ : A → M s + of the homomorphism ρ : A → E s + . By the universal property of the tensor extension, one gets an homomorphism ρ * : T A → M s + compatible with the ideals: [11]). Then for any odd integer n one constructs a chain map χ n from the (b + B)-complex of noncommutative differential forms over M s + , to the X-complex of M as follows. χ n has two components χ n 0 : Ω n M s + → R n and χ n 1 : Ω n+1 M s + → ♮(R n dM ) given by 0 / / JA / / ρ * T A / / ρ * A / / ρ ℓ } } 0 0 / / N s + / / M s + / / E s + / / 0 (21) Define F = ε⊗ 1 0 0 −1 acting on M s as a multiplier of odd degree. Hence F 2 = 1 and the commutator [F, M s + ] coincides with the subspace ε ⊗ 0 R R 0 ⊂ M s . Denote by tr s the supertrace of odd degree C 1 ⊗ M 2 (C) → C, sending the matrix ε ⊗ a b c d to − √ 2i(a + d) and 1 ⊗ a b c d to 0 (seeχ n 0 (x 0 dx 1 . . . dx n ) = − Γ(1 + n 2 ) (n + 1)! λ∈Λn+1 ± tr s (x λ(0) [F, x λ(1) ] . . . [F, x λ(n) ]) (22) χ n 1 (x 0 dx 1 . . . dx n+1 ) = − Γ(1 + n 2 ) (n + 1)! n+1 i=1 tr s ♮(x 0 [F, x 1 ] . . . dx i . . . [F, x n+1 ]) where Λ n+1 is the cyclic permutation group of n + 1 elements and ± denotes the signature of permutation λ. The overall minus sign is conventional (it is cancelled by the other minus coming from the supertrace). χ n is actually defined on the direct product space ΩM s + = k≥0 Ω k M s + because it vanishes on differential forms of degree > n + 1, and its image lies in the subcomplex F 2n−1 R X(M ). It clearly extends to a chain map χ n : Ω M s + → F 2n−1 R X( M ) where M s + is the N s + -adic completion of M s + . The bivariant Chern character of degree n (odd) associated to the quasihomomorphism ρ is the composition of chain maps ch n (ρ) : X( T A ) γ −→ Ω T A ρ * −→ Ω M s + b χ n −→ F 2n−1 R X( M ) ,(23) where γ is the generalized Goodwillie equivalence of Cuntz-Quillen (see [11]) and the middle arrow is the map of (b+B)-complexes induced by the homomorphism ρ * : T A → M s + . Hence if τ is a cocycle over F 2n−1 R X( M ) , the composite τ ch n (ρ) defines a periodic cyclic cohomology class over A . Proposition 3.2 Let (E) : 0 → B → E → A → 0 be an invertible extension with lifting 0 → R → M → P → 0, and let ρ : A → E s ⊲ I s ⊗ B be the associated quasihomomorphism. Then for any cocycle τ over F 2n−1 R X( M ) representing a class [τ ] ∈ HP i (B), where n is odd, the equality τ ch n (ρ) = √ 2πi E * ([τ ])(24) holds in HP i+1 (A ). Proof: We shall relate τ ch n (ρ) to E * ([τ ]) via the eta-cochain and its renormalization introduced in [11,12]. The eta-cochain of degree n + 1 has two components η n+1 0 : Ω n+1 M s + → R n+1 and η n+1 1 : Ω n+2 M s + → ♮(R n+1 dM ) given by η n+1 0 (x 0 dx 1 . . . dx n+1 ) = Γ( n 2 + 1) (n + 2)! 1 2 tr s F x 0 [F, x 1 ] . . . [F, x n+1 ]+ n+1 i=1 (−) (n+1)i [F, x i ] . . . [F, x n+1 ]F x 0 [F, x 1 ] . . . [F, x i−1 ] η n+1 1 (x 0 dx 1 . . . dx n+2 ) = Γ( n 2 + 1) (n + 3)! n+2 i=1 1 2 tr s ♮(ix 0 F + (n + 3 − i)F x 0 )[F, x 1 ] . . . dx i . . . [F, x n+2 ] . These components extend as above to a linear map η n+1 : Ω M s + → F 2n+1 R X( M ). The eta-cochain makes the connection between the chain maps χ n and χ n+2 for any odd integer n. Indeed let ∂ and (b + B) denote the boundaries on the complexes X( M ) and Ω M s + respectively. The following transgression relation holds ( [11]): χ n − χ n+2 = ∂ • η n+1 − η n+1 • (b + B) .(25) Now let τ be a cocycle over the subcomplex Then χ R is a (b + B)-cocycle cohomologous to τ χ n . Indeed using the transgressions (25) one gets τ χ n − χ R = k odd <n τ R η k+1 • (b + B) . Hence composition with the chain map ρ * γ : X( T A ) → Ω M s + yields the equality of cyclic cohomology classes τ ch n (ρ) ≡ χ R ρ * γ in HP i+1 (A ). It remains to compare χ R ρ * γ and E * ([τ ]). We produce a deformation of the homomorphism ρ * : T A → M s + as follows. Recall that ρ * is induced by a linear lifting ℓ : A → M s + of the homomorphism ρ : A → E s + . In matrix form we can write ℓ = σ λ µ σ , where σ : A → M is a linear splitting of the projection homomorphism M → A as in section 2, and λ, µ are linear maps from A to R. Consider the linear homotopy of linear maps ℓ t : A → M s + , defined for any t ∈ [0, 1] by ℓ t = σ tλ tµ σ . In particular ℓ 0 = σ 0 0 σ is a diagonal matrix, and ℓ 1 = ℓ. Then observe that ℓ t followed by the projection M s + = M R R M → A 0 0 A = A ⊕ A yields an algebra homomorphism (independent of t). Hence one gets as usual an homomorphism ρ t * : T A → M s + by means of the diagram 0 / / JA / / ρ t * T A / / ρ t * A / / ℓ t z z 0 0 / / M 2 (R) + N s + / / M s + / / A ⊕ A / / 0 where the ideal M 2 (R) + N s + = R+N R R R+N is the kernel of the projection M s + → A ⊕ A . Thus ρ t * extends to an homomorphism from T A to the proalgebra M s + = lim ← −m M s + /(M 2 (R) + N s + ) m for any t ∈ [0, 1] . This provides a homotopy between ρ 0 * = σ * 0 0 σ * and ρ 1 * = ρ * . One knows that τ R ∂ vanishes on F 2n−1 R X( M ). Hence χ R descends to a (b + B)-cocycle over Ω M s + , and by homotopy invariance of periodic cyclic cohomology [5], the composite maps χ R ρ t * γ : X( T A ) → C define the same cohomology class for all t. We deduce the equality of cyclic cohomology classes τ ch n (ρ) ≡ χ R ρ * γ = χ R ρ 1 * γ ≡ χ R ρ 0 * γ = k odd <n −τ R ∂ η k+1 ρ 0 * γ in HP i+1 (A ). Since ρ 0 * is a diagonal matrix, [F, ρ 0 * (x)] = 0 for any x ∈ T A and the map η k+1 ρ 0 * vanishes unless k = −1. Hence only the term containing η 0 ρ 0 * survives in the sum over k. One has η 0 0 ρ 0 * (x) = Γ(1/2) 1 2 tr s (F ρ 0 * (x)) = − √ 2πi 1 2 (σ * (x) − σ * (x)) , η 0 1 ρ 0 * (xdy) = Γ(1/2) 1 2 tr s ♮(F ρ 0 * (x)dρ 0 * (y)) = − √ 2πi 1 2 ♮(σ * (x)dσ * (y) − σ * (x)dσ * (y)) , which shows that τ ch n (ρ) is cohomologous to √ 2πi τ R ∂ • 1 2 (σ * − σ * ), where σ * and σ * are viewed as chain maps from X( T A ) to X( M ). Finally consider ρ t * as a family of homomorphisms T A → M 2 ( M ). The cup-product of the cocycle τ R ∂ over X( M ) with the usual trace over M 2 (C) yields a cocycle τ R ∂#tr over X(M 2 ( M )). Using homotopy invariance, one has the equality of periodic cyclic cohomology classes τ R ∂σ * + τ R ∂σ * = (τ R ∂#tr)ρ 0 * ≡ (τ R ∂#tr)ρ 1 * = (τ R ∂#tr)ρ * ≡ 0 where the last equality comes from Proposition 2.3 and the fact that ρ : A → M 2 (E ) is an homomorphism. Hence τ R ∂ • 1 2 (σ * − σ * ) ≡ τ R ∂σ * = E * ([τ ]), and τ ch n (ρ) coincides with √ 2πi E * ([τ ]) in HP i+1 (A ) as wanted.G = 0 1 −1 P 1 0 Q 1 1 −P 0 1 = Q 1 − QP P Q − 1 P + P (1 − QP )(26) Before showing that the index map is adjoint to the connecting morphism in periodic cyclic cohomology E * : HP 0 (B) → HP 1 (A ), let us explain how Connes' pairing HP 0 (B)×K 0 (B) → C is computed when the cyclic cohomology of B is represented by cocycles over the complexes F n R X( M ) as in section 2. We will use the formulas established in [11] §4 in connection with the Chern-Connes character in cyclic homology. Consider a lifting 0 → R → M → P → 0 of the extension (E): 0 0 0 0 / / J / / N / / Q / / 0 0 / / R / / M / / P / / 0 0 / / B / / E / / A / / 0 0 0 0(28) For any n ≥ 1, the intersection R ∩ N n is an ideal in R and the quotient R/(R ∩ N n ) is a nilpotent extension of B. Take R as the projective limit lim ← −n R/(R ∩ N n ). Proceeding as in [5], any idempotent e ∈ M ∞ (B + ) such that e − p N ∈ M ∞ (B) can be lifted to an idempotentê ∈ M ∞ ( R + ). The latter is defined up to similarity in the matrix algebra over R + . One hasê − p N ∈ M ∞ ( R) and the trace tr(ê − p N ) ∈ R defines a cycle of even degree in the subcomplex F 1 R X( M ) ⊂ X( M ). Thus if τ is a cocycle of even degree over Proof: We have to show that τ #tr (ê − p N ) 2n+1 does not depend on n (sufficiently large), and that it is invariant whenê is conjugated by an invertible matrix u such that u − 1 ∈ M ∞ ( R + ) and u −1ê u ≡ p N mod M ∞ ( R). For convenience we rewrite the pairing using the Z 2 -graded algebra of 2 × 2 matrices over M ∞ ( R + ), with grading induced by the decomposition of matrices into diagonal/off diagonal form: consider the odd element F = 0 1 1 0 such that F 2 = 1, and set f = ê 0 0 pN as an idempotent of even degree. Then if tr s denotes the supertrace on M 2 (M ∞ (C)) one has τ #tr (ê − p N ) 2n+1 = τ #tr s F ([F, f ]) 2n+1 . The right-hand-side is recognized as a Chern-Connes pairing [1] and has wellknown properties. In particular it does not depend on n provided F ([F, f ]) 2n+1 remains in the domain of the supertrace τ #tr s , and it is invariant with respect to homotopies of f preserving the condition [F, f ] ∈ M 2 (M ∞ ( R)). Now let u be an invertible matrix such that u − 1 ∈ M ∞ ( R + ) and u −1ê u ≡ p N mod M ∞ ( R). Let v be the image of u under the projection R + = R ⊕ C → C. Then v is an invertible matrix such that v − 1 ∈ M ∞ (C) and v −1 p N v = p N . The invertible matrix of even degree g = u 0 0 v conjugates f to g −1 f g = u −1ê u 0 0 pN , and fulfills the commutation relation [F, g] ∈ M 2 (M ∞ ( R)). This allows to construct a (stable) homotopy between f and g −1 f g by a standard procedure using rotation matrices. Remark 4.2 In general the definition of R given here does not coincide with the pro-algebra lim ← −n R/J n . It does coincide under strong conditions, for example when the equality R ∩ N n = J n holds for all n. The latter condition was implicitely assumed in [11], where the cycle tr (ê − p N ) 2n+1 was taken as the definition of the Chern character in K-theory. Finally recall Connes' pairing HP 1 (A ) × K 1 (A ) → C in the Cuntz-Quillen formalism [5]. Let [ϕ] ∈ HP 1 (A ) be a cyclic cohomology class represented by a cocycle of odd degree ϕ : X( T A ) → C, where T A is the JA -adic completion of the tensor algebra T A . Let [g] ∈ K 1 (A ) be represented by an invertible element g ∈ GL ∞ (A ). Then g can be lifted to an invertible elementĝ ∈ GL ∞ ( T A ), and the one-form ♮(ĝ −1 dĝ) ∈ Ω 1 T A ♮ is a cycle of odd degree in the complex X( T A ) whose homology class is independent of the choice of lifting. The pairing is defined as [ϕ], [g] = 1 √ 2πi ϕ ĝ −1 dĝ .(30) One can think of the normalization factor 1/ √ 2πi as a pure convention. However note that this normalization is uniquely determined by the compatibility of the bivariant Chern character with Bott periodicity, see for example [11]. [τ ], Ind E ([g]) = √ 2πi E * ([τ ]), [g] .(31) Proof: We will not use excision since we assume from the beginning that the cyclic cohomology class [τ ] is represented by a cocycle over F 4n+1 R X( M ). Let g ∈ GL N (A ) be an invertible element representing [g]. Thus in particular g − 1 N ∈ M N (A ). We shall replace (E) with an invertible extension as follows. Denote by C[z, z −1 ] the commutative algebra of Laurent polynomials in the indeterminate variable z, and let C be the subalgebra of polynomials f ∈ C[z, z −1 ] such that f (1) = 0. Equivalently, C is the (non-unital) commutative algebra generated by two elements u, v with relations uv = vu = −u − v. The inclusion of C into C[z, z −1 ] is recovered by setting z = 1 + u and z −1 = 1 + v. The geometric picture is that of the algebra of trigonometric functions over the unit circle, vanishing at point z = 1. Hence C is a suitable algebraic definition of a suspension algebra. We define a homomorphism α : C → M N (A ) by setting α(u) = g − 1 and α(v) = g −1 − 1. Equivalently we may extend α to a unital homomorphism from C + = C[z, z −1 ] to M N (A + ) and set α(z) = g, α(z −1 ) = g −1 . Thus α carries the "Bott element" [z] ∈ K 1 (C ) to [g] ∈ K 1 (A ). Define (F ) as the pullback extension of (E) (tensored with M N (C)) induced by α, that is, (F ) is the first row in the commutative diagram 0 / / M N (B) / / F / / β C / / α 0 0 / / M N (B) / / M N (E ) π / / M N (A ) / / 0 Explicitely F = {(h, f ) ∈ M N (E ) ⊕ C | π(h) = α(f )}. The homomorphisms F → M N (E ) and F → C are induced respectively by the projections onto the first and second summand in M N (E )⊕C . It is immediate from the construction of the index map that the equality which may be pulled back to a cocycle β * (τ ) over F 4n+1 MN (R) X( G ). One has [τ ], Ind E ([g]) = β * ([τ ]), Ind F ([z]) . Moreover the homomorphism α : C → A induces a pullback in cyclic cohomology α * : HP * (A ) → HP * (C ). The pair α * , β * intertwines the action of the connecting morphisms E * and F * in the sense that α * • E * = F * • β * . Therefore one has E * ([τ ]), [g] = E * ([τ ]), α([z]) = α * • E * ([τ ]), [z] = F * • β * ([τ ]), [z] and the equality [τ ], Ind E ([g]) = √ 2πi E * ([τ ]), [g] would follow from the equality [τ ′ ], Ind F ([z]) = √ 2πi F * ([τ ′ ]), [z] for the cocycle τ ′ = β * (τ ). We decided to replace the extension (E) with the extension (F ) because the latter is invertible. Indeed choose arbitrary liftings U, V ∈ F of u, v ∈ C and set Q = 1 + U , P = 1 + V in F + . Then 1 − QP and 1 − P Q sit in the ideal M N (B), and the map ρ defined on generators by ρ(u) = U 1 − QP P Q − 1 V + P (1 − QP ) , ρ(v) = V + P (1 − QP ) P Q − 1 1 − QP U extends to a homomorphism ρ : C → F s + . Passing to the unitalized algebra C + , the map ρ carries z to the invertible Its bivariant Chern character ch 2n+1 (ρ) is a chain map X( T C ) → F 4n+1 MN (R) X( G ). By Proposition 3.2, the composite τ ′ ch 2n+1 (ρ) represents a cyclic cohomology class over C which coincides with √ 2πi F * ([τ ′ ]). Let us calculate explicitely the pairing τ ′ ch 2n+1 (ρ), [z] . We know that ρ lifts to a homomorphism ρ * : T C → G s + = lim ← −k G s + /(H s + ) k . Choose an invertible liftingẑ ∈ ( T C ) + of z. The idempotentê = ρ * (ẑ) −1 1 0 0 0 ρ * (ẑ) ∈ M 2 M N (R) + , where M N (R) = lim ← −k M N (R)/(M N (R) ∩ H k ),τ ′ ch 2n+1 (ρ), [z] = τ ′ #tr (ê − p 1 ) 2n+1 = [τ ′ ], Ind F ([z]) where p 1 is the 2 × 2 matrix [τ ], Ind E ([g]) = τ R ∂#tr σ * (ĝ) −1 dσ * (ĝ) for any invertible liftingĝ ∈ GL ∞ ( T A ). Then by [5] §12 the homology class of ♮σ * (ĝ) −1 dσ * (ĝ) remains unchanged if σ * (ĝ) is replaced by any other invertible liftingg ∈ GL ∞ ( M ) of g. Pseudodifferential operators B = C ∞ c (B, CL −1 ) , E = C ∞ c (B, CL 0 ) , A = C ∞ c (B, LS 0 )(33) thus lead to an extension (E) : 0 → B → E → A → 0. The index morphism Ind E : K 1 (A ) → K 0 (B) maps a family of elliptic symbols g ∈ GL ∞ (A ) to an idempotent in M ∞ (B + ) representing a K-theory class (index bundle) of the base manifold B. Our aim is to evaluate the image of Ind E on certain cyclic cohomology classes [τ ] ∈ HP 0 (B) associated to closed currents over B. As explained before this requires to work with suitable extensions of the algebras B, E , A . Inspired by Cuntz and Quillen [5], the basic idea is to replace the algebra of smooth functions C ∞ (B) with a Fedosov-type deformation of the algebra of (ordinary) differential forms Ω(B). First we consider the graded space Ω(B, E) = n≥0 Ω n (B, E) , Ω 0 (B, E) = C ∞ (B, E) ,(34) of smooth E-valued differential forms over B. One has an isomorphism of vector spaces Ω n (B, E) ∼ = C ∞ (M, π * (Λ n T * B)), where π * (Λ n T * B) is the pullback of the vector bundle Λ n T * B → B on the total space of the submersion. Ω(B, E) is a right Ω(B)-module, for the usual exterior product of differential forms. The graded algebra of differential forms with values in fiberwise pseudodifferential operators is defined analogously: for any vector fields X 0 , . . . , X n over B and ξ ∈ Ω n (B, E). ∇ is a derivation of right Ω(B)-module: ∇(ξω) = (∇ξ)ω + (−) n ξdω for any ξ ∈ Ω n (B, E) and ω ∈ Ω(B). The curvature ∇ 2 is the endomorphism θ ∈ Ω 2 (B, CL 1 ) mapping two basic vector fields X, Y ∈ C ∞ (B, T B) to the vertical vector field Ω(B, CL) = n≥0 Ω n (B, CL) , CL = k∈Z CL k .(35)θ(X, Y ) = [h(X), h(Y )] − h([X, Y ]) ∈ C ∞ (M, Ker(π * )) .(37) For any n ∈ N let the quotient LS n = CL n /CL n−1 denote the space of leading symbols of order n. We introduce the following algebras of operator-valued differential forms with compact support on B: R 0 = n≥0 Ω n c (B, CL n−1 ) , M 0 = n≥0 Ω n c (B, CL n ) , P 0 = n≥0 Ω n c (B, LS n ) M 0 acts naturally by endomorphisms on Ω(B, E), R 0 is a two-sided ideal in M 0 and P 0 is the quotient algebra. Observe that B, E , A are exactly the subalgebras of degree zero forms in R 0 , M 0 , P 0 respectively. Now, the connection acts on endomorphisms by the odd derivation δ = [∇, ]. It is easy to see that δ leaves R 0 and M 0 globally invariant, hence it acts on P 0 also. Moreover the curvature θ ∈ Ω 2 (B, CL 1 ) is a multiplier of R 0 and M 0 , hence of P 0 . The derivation δ is not a differential since δ 2 = [θ, ] = 0. Using a trick of Connes ([2] pp. 229), we shall enlarge M 0 by adding a multiplier v such that v 2 = θ and ω 1 vω 2 = 0 for any ω 1 , ω 2 ∈ M 0 . One can think of v as having form degree one. Hence the resulting graded algebra M 0 [v] is the set of elements α = ω 11 + ω 12 v + vω 21 + vω 22 v , ω ij ∈ M 0 .(38) The crucial fact is that M 0 [v] is provided with a differential d of degree one defined by the relations dω = δω + vω + (−) n ωv if ω ∈ M 0 has degree n, and dv = 0. One checks that d 2 = 0 which turns M 0 [v] into a differential graded algebra. We denote by M the even degree subspace of M 0 [v] endowed with the Fedosov product α 1 ⊙ α 2 = α 1 α 2 − dα 1 dα 2 .(39) M is an associative (trivially graded) algebra. The map M → C ∞ c (B, CL 0 ), which projects an element α = ω 11 + ω 12 v + vω 21 + vω 22 v to its component of degree zero (equivalently the component of degree zero of the differential form ω 11 ), defines a linearly split homomorphism M → E . Hence M is an extension of E . One proceeds similarly with R 0 and P 0 : the algebras R and P are defined as the even subspaces of R 0 [v] and P 0 [v] respectively, endowed with the Fedosov product. Again the projections onto the degree zero components induce surjective homomorphisms R → B and P → A . Moreover the extension 0 → R → M → P → 0 is a lifting of (E). Putting everything together we have built a diagram of extensions 0 0 0 0 / / J / / trace for the Fedosov product. The factor m!/(2m)! is (up to a sign) the correct normalization needed for passing from the X-complex to the de Rham complex ( [5]). Thus τ is a cocycle of even degree over the subcomplex F 2n+1 R X(M ) of the R-adic filtration of X(M ), provided n = dim(M/B) + dim C. The next step is to extend τ to a linear map τ R : M → C and view it as a cochain over the whole complex X(M ). We use zeta-function renormalization [12]. Fix a smooth family of fiberwise elliptic positive pseudodifferential operators D of order one. For example, D may be taken as the square root of a fiberwise Laplacian associated to some smooth family of Riemannian metrics on the fibers of the submersion. For ω ∈ Ω c (B, CL) of any pseudodifferential order, the zeta-function z ∈ C → Tr(ωD −z ) ∈ Ω c (B) is holomorphic on a half-plane Re(z) ≫ 0 and admits a meromorphic extension to the entire plane with only simple poles. Taking the finite part of this function at z = 0 thus defines a renormalization of the operator trace: Pf τ R : M → C τ R (ω 11 + ω 12 v + vω 21 + vω 22 v) = m! (2m)! C Pf z=0 Tr (ω 11 − ω 22 θ)D −z(46) for any ω 11 + ω 12 v + vω 21 + vω 22 v ∈ M . Of course τ R is not a trace on M because the insertion of the operator D −z destroys the cyclicity of the operator trace. It is well-known however that the obstruction for the zeta-renormalized trace to be a true trace on the algebra of pseudodifferential operators is expressed in terms of the (fiberwise) Wodzicki residue [15]. Indeed if ω 1 , ω 2 ∈ Ω c (B, CL) are differential forms with values in pseudodifferential operators of any order, the zeta-renormalized operator trace applied to their graded commutator yields a residue (see e.g. [9]) Pf z=0 Tr([ω 1 , ω 2 ]D −z ) = Res z=0 Tr(ω 1 [ln D, ω 2 ]D −z ) ,(47) The logarithm ln D does not belong to the algebra of classical pseudodifferential operators, but the commutator [ln D, ω] is a well-defined element of Ω c (B, CL) modulo smoothing operators. Indeed it admits the asymptotic expansion We define a linear functional Ω c (B, CL) → C by integration of the fiberwise Wodzicki residue over the cycle C: − C ω := C Res z=0 Tr(ωD −z ) = C S * M/B s(ω) −n η(dη) n−1(49) where s(ω) is the complete symbol of ω, n = dim(M/B), η is the canonical oneform on the cotangent bundle T * M/B of the submersion fibers, and S * M/B denotes integration along the cotangent sphere bundle. It follows from the properties of the Wodzicki residue that (49) does not depend on the choice of (elliptic, positive, order one) operator D, and defines a δ-closed graded trace over Ω c (B, CL). This allows to express the boundary of τ R viewed as a cochain of even degree over X(M ). Indeed the boundary map ∂ : Ω 1 M ♮ → M is given by the Fedosov commutator ∂(α 1 dα 2 ) = α 1 ⊙ α 2 − α 2 ⊙ α 1 , so that the composition τ R ∂(α 1 dα 2 ) = τ R (α 1 ⊙ α 2 − α 2 ⊙ α 1 ) has to be a sum of Wodzickitype residues. For simplicity we shall only evaluate τ R ∂ on the range of the chain map σ * : X(T A ) → X(M ). In odd degree the range is linearly generated by elements of type ♮((σ 1 ⊙ . . . ⊙ σ n )dσ n+1 ) ∈ Ω 1 M ♮ where σ i = σ(a i ) for some a i ∈ A . Writing the Fedosov products by means of differential forms, this is equivalent to the linear span of elements of type ♮σ 0 dσ 1 . . . dσ 2n dσ 2n+1 and ♮dσ 1 . . . dσ 2n dσ 2n+1 . Notice that the derivative δ ln D = [∇, ln D] is a pseudodifferential operator with asymptotic expansion δ ln D ∼ δDD −1 − 1 2 [D, δD]D −2 + 1 3 [D, [D, δD]]D −3 − . . .(50)+ (n + 1)! (2n + 2)! − C (σ 0 dσ 1 . . . dσ 2n+1 − σ 2n+1 dσ 0 . . . dσ 2n ) 11 δ ln D τ R ∂(dσ 1 . . . dσ 2n dσ 2n+1 ) = n! (2n)! − C (dσ 1 . . . dσ 2n ) 11 [ln D, σ 2n+1 ](51) where for any α = ω 11 + ω 12 v + vω 21 + vω 22 v ∈ M 0 [v] the bracket (α) 11 means projection onto the component ω 11 . If k = 2n one finds (σ 0 dσ 1 . . . dσ k ) 11 = σ 0 δσ 1 . . . . . . . . . δσ k + k−1 i=1 σ 0 δσ 1 . . . σ i θσ i+1 . . . δσ k(53)+ k−3 i=1 k−1 j=i+2 σ 0 δσ 1 . . . σ i θσ i+1 . . . σ j θσ j+1 . . . δσ k . . . + σ 0 σ 1 θσ 2 . . . σ k−1 θσ k , whereas if k = 2n + 1 the last line is k i=1 σ 0 σ 1 θσ 2 . . . δσ i . . . σ k−1 θσ k . Similarly with (dσ 1 . . . dσ 2n ) 11 . Proof: By definition the boundary map ∂ = b : Ω 1 M ♮ → M carries an element ♮αdβ to the Fedosov commutator α⊙β−β⊙α = [α, β]−dαdβ+dβdα. Therefore τ R ∂(σ 0 dσ 1 . . . dσ 2n dσ 2n+1 ) = τ R ([σ 0 dσ 1 . . . dσ 2n , σ 2n+1 ]) +τ R (dσ 2n+1 dσ 0 . . . dσ 2n − dσ 0 . . . dσ 2n+1 ) τ R ∂(dσ 1 . . . dσ 2n dσ 2n+1 ) = τ R ([dσ 1 . . . dσ 2n , σ 2n+1 ]) . Let us first evaluate τ R on a commutator [α, σ] where α = ω 11 + ω 12 v + vω 21 + vω 22 v is an element of degree 2n and σ is of form degree zero. One has ασ = ω 11 σ + vω 21 σ and σα = σω 11 + σω 12 v, hence τ R ([α, σ]) = n! (2n)! C Pf z=0 Tr([ω 11 , σ]D −z ) = n! (2n)! − C ω 11 [ln D, σ] . Applying this to the forms α = σ 0 dσ 1 . . . dσ 2n or α = dσ 1 . . . dσ 2n and σ = σ 2n+1 yields the first terms in (51, 52). Then we evaluate τ R on a coboundary dα where α = ω 11 + ω 12 v is an odd element of form degree 2n + 1. One has dα = δω 11 + ω 12 θ + (δω 12 − ω 11 )v + vω 11 + vω 12 v, so that τ R (dα) = (n + 1)! (2n + 2)! C Pf z=0 Tr (δω 11 + ω 12 θ − ω 12 θ)D −z = (n + 1)! (2n + 2)! C Pf z=0 Tr(ω 11 δD −z ) = − (n + 1)! (2n + 2)! − C ω 11 δ ln D where we used an integration by parts in the second equality (remark that the form ω 11 is odd), and the third equality can be found for example in [9]. Applying this to the form α = σ 2n+1 dσ 0 . . . dσ 2n − σ 0 dσ 1 . . . dσ 2n+1 yields the second term in (51). Formula (53) is straightforward using dσ i = δσ i + vσ i + σ i v and v 2 = θ. One knows that τ R ∂ vanishes on the subcomplex F n+1 X(M ) for n = dim(M/B) + dim C, hence extends to a cocycle over X( M ). Finally the cyclic cohomology class E * ([τ ]) ∈ HP 1 (A ) is represented by the composition of chain maps X( T A ) σ * −→ X( M ) τR∂ −→ C .(54) Here we can interpret the fact that τ R ∂ extends to a cocycle over X( M ) by the property of the fiberwise Wodzicki residue that ignores the pseudodifferential operators of low order, that is, the high powers of the ideal R ⊂ M . It remains to evaluate the pairing of E * ([τ ]) with an elliptic symbol class [g] ∈ K 1 (A ). Let g ∈ GL ∞ (A ) be a representative of [g]. For notational simplicity we shall forget the stabilization by matrices and suppose that g ∈ GL 1 (A ) ⊂ A + . Then σ(g) and σ(g −1 ) are two families of elliptic pseudodifferential operators over B such that 1 − σ(g)σ(g −1 ) ∈ C ∞ c (B, CL −1 ) ⊂ R . Let Q be the image of σ(g) under the natural map M + → M + . Then Q ∈ GL 1 ( M ). Indeed, 1 − σ(g) ⊙ σ(g −1 ) = 1 − σ(g)σ(g −1 ) + dσ(g)dσ(g −1 ) is in the ideal R + N by virtue of (55), and one can easily show that the inverse of Q (for the Fedosov product) is given by the series Q −1 = ∞ n=0 σ(g −1 ) ⊙ (1 − σ(g) ⊙ σ(g −1 )) ⊙n ∈ M + .(56) There is an equivalent description of the Fedosov inverse involving the parametrix P ≡ Q −1 mod N of Q. This allows to write Q −1 in terms of differential forms: Q −1 = [ dim B 2 ] n=0 P (dQdP ) n , P = ∞ n=0 σ(g −1 )(1 − σ(g)σ(g −1 )) n . , Ind E ([g]) in terms of Q, its parametrix P and the fiberwise Wodzicki residue. The computation is tedious but straightforward. We shall state the result in an elegant way using Chern-Simons forms. Introduce an infinitesimal parameter ε of odd degree, which means ε 2 = 0. The superconnection ∇ ε D := ∇ + ε ln D acts on the algebra Ω c (B, CL)[ε] = Ω c (B, CL) ⊕ ε Ω c (B, CL) by graded commutator. Its curvature is (∇ + ε ln D) 2 = θ − εδ ln D. The "adjoint" action of Q gives a new superconnection P ∇ ε D Q. Now if ∇ 0 and ∇ 1 are two superconnections, we let ∇ t = (1 − t)∇ 0 + t∇ 1 be the linear interpolation for t ∈ [0, 1]. The associated Chern-Simons form is the following element of even degree in Ω c (B, CL) (defined as always modulo smoothing operators, due to the presence of ln D): cs(∇ 0 , ∇ 1 ) = 1 0 dt (∇ 1 − ∇ 0 )e ∇ 2 t | ε ,(59) where | ε means that we only take the ε-component in Ω c (B, CL)[ε]. Since ∇ is of form degree one and ε is nilpotent, the exponential is actually a polynomial in the curvature ∇ 2 t . Applying this to ∇ 0 = ∇ ε D and ∇ 1 = P ∇ ε D Q one has where ∇ ε D is the superconnection ∇ + ε ln D, and D is any family of elliptic positive pseudodifferential operators of order one. Let us display some useful formulas in low dimension. If C is just a point in the base manifold B, the above pairing calculates the index of the elliptic operator Q at point C. The formula amounts to a Wodzicki residue on one fiber: [τ ], Ind E ([g]) = − C P [ln D, Q] . One recognizes the Radul cocycle [14] evaluated on Q and its parametrix. It is instructive to check that the number (61) does not depend on the choice of D. Indeed if D ′ is another elliptic operator of order one, the difference ln D ′ −ln D is a classical pseudodifferential operator, hence the commutator with ln D ′ − ln D is an inner derivation. It follows from the trace property of the Wodzicki residue that (61) remains unchanged. Now if C is a two-dimensional cycle then the above pairing computes the evaluation of C on the first Chern class of the index bundle associated to the elliptic family Q. One obtains In higher dimensions the formulas involve increasing powers of δQδP and of the curvature θ. Corollary 2. 6 6Given any extension (E) : 0 → B → E → A → 0 the renormalization procedure of Proposition 2.3 yields a transformation F 2n− 1 R 1X( M ), and choose any renormalization τ R : X( M ) → C as in section 2. Hence the composite map τ R ∂ vanishes on F 2n−1 R X( M ) but not on X( M ). Define the cochain χ R : Ω M s + → C by χ R := k odd <n −τ R ∂ η k+1 . Remark 3. 3 ∈ 3The above proof relates the connecting morphism of an invertible extension to the boundary of the renormalized eta-cochain, a technique introduced in[12] as a way of building local representatives for the bivariant Chern character of quasihomomorphisms. It was shown that this procedure is intimately related to anomalies in quantum field theory. Thus we may consider the cyclic cocycle τ R ∂σ * representing E * ([τ ]) as a kind of "anomaly formula" adapted to extensions.4 Index theoremFirst recall the definition of the algebraic K-theory groups of a (non-unital) algebra A in low degrees. As usual denote by M ∞ (A ) = lim − →N M N (A ) the inductive limit of matrix algebras with entries in A , under the inclusion maps a → a 0 0 0 . Let A + = A ⊕C be the unitalization of A . An equivalence relation is defined on the set of idempotents in M ∞ (A + ) as follows: two idempotents e and e ′ are equivalent if there exists an invertible matrix g with entries in A + , such that g − 1 ∈ M ∞ (A + ) and e ′ = g −1 eg (similarity). The set of equivalence classes of idempotents forms a semigroup for the direct sum of matrices. Denote by K 00 (A ) its Grothendieck group and let K 0 (A ) be the kernel of the morphism K 00 (A + ) → K 00 (C) = Z. The elements of K 0 (A ) are represented by formal differences [e] − [f ] of equivalence classes of idempotents e, f ∈ M ∞ (A + ) such that e ≡ f mod M ∞ (A ). Any such class can be further reduced to a difference [e] − [p N ] where p N is the diagonal matrix 1N 0 0 0 with N units on the diagonal. For any integer N denote by GL N (A ) the group of invertible matrices g ∈ M N (A + ) such that g ≡ 1 N mod M N (A ), and by GL ∞ (A ) the inductive limit of the groups GL N (A ) under the inclusions a → a 0 0 1 . Then K 1 (A ) is the abelianization of GL ∞ (A ), i.e. its quotient by the commutator subgroup [GL ∞ (A ), GL ∞ (A )].Now let (E) : 0 → B → E → A → 0 be an extension. The connecting map for the algebraic K-theory in low degree is constructed as follows[7]. Take a class [g] ∈ K 1 (A ) represented by an invertible matrix g ∈ M N (A + ) with g − 1 ∈ M N (A )M 2N (A + ) can be lifted to an invertible matrix in M 2N (E + ). Indeed, choose any preimages Q, P ∈ M N (E + ) of g, g −1 respectively. Then is an invertible matrix in M 2N (E + ). Moreover the image of 1 − QP vanishes in M N (A + ), hence 1 − QP ∈ M N (B) and similarly for P Q − 1. Thus G is an invertible lifting of g 0 0 g −1 . Viewing p N = 1N 0 0 0 as a 2N × 2N matrix, the idempotent e = G −1 p N G fulfills the property e ≡ p N mod M 2N (B) and consequently [e] − [p N ] determines a class in K 0 (B). The latter is independent of the choice of lifting G. Since by definition any additive map from GL ∞ (A ) to an abelian group factors through K 1 (A ), we obtain the index morphism of the extension (E) Ind E : K 1 (A ) → K 0 (B) . F 1 R 1X( M ), the pairing τ #tr(ê−p N ) is defined. More generally, tr (ê−p N ) 2n+1 is a cycle of even degree in F 4n+1 R X( M ) and hence can be paired with any cocycle τ : F 4n+1 R X( M ) → C. Lemma 4. 1 1Let τ : F 4n+1 R X( M ) → C be a cocycle of even degree. Let e ∈ M ∞ (B + ) be an idempotent such that e − p N ∈ M ∞ (B), and choose an idempotent liftingê ∈ M ∞ ( R + ) of e. Then the formula [τ ], [e] = τ #tr (ê − p N ) 2n+1 (29) descends to a well-defined pairing lim − →n H 0 F 4n+1 R X( M ) × K 0 (B) → C. Theorem 4. 3 3Let (E) : 0 → B → E → A → 0 be an extension with lifting 0 → R → M → P → 0. Let τ : F 4n+1 R X( M ) → C ce a cocycle representing an element [τ ] ∈ HP 0 (B), and take any [g] ∈ K 1 (A ). Then P +P (1−QP ) and z −1 to its inverse P +P (1−QP ) P Q−1 1−QP Q . Observe that the K-theory class Ind F ([z]) is represented by the idempotent e = ρ(z) −1 1 0 0 0 ρ(z) ∈ M 2 (M N (B) + ). As explained in section 3 the invertible extension (F ) determines a quasihomomorphism ρ : C → F s ⊲ I s ⊗ M N (B) . is a lifting of the idempotent e = Ind F (z). According to Lemma 4.1 the pairing [τ ′ ], Ind F ([z]) is given by τ ′ #tr (ê − p 1 ) 2n+1 . On the other hand, the calculation performed in the proof of [11] Theorem 6.3 part III) applies verbatim and yields . Hence we conclude that [τ ′ ], Ind F ([z]) = √ 2πi F * ([τ ′ ],[z] as wanted. Corollary 4. 4 4Let (E) : 0 → B → E → A → 0 be an extension with lifting 0 → R → M → P → 0. Then for any class [τ ] ∈ HP 0 (B) represented by a cocycle of even degree τ : F 4n+1 R X( M ) → C and any class [g] ∈ K 1 (A ) represented by an invertible g ∈ GL ∞ (A ), one has[τ ], Ind E ([g]) = τ R ∂#tr g −1 dg .(32)Hereτ R : X( M ) → C is any renormalization of τ , andg ∈ GL ∞ ( M ) isany invertible lifting of g in the pro-algebra M = lim ← −k M /(R + N ) k . Proof: Choose a linear section σ : A → M of the projection homomorphism. By definition the cyclic cohomology class E * ([τ ]) ∈ HP 1 (A ) is represented by τ R ∂ • σ * : X( T A ) → X( M ) → C, where σ * is the homomorphism T A → M induced by σ (section 2). Hence by Theorem 4.3, evaluating this cocycle on [g] yields the formula π : M → B be a proper submersion of smooth manifolds without boundary. Hence at any point b ∈ B the fiber π −1 (b) = (M/B) b is a compact submanifold of M . Define E → B as the infinite-dimensional Fréchet bundle whose fiber at a point b is the space of smooth functions C ∞ ((M/B) b ). The space of smooth sections C ∞ (B, E) is thus isomorphic to C ∞ (M ). For any k ∈ Z we denote by CL k b = CL k ((M/B) b ) the space of classical pseudodifferential operators of order k on the manifold (M/B) b . In particular CL −1 b is a two-sided ideal in the algebra CL 0 b and the quotient CL 0 b /CL −1 b = LS 0 b is isomorphic to the commutative algebra of smooth functions C ∞ (S * (M/B) b ) on the cotangent sphere bundle of (M/B) b . The projection homomorphism CL 0 b → LS 0 b is the map which carries a pseudodifferential operator of order zero to its leading symbol. The algebra of smooth families of fiberwise pseudodifferential operators C ∞ (B, CL 0 ), parametrized by the base manifold B, naturally acts on C ∞ (E) by endomorphisms. The following algebras of smooth families of operators with compact support on B It acts naturally as (left) endomorphisms on the module Ω(B, E). Then we need some extra structure in order to define a connection on E. Recall that the set of vertical vector fields on M is the kernel of the tangent map π * : T M → T B, or equivalently the subbundle of T M tangent to the submanifolds (M/B) b , b ∈ B. Then choose a horizontal distribution H ⊂ T M , i.e. a direct summand for the vertical vector fields: T M = Ker(π * ) ⊕ H. This provides a lifting h : C ∞ (B, T B) → C ∞ (M, H) of the vector fields from the base to the total space of the submersion. A connection ∇ : Ω n (B, E) → Ω n+1 (B, E) is given by the usual formula (∇ξ)(X 0 , . . . , X n ) = n i=0 (−) i h(X i ) · ξ(X 0 , . . . ,X i , . . . , X n ) (36) + i<j (−) i+j ξ([X i , X j ], X 0 , . . . ,X i , . . . ,X j , . . . , X n ) Tr(ωD −z ) ∈ Ω c (B) , ∀ω ∈ Ω c (B, CL) .(45)Definition 5.2 Let C be a cycle of dimension 2m in B and let τ be the associated trace over R n+1 , n = dim(M/B) + 2m. Choose an elliptic and positive family of pseudodifferential operators D ∈ C ∞ (B, CL 1 ). The zeta-function renormalization of τ is the linear map , [D, [D, ω]]]D −3 − . . . (48) where the commutator [D, ] does not increase the order of operators since D is of order one. The residue Res z=0 Tr(ωD −z ) is a differential form over B which depends on the complete symbol of ω only and hence kills all smoothing operators: it is the integral, over the cotangent sphere bundle S * M/B, of the order − dim(M/B) component in the asymptotic expansion of the symbol [15]. Proposition 5. 3 3Let ♮σ 0 dσ 1 . . . dσ 2n dσ 2n+1 and ♮dσ 1 . . . dσ 2n dσ 2n+1 be generic odd chains in X(M ) such that all σ i 's are in the image of the linear splitting σ : A → M . Let C be a cycle of even dimension with associated trace τ . Then the boundary of the renormalized trace τ R reads τ R ∂(σ 0 dσ 1 . . . dσ 2n dσ 2n+1 ) = n! (2n)! − C (σ 0 dσ 1 . . . dσ 2n ) 11 [ln D, σ 2n+1 ] Now we can calculate the pairing as E * ([τ ]), [g] = 1 √ 2πi τ R ∂(Q −1 dQ). Combining this with the explicit formula of Proposition 5.3 and the Index Theorem 4.3 gives an expression of [τ ] Corollary 5. 4 4Let M → B be a proper submersion with connection ∇. Let [g] ∈ K 1 (A ) be the symbol class of an elliptic family of fiberwise pseudodifferential operators Q with parametrix P . Let C be a cycle of even dimension in the base manifold B, and [τ ] ∈ HP 0 (B) the associated cyclic cohomology class. The evaluation of [τ ] on the index class Ind E ([g]) ∈ K 0 (B) is given by the fiberwise residue [τ ], Ind E ([g] [ln D, Q] + P (δQδ ln D − δ ln DδQ) + (θP + P θ)[ln D, Q] This diagram is naturally mapped to (28) (tensored with M N (C)) under the homomorphism β * : G → M N (M ). If τ is any cocycle over F 4n+1 R X( M ), its cupproduct with the trace of matrices yields a cocycle τ #tr over F 4n+1There is a concrete description of the first line. The ideal N ⊂ M is the set of elements α = ω 11 + ω 12 v + vω 21 + vω 22 v, with ω 11 , ω 22 ∈ M 0 of even degree, ω 12 , ω 21 ∈ M 0 of odd degree, and ω 11 has no component of degree zero. Hence α has an overall degree ≥ 2, which means that the algebra N is nilpotent. Similarly with J and Q. Hence according to Remark 2.4 this implies M = M and any trace over R n+1 , n ≥ 1, determines a class in HP 0 (B).We shall now construct the connecting morphism of the extension (E) as explained in section 2. A linear splitting σ : A → M is obtained as follows: first choose a "quantization map" q : A → E , which associates to any function a ∈ C ∞ c (B, LS 0 ) a family of pseudodifferential operators q(a) ∈ C ∞ c (B, CL 0 ) with leading symbol a. Then map E to the degree zero subspace of M . This 20 yields the desired linear splitting σ for the extensionwhence a classifying homomorphism σ * : T A → M which intertwines the tensor product and the Fedosov product: σ * (a 1 ⊗. . .⊗a n ) = σ(a 1 )⊙. . .⊙σ(a n ) for any element a 1 ⊗ . . . ⊗ a n ∈ T A . By construction σ * restricts to a homomorphism JA → R + N . To see this, recall that the ideal JA = Ker(T A → A ) is generated by the differences a 1 a 2 − a 1 ⊗ a 2 for any pair a 1 , a 2 ∈ A . Then= σ(a 1 a 2 ) − σ(a 1 )σ(a 2 ) + dσ(a 1 )dσ(a 2 ) .The first term of the r.h.s. σ(a 1 a 2 ) − σ(a 1 )σ(a 2 ) ∈ C ∞ c (B, CL −1 ) lies in the degree zero subspace of R, whereas the two-form dσ(a 1 )dσ(a 2 ) lies in N . Hence JA is mapped to the ideal R + N as claimed, and σ * extends to a homomorphism of pro-algebras (recall that N is nilpotent)Now consider the classes in HP 0 (B) represented by traces on the powers of R.We shall construct such traces by combining closed currents (cycles) in the base manifold B with the ordinary (fiberwise) trace of pseudodifferential operators. Indeed if ω ∈ Ω c (B, CL k ) takes values in the space of pseudodifferential operators of order k < − dim(M/B), the operator trace is well-defined and yields a smooth differential form Tr(ω) ∈ Ω c (B). Proof: The algebra R 0 is the direct sum of the spaces Ω k c (B, CL k−1 ). Let us take the n-fold (ordinary) product of operator-valued differential forms:Integration over C will retain the case k 1 + . . . + k n = dim C only. Hence the pseudodifferential order k 1 + . . . + k n − n is dim C − n. If moreover one chooses n > dim(M/B) + dim C, the order is < − dim(M/B) and the corresponding pseudodifferential operators are trace-class. This estimate does not change if one takes further products with the multiplier θ ∈ Ω 2 (B, CL 1 ), or if the ordinary product is replaced by the Fedosov product (which increases the form degree by two and the order by at most one). One concludes that the linear functional τ is well-defined on R n+1 provided one chooses n = dim(M/B) + dim C.Then one checks as in[2]pp. 229 that τ (α) vanishes if α is the graded commutator of elements in the DG algebra R 0 [v], or if α = dβ is closed. Hence τ is a Non-commutative differential geometry. A Connes, Publ. Math. IHES. 62A. Connes: Non-commutative differential geometry, Publ. Math. IHES 62 (1986) 41-144. A Connes, Non-commutative geometry. New-YorkAcademic PressA. Connes: Non-commutative geometry, Academic Press, New-York (1994). A Connes, H Moscovici, The local index formula in non-commutative geometry. 5A. Connes, H. Moscovici: The local index formula in non-commutative ge- ometry, GAFA 5 (1995) 174-243. Universal extensions and cyclic cohomology. J Cuntz, C. R. Acad. Sci. Paris. 309J. Cuntz: Universal extensions and cyclic cohomology, C. R. Acad. Sci. Paris 309 Série I (1989) 5-8. J Cuntz, D Quillen, Cyclic homology and nonsingularity. 8J. Cuntz, D. Quillen: Cyclic homology and nonsingularity, JAMS 8 (1995) 373-442. Excision in bivariant periodic cyclic cohomology. J Cuntz, D Quillen, Invent. Math. 127J. Cuntz, D. Quillen: Excision in bivariant periodic cyclic cohomology, In- vent. Math. 127 (1997) 67-98. Algebraic K-theory. J Milnor, Ann. Math. Studies. 72Princeton University pressJ. Milnor: Algebraic K-theory, Ann. Math. Studies 72 Princeton University press (1974). Higher index theorems and the boundary map in cyclic cohomology. V Nistor, Doc. Math. J. DMV. 2V. Nistor: Higher index theorems and the boundary map in cyclic cohomol- ogy, Doc. Math. J. DMV 2 (1997) 263-295. Chern-Weil forms associated with superconnections, Analysis, geometry and topology of elliptic operators. S Paycha, S Scott, World Sci. Publ. S. Paycha, S. Scott: Chern-Weil forms associated with superconnections, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hack- ensack, NJ (2006) 79-104 Anomalies and noncommutative index theory, lectures given at Villa de Leyva. D Perrot, Contemp. Math. S. Paycha and B. Uribe Ed.434D. Perrot: Anomalies and noncommutative index theory, lectures given at Villa de Leyva, Colombia (2005), S. Paycha and B. Uribe Ed., Contemp. Math. 434 (2007) 125-160. Secondary invariants for Fréchet algebras and quasihomomorphisms. D Perrot, Doc. Math. J. DMV. 13D. Perrot: Secondary invariants for Fréchet algebras and quasihomomor- phisms, Doc. Math. J. DMV 13 (2008) 275-363. D Perrot, arXiv:0804.1048Quasihomomorphisms and the residue Chern character. preprintD. Perrot: Quasihomomorphisms and the residue Chern character, preprint arXiv:0804.1048. Localization over complex-analytic groupoids and conformal renormalization. D Perrot, J. Noncommut. Geom. 3D. Perrot: Localization over complex-analytic groupoids and conformal renormalization, J. Noncommut. Geom. 3 (2009) 289-325. Lie algebras of differential operators, their central extensions and W -algebras. A O , Russian) Funktsional. Anal. i Prilozhen. 25Funct. Anal. Appl.A. O. Radul: Lie algebras of differential operators, their central extensions and W -algebras. (Russian) Funktsional. Anal. i Prilozhen. 25 (1991) 33-49; translation in Funct. Anal. Appl. 25 (1991) 25-39 M Wodzicki, Non-commutative residue. Springer Verlag1283M. Wodzicki: Non-commutative residue, Lect. Notes in Math. 1283, Springer Verlag (1987).
[]
[ "Cardiologist-Level Arrhythmia Detection with Convolutional Neural Networks", "Cardiologist-Level Arrhythmia Detection with Convolutional Neural Networks" ]
[ "Pranav Rajpurkar [email protected] ", "Awni Y Hannun ", "Masoumeh Haghpanahi [email protected] ", "Codie Bourn [email protected] ", "Andrew Y Ng " ]
[]
[]
We develop an algorithm which exceeds the performance of board certified cardiologists in detecting a wide range of heart arrhythmias from electrocardiograms recorded with a single-lead wearable monitor. We build a dataset with more than 500 times the number of unique patients than previously studied corpora. On this dataset, we train a 34-layer convolutional neural network which maps a sequence of ECG samples to a sequence of rhythm classes. Committees of boardcertified cardiologists annotate a gold standard test set on which we compare the performance of our model to that of 6 other individual cardiologists. We exceed the average cardiologist performance in both recall (sensitivity) and precision (positive predictive value). * Authors contributed equally. Project website at https://stanfordmlgroup. github.io/projects/ecg
null
[ "https://arxiv.org/pdf/1707.01836v1.pdf" ]
11,797,512
1707.01836
a679c01709fc379d9ca464f252ff1854bbe46d53
Cardiologist-Level Arrhythmia Detection with Convolutional Neural Networks Pranav Rajpurkar [email protected] Awni Y Hannun Masoumeh Haghpanahi [email protected] Codie Bourn [email protected] Andrew Y Ng Cardiologist-Level Arrhythmia Detection with Convolutional Neural Networks We develop an algorithm which exceeds the performance of board certified cardiologists in detecting a wide range of heart arrhythmias from electrocardiograms recorded with a single-lead wearable monitor. We build a dataset with more than 500 times the number of unique patients than previously studied corpora. On this dataset, we train a 34-layer convolutional neural network which maps a sequence of ECG samples to a sequence of rhythm classes. Committees of boardcertified cardiologists annotate a gold standard test set on which we compare the performance of our model to that of 6 other individual cardiologists. We exceed the average cardiologist performance in both recall (sensitivity) and precision (positive predictive value). * Authors contributed equally. Project website at https://stanfordmlgroup. github.io/projects/ecg Introduction We develop a model which can diagnose irregular heart rhythms, also known as arrhythmias, from single-lead ECG signals better than a cardiologist. Key to exceeding expert performance is a deep convolutional network which can map a sequence of ECG samples to a sequence of arrhythmia annotations along with a novel dataset two orders of magnitude larger than previous datasets of its kind. Many heart diseases, including Myocardial Infarction, AV Block, Ventricular Tachycardia and Atrial Fibrillation can all be diagnosed from ECG signals with an estimated 300 million ECGs recorded annually (Hedén et al., 1996). We investigate the task of arrhythmia detection from the ECG record. This is known to be a challenging task for computers but can usually be determined by an expert from a single, well-placed lead. Arrhythmia detection from ECG recordings is usually performed by expert technicians and cardiologists given the high error rates of computerized interpretation. One study found that of all the computer predictions for non-sinus rhythms, only about 50% were correct (Shah & Rubin, 2007); in another study, only 1 out of every 7 presentations of second degree AV block were correctly recognized by the algorithm (Guglin & Thatai, 2006). To automatically detect heart arrhythmias in an ECG, an algorithm must implicitly recognize the distinct wave types and discern the complex relationships between them over time. This is difficult due to the variability in wave morphology between patients as well as the presence of noise. We train a 34-layer convolutional neural network (CNN) to detect arrhythmias in arbitrary length ECG time-series. Figure 1 shows an example of an input to the model. In addition to classifying noise and the sinus rhythm, the network learns to classify and segment twelve arrhythmia types present in the time-series. The model is trained end-to-end on a single-lead ECG signal sampled at 200Hz and a sequence of annotations for every second of the ECG as supervision. To make the optimization of such a deep model tractable, we use residual connections and batchnormalization (He et al., 2016b;Ioffe & Szegedy, 2015). The depth increases both the non-linearity of the computation as well as the size of the context window for each classification decision. We construct a dataset 500 times larger than other datasets of its kind (Moody & Mark, 2001;Goldberger et al., 2000). One of the most popular previous datasets, the MIT-BIH corpus contains ECG recordings from 47 unique patients. In contrast, we collect and annotate a dataset of about 30,000 unique patients from a pool of nearly 300,000 patients who have used the Zio Patch monitor 1 (Turakhia et al., 2013). We intentionally select patients exhibiting abnormal rhythms in order to make the class balance of the dataset more even and thus the likelihood of observing unusual heart-activity high. We test our model against board-certified cardiologists. A committee of three cardiologists serve as gold-standard annotators for the 336 examples in the test set. Our model exceeds the individual expert performance on both recall (sensitivity), and precision (positive predictive value) on this test set. Model Problem Formulation The ECG arrhythmia detection task is a sequence-tosequence task which takes as input an ECG signal X = [x 1 , ..x k ], and outputs a sequence of labels r = [r 1 , ...r n ], such that each r i can take on one of m different rhythm classes. Each output label corresponds to a segment of the input. Together the output labels cover the full sequence. For a single example in the training set, we optimize the cross-entropy objective function L(X, r) = 1 n n i=1 log p(R = r i | X) where p(·) is the probability the network assigns to the i-th output taking on the value r i . Model Architecture and Training We use a convolutional neural network for the sequence-tosequence learning task. The high-level architecture of the network is shown in Figure 2. The network takes as input a time-series of raw ECG signal, and outputs a sequence of label predictions. Figure 2. The architecture of the network. The first and last layer are special-cased due to the pre-activation residual blocks. Overall, the network contains 33 layers of convolution followed by a fully-connected layer and a softmax. sampled at 200Hz, and the model outputs a new prediction once every second. We arrive at an architecture which is 33 layers of convolution followed by a fully connected layer and a softmax. In order to make the optimization of such a network tractable, we employ shortcut connections in a similar manner to those found in the Residual Network architecture (He et al., 2015b). The shortcut connections between neuralnetwork layers optimize training by allowing information to propagate well in very deep neural networks. Before the input is fed into the network, it is normalized using a robust normalization strategy. The network consists of 16 residual blocks with 2 convolutional layers per block. The convolutional layers all have a filter length of 16 and have 64k filters, where k starts out as 1 and is incremented every 4-th residual block. Every alternate residual block subsamples its inputs by a factor of 2, thus the original input is ultimately subsampled by a factor of 2 8 . When a residual block subsamples the input, the corresponding shortcut connections also subsample their input using a Max Pooling operation with the same subsample factor. Before each convolutional layer we apply Batch Normalization (Ioffe & Szegedy, 2015) and a rectified linear activation, adopting the pre-activation block design (He et al., 2016a). The first and last layers of the network are specialcased due to this pre-activation block structure. We also apply Dropout (Srivastava et al., 2014) between the convolutional layers and after the non-linearity. The final fully connected layer and softmax activation produce a distribution over the 14 output classes for each time-step. We train the networks from scratch, initializing the weights of the convolutional layers as in (He et al., 2015a). We use the Adam (Kingma & Ba, 2014) optimizer with the default parameters and reduce the learning rate by a factor of 10 when the validation loss stops improving. We save the best model as evaluated on the validation set during the optimization process. [ht] 3. Data Training We collect and annotate a dataset of 64,121 ECG records from 29,163 patients. The ECG data is sampled at a frequency of 200 Hz and is collected from a single-lead, noninvasive and continuous monitoring device called the Zio Patch which has a wear period up to 14 days (Turakhia et al., 2013). Each ECG record in the training set is 30 seconds long and can contain more than one rhythm type. Each record is annotated by a clinical ECG expert: the expert highlights segments of the signal and marks it as corresponding to one of the 14 rhythm classes. The 30 second records were annotated using a web-based ECG annotation tool designed for this work. Label annotations were done by a group of Certified Cardiographic Technicians who have completed extensive training in arrhythmia detection and a cardiographic certification examination by Cardiovascular Credentialing International. The technicians were guided through the interface before they could annotate records. All rhythms present in a strip were labeled from their corresponding onset to offset, resulting in full segmentation of the input ECG data. To improve labeling consistency among different annotators, specific rules were devised regarding each rhythm transition. We split the dataset into a training and validation set. The training set contains 90% of the data. We split the dataset so that there is no patient overlap between the training and validation sets (as well as the test set described below). Testing We collect a test set of 336 records from 328 unique patients. For the test set, ground truth annotations for each record were obtained by a committee of three boardcertified cardiologists; there are three committees responsible for different splits of the test set. The cardiologists discussed each individual record as a group and came to a consensus labeling. For each record in the test set we also collect 6 individual annotations from cardiologists not participating in the group. This is used to assess performance of the model compared to an individual cardiologist. Rhythm Classes We identify 12 heart arrhythmias, sinus rhythm and noise for a total of 14 output classes. The arrhythmias are characterized by a variety of features. Table 2 in the Appendix shows an example of each rhythm type we classify. The noise label is assigned when the device is disconnected from the skin or when the baseline noise in the ECG makes identification of the underlying rhythm impossible. The morphology of the ECG during a single heart-beat as well as the pattern of the activity of the heart over time determine the underlying rhythm. In some cases the distinction between the rhythms can be subtle yet critical for treatment. ical, requiring immediate attention (Dubin, 1996). Table 2 in the Appendix also shows the number of unique patients in the training (including validation) set and test set for each rhythm type. Results Evaluation Metrics We use two metrics to measure model accuracy, using the cardiologist committee annotations as the ground truth. Sequence Level Accuracy (F1): We measure the average overlap between the prediction and the ground truth sequence labels. For every record, a model is required to make a prediction approximately once per second (every 256 samples). These predictions are compared against the ground truth annotation. Set Level Accuracy (F1): Instead of treating the labels for a record as a sequence, we consider the set of unique arrhythmias present in each 30 second record as the ground truth annotation. Set Level Accuracy, unlike Sequence Level Accuracy, does not penalize for time-misalignment within a record. We report the F1 score between the unique class labels from the ground truth and those from the model prediction. In both the Sequence and the Set case, we compute the F1 score for each class separately. We then compute the overall F1 (and precision and recall) as the class-frequency weighted mean. Model vs. Cardiologist Performance We assess the cardiologist performance on the test set. Recall that each of the records in the test set has a ground truth label from a committee of three cardiologists as well as individual labels from a disjoint set of 6 other cardiologists. To assess cardiologist performance for each class, we take the average of all the individual cardiologist F1 scores using the group label as the ground truth annotation. Table 1 shows the breakdown of both cardiologist and model scores across the different rhythm classes. The model outperforms the average cardiologist performance on most rhythms, noticeably outperforming the cardiologists in the AV Block set of arrhythmias which includes Mobitz I (Wenckebach), Mobitz II (AVB Type2) and complete heart block (CHB). This is especially useful given the severity of Mobitz II and complete heart block and the importance of distinguishing these two from Wenckebach which is usually considered benign. Analysis The model outperforms the average cardiologist score on both the sequence and the set F1 metrics. Figure 4 shows a confusion matrix of the model predictions on the test set. Many arrhythmias are confused with the sinus rhythm. We expect that part of this is due to the sometimes ambiguous location of the exact onset and offset of the arrhythmia in the ECG record. Often the mistakes made by the model are understandable. For example, confusing Wenckebach and AVB Type2 makes sense given that the two rhythms in general have very similar ECG morphologies. Similarly, Supraventricular Tachycardia (SVT) and Atrial Fibrillation (AFIB) are often confused with Atrial Flutter (AFL) which is understandable given that they are all atrial arrhythmias. We also note that Idioventricular Rhythm (IVR) is sometimes mistaken as Ventricular Tachycardia (VT), which again makes sense given that the two only differ in heart-rate and are difficult to distinguish close to the 100 beats per minute delineation. One of the most common confusions is between Ectopic Atrial Rhythm (EAR) and sinus rhythm. The main distinguishing criteria for this rhythm is an irregular P wave. This can be subtle to detect especially when the P wave has a small amplitude or when noise is present in the signal. AFIB Related Work Automatic high-accuracy methods for R-peak extraction have existed at least since the mid 1980's (Pan & Tompkins, 1985). Current algorithms for R-peak extraction tend to use wavelet transformations to compute features from the raw ECG followed by finely-tuned threshold based classifiers (Li et al., 1995;Martínez et al., 2004). Because accurate estimates of heart rate and heart rate variability can be extracted from R-peak features, feature-engineered algorithms are often used for coarse-grained heart rhythm classification, including detecting tachycardias (fast heart rate), bradycardias (slow heart rate), and irregular rhythms. However, such features alone are not sufficient to distinguish between most heart arrhythmias since features based on the atrial activity of the heart as well as other features pertaining to the QRS morphology are needed. Much work has been done to automate the extraction of other features from the ECG. For example, beat classification is a common sub-problem of heart-arrhythmia classification. Drawing inspiration from automatic speech recognition, Hidden Markov models with Gaussian observation probability distributions have been applied to the task of beat detection (Coast et al., 1990). Artificial neural networks have also been used for the task of beat detection (Melo et al., 2000). While these models have achieved high-accuracy for some beat types, they are not yet sufficient for high-accuracy heart arrhythmia classification and segmentation. For example, (Artis et al., 1991) train a neural network to distinguish between Atrial Fibrillation and Sinus Rhythm on the MIT-BIH dataset. While the network can distinguish between these two classes with high-accuracy, it does not generalize to noisier single-lead recordings or classify among the full range of 15 rhythms available in MIT-BIH. This is in part due to insufficient training data, and because the model also discards critical information in the feature extraction stage. The most common dataset used to design and evaluate ECG algorithms is the MIT-BIH arrhythmia database (Moody & Mark, 2001) which consists of 48 half-hour strips of ECG data. Other commonly used datasets include the MIT-BIH Atrial Fibrillation dataset (Moody & Mark, 1983) and the QT dataset (Laguna et al., 1997). While useful benchmarks for R-peak extraction and beat-level annotations, these datasets are too small for fine-grained arrhythmia classification. The number of unique patients is in the single digit hundreds or fewer for these benchmarks. A recently released dataset captured from the AliveCor ECG monitor contains about 7000 records (Clifford et al., 2017). These records only have annotations for Atrial Fibrillation; all other arrhythmias are grouped into a single bucket. The dataset we develop contains 29,163 unique patients and 14 classes with hundreds of unique examples for the rarest arrhythmias. Machine learning models based on deep neural networks have consistently been able to approach and often exceed human agreement rates when large annotated datasets are available (Amodei et al., 2016;Xiong et al., 2016;He et al., 2015c). These approaches have also proven to be effective in healthcare applications, particularly in medical imaging where pretrained ImageNet models can be applied (Esteva et al., 2017;Gulshan et al., 2016). We draw on work in automatic speech recognition for processing time-series with deep convolutional neural networks and recurrent neural networks (Hannun et al., 2014;Sainath et al., 2013), and techniques in deep learning to make the optimization of these models tractable (He et al., 2016b;c;Ioffe & Szegedy, 2015). Conclusion We develop a model which exceeds the cardiologist performance in detecting a wide range of heart arrhythmias from single-lead ECG records. Key to the performance of the model is a large annotated dataset and a very deep convolutional network which can map a sequence of ECG samples to a sequence of arrhythmia annotations. On the clinical side, future work should investigate extending the set of arrhythmias and other forms of heart disease which can be automatically detected with high-accuracy from single or multiple lead ECG records. For example we do not detect Ventricular Flutter or Fibrillation. We also do not detect Left or Right Ventricular Hypertrophy, Myocardial Infarction or a number of other heart diseases which do not necessarily exhibit as arrhythmias. Some of these may be difficult or even impossible to detect on a single-lead ECG but can often be seen on a multiple-lead ECG. Given that more than 300 million ECGs are recorded annually, high-accuracy diagnosis from ECG can save expert clinicians and cardiologists considerable time and decrease the number of misdiagnoses. Furthermore, we hope that this technology coupled with low-cost ECG devices enables more widespread use of the ECG as a diagnostic tool in places where access to a cardiologist is difficult. Table 2. A list of all of the rhythm types which the model classifies. For each rhythm we give the label name, a more descriptive name and an example chosen from the training set. We also give the total number of patients with each rhythm for both the training and test sets. Figure 1 . 1Our trained convolutional neural network correctly detecting the sinus rhythm (SINUS) and Atrial Fibrillation (AFIB) from this ECG recorded with a single-lead wearable heart monitor. Figure 3 . 3Evaluated on the test set, the model outperforms the average cardiologist score on both the Sequence and the Set F1 metrics. Figure 4 . 4A confusion matrix for the model predictions on the test set. Many of the mistakes the model makes are not surprising. For example, confusing second degree AV Block (Type 2) with Wenckebach makes sense given the often similar expression of the two arrhythmias in the ECG record. Table 1. The top part of the table gives a class-level comparison of the expert to the model F1 score for both the Sequence and the Set metrics. The bottom part of the table shows aggregate results over the full test set for precision, recall and F1 for both the Sequence and Set metrics.For example two forms of second degree AV Block, Mobitz I (Wenckebach) and Mobitz II (here referred to as AVB TYPE2) can be difficult to distinguish. Wenckebach is considered benign and Mobitz II is considered patholog- Seq Set Model Cardiol. Model Cardiol. Class-level F1 Score AFIB 0.604 0.515 0.667 0.544 AFL 0.687 0.635 0.679 0.646 AVB TYPE2 0.689 0.535 0.656 0.529 BIGEMINY 0.897 0.837 0.870 0.849 CHB 0.843 0.701 0.852 0.685 EAR 0.519 0.476 0.571 0.529 IVR 0.761 0.632 0.774 0.720 JUNCTIONAL 0.670 0.684 0.783 0.674 NOISE 0.823 0.768 0.704 0.689 SINUS 0.879 0.847 0.939 0.907 SVT 0.477 0.449 0.658 0.556 TRIGEMINY 0.908 0.843 0.870 0.816 VT 0.506 0.566 0.694 0.769 WENCKEBACH 0.709 0.593 0.806 0.736 Aggregate Results Precision (PPV) 0.800 0.723 0.809 0.763 Recall (Sensitivity) 0.784 0.724 0.827 0.744 F1 0.776 0.719 0.809 0.751 Table 1 1also compares the aggregate precision, recall and F1 for both model and cardiologist compared to the ground truth annotations. The aggregate scores for the cardiologist are computed by taking the mean of the individual cardiologist scores. The model outperforms the cardiologist average in both precision and recall. AcknowledgementsWe thank Geoffrey H. Tison MD, MPH of UCSF for helpful feedback on the experiments and references.Appendix Deep speech 2: End-to-end speech recognition in english and mandarin. Dario Amodei, Anubhai, Rishita, Battenberg, Eric, Case, Carl, Jared Casper, Catanzaro, Bryan, Chen, Jingdong, Chrzanowski, Mike, Coates, Adam, Greg Diamos, Proceedings of The 33rd International Conference on Machine Learning. The 33rd International Conference on Machine LearningAmodei, Dario, Anubhai, Rishita, Battenberg, Eric, Case, Carl, Casper, Jared, Catanzaro, Bryan, Chen, JingDong, Chrzanowski, Mike, Coates, Adam, Diamos, Greg, et al. Deep speech 2: End-to-end speech recognition in english and mandarin. In Proceedings of The 33rd International Conference on Machine Learning, pp. 173-182, 2016. Detection of atrial fibrillation using artificial neural networks. In Computers in Cardiology. Shane G Artis, R G Mark, G B Moody, Proceedings. IEEEArtis, Shane G, Mark, RG, and Moody, GB. Detection of atrial fibrillation using artificial neural networks. In Computers in Cardiology 1991, Proceedings., pp. 173- 176. IEEE, 1991. Af classification from a short single lead ecg recording: The physionet computing in cardiology challenge. G D Clifford, C Y Liu, B Moody, L Lehman, I Silva, Q Li, Aew Johnson, R G Mark, Clifford, GD, Liu, CY, Moody, B, Lehman, L, Silva, I, Li, Q, Johnson, AEW, and Mark, RG. Af classification from a short single lead ecg recording: The physionet comput- ing in cardiology challenge 2017. 2017. An approach to cardiac arrhythmia analysis using hidden markov models. Douglas A Coast, Richard M Stern, Cano, G Gerald, Briller, A Stanley, IEEE Transactions on biomedical Engineering. 379Coast, Douglas A, Stern, Richard M, Cano, Gerald G, and Briller, Stanley A. An approach to cardiac arrhythmia analysis using hidden markov models. IEEE Transac- tions on biomedical Engineering, 37(9):826-836, 1990. Rapid Interpretation of EKG's. USA. Dale Dubin, Cover Publishing CompanyDubin, Dale. Rapid Interpretation of EKG's. USA: Cover Publishing Company, 1996, 1996. Dermatologist-level classification of skin cancer with deep neural networks. Andre Esteva, Kuprel, Brett, Roberto A Novoa, Justin Ko, Susan M Swetter, Blau, M Helen, Sebastian Thrun, Nature. 5427639Esteva, Andre, Kuprel, Brett, Novoa, Roberto A, Ko, Justin, Swetter, Susan M, Blau, Helen M, and Thrun, Se- bastian. Dermatologist-level classification of skin cancer with deep neural networks. Nature, 542(7639):115-118, 2017. Ary L Goldberger, Amaral, A N Luis, Glass, Leon, Jeffrey M Hausdorff, Plamen Ivanov, Ch, Roger G Mark, Joseph E Mietus, George B Moody, Peng, Stanley Chung-Kang, H Eugene, Physiobank, physiotoolkit, and physionet components of a new research resource for complex physiologic signals. 101Goldberger, Ary L, Amaral, Luis AN, Glass, Leon, Haus- dorff, Jeffrey M, Ivanov, Plamen Ch, Mark, Roger G, Mietus, Joseph E, Moody, George B, Peng, Chung- Kang, and Stanley, H Eugene. Physiobank, phys- iotoolkit, and physionet components of a new research resource for complex physiologic signals. Circulation, 101(23):e215-e220, 2000. Common errors in computer electrocardiogram interpretation. Maya E Guglin, Deepak Thatai, International journal of cardiology. 1062Guglin, Maya E and Thatai, Deepak. Common errors in computer electrocardiogram interpretation. Interna- tional journal of cardiology, 106(2):232-237, 2006. Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs. Gulshan, Varun, Peng, Lily, Coram, Marc, Martin C Stumpe, Wu, Derek, Narayanaswamy, Arunachalam, Venugopalan, Subhashini, Widner, Kasumi, Madams, Tom, Jorge Cuadros, JAMA. 31622Gulshan, Varun, Peng, Lily, Coram, Marc, Stumpe, Mar- tin C, Wu, Derek, Narayanaswamy, Arunachalam, Venu- gopalan, Subhashini, Widner, Kasumi, Madams, Tom, Cuadros, Jorge, et al. Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs. JAMA, 316 (22):2402-2410, 2016. Deep speech: Scaling up end-toend speech recognition. abs/1412. Awni Y Hannun, Case, Carl, Jared Casper, Catanzaro, Bryan, Diamos, Greg, Elsen, Erich, Prenger, Ryan, Satheesh, Sanjeev, Sengupta, Shubho, Adam Coates, Andrew Y Ng, 5567Hannun, Awni Y., Case, Carl, Casper, Jared, Catanzaro, Bryan, Diamos, Greg, Elsen, Erich, Prenger, Ryan, Satheesh, Sanjeev, Sengupta, Shubho, Coates, Adam, and Ng, Andrew Y. Deep speech: Scaling up end-to- end speech recognition. abs/1412.5567, 2014. URL http://arxiv.org/abs/1412.5567. Delving deep into rectifiers: Surpassing humanlevel performance on imagenet classification. He, Kaiming, Zhang, Xiangyu, Shaoqing Ren, Jian Sun, abs/1502.01852He, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Delving deep into rectifiers: Surpassing human- level performance on imagenet classification. CoRR, abs/1502.01852, 2015a. URL http://arxiv.org/ abs/1502.01852. Deep residual learning for image recognition. CoRR, abs/1512.03385. He, Kaiming, Zhang, Xiangyu, Shaoqing Ren, Jian Sun, He, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Deep residual learning for image recogni- tion. CoRR, abs/1512.03385, 2015b. URL http: //arxiv.org/abs/1512.03385. Delving deep into rectifiers: Surpassing humanlevel performance on imagenet classification. He, Kaiming, Zhang, Xiangyu, Shaoqing Ren, Jian Sun, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionHe, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Delving deep into rectifiers: Surpassing human- level performance on imagenet classification. In Pro- ceedings of the IEEE international conference on com- puter vision, pp. 1026-1034, 2015c. Identity mappings in deep residual networks. CoRR, abs/1603.05027. He, Kaiming, Zhang, Xiangyu, Shaoqing Ren, Jian Sun, He, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Identity mappings in deep residual net- works. CoRR, abs/1603.05027, 2016a. URL http: //arxiv.org/abs/1603.05027. Deep residual learning for image recognition. He, Kaiming, Zhang, Xiangyu, Shaoqing Ren, Jian Sun, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionHe, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vi- sion and Pattern Recognition, pp. 770-778, 2016b. Identity mappings in deep residual networks. He, Kaiming, Zhang, Xiangyu, Shaoqing Ren, Jian Sun, European Conference on Computer Vision. SpringerHe, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Identity mappings in deep residual networks. In European Conference on Computer Vision, pp. 630-645. Springer, 2016c. Detection of frequently overlooked electrocardiographic lead reversals using artificial neural networks. Bo Hedén, Ohlsson, Mattias, Holst, Holger, Mjöman, Mattias, Rittner, Ralf, Pahlm, Olle, Carsten Peterson, Lars Edenbrandt, The American journal of cardiology. 785Hedén, Bo, Ohlsson, Mattias, Holst, Holger, Mjöman, Mat- tias, Rittner, Ralf, Pahlm, Olle, Peterson, Carsten, and Edenbrandt, Lars. Detection of frequently overlooked electrocardiographic lead reversals using artificial neu- ral networks. The American journal of cardiology, 78 (5):600-604, 1996. Batch normalization: Accelerating deep network training by reducing internal covariate shift. Sergey Ioffe, Christian Szegedy, arXiv:1502.03167arXiv preprintIoffe, Sergey and Szegedy, Christian. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015. Diederik Kingma, Jimmy Ba, Adam, arXiv:1412.6980A method for stochastic optimization. arXiv preprintKingma, Diederik and Ba, Jimmy. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. A database for evaluation of algorithms for measurement of qt and other waveform intervals in the ecg. Pablo Laguna, Roger G Mark, A Goldberg, George B Moody, Computers in Cardiology. IEEELaguna, Pablo, Mark, Roger G, Goldberg, A, and Moody, George B. A database for evaluation of algorithms for measurement of qt and other waveform intervals in the ecg. In Computers in Cardiology 1997, pp. 673-676. IEEE, 1997. Detection of ECG characteristic points using wavelet transforms. Cuiwei Li, Chongxun Zheng, Changfeng Tai, IEEE Transactions on biomedical Engineering. 421Li, Cuiwei, Zheng, Chongxun, and Tai, Changfeng. De- tection of ECG characteristic points using wavelet trans- forms. IEEE Transactions on biomedical Engineering, 42(1):21-28, 1995. A waveletbased ECG delineator: evaluation on standard databases. Juan Martínez, Pablo, Almeida, Rute, Olmos, Salvador, Ana Rocha, Paula, Pablo Laguna, IEEE Transactions on biomedical engineering. 514Martínez, Juan Pablo, Almeida, Rute, Olmos, Salvador, Rocha, Ana Paula, and Laguna, Pablo. A wavelet- based ECG delineator: evaluation on standard databases. IEEE Transactions on biomedical engineering, 51(4): 570-581, 2004. Arrhythmia analysis using artificial neural network and decimated electrocardiographic data. S L Melo, L P Caloba, J Nadal, Computers in Cardiology. IEEEMelo, SL, Caloba, LP, and Nadal, J. Arrhythmia analysis using artificial neural network and decimated electrocar- diographic data. In Computers in Cardiology 2000, pp. 73-76. IEEE, 2000. A new method for detecting atrial fibrillation using RR intervals. George B Moody, Roger G Mark, Computers in Cardiology. 101Moody, George B and Mark, Roger G. A new method for detecting atrial fibrillation using RR intervals. Comput- ers in Cardiology, 10(1):227-230, 1983. The impact of the MIT-BIH arrhythmia database. George B Moody, Roger G Mark, IEEE Engineering in Medicine and Biology Magazine. 203Moody, George B and Mark, Roger G. The impact of the MIT-BIH arrhythmia database. IEEE Engineering in Medicine and Biology Magazine, 20(3):45-50, 2001. A real-time QRS detection algorithm. Jiapu Pan, Willis J Tompkins, IEEE transactions on biomedical engineering. 3Pan, Jiapu and Tompkins, Willis J. A real-time QRS detec- tion algorithm. IEEE transactions on biomedical engi- neering, (3):230-236, 1985. Deep convolutional neural networks for lvcsr. Tara N Sainath, Mohamed, Brian Kingsbury, Ramabhadran, Bhuvana, Acoustics, speech and signal processing (ICASSP), 2013 IEEE international conference on. IEEESainath, Tara N, Mohamed, Abdel-rahman, Kingsbury, Brian, and Ramabhadran, Bhuvana. Deep convolutional neural networks for lvcsr. In Acoustics, speech and sig- nal processing (ICASSP), 2013 IEEE international con- ference on, pp. 8614-8618. IEEE, 2013. Errors in the computerized electrocardiogram interpretation of cardiac rhythm. Atman P Shah, Stanley A Rubin, Journal of electrocardiology. 405Shah, Atman P and Rubin, Stanley A. Errors in the computerized electrocardiogram interpretation of car- diac rhythm. Journal of electrocardiology, 40(5):385- 390, 2007. Dropout: a simple way to prevent neural networks from overfitting. Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, Ruslan Salakhutdinov, Journal of Machine Learning Research. 151Srivastava, Nitish, Hinton, Geoffrey E, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Dropout: a simple way to prevent neural networks from overfit- ting. Journal of Machine Learning Research, 15(1): 1929-1958, 2014. Diagnostic utility of a novel leadless arrhythmia monitoring device. Mintu P Turakhia, Donald D Hoang, Zimetbaum, Peter, Jared D Miller, Froelicher, F Victor, Kumar, N Uday, Xu, Xiangyan, Felix Yang, Paul A Heidenreich, The American journal of cardiology. 1124Turakhia, Mintu P, Hoang, Donald D, Zimetbaum, Peter, Miller, Jared D, Froelicher, Victor F, Kumar, Uday N, Xu, Xiangyan, Yang, Felix, and Heidenreich, Paul A. Diagnostic utility of a novel leadless arrhythmia moni- toring device. The American journal of cardiology, 112 (4):520-524, 2013. Achieving human parity in conversational speech recognition. Wayne Xiong, Droppo, Jasha, Huang, Xuedong, Seide, Frank, Seltzer, Mike, Andreas Stolcke, Dong Yu, Geoffrey Zweig, arXiv:1610.05256arXiv preprintXiong, Wayne, Droppo, Jasha, Huang, Xuedong, Seide, Frank, Seltzer, Mike, Stolcke, Andreas, Yu, Dong, and Zweig, Geoffrey. Achieving human parity in conversational speech recognition. arXiv preprint arXiv:1610.05256, 2016.
[]
[ "Few-body reference data for multicomponent formalisms: Light nuclei molecules", "Few-body reference data for multicomponent formalisms: Light nuclei molecules" ]
[ "Ilkka Kylänpää \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801IllinoisUSA\n", "Tapio T Rantala \nDepartment of Physics\nTampere University of Technology\nP.O. Box 692FI-33101TampereFinland\n", "David M Ceperley \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801IllinoisUSA\n" ]
[ "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801IllinoisUSA", "Department of Physics\nTampere University of Technology\nP.O. Box 692FI-33101TampereFinland", "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801IllinoisUSA" ]
[]
We present full quantum statistical energetics of some electron-light nuclei systems. This is accomplished with the path integral Monte Carlo method. The effects on energetics arising from the change in the nuclear mass are studied. The obtained results may serve as reference data for the multicomponent density functional theory calculations of light nuclei system. In addition, the results reported here will enable better fitting of todays electron-nuclear energy functionals, for which the description of light nuclei is most challenging, in particular.
10.1103/physreva.86.052506
[ "https://arxiv.org/pdf/1208.2216v1.pdf" ]
14,317,097
1208.2216
263a711ae442d5f2d7a21aa3783208c3faf76b27
Few-body reference data for multicomponent formalisms: Light nuclei molecules 10 Aug 2012 (Dated: May 2, 2014) Ilkka Kylänpää Department of Physics University of Illinois at Urbana-Champaign 61801IllinoisUSA Tapio T Rantala Department of Physics Tampere University of Technology P.O. Box 692FI-33101TampereFinland David M Ceperley Department of Physics University of Illinois at Urbana-Champaign 61801IllinoisUSA Few-body reference data for multicomponent formalisms: Light nuclei molecules 10 Aug 2012 (Dated: May 2, 2014) We present full quantum statistical energetics of some electron-light nuclei systems. This is accomplished with the path integral Monte Carlo method. The effects on energetics arising from the change in the nuclear mass are studied. The obtained results may serve as reference data for the multicomponent density functional theory calculations of light nuclei system. In addition, the results reported here will enable better fitting of todays electron-nuclear energy functionals, for which the description of light nuclei is most challenging, in particular. Density functional theory (DFT) is among the most succesful approaches to calculate the electronic structure of atoms, molecules and solids. A similar approach, however, including more degrees of freedom was introduced in 2001 by Kreibich and Gross [1], and is called as multicomponent density-functional theory (MCDFT). In contrast to original form of the DFT, MCDFT enables the complete quantum treatment of many particle systems consisting of electrons and nuclei. As is well known, the original form of DFT incorporates the Born-Oppenheimer approximation for the nuclei [2,3]. With the MCDFT approach it is possible to extend the success of DFT into an entirely new field of applications, such as first-principles calculation of electronphonon coupling in solids [4,5], polaronic motion [6] and positron scattering and annihilation [7][8][9]. That is, with MCDFT physical phenomena that depend on a strong coupling between electronic and nuclear motion can be evaluated from first principles. The original DFT is also known for its need of good functional forms, especially for the exchange and correlation functional. One of the most widely employed functional is the so-called local density approximation (LDA), which uses the Monte Carlo data of the free electron gas [10] as a basic input. Proper functional forms are also needed in the MCDFT scheme, for the electron-nuclear energy functional [11,12], in particular. For the present, the absence of good multicomponent reference data is slowing down the development of new functional forms for the MCDFT. The main difficulties are encountered in the description of light nuclei. In this brief report, we will provide few-body reference data for light nuclei systems, which can be used in the development of better MCDFT functionals and improving the present fits. This is accomplished with full quantum statistical simulations using path integral Monte Carlo (PIMC) approach [13]. The nuclear mass is given values ranging from that of a positron to that of a proton described by the following processes: x + e − 2 , x + 2 e − , x + 2 e − 2 and x + p + e − 2 , where x + goes from positron (e + ) to proton (p + ). A more detailed description of our approach is given in Ref. [14]. According to the Feynman formulation of the quantum statistical mechanics [15] the partition function for interacting distinguishable particles is given by the trace of the density matrix: Z = Trρ(β) = dR 0 dR 1 . . . dR M−1 M−1 i=0 e −S(Ri,Ri+1;τ ) , whereρ(β) = e −βĤ , S is the action, β = 1/k B T , τ = β/M , R M = R 0 and M is called the Trotter number. In this paper, we use the pair approximation in the action [13,16] for the Coulomb interaction of charges. Sampling in the configuration space is carried out using the Metropolis procedure [17] with multilevel bisection moves [18]. The total energy is calculated using the virial estimator [19]. In the following we use atomic units, where the lengths, energies and masses are given in units of the Bohr radius (a 0 ), hartree (E h ) and free electron mass (m e ), respectively. The statistical standard error of the mean (SEM) with 2SEM limits is used as an error estimate for the observables. In our model, all the particles are described as "boltzmannons", i.e. they obey the Boltzmann statistics. For the present study the particles involved can be treated accurately as distinguishable particles. This is possible by assigning spin-up to one electron and spin-down to the other one, and applying the same for the positive particles. This is accurate enough, as long as the thermal energy is well below that of the lowest electronic triplet excitation, ∆E st . For the systems in consideration ∆E st > 0.18E h , the smallest being that of the Ps 2 molecule [20,21]. For more details on our model, see Ref. [14]. In the simulations we use m e = 1 = m e + as the mass of the electrons and the positron, and for the protons we use In Figs. 1 and 2 we show the total energy as a function of mass of the nuclei, i.e. the positive particles: On the left x + is equal to a positron, on the right x + corresponds to a proton, and in the middle region we assign ten different masses for the x + particle. The total energies are also given in Table I. The timestep error affects mainly the fourth decimal in the total energies, which can be validated by comparing the end-point values in Table I to high-accuracy zero Kelvin results. The comparison shows that the difference between high accuracy results and our PIMC values is less than 0.00094E −1 h , which also confirms that the order of the time-step error is O(τ 3 ). Since the fourth decimal is also uncertain due to statistical error estimate, the present time-step error is considered acceptable. All energies given in Table I are from separate long enough simulations. Due to the finite temperature present in our simulations there is a small possibility for these molecules to dissociate even at the temperature of 300 K, however, none of our simulations experienced dissociation. The main difficulties in the MCDFT are related to the description of light nuclei. Protons in small systems are already treated reasonably. However, there definitely is room for improvement in that case, and especially in case of positronic systems. The data presented in Table I will serve as a good reference data in the development and fitting of electron-nuclear energy functionals. It enables one to gradually go towards proper description of the lightest and most difficult "nucleus", i.e. the positron. It should be pointed out, that proper density dependent reference data will be essential for the success of MCDFT. Obtaining such results is computationally demanding, however, the authors of this paper are already Table I: Total energies at different nuclear masses, see also Figs. 1 and 2. Energies are given in units of hartree with 2SEM error estimates. Reading from up to down the x + in the table goes from positron to proton -the mass of the particle increases. working on it. For now, the results of this paper give useful complementary information on the energetics of small light nuclei systems, which can be used in the finding better fits for the functionals. We acknowledge CSC -IT Center for Science Ltd. and TCSC -Tampere Center for Scientific Computing for the allocation of computational resources. For financial support we thank the Finnish Cultural Foundation and the Physics Department of the University of Illinois at Urbana-Champaign. log 2 (mx/me) x + e − 2 x + 2 e − x + 2 e − 2 x + p + e − m p = 1836.1527m e . The simulations are carried out at 300 K temperature, and for the Trotter number we have chosen M = 8192. This leads to "time-step" τ = β/M ≈ 0.1285E −1 h , which ensures good enough accuracy in the case of light nuclei -the error is of order O(τ 3 ). The simulations apply the minimum image convention and a cubic simulation cell, V = (300a 0 ) 3 . Figure 1 : 1Total energy as a function of mass of the nuclei, i.e. positive particles. Blue circles show the energy for x + e − 2 , and red down-triangles that for x + 2 e − , where x + goes from e + to p + . The reference energies are given as solid line, dashed line and dash-dotted line corresponding to Ps − (as well as Ps + 2 ), H − and H + 2 , respectively. Figure 2 : 2Total energy as a function of mass of the nuclei, i.e. positive particles. Blue up-triangles show the energy for x + 2 e − 2 , and red squares that for x + p + e − 2 , where x + goes from e + to p + . The reference energies are given as solid line, dashed line and dash-dotted line corresponding to Ps2, PsH and H2, respectively. 0.5258(2) -0.5866(2) -1.1458(3) -1.1533(3) 9.0000 -0.5269(2) -0.5913(2) -1.1549(3) -1.1588(3) 10.0000 -0.5275(2) -0.5944(2) -1.1612(3) -1.1627(3) 10.8425 -0.5277(3) -0.5962(2) -1.1646(3) -1.1647(3) . T Kreibich, E K U Gross, Phys. Rev. Lett. 862984T. Kreibich and E. K. U. Gross, Phys. Rev. Lett. 86, 2984 (2001). . P Hohenberg, W Kohn, Phys. Rev. 136864P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). . W Kohn, L J Sham, Phys. Rev. 1401133W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). . R Van Leeuwen, Phys. Rev. B. 69115110R. van Leeuwen, Phys. Rev. B 69, 115110 (2004). . R V Leeuwen, Phys. Rev. B. 69199901R. v. Leeuwen, Phys. Rev. B 69, 199901 (2004). . K Hannewald, P A Bobbert, Phys. Rev. B. 6975212K. Hannewald and P. A. Bobbert, Phys. Rev. B 69, 075212 (2004). . G F Gribakin, J A Young, C M Surko, Rev. Mod. Phys. 822557G. F. Gribakin, J. A. Young, and C. M. Surko, Rev. Mod. Phys. 82, 2557 (2010). . H R J Walters, Science. 330762H. R. J. Walters, Science 330, 762 (2010). . S J Brawley, S Armitage, J Beale, D E Leslie, A I Williams, G Laricchia, Science. 330789S. J. Brawley, S. Armitage, J. Beale, D. E. Leslie, A. I. Williams, and G. Laricchia, Science 330, 789 (2010). . D M Ceperley, B J Alder, Phys. Rev. Lett. 45566D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). . T Kreibich, R Van Leeuwen, E K U Gross, Phys. Rev. A. 7822501T. Kreibich, R. van Leeuwen, and E. K. U. Gross, Phys. Rev. A 78, 022501 (2008). . A Chakraborty, M V Pak, S Hammes-Schiffer, Phys. Rev. Lett. 101153001A. Chakraborty, M. V. Pak, and S. Hammes-Schiffer, Phys. Rev. Lett. 101, 153001 (2008). . D M Ceperley, Rev. Mod. Phys. 67279D. M. Ceperley, Rev. Mod. Phys 67, 279 (1995). . I Kylänpää, T T Rantala, J. Chem. Phys. 135104310I. Kylänpää and T. T. Rantala, J. Chem. Phys. 135, 104310 (2011). R P Feynman, Statistical Mechanics. Reading, MAPerseus BooksR. P. Feynman, Statistical Mechanics (Perseus Books, Reading, MA, 1998). . R G Storer, J. Math. Phys. 9964R. G. Storer, J. Math. Phys. 9, 964 (1968). . N Metropolis, A W Rosenbluth, M N Rosenbluth, A H Teller, E Teller, J. Chem. Phys. 211087N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). . C Chakravarty, M C Gordillo, D M Ceperley, J. Chem. Phys. 1092123C. Chakravarty, M. C. Gordillo, and D. M. Ceperley, J. Chem. Phys. 109, 2123 (1998). . M F Herman, E J Bruskin, B J Berne, J. Chem. Phys. 765150M. F. Herman, E. J. Bruskin, and B. J. Berne, J. Chem. Phys. 76, 5150 (1982). . J Usukura, Y Suzuki, Phys. Rev. A. 6610502J. Usukura and Y. Suzuki, Phys. Rev. A 66, 010502 (1998). . J Mitroy, M W J Bromley, Phys. Rev. A. 7352712J. Mitroy and M. W. J. Bromley, Phys. Rev. A 73, 052712 (2006).
[]
[ "A Bethe Ansatz type formula for the superconformal index", "A Bethe Ansatz type formula for the superconformal index" ]
[ "Francesco Benini [email protected] \nSISSA\nVia Bonomea 26534136TriesteItaly\n\nINFN\nSezione di Trieste\nVia Valerio 234127TriesteItaly\n\nICTP\nStrada Costiera 1134151TriesteItaly\n", "Paolo Milan [email protected] \nSISSA\nVia Bonomea 26534136TriesteItaly\n\nINFN\nSezione di Trieste\nVia Valerio 234127TriesteItaly\n" ]
[ "SISSA\nVia Bonomea 26534136TriesteItaly", "INFN\nSezione di Trieste\nVia Valerio 234127TriesteItaly", "ICTP\nStrada Costiera 1134151TriesteItaly", "SISSA\nVia Bonomea 26534136TriesteItaly", "INFN\nSezione di Trieste\nVia Valerio 234127TriesteItaly" ]
[]
Inspired by recent work by Closset, Kim and Willett, we derive a new formula for the superconformal (or supersymmetric) index of 4d N = 1 theories. Such a formula is a finite sum, over the solution set of certain transcendental equations that we dub Bethe Ansatz Equations, of a function evaluated at those solutions.
10.1007/s00220-019-03679-y
[ "https://arxiv.org/pdf/1811.04107v3.pdf" ]
85,530,619
1811.04107
c56b6e153d505ef8d7ea622b1edb793418f14d0a
A Bethe Ansatz type formula for the superconformal index 26 Mar 2019 Francesco Benini [email protected] SISSA Via Bonomea 26534136TriesteItaly INFN Sezione di Trieste Via Valerio 234127TriesteItaly ICTP Strada Costiera 1134151TriesteItaly Paolo Milan [email protected] SISSA Via Bonomea 26534136TriesteItaly INFN Sezione di Trieste Via Valerio 234127TriesteItaly A Bethe Ansatz type formula for the superconformal index 26 Mar 2019 Inspired by recent work by Closset, Kim and Willett, we derive a new formula for the superconformal (or supersymmetric) index of 4d N = 1 theories. Such a formula is a finite sum, over the solution set of certain transcendental equations that we dub Bethe Ansatz Equations, of a function evaluated at those solutions. Introduction and summary In supersymmetric quantum field theories there are many classes of observables that can be computed exactly and non-perturbatively, making supersymmetry an appealing testing ground for general ideas about quantum field theory and, through holography, also quantum gravity. One of those observables is the superconformal index [1][2][3] which-in theories with superconformal invariance-counts with signs the number of local operators in short representations of the superconformal algebra. This counting can be done keeping track of the spin and other charges of the operators. Despite its simplicity, the superconformal index is an observable that contains a lot of information about the theory, and indeed it has been studied in all possible dimensions (i.e. up to six) and under so many angles (for reviews see [4]). In this note we focus on the four-dimensional superconformal index. Because the index does not depend on continuous deformations of the theory, and a suitable supersymmetric generalization thereof does not depend on the RG flow, it follows that in theories that are part of a conformal manifold and have a weakly-coupled point on it, and in theories that are asymptotically free, the evaluation of the index can be reduced to a weak coupling computation. 1 This amounts to counting all possible local operators in short representations one can write down, and then restricting to the gauge-invariant ones. In the language of radial quantization, one counts all multi-particle states in short representations on the sphere, and then imposes Gauss law. In the case of the 4d N = 1 superconformal (or supersymmetric) index of a gauge theory with gauge group G and matter chiral multiplets in representation R, the counting is captured by the standard formula [1,2,5]: I(p, q; v) = (p; p) rk(G) ∞ (q; q) rk(G) ∞ |W G | T rk(G) ρa∈R Γ (pq) ra/2 z ρa v ωa ; p, q α∈∆ Γ(z α ; p, q) rk(G) i=1 dz i 2πiz i . (1.1) Here, briefly, p, q are the (complex) fugacities associated to the angular momentum, v collectively indicates the fugacities for flavor symmetries, z indicates the fugacities for the gauge symmetry, r a are the R-charges, and Γ is the elliptic gamma function. All the details will be reviewed in Section 2. In this note, inspired by recent work of Closset, Kim and Willett [6,7], we show that when the fugacities for the angular momentum satisfy q a = p b (1.2) for some coprime positive integers a, b, then one can derive an alternative, very different formula for the 4d superconformal index. The condition (1.2) can be rewritten as p = h a , q = h b (1.3) for some fugacity h and coprime a, b ∈ N. The new formula, that we will explain in great detail in Section 3, is a finite sum over the solution set M BAE to certain transcendental equations-that we dub Bethe Ansatz equations (BAEs)-of a function, closely related to the integrand in (1.1), evaluated at those solutions. Very schematically, we prove that I(p, q; v) = (p; p) rk(G) ∞ (q; q) rk(G) ∞ |W G | z ∈ M BAE ab {m i }=1 Z z/h m , p, q, v H(z, p, q, v) −1 . (1.4) Here the function Z is the integrand in the standard formula (1.1); M BAE is the set of solutions, on a torus of exponentiated modular parameter h, to the BAEs which take the schematic form Q i (z, p, q, v) = 1 for i = 1, . . . , rk(G) (1.5) in terms of functions Q i defined in (3.3); the function H is a "Jacobian" H(z, p, q, v) = det ij ∂Q i (z, p, q, v) ∂ log z j . (1.6) The precise expressions (in which we use chemical potentials instead of fugacities, in order to deal with single-valued functions) can be found at the beginning of Section 3. A special case of this formula when p = q, namely a = b = 1, was derived in [6]. The condition (1.2) limits the applicability of the Bethe Ansatz (BA) formula (1.4) in the space of complex fugacities. Yet, as we discuss in Section 3.1, the domain of the formula is rich enough to uniquely fix the index as a continuous function (with poles) of general fugacities. We offer two arguments, one that uses holomorphy of the index and one that just uses continuity. Roughly, the reason is that the set of pairs (p, q) satisfying (1.2) is dense in the space of general complex fugacities (see Appendix B). In a separate publication [8] we will use the BA formula (1.4) to address the large N limit of the index of a specific theory, namely N = 4 Super-Yang-Mills, finding some differences with previous literature. We will connect the large N limit of the index to the entropy of BPS black holes in AdS 5 . 2 In this way, we will extend the success of the counting of microstates of dyonic black hole in AdS 4 [11][12][13][14] to the case of electric rotating black holes in AdS 5 [15][16][17][18]. More generally, the new BA formula is much easier to deal with, compared to the standard integral formula, when performing numerical computations. We thus hope that it will be useful in a wider context. The BA formula (1.4) can be thought of, in some sense, as the "Higgs branch localization" partner of the standard "Coulomb branch localization" integral formula (1.1), using the terminology of [19,20]. More precisely, the existence of a formula as (1.4) can be justified along the lines of [11,13,21,22,6,7]. The superconformal index can be defined as the partition function of the Euclidean theory on S 3 × S 1 , with suitable flat connections along S 1 and a suitable complex structure that depends on p, q, and with the Casimir energy [23,24] stripped off. 3 The standard localization computation of the partition function leads to (1.1). However, when p, q satisfy (1.2) the geometry is also a Seifert torus fibration over S 2 . Along the lines of [7], one expects to be able to reduce to the computation of a correlator in an A-twisted theory on S 2 [25], which should give an expression as in (1.4). In any case, we have derived the BA formula (1.4) by standard manipulations of the integral expression and thus we do not rely on any such putative 2d reduction. The note is organized as follows. In Section 2 we review the standard formula for the 4d superconformal index, carefully stressing its regime of applicability. In Section 3 we present our new BA formula in great detail, and then we derive it in Section 3.2. The 4d superconformal index In order to fix our notation, let us review the standard formulation of the superconformal index [1,2], which counts local operators in short representations of the 4d N = 1 superconformal algebra (SCA) su(2, 2|1). Going to radial quantization, this is the same as counting (with sign) 1 4 -BPS states of the theory on S 3 . The bosonic part of the superconformal algebra is su(2, 2)⊕u(1) R , where the first factor is the 4d conformal algebra and the second one is the R-symmetry. We pick on S 3 one Poincaré supercharge, specifically Q = Q − , and its conjugate conformal supercharge Q † = S + . Together with ∆ = 1 2 {Q, Q † } they form an su(1|1) superalgebra. The superconformal index is then equal to the Witten index I(t) = Tr H[S 3 ] (−1) F e −β∆ k t J k k , (2.1) where J k are Cartan generators of the commutant of su(1|1) in the full SCA and t k are the associated complex fugacities. By standard arguments [26], I(t) counts only states with ∆ = 0, i.e. annihilated by both Q and Q † , and thus it does not depend on β. On the other hand, it is holomorphic in the fugacities t k , which serve both as regulators and as refinement parameters. To be more precise, the states counted by (2.1) have ∆ = E − 2j + − 3 2 r = 0, where E is the conformal Hamiltonian or dimension, j ± are the Cartan generators of the angular momentum su(2) + ⊕su(2) − ⊂ su(2, 2), and r is the superconformal U(1) R charge. Moreover, the subalgebra of su(2, 2|1) which commutes with su(1|1) has Cartan generators E + j + and j − . Therefore we write I(p, q) = Tr ∆=0 (−1) F p 1 3 (E+j + )+j − q 1 3 (E+j + )−j − = Tr ∆=0 (−1) F p j 1 + r 2 q j 2 + r 2 ,(2.2) where j 1,2 = j + ± j − parametrize the rotated frame u(1) 1 ⊕ u(1) 2 ⊂ su(2) + ⊕ su(2) − and p, q are the associated fugacities (up to a shift by r/2). Whenever the theory enjoys flavor symmetries, one can introduce fugacities v α for the Cartan generators of the flavor group. Then, the index will depend holomorphically also on v α . The trace formula (2.2) can be exactly evaluated at all regimes in the couplings. Indeed, since I is invariant under any continuous deformation of the theory, one can explicitly account for the contribution of every gauge-invariant state with ∆ = 0 in the free regime [27,28,2]. In particular, the contributions of all the multi-particle states are simply encoded in the plethystic exponential [29] of the "single-letter partition functions", whereas the restriction to the gauge-invariant sector is done by integrating the latter contributions over the gauge group. This procedure yields a finite-dimensional integral formula for the superconformal index, which can be expressed as an elliptic hypergeometric integral [5]. 4 For concreteness, we consider a generic N = 1 gauge theory with semi-simple gauge group G, flavor symmetry group G F and non-anomalous U(1) R R-symmetry. We assume that the theory flows in the IR to a non-trivial fixed point and we parametrize U(1) R with the superconformal R-charge sitting in the SCA of the IR CFT (assuming this is visible in the UV). Furthermore, the matter content consists of n χ chiral multiplets Φ a in representations R a of G, carrying flavor weights ω a in some representations R F of G F and with superconformal R-charges r a . Additionally, we turn on flavor fugacities v α , with α = 1, . . . , rk(G F ), parametrizing the maximal torus of G F . The integral representation of the superconformal index is given by I(p, q; v) = (p; p) rk(G) ∞ (q; q) rk(G) ∞ |W G | T rk(G) nχ a=1 ρa∈Ra Γ (pq) ra/2 z ρa v ωa ; p, q α∈∆ Γ(z α ; p, q) rk(G) i=1 dz i 2πiz i . (2. 3) The integration variables z i parametrize the maximal torus of G, and the integration contour is the product of rk(G) unit circles. Then ρ a are the weights of the representation R a , α parametrizes the roots of G and |W G | is the order of the Weyl group. Moreover, we have introduced the notation z ρa = rk(G) i=1 z ρ i a i and v ωa = rk(G F ) α=1 v ω α a α , whereas Γ(z; p, q) = ∞ m,n=0 1 − p m+1 q n+1 /z 1 − p m q n z , |p| < 1 , |q| < 1 (2.4) is the elliptic gamma function [33] and (z; q) ∞ = ∞ n=0 (1 − zq n ) , |q| < 1 (2.5) is the q-Pochhammer symbol. This representation makes manifest the holomorphic dependence of the index on p, q, v α . It is important to stress that the expression (2.3), which is a contour integral along rk(G) unit circles, is only valid as long as the fugacities stay within the following Domain: |p|, |q| < 1 , |pq| < (pq) ra/2 v ωa < 1 , ∀a . (2.6) These conditions descend from the requirement of convergence of the plethystic representation of the index, from which (2.3) is derived. The plethystic expansion of the elliptic gamma 4 An alternative way to obtain the integral formula is to use supersymmetric localization [30]. Indeed, the supersymmetric partition function Z of the theory on a primary Hopf surface H p,q ≃ S 1 × S 3 can be computed with localization [31,32] and it is related to the superconformal index through Z = e −ESUSY I, where E SUSY is the supersymmetric Casimir energy [23,24]. function, Γ(z; p, q) = exp ∞ m=1 1 m z m − (pq) m z −m (1 − p m )(1 − q m ) , (2.7) converges for |pq| < |z| < 1 and |p|, |q| < 1 . (2.8) The domain (2.6) then follows from requiring the integrand of (2.3) to have a convergent expansion. Indeed, within the domain of convergence (2.8), the elliptic gamma function is a single-valued analytic function with no zeros, poles nor branch cuts. Both Γ(z; p, q) and (z; q) ∞ can be analytically continued to z ∈ C. However, when we analytically continue the integral (2.3) outside the domain (2.6), the integration contour must be continuously deformed in order to take into account the movement of the various poles of the integrand in the complex plane, in such a way that the poles do not cross the contour. As a result, for generic fugacities the integration contour is not as simple as a product of unit circles. To avoid this complication, throughout this paper we will always work within (2.6)-and perform analytic continuation only at the end, if needed. It will be useful to set some new notation. We define a set of chemical potentials through p = e 2πiτ , q = e 2πiσ , v α = e 2πiξα , z i = e 2πiu i ,(2.9) as well as a fictitious chemical potential ν R for the R-symmetry, whose value is fixed to ν R = 1 2 (τ + σ) (2.10) by supersymmetry. Moreover, we redefine the elliptic gamma function as a (periodic) function of the chemical potentials: Γ(u, τ, σ) = Γ e 2πiu ; e 2πiτ , e 2πiσ , (2.11) so that the integrand of (2.3) can be expressed as Z(u; ξ, ν R , τ, σ) = nχ a=1 ρa∈Ra Γ ρ a (u) + ω a (ξ) + r a ν R ; τ, σ α∈∆ Γ α(u); τ, σ . (2.12) At last, we define κ G = (p; p) rk(G) ∞ (q; q) rk(G) ∞ |W G | . (2.13) The integral representation of the index takes then the following compact form: I(p, q; v) = κ G T rk(G) Z(u; ξ, ν R , τ, σ) d rk(G) u . (2. 14) The integration contour T rk(G) is represented on the u-plane by a product of straight segments of length one on the real axes. In terms of the chemical potentials, the domain (2.6) can be rewritten as: Im τ, Im σ > 0 , 0 < Im ω a (ξ) < Im(τ + σ) , ∀a . (2.15) The integral formula (2.14) is the starting point of our analysis. In the next Section we will focus our attention to the case where τ /σ is a rational number to derive-from (2.14)-a new formula that expresses the index as a finite sum. A new Bethe Ansatz type formula The integral representation (2.14) of the superconformal index is valid for generic complex values of the chemical potentials within the domain (2.15). However, if we restrict to a case where τ /σ ∈ Q + , (3.1) we can prove an alternative formula describing the index as a finite sum over the set of solutions to certain transcendental equations, which we call Bethe Ansatz Equations (BAEs). We will first present the formula in detail, and then provide a proof. In Section 3.1 we will also discuss the properties of the set of pairs (τ, σ) satisfying (3.1). Let us take τ = aω , σ = bω with a, b ∈ N such that gcd(a, b) = 1 (3.2) and Im ω > 0. This implies (1.2). We can set p = h a and q = h b with h = e 2πiω , although we will mostly work with chemical potentials. We introduce the BAEs as the set of equations Q i (u; ξ, ν R , ω) = 1 , ∀ i = 1, . . . , rk(G) ,(3.3) written in terms of "BA operators" defined as Q i (u; ξ, ν R , ω) = nχ a=1 ρa∈ Ra P ρ a (u) + ω a (ξ) + r a ν R ; ω ρ i a . (3.4) The basic BA operator is P (u; ω) = e −πi u 2 ω +πiu θ 0 (u; ω) , (3.5) where θ 0 (u; ω) = (z; h) ∞ (z −1 h; h) ∞ with z = e 2πiu and h = e 2πiω . The BA operators satisfy three important properties. First, they are doubly-periodic in the gauge chemical potentials: Q i (u + n + mω; ξ, ν R , ω) = Q i (u; ξ, ν R , ω) , ∀ n i , m i ∈ Z , i = 1, . . . rk(G) . (3.6) Second, they are invariant under SL(2, Z) modular transformations of ω: Q i (u; ξ, ν R , ω) = Q i (u; ξ, ν R , ω + 1) = Q i u ω ; ξ ω , ν R ω , − 1 ω = Q i (−u; −ξ, −ν R , ω) . (3.7) The last equality represents invariance under the center of SL(2, Z). Third, they capture the quasi-periodicity of the index integrand: Q i (u; ξ, ν R , ω) Z(u; ξ, ν R , aω, bω) = Z(u − δ i abω; ξ, ν R , aω, bω) , (3.8) valid ∀ i and where δ i = (δ ij ) rk(G) j=1 so that (u − δ i abω) j = u j − δ ij abω. Because of the double-periodicity of Q i , the actual number of solutionsû i to the system of BAEs (3.3) is infinite. However, the solutions can be grouped into a finite number of equivalence classes [û i ] such thatû i ∼û i + 1 ∼û i + ω. In other words, the equations and their solutions are well-defined on a torus T 2rk(G) which is the product of rk(G) identical complex tori of modular parameter ω, and the number of solutions on the torus is finite. The modular invariance (3.7) confirms that the equations are well-defined on the torus. We define M BAE = [û i ] , i = 1, . . . , rk(G) Q i [û]; ξ, ν R , ω = 1 , w · [û] = [û] ∀ w ∈ W G (3.9) as the set of solutions (on the torus) that are not fixed by non-trivial elements of the Weyl group. For definiteness we can choose, as representatives, the elements living in a fundamental domain of the torus with modulus ω, i.e. with 0 ≤ Reû i < 1 and 0 ≤ Imû i < Im ω. Notice that, because of (3.7), the solutions must organize into representations of SL(2, Z). As we prove below, thanks to the properties of the BA operators, we can rewrite the superconformal index as a sum over solutions to the BAEs in the following way: I(p, q; v) = κ G û ∈ M BAE Z tot (û; ξ, ν R , aω, bω) H(û; ξ, ν R , ω) −1 . (3.10) Here Z tot (u; ξ, ν R , aω, bω) = ab {m i }=1 Z(u − mω; ξ, ν R , aω, bω) , (3.11) where Z is precisely the integrand defined in (2.12) and H(u; ξ, ν R , ω) = det ij 1 2πi ∂Q i (u; ξ, ν R , ω) ∂u j (3.12) is the contribution from the Jacobian of the change of variables u i → Q i (u). Notice that both the function H, and the function Z tot evaluated on the solutions to the BAEs, are doubly-periodic on the product of complex tori of modular parameter ω. A specialization of this formula to the case τ = σ was derived in [6], while a threedimensional analog was derived in [7]. In the next Section we will spell out in detail how the BA formula uniquely fixes the index for all values of the complex fugacities, using either holomorphy or continuity. In Section 3.2 we will derive the final formula (3.10), starting from the integral representation (2.14). The proof is rather technical and it does not give new physical insights on the main result. Therefore, uninterested readers may stop here. Continuation to generic fugacities Our BA formula (3.10) can only be applied for special values of the angular fugacities that satisfy (1.2). We will offer two arguments, one based on holomorphy and the other based on just continuity, that this is enough to completely determine the index for all values of the complex fugacities. Using the standard definition (2.2), the index is not a single-valued function of the angular fugacities p, q-unless the R-charges of chiral multiplets are all even. This is also apparent from the integral formula (2.3). On the other hand, regarded as a function of chemical potentials τ, σ each living on the upper half-plane H, the index is single-valued and holomorphic. Keeping the flavor fugacities fixed in the argument that follows, the BA formula applies to points (τ, σ) ∈ H 2 such that τ /σ ∈ Q + . Such a set is dense in a hyperplane J ∼ = R 3 of real codimension one in H 2 defined as J = (τ, σ) τ /σ ∈ R + . Thus, the BA formula determines the index on J by continuity. On the other hand, we know that the index is a holomorphic function on H 2 , therefore its restriction to J completely fixes the function on H 2 by analytic continuation. It turns out that we can refine the argument in such a way that we only use continuity, and not holomorphy, of the index. This is because if we think in terms of angular fugacities p, q each living in the open unit disk D, then the set of points (p, q) ∈ D 2 such that q a = p b for coprime a, b ∈ N is dense in D 2 . This fact is not completely obvious, and we show it in Appendix B. Unfortunately, the index (2.3) is not a single-valued function of p, q if we keep the flavor fugacities v α fixed, unless the R-charges are all even. However, it is always possible to find a change of variables which expresses I as a single-valued function of a set of new fugacities. The latter is defined by ∆ a = ω a (ξ) + r a ν R ⇒ y a = e 2πi∆a = v ωa (pq) ra 2 , ∀ a = 1, . . . , n χ . (3.13) This gives us a set of (redundant) chemical potentials ∆ a , one for each chiral multiplet present in the theory, which must satisfy some linear constraint, following the requirement of invariance of the theory under flavor and R-symmetry. Suppose, indeed, the theory has a superpotential given by W (Φ) = A W A (Φ) ,(3.14) where each W A (Φ) is a gauge-invariant homogeneous polynomial of degree n A . Then, for each term in (3.14), the following linear constraints must be satisfied: a∈A r a = 2 , a∈A ω α a = 0 , ∀ α = 1, . . . , rk(G) ,(3.15) where we used a ∈ A to indicate the chiral components Φ a which are present in W A . The first equation imposes that the superpotential has R-charge 2. The second equation constrains W to be invariant under G F . Indeed, ω a = (ω α a ) rk(G F ) α=1 are the flavor weights carried by Φ a . A similar role is played by ABJ anomalies. Translating (3.15) to the definition of ∆ a , we obtain a∈A ∆ a = 2ν R = τ + σ ∀ A . (3.16) In such a new set of variables we have Z(u; ∆, τ, σ) = nχ a=1 ρa∈Ra Γ(ρ a (u) + ∆ a ; τ, σ) α∈∆ Γ(α(u); τ, σ) , (3.17) showing that the index is now a well-defined, single-valued and continuous function (in fact, also holomorphic) of the fugacities p, q, y a . Indeed, recall that the elliptic gamma function is a single-valued function of its arguments, and notice that the constraints (3.16) always involve integer combinations of τ , σ, thus never introducing non-trivial monodromies under integer shifts. Once again, the BA formula can be applied whenever q a = p b and for generic values of y a . Since such a set of points is dense in the space of generic fugacities, we conclude that the BA formula fixes the index completely. Proof of the formula We prove the formula (3.10) in three steps. First we verify the properties (3.6) and (3.8) of the BA operators. Then we use them to modify the contour of the integral (2.14) and to reduce it to a sum of simple residues. Finally we prove that the only poles that contribute to the residue formula are determined by the BAEs, thus obtaining (3.10). Properties of the BA operators First, we prove the identities (3.6) and (3.8). For later convenience, let us briefly recall the anomaly cancellation conditions that are required to have a well-defined four-dimensional theory. These requirements can be expressed in terms of the anomaly coefficients. In particular, let i = (i, α) collectively denote the Cartan indices of the gauge × flavor group, where i = 1, . . . , rk(G) are the gauge indices and α = 1, . . . , rk(G F ) are the flavor indices. Moreover, define a = (a, ρ a ) as running over all chiral multiplets components, where ρ a are the weights of the gauge representation R a . Then the anomaly coefficients for gauge/flavor symmetries are defined by A ijk = a Q i a Q j a Q k a , A ij = a Q i a Q j a , A i = a Q i a , (3.18) where Q i a = Q i (a,ρa) = (ρ i a , ω α a ) are the components of the gauge × flavor weights carried by the chiral multiplets. The first and the last coefficient in (3.18) are associated with the gauge 3 and mixed gauge-gravitational 2 perturbative anomalies. The second term-sometimes called pseudo-anomaly coefficient-describes the non-perturbative or global anomaly [34][35][36] when the corresponding perturbative anomaly vanishes. Similarly, the perturbative anomaly coefficients involving the R-symmetry are defined by A ijR = a Q i a Q j a (r a − 1) + δ ij,ij α∈∆ α i α j A iRR = a Q i a (r a − 1) 2 A RRR = a (r a − 1) 3 + dim G A R = a (r a − 1) + dim G ,(3.19) whereas the pseudo R-anomaly coefficients are A iR = a Q i a (r a − 1) A RR = a (r a − 1) 2 + dim G . (3.20) Anomaly cancellation is realized by a set of conditions on the coefficients defined above, that a well-defined quantum gauge theory must satisfy. We will also restrict to the case that the gauge group G is semi-simple. The conditions for the cancellation of the gauge and gravitational anomaly are A ijk = A i = 0 and A ij ∈ 4Z for G semi-simple . (3.21) The conditions for the cancellation of the ABJ anomalies of G F and U(1) R , namely that those are global symmetries of the quantum theory, are A ijα = A ijR = 0 . (3.22) Finally, A iαβ = A iαR = A iRR = 0 and A iα = A iR = 0 (3.23) simply follow from the restriction to semi-simple gauge group G. We now focus on describing some properties of the basic BA operator P (u; ω) = e −πi u 2 ω +πiu θ 0 (u; ω) . (3.24) First, consider the function θ 0 (u; ω) = (z; h) ∞ (z −1 h; h) ∞ = ∞ k=0 (1 − zh k )(1 − z −1 h k+1 ) , z = e 2πiu , h = e 2πiω (3.25) which is holomorphic in z and h, and satisfies the following properties: θ 0 (u + n + mω; ω) = (−1) m e −2πimu−πim(m−1)ω θ 0 (u; ω) ∀ n, m ∈ Z θ 0 (−u; ω) = θ 0 (u + ω; ω) = −e −2πiu θ 0 (u; ω) . (3.26) They immediately imply P (−u; ω) = −P (u; ω) P (u + n + mω; ω) = (−1) n+m e − πi ω (2nu+n 2 ) P (u; ω) ∀ n, m ∈ Z . (3.27) It turns out that the basic BA operator has also nice modular transformation properties: P (u; ω + 1) = e πi u 2 ω(ω+1) P (u; ω) , P u ω ; − 1 ω = e πi u 2 ω − ω 6 − 1 6ω + 1 2 P (u; ω) . (3.28) In order to prove (3.8), we also need to show that P (u + rν R ; ω) m Γ(u + rν R ; aω, bω) = (−1) Here r ∈ R mimics the contribution from the R-charge of a generic multiplet in the theory. Notice that all factors in front of Γ in the r.h.s. of (3.29) explicitly depend on the fermion R-charge r − 1. This will be crucial to ensure anomaly cancellation in the full BA operator. Proof. The identity (3.29) follows from the properties of the elliptic gamma function. Indeed, for generic τ and σ, we have that Γ(u + τ ; τ, σ) = θ 0 (u; σ) Γ(u; τ, σ) , Γ(u + σ; τ, σ) = θ 0 (u; τ ) Γ(u; τ, σ) . Now, by enforcing the assumption that gcd(a, b) = 1, we can use the properties of numerical semigroups (see Appendix A for more details) to reduce the periods of the theta functions from abω to ω. In order to do so, let us introduce some notation. We call R(a, b) the set of non-negative integer linear combinations of a, b: w(a, b). It is a classic result in mathematics that, in terms of a, b, the latter read R(a, b) = {am + bn | m, n ∈ Z ≥0 }.χ(a, b) = (a − 1)(b − 1) 2 , w(a, b) = (a − 1)(b − 1)(2ab − a − b − 1) 12 . (3.35) Thanks to the properties of these objects, we can use the following identities (proved in Appendix A): We now turn to analyzing the full BA operators. Notice that, in the definition (3.4), Q i receive contribution only from the chiral multiplets of the theory. The vector multiplets do not appear in (3.4) because their contribution simply amounts to α∈∆ P α(u); ω a−1 r=0 b−1 s=0 (zh as+br ; h ab ) ∞ = (z; h) ∞ k∈R(a,b) (1 − zh k ) a−1 r=0 b−1 s=0 (z −1 h ab−as−br ; h ab ) ∞ = (z −1 h; h) ∞ k∈R(a,b) (1 − z −1 h −k ) ,−α i = α>0 P − α(u); ω P α(u); ω α i = (−1) α>0 α i = 1 ,(3.39) which holds true if G is semi-simple, as in this case the sum of positive roots is always an even integer. Despite this fact, as far as the proof of (3.10) is concerned, we find it more convenient to write Q i (u; ξ, ν R , ω) = nχ a=1 ρa∈Ra P ρ a (u) + ω a (ξ) + r a ν R ; ω ρ i a × α∈∆ P α(u); ω −α i (3.40) without simplifying the vector multiplet contribution. At this point, using (3.27) we can show that Q i satisfy: Applying (3.29), the latter equation reduces to Q i (u + n; ξ, ν R , ω) = (−1) A ij n j e − πi ω (A ijk n j (2u k +n k )+2A ijα n j ξα+2A ijR n j ω) Q i (u; ξ, ν R , ω) Q i (u + mω; ξ, ν R , ω) = (−1) A ij m j Q i (u; ξ, ν R , ω) ,(3.Q i (u; ξ, ν R , ω) Z(u; ξ, ν R , aω, bω) = (−1) ab 2 A ii + a+b−1 2 A i e πiab(A iij u j +A iiα ξα) × × e − πi ω (A ijk u j u k +A iαβ ξαξ β +2A ijα u j ξα) e −πi(a+b)(A ijR u j +A iαR ξα)+ πiab(a+b) 2 A iiR ω × × e − πi(a+b) 2 4 A iRR ω− πia 2 b 2 3 A iii ω+ πi(a 2 +b 2 ) 12 A i ω Z(u − δ i abω; ξ, ν R , ω) , (3.43) which, by anomaly cancellation, reduces to (3.8). Residue formula We now use the BA operators and their properties to modify the contour of integration of the index in (2.14). For our purposes, it is sufficient to implement the following trivial relation: I(p, q; v) = κ G Z(u; ξ, ν R , aω, bω) d rk(G) u = κ G rk(G) i=1 1 − Q i (u; ξ, ν R , ω) rk(G) i=1 1 − Q i (u; ξ, ν R , ω) Z(u; ξ, ν R , aω, bω) d rk(G) u . (3.44) The numerator of the integrand can be expanded as rk(G) i=1 1 − Q i (u; ξ, ν R , ω) × Z(u; ξ, ν R , aω, bω) = = rk(G) n=0 (−1) n n! rk(G) i 1 =··· =in Q i 1 (u; ξ, ν R , ω) . . . Q in (u; ξ, ν R , ω) Z(u; ξ, ν R , aω, bω) = rk(G) n=0 (−1) n n! rk(G) i 1 =··· =in Z u − (δ i 1 + . . . + δ in )abω; ξ, ν R , aω, bω ,I(p, q; v) = κ G rk(G) n=0 (−1) n n! rk(G) i 1 =··· =in I i 1 ...in (p, q; v) ,(3.46) with I i 1 ...in (p, q; v) = T rk(G) Z u − (δ i 1 + . . . + δ in )abω; ξ, ν R , aω, bω rk(G) i=1 1 − Q i (u; ξ, ν R , ω) d rk(G) u = C i 1 ...in Z u; ξ, ν R , aω, bω rk(G) i=1 1 − Q i (u; ξ, ν R , ω) d rk(G) u (3.47) and where C i 1 ...in = T rk(G)−n × n k=1 |z i k | = |h| −ab ; . (3.48) This is a contour where z i 1 , . . . , z in live on circles of radius |h| −ab , whereas the other variables z j parametrize the unit circles in T rk(G)−n . The second line in (3.47) has been obtained by implementing the change of variables u i k → u i k + abω for k = 1, . . . , n and using the periodicity (3.6). The series of integrals in (3.46) can be resummed to a unique integral over a composite contour: I(p, q; v) = κ G C Z u; ξ, ν R , aω, bω rk(G) i=1 1 − Q i (u; ξ, ν R , ω) d rk(G) u , (3.49) where C = rk(G) n=0 (−1) n n! rk(G) i 1 =··· =in C i 1 ...in ≃ rk(G) i=1 |z i | = 1; ∪ |z i | = |h| −ab ; (3.50) is a contour encircling the annulus A = u i 1 < |z i | < |h| −ab , i = 1, . . . , rk(G) . We now apply the residue theorem to (3.49). The integrand has simple poles coming from the denominator, whose positions are precisely described by the BAEs (3.3). Obviously, only the poles that lie inside the annulus A contribute to the contour integral. Moreover, as we do in Appendix C, one can show that whenever a particular solution [û] to the BAEs (3.3) is fixed (on the torus) by a non-trivial element of the Weyl group W G , namely w · [û] = [û], then the numerator Z(û; ξ, ν R , aω, bω) is such that cancelations take place and there is no contribution to the integral-more precisely, the function Z tot (û; ξ, ν R , aω, bω) defined in (3.11) vanishes. 5 Hence, we define the set of relevant poles by: M index = û i [û i ] ∈ M BAE and 1 < |ẑ i | < |h| −ab , i = 1, . . . , rk(G) . (3.51) This includes all points inside the annulus A such that their class belongs to M BAE . In particular, the same equivalence class [û i ] ∈ M BAE appears in M index as many times as the number of its representatives living in A. For this reason, we employ the following alternative description: M index = û ( m) i = [û i ] − m i ω [û i ] ∈ M BAE , m i = 1, . . . , ab , i = 1, . . . , rk(G) (3.52) where, we some abuse of notation, we have denoted as [û i ] the representative in the fundamental domain of the torus as after (3.9). In addition, the numerator Z has other poles coming from the elliptic gamma functions. As we show below, as long as the fugacities v α , p, q are taken within the domain (2.6)-which is necessary in order for the standard contour integral representation (2.3) to be valid-those other poles either lie outside the annulus A or are not poles of the integrand (because the denominator has a pole of equal or higher degree) and thus do not contribute to the integral. Therefore, working within the domain (2.6), we can rewrite the index as I(p, q; v) = (−2πi) rk(G) κ G û ( m) ∈ M index Res u=û ( m) Z(u; ξ, ν R , aω, bω) rk(G) i=1 1 − Q i (u; ξ, ν R , ω) d rk(G) u . (3.53) Computing the residues produces the final expression for the supersymmetric index: I(p, q; v) = κ G û ( m) ∈ M index Z(û ( m) ; ξ, ν R , aω, bω) H(û ( m) ; ξ, ν R , ω) −1 ,(3.54) where H is defined in (3.12). The residue formula (3.54) can be rewritten, more elegantly, in the final form: I(p, q; v) = κ G û ∈ M BAE Z tot (û; ξ, ν R , aω, bω) H(û; ξ, ν R , ω) −1 , (3.55) where Z tot (u; ξ, ν R , aω, bω) = ab {m i }=1 Z(u − mω; ξ, ν R , aω, bω) . (3.56) To obtain this expression we have split the sum over the poles in M index into a sum over the inequivalent solutions to the BAEs, described by the elements of M BAE , and the sum over the "repetitions" of these elements in the annulus A. Moreover, we have used the double-periodicity of the Jacobian H(u; ξ, ν R , ω) to pull the latter sum inside the definition of Z tot . Analysis of the residues The last step consists in showing that the only residues contributing to (3.49) come from zeros of the denominator. In particular we need to show that, remaining within the domain (2.6), all poles in (3.49) which are not given by the BAEs live outside the annulus A and thus do not contribute to the integral. We concretely do so by proving that every pole of Z inside A is also a pole of the denominator i (1 − Q i ) with a high enough degree that the integrand of (3.49) is non-singular at those points. We begin by classifying the poles of Z. Using we can rewrite Z as Z(u; ξ, ν R , aω, bω) = α>0 θ 0 α(u); aω θ 0 −α(u); bω × a,ρa Γ ρ a (u) + ω a (ξ) + r a ν R ; aω, bω . (3.58) Since θ 0 (u; ω) has no poles for finite u, the only singularities of Z come from the elliptic gamma functions related to the chiral multiplets. These can be read off the product expansion: Γ(u; aω, bω) = ∞ m=0 a−1 r=0 b−1 s=0 1 − h ab(m+2)−as−br z −1 1 − h abm+as+br z m+1 (3.59) that follows from (3.31), and so they are given by z ρa = v −ωa h −ra(a+b)/2−abm−as−br (3.60) for 0 ≤ r ≤ a − 1, 0 ≤ s ≤ b − 1 and m ≥ 0. The multiplicity of each pole is µ a m = m + 1. 6 Notice that one could also write z ρa = v −ωa h −ra(a+b)/2−k for k ∈ R(a, b). We now turn to analyzing the denominator. More specifically, we need to find the singularities of rk(G) i=1 1 − Q i (u) . From (3.4) and (3.5) we see that Q i has a pole whenever θ 0 ρ a (u) + ω a (ξ) + r a ν R ; ω = 0 and ρ i a > 0. Therefore, the singularities of the denominator are given by z ρa = v −ωa h −ra(a+b)/2+n for n ∈ Z , (3.61) all with the same multiplicity ν a = i ∈ D + a ρ i a . Here D ± a represents the set of indices such that ρ i a > 0, resp. ρ i a < 0, thus ν a is the sum of the positive components of ρ a . We notice that the denominator poles in (3.61) with −n ∈ R(a, b) coincide with the numerator poles. Therefore, the actual singularities of the integrand in (3.49) are only those points in (3.60) such that µ a m > ν a , or more explicitly m ≥ i ∈ D + a ρ i a . (3.62) We now want to show that, when the fugacities satisfy (2.6), the set of actual singularities is always living outside the annulus A. Therefore, we first study the conditions for which (3.60) belong to the annulus A. By imposing that 1 < |z i | < |h| −ab , we obtain that |h| −ab i∈D − a ρ i a < |z ρa | < |h| −ab i∈D + a ρ i a , ∀ a . (3.63) 6 In counting the multiplicity one may worry that there could be different choices of r, s that give the same abm+as+br for fixed m. This is equivalent to finding non-trivial solutions to the equation as+br = as ′ +br ′ . However, it is easy to see that, as long as 0 ≤ r, r ′ ≤ a − 1 and 0 ≤ s, s ′ ≤ b − 1, such an equation has no non-trivial solution in Z. Then we determine the constraints imposed on (3.60) by requiring (2.6). In the rational case, the latter conditions are expressed by |h| a+b < |v ωa h ra(a+b)/2 | < 1, ∀ a. These inequalities, together with 0 ≤ as + br ≤ 2ab − a − b, imply that |h| −abm ≤ |h| −abm−as−br < |z ρa | < |h| −abm−a(s+1)−b(r+1) ≤ |h| −ab(m+2) . (3.64) Furthermore, requiring (3.62) to be satisfied, we obtain that |z ρa | > |h| −ab i∈D + a ρ i a , ∀ a , (3.65) which is satisfied by all the singularities of (3.49) coming from the numerator Z. At this point, we immediately notice that the intersection between (3.63) and (3.65) is empty. This means that, if the flavor fugacities satisfy (2.6), all poles of the integrand (3.49) that come from poles of the numerator Z live outside the annulus A, and so the only residues contributing to the integral are those given by the BAEs. This completes the proof of (3.10). Acknowledgments F.B. is supported in part by the MIUR-SIR grant RBSI1471GJ "Quantum Field Theories at Strong Coupling: Exact Computations and Applications". A Numerical semigroups and the Fröbenius problem Given a set of non-negative integer numbers {a 1 , . . . , a r }, the Fröbenius problem consists in classifying which integers can (or cannot) be written as non-negative integer linear combinations of those. This problem has deep roots in the theory of numerical semigroups. A semigroup is an algebraic structure R endowed with an associative binary operation. Analogously to groups, we denote it as (R, * ). On the other hand, differently from the case of a group, no requirement on the presence of identity and inverse elements is made. A numerical semigroup is an additive semigroup (R, +), where R consists of all non-negative integers Z ≥0 except for a finite number of positive elements (thus 0 ∈ R). The set {n 1 , . . . , n t } is called a generating set for (R, +) if all elements of R can be written as non-negative integers linear combinations of n 1 , . . . , n t . We then denote the semigroup with the presentation R = n 1 , . . . , n t . (A.1) Among all possible presentations of R, there exists a unique minimal presentation, which contains the minimal number of generators. Such a number is called the embedding dimension e(R) of the semigroup. We now define other important quantities associated with numerical semigroups: • The multiplicity m(R) is the smallest non-zero element of R. • The set of gaps R = N \ R is the set of positive integers which are not contained in R. Equivalently, the gaps are defined as all natural numbers which cannot be written as non-negative integer linear combination of the generators n 1 , . . . , n t of R. • The set of gaps R is always a finite set. Its largest element is the Fröbenius number F (R). Alternatively, given a presentation n 1 , . . . , n t , the Fröbenius number is defined as the largest integer which cannot be written as a non-negative integer linear combination of the generators. • The genus χ(R) is the number of gaps, i.e. it is the order of the set of gaps: χ(R) = R . • The weight w(R) is the sum of all gaps: w(R) = k∈R k. • The following inequalities hold: e(R) ≤ m(R) F (R) ≤ 2χ(R) − 1. (A.2) In particular, if x ∈ R, then F (R) − x / ∈ R. We now study the case where the embedding dimension is e(R) = 2, i.e. the minimal presentation is defined by two positive integers a, b with gcd(a, b) = 1. The associated numerical semigroup is denoted by R(a, b) = a, b and the set of gaps is R(a, b) = N\R(a, b). The multiplicity is simply m(a, b) = min{a, b}, whereas the Fröbenius number is given by F (a, b) = ab − a − b . (A.3) The genus and the weight are χ(a, b) = (a − 1)(b − 1) 2 w(a, b) = (a − 1)(b − 1)(2ab − a − b − 1) 12 . (A.4) Thanks to the properties of R(a, b), one can prove the following identities: a−1 r=0 b−1 s=0 (zh as+br ; h ab ) ∞ = (z; h) ∞ k∈R(a,b) (1 − zh k ) a−1 r=0 b−1 s=0 (z −1 h ab−as−br ; h ab ) ∞ = (z −1 h; h) ∞ k∈R(a,b) (1 − z −1 h −k ) . (A.5) Proof. We begin with the first identity. Using the definition of the q-Pochhammer symbol we can write: a−1 r=0 b−1 s=0 (zh as+br ; h ab ) ∞ = ∞ n=0 a−1 r=0 b−1 s=0 (1 − zh abn+as+br ) . (A.6) Using that a, b are coprime, the set of integers as + br r = 0, . . . , a − 1, s = 0, . . . , b − 1 covers once and only once every class modulo ab. It follows that the set of exponents {abn + as + br} is precisely R(a, b). Then a−1 r=0 b−1 s=0 (zh as+br ; h ab ) ∞ = k∈R(a,b) (1 − zh k ) = ∞ k=0 (1 − zh k ) k∈R(a,b) (1 − zh k ) = (z; h) ∞ k∈R(a,b) (1 − zh k ) , (A.7) which proves the first equality in (3.36). The proof of the second identity is a bit trickier. The key point is to notice that the set {as + br} does not contain any element of R(a, b) and thus {as + br} = k + ∆ k ab k = 0, . . . , ab − 1 with ∆ k =    0 if k ∈ R(a, b) 1 if k ∈ R(a, b(1 − z −1 h −k ) × ∞ k=1 (1 − z −1 h k ) = (z −1 h; h) ∞ k∈R(a,b) (1 − z −1 h −k ) . (A.10) This completes the proof of (3.36). Thanks to the definition of θ 0 (u; ω), we can apply (3.36) and we obtain that a−1 r=0 b−1 s=0 θ 0 u + (as + br)ω; ω = k∈R(a,b) (1 − z −1 h −k ) (1 − zh k ) θ 0 (u; ω) = 1 (−z) χ(a,b) h w(a,b) θ 0 (u; ω) . (A.11) B A dense set Here we show that the set of points (p, q) such that q a = p b for coprime a, b ∈ N (B.1) is dense in |p| < 1, |q| < 1 . We write the fugacities in terms of chemical potentials, p = e 2πiσ and q = e 2πiτ with Im σ, Im τ > 0, and for the sake of this argument we choose the determination on the "strip" 0 ≤ Re σ, Re τ < 1. Then the condition (B.1) is equivalent to a(τ + n) = b(σ + m) (B.2) for some m, n ∈ Z and a, b ∈ N coprime. We choose an arbitrary point (τ 0 , σ 0 ) in the strip and ask if we can find another point (τ, σ), arbitrarily close, that satisfies (B.2). Consider a straight line in the complex plane that starts from 0 and goes through τ 0 + n for some integer n. When winding once around the strip, this line has an imaginary excursion ∆y = Im τ 0 Re τ 0 + n . (B.3) We can make this quantity arbitrarily small by choosing n sufficiently large. We define σ ′ as the closest point to σ 0 that lies on the image of the line on the strip modulo 1, and has Re σ ′ = Re σ 0 . It is clear that |σ ′ − σ 0 | = Im σ ′ − Im σ 0 ≤ ∆y/2 , (B.4) and, by construction, (σ ′ + m) = t(τ 0 + n) for some m, n ∈ Z and t ∈ R + . We see that |σ ′ − σ 0 | can be made arbitrarily small by increasing n. Next, we approximate t by a fraction a/b ∈ Q + . This, for a/b sufficiently close to t, defines a point σ in the strip by (σ + m) = a b (τ 0 + n) . (B.5) It is clear that σ can be made arbitrarily close to σ ′ by approximating t sufficiently well with a/b. We have thus found a pair (σ, τ = τ 0 ), arbitrarily close to (σ 0 , τ 0 ), that satisfies the constraint (B.2). C Weyl group fixed points In this appendix we prove that Z tot (u; ξ, ν R , aω, bω) vanishes when evaluated at a pointû which is fixed, on a torus of modular parameter ω, by a non-trivial element w of the Weyl group W G : w · [û] = [û] . (C.1) This implies that the solutions to the BAEs (3.9) which are fixed points on the torus of an element of the Weyl group, can be excluded from the set M BAE -as is done in (3.9)-because they do not contribute to the BA formula (3.10) for the superconformal index. C.1 The rank-one case Let us first consider the case that the gauge group G has rank one, i.e., that g = su (2). Then there are only two roots, α and −α, and the Weyl group is W G = {1, s α } ∼ = Z 2 where s α is the unique non-trivial Weyl reflection along the root α: s α (u) = −u ∀ u ∈ h . (C.2) We choose a basis element {H} for the Cartan subalgebra h such that ρ(H) ≡ ρ ∈ Z for any weight ρ ∈ Λ weight . In this canonical basis α = 2 (while the fundamental weight is λ = 1). The solutions to s α · [û] = [û] are given by 7 u = p + qω 2 with p, q ∈ Z . (C.3) Choosing a representative for [û] in the fundamental domain of the torus, the inequivalent solutions are with p = 0, 1 and q = 0, 1. The representations of su(2) are labelled by a half-integer spin j ∈ N/2 and their weights are ρ ∈ ℓα ℓ = −j, −j + 1, . . . , j − 1, j . Therefore, exploiting the expression in (3.58), the function Z reduces to Z(u; ξ, ν R , aω, bω) = = θ 0 α(u); aω θ 0 −α(u); bω a ja ℓa=−ja Γ ℓ a α(u) + ω a (ξ) + r a ν R ; aω, bω . (C.4) Moreover, the function Z tot defined in (3.11) is a single sum over m = 1, . . . , ab. We want to prove that Z tot (û; ξ, ν R , aω, bω) = 0. To do that, we construct an involutive map γ : m → m ′ acting on the set of integers {1, . . . , ab} according to m ′ = m mod b , m ′ = q − m mod a , (C.5) which define m ′ uniquely. It will be convenient to introduce the numbers r, s ∈ Z such that m ′ = m + sb = q − m + ra. The map γ has the property that m ′ − q/2 =    m − q/2 mod b , −(m − q/2) mod a , = m − q/2 + sb = −(m − q/2) + ra . (C.6) We will prove that Z û − m ′ ω; ξ, ν R , aω, bω = −Z û − mω; ξ, ν R , aω, bω . (C.7) In particular, the sum over m inside Z tot splits into a sum over the fixed points of γ and a sum over the pairs of values related by γ. The property (C.7) guarantees that each term in those sums vanishes, implying that Z tot vanishes. Let us adopt the notation Z m ≡ Z(û − mω; ξ, ν R , aω, bω) = Z p/2 − (m − q/2)ω; ξ, ν R , aω, bω . (C.8) We define the vector multiplet and the chiral multiplet contribution, respectively, as A m = θ 0 α(p/2) − α(m − q/2)ω; aω θ 0 −α(p/2) + α(m − q/2)ω; bω B m = a ja ℓa=−ja Γ ℓ a α(p/2) − ℓ a α(m − q/2)ω + ω a (ξ) + r a ν R ; aω, bω , (C.9) such that Z m = A m B m . Then Z tot evaluated onû can be expressed as Z tot p + qω 2 ; ξ, ν R , aω, bω = ab m=1 : m ′ =m Z m + (m,m ′ ) : m ′ =m Z m + Z m ′ . (C.10) Our goal is to show that Z m ′ = −Z m . We begin by considering the contribution of A m . Using (C.6) we can write A m ′ = θ 0 p + (2m − q)ω − 2raω; aω θ 0 −p + (2m − q)ω + 2sbω; bω = θ 0 −p − (2m − q)ω + (2r + 1)aω; aω θ 0 −p + (2m − q)ω + 2sbω; bω . (C.11) In the second equality we used the second relation in (3.26). Using the first relation in (3.26), the identity 2m − q − ra + sb = 0 and reinstating α, with some algebra we obtain A m ′ = − e −2πi α(r) α(s) ν R A m . (C.12) Then we turn to B m and, using (C.6), write B m ′ = a ja ℓa=−ja Γ ℓ a p + ℓ a (2m − q)ω + ω a (ξ) + r a ν R − 2ℓ a raω; aω, bω = a ja ℓa=−ja Γ ℓ a p − ℓ a (2m − q)ω + ω a (ξ) + r a ν R + 2ℓ a raω; aω, bω . (C. 13) We recall that j a can be integer or half-integer. In the second equality we simply redefined ℓ a → −ℓ a and shifted the argument by the integer 2ℓ a p. Using the identity (3.30) repeatedly and distinguishing the cases ℓ a ≶ 0, we obtain B m ′ = Θ × B m (C.14) where the factor Θ equals Θ = a ja ℓa>0 2ℓar−1 k=0 θ 0 ℓ a p − ℓ a (2m − q)ω + ω a (ξ) + r a ν R + kaω; bω θ 0 −ℓ a p + ℓ a (2m − q)ω + ω a (ξ) + r a ν R + (k − 2ℓ a r)aω; bω . (C.15) The second product starts from 1 or 1 2 depending on j a being integer or half-integer. Using 2m − q − ra + sb = 0 at denominator and shifting the arguments by integers, we rewrite Reinstating the root α, this factor can be written as Θ = a ja ℓa>0 2ℓar−1 k=0 θ 0 ℓ a p − ℓ a (2m − q)ω + ω a (ξ) + r a ν R + kaω; bω θ 0 ℓ a p − ℓ a (2m − q)ω + ω a (ξ) + r a ν R + kaω −Θ = a ja ℓa>0 (−1) ℓ 3 a α(r)α(s)α(p) × a, ρa∈Ra e πiρa(r)ρa(s)( 1 2 −ωa(ξ)−(ra−1)ν R ) . (C.18) Combining with (C.12), the factor picked up by Z can be expressed in terms of the anomaly coefficients (3.18) and (3.19): Z m ′ = − e 2πiφ e πirs( 1 2 A ii −A iiα ξα−A iiR ν R) Z m . (C.19) Here i is the gauge index taking a single value. We recall the anomaly cancelation conditions A iiα = A iiR = 0 and A ii ∈ 4Z, implying that the second exponential equals 1. In the first exponential we defined φ = 1 2 α(r) α(s) α(p) a ja ℓa>0 ℓ 3 a = 4rsp a ja ℓa>0 ℓ 3 a . (C.20) It remains to show that φ ∈ Z, so that also the first exponential equals 1. For each chiral multiplet in the theory, indicized by a, in order to evaluate the second sum in (C.20) we should distinguish different cases: ψ j ≡ 4 j ℓ>0 ℓ 3 =          j 2 (j + 1) 2 ∈ 4Z if j ∈ Z 2(k + 1) 2 (8k 2 + 16k + 7) ∈ 2Z if j = 2k + 3 2 ∈ 2Z + 32 1 2 (2k + 1) 2 (8k 2 + 8k + 1) ∈ 4Z + 1 2 if j = 2k + 1 2 ∈ 2Z + 1 2 . (C.21) Therefore, chiral multiplets whose gauge representation has spin j ∈ Z or j ∈ 2Z + 3 2 give integer contribution to φ. On the other hand, chiral multiplets with j ∈ 2Z + 1 2 can give halfinteger contribution. However, because of the Witten anomaly [34], the total number of such multiplets must be even. This is reproduced by the condition (3.21) on the pseudo-anomaly coefficient A ii . Indeed, the contribution of a chiral multiplet to the pseudo-anomaly is A ii (j) = j ℓ=−j (2ℓ) 2 = 4 3 j(j + 1)(2j + 1) ∈    4Z if j ∈ Z or j ∈ 2Z + 3 2 4Z + 2 if j ∈ 2Z + 1 2 , (C. 22) and the condition A ii ∈ 4Z requires that the total number of chiral multiplets with j ∈ 2Z+ 1 2 be even. This implies that φ ∈ Z, and thus that Z m ′ = −Z m . In turn, using (C.10), this implies that Z tot (û; ξ, ν R , aω, bω) = 0 (C. 23) wheneverû is fixed on the torus by the non-trivial element s α of the Weyl group of su(2). C.2 The higher-rank case Let us now move to the case of a generic semi-simple gauge algebra g of rank rk(G). The Weyl group W G is a finite group generated by the Weyl reflections s α (u) = u − 2 α(u) (α, α) α ∀ u ∈ h , (C.24) where α is the image of the root α under the isomorphism h * → h induced by the nondegenerate scalar product (·, ·) on h * . Suppose that there exists a non-trivial element w of W G such that w ·û =û. It is a standard theorem that the Weyl group acts freely and transitively on the set of Weyl chambers. Therefore,û cannot belong to a Weyl chamber but must instead lie on a boundary between two or more chambers. Such boundaries are the hyperplanes fixed by the Weyl reflections, {u|s α (u) = u}, and their intersections. We conclude that there must exist at least one rootα such that sα(û) =û. On the other hand, we are interested in pointsû such that their equivalence class on the torus is fixed by a non-trivial element of the Weyl group, w · [û] = [û]. In this case, for each w we can always identify (at least) one rootα such that sα[û] = [û], and moreover we can choose a set of simple roots that containsα. Let us fix a basis of simple roots {α l } l=1,...,rk(G) for g that containsα. The fundamental weights λ l are defined by 2 (λ k , α l ) (α l , α l ) = δ kl . (C.25) We choose a basis {H i } for the Cartan subalgebra h such that the fundamental weights have components λ l i = λ l (H i ) = δ i l . In this basis ρ(H i ) ≡ ρ i ∈ Z for any weight ρ ∈ Λ weight . Moreover, the double periodicity of the gauge variables u = u i H i is u i ∼ u i + 1 ∼ u i + ω. From (C.24), the fixed points should satisfy 2α (û) (α,α) α = p + qω for p = p i H i , q = q i H i and p i , q i ∈ Z . (C.26) Here α is dual toα. It is clear that p, q should be aligned with α, therefore we set p = 2p (α,α) α , q = 2q (α,α) α , withp,q ∈ Z . (C.27) In the basis {H i } we have choosen, the components of α are (λ i ,α) = δ il (α,α)/2, where l is such thatα = α l and we have used (C.25). Only one component of α is non-zero, which implies that the integer components of p, q are p i =p δ il and q i =q δ il . This proves integrality ofp,q. The general solution to (C.26) can then be written aŝ u =û 0 + p + qω 2 , (C.28) whereû 0 is such thatα(û 0 ) = 0. Now, consider the explicit expression (3.11) for Z tot , in terms of Z given in (2.12). Given any representation R of g, we can always decompose it into irreducible representations of the su(2)α subalgebra associated withα. The set of weights (with multiplicities) Λ R corresponding to R can be organized as a union Λ R = ∪ I Λ R,I of subsets Λ R,I , each corresponding to a representation of su(2)α. Concretely, each Λ R,I is associated to a representation of su(2)α of spin j I , so that its elements can be expressed as anα-chain: Λ R,I = ρ I + ℓ Iα ℓ I = −j I , −j I + 1, . . . , j I − 1, j I . (C. 29) Hereρ I is the central point, which is orthogonal toα, i.e. such that (ρ I ,α) = 0. Notice that, in general,ρ I is not a weight. 8 The product over all weights ρ of the representation R can then be expressed as a product over the representations of su(2)α contained in R. In particular we can write a ρa∈Ra Γ ρ a (u) + ω a (ξ) + r a ν R ; aω, bω = = a,I j aI ℓ aI =−j aI Γ ρ aI (u) + ℓ aIα (u) + ω a (ξ) + r a ν R ; aω, bω . (C. 30) When specifying R to the adjoint representation, we obtain a similar decomposition for the roots of g. Besides the rootsα and −α of su(2)α, the other roots organize intoα-chains that we indicate as Λ roots,J = β J + ℓ Jα ℓ J = −j J , −j J + 1, . . . , j J − 1, j J , (C. 31) whereβ J is the non-vanishing central point orthogonal toα (once again,β J is in general not a weight). Notice that, for each subset Λ roots,J of the set of roots, there is a disjoint conjugate subset Λ roots,J with the same spin j J but opposite central point −β J . 9 For this reason, we have that Z(u; ξ, ν R , aω, bω) = θ 0 α(u); aω θ 0 −α(u); bω × × a,I j aI ℓ aI =−j aI Γ ρ aI (u) + ℓ aIα (u) + ω a (ξ) + r a ν R ; aω, bω Similarly to the rank one case, we want to prove that Z tot (û; ξ, ν R , aω, bω) = 0 forû in (C.28). Thus, we construct an involutive map γ : m → m ′ , acting on the set M of vectors m = m i H i with integer components 1 ≤ m i ≤ ab. The map is constructed in such a way that it leaves m invariant along the directions orthogonal to α, whereas it shifts the component parallel to α by an integer amount. To be precise, take two vectors r, s ∈ h such that r = 2r (α,α) α , s = 2ŝ (α,α) α , withr,ŝ ∈ Z , (C. 33) meaning that r, s are parallel to α and have integer components r i =r δ il , s i =ŝ δ il . Then, we construct m ′ as m ′ = m + s b , (C. 34) which implies that m ′ differs from m only by integer shifts along the direction of α. Forŝ we take the unique integer such that m ′ ∈ M and α(m ′ ) =α(m) +α(s) b =α(q − m) +α(r) a . (C. 35) Indeed, consider the following equation in r and s: 2α(m) −α(q) =α(r) a −α(s) b. Using (C.27) and(C.33), it reduces toα(m) −q =ra −ŝb. Since a, b are coprime, this equation always admits an infinite number of solutions in the pair (r,ŝ), which can be parametrized as (r 0 + kb,ŝ 0 + ka) with k ∈ Z. There is however one and only one solution such that m ′ has components 1 ≤ m ′ i ≤ ab. We define γ(m) = m ′ in such a way. One can easily check that it is an involution. As in the rank-one case, we adopt the notation Z m ≡ Z û − mω; ξ, ν R , aω, bω = Z û 0 + p/2 − (m − q/2)ω; ξ, ν R , aω, bω , (C. 36) and, for later convenience, split Z into the vector multiplet and chiral multiplet contributions: A m = θ 0 α(p/2) −α(m − q/2)ω; aω θ 0 −α(p/2) +α(m − q/2)ω; bω (C.37) C ± m = J j J ℓ J =−j J Γ ±β J (û 0 − mω) + ℓ Jα (p/2) − ℓ Jα (m − q/2)ω; aω, bω) B m = a,I j aI ℓ aI =−j aI Γ ρ aI (û 0 − mω) + ℓ aIα (p/2) − ℓ aIα (m − q/2)ω + ω a (ξ) + r a ν R ; aω, bω such that Z m = A m B m /C + m C − m . We will prove that Z m ′ = −Z m , which implies that Z tot (û) vanishes because γ is an involution. We begin by considering the contribution of A m ′ . Following the same steps as in (C.11) and using (C.35) and (3.26), we can show A m ′ = −e −2πiα(r)α(s) ν R A m . (C.38) We also used thatα(r),α(s),α(q) ∈ 2Z, which is guaranteed by (C.27) and (C.33). We now turn to B m ′ . Eqn. (C.34) implies thatρ aI (m ′ ) =ρ aI (m) for anyρ aI orthogonal toα. Using the identity (3.30) repeatedly and distinguishing the cases ℓ aI ≶ 0, we obtain Combining (C.38) with the latter, we obtain that the vector-multiplet contribution is A m ′ /C + m ′ C − m ′ = −e −2πi α∈∆ α(r)α(s)ν R A m /C + m C − m . (C.41) We usedα(r)α(s) ∈ 4Z, as well as α∈∆ α(r)α(s) ∈ 4Z for any semi-simple Lie algebra g, and that 2ℓ 3 Jα (r)α(s)α(p) ∈ 2Z for any integer or half-integer spin. Including now also the contribution from B m , the factor picked up by Z can be expressed in terms of the anomaly coefficients (3.18) and (3.19): Z m ′ = − e 2πiφ e πi r i s j ( 1 2 A ij −A ijk (û 0 −mω) k −A ijα ξα−A ijR ν R) Z m . (C.42) The anomaly cancelation conditions A ijk = A ijα = A ijR = 0 and A ij ∈ 4Z imply that the second exponential equals 1. In the first exponential we defined Γ (u + rν R − mabω; aω, bω) .(3.29) × a, b ∈ Z (not necessarily coprime). Using both (3.30) and (3.31) we obtain the identity Γ(u + abω; aω, bω) u + (as + br)ω; abω × Γ(u; aω, bω)(3.32) and its generalizations to m ∈ Z, given by Γ(u + mabω; aω, bω) = (−z) − abm(Γ(u; aω, bω) .(3.33) (a, b) forms a numerical semigroup, which can be thought of as a subset of Z ≥0 , closed under addition, with only a finite number of excluded non-vanishing elements. The latter elements form the so-called set of gaps R(a, b) = N \ R(a, b). The highest element of R(a, b) is the Fröbenius number F (a, b) = ab − a − b, whereas the order of R(a, b) is called the genus χ(a, b) and the sum of all its elements is the weight (u; ω) m Γ(u; aω, bω) .(3.38) Finally, applying (3.38) to the l.h.s. of (3.29) proves the latter identity. 41) which, thanks to (3.21)-(3.23), reduce to (3.6) ∀ n i , m i ∈ Z in an anomaly-free theory. Similarly, (3.27) and (3.28) together with the anomaly cancelation conditions (3.21)-(3.23) imply (3.7). Moreover, using (3.40), we can write Q i (u; ξ, ν R , ω) Z(u; ξ, ν R , aω, bω) = = a,ρa P ρ a (u) + ω a (ξ) + r a ν R ; ω ρ i a Γ ρ a (u) + ∆ a ; aω, bω α∈∆ P α(u); ω α i Γ α(u); aω, bω . (3.42) in the last line, we have used the shift property (3.8). Plugging the last equation back in (3.44) gives: u; τ, σ) = 1 Γ(τ + σ − u; τ, σ),(3.57) that {ab − as − br} = −k + (1 − ∆ k )ab k = 0, . . . , ab − 1 . Finally, including the freedom of choosing n ≥ 0, we find that the set of exponents is {abn + ab − as − ar} = (−R) ∪ Z >0 . −1 h ab−as−br ; h ab ) a rsp e −8πiℓ 2 a rs ωa(ξ) e −8πiℓ 2 a rs(ra−1)ν R . (C.17) ℓ J =−j J Γ β J (u) + ℓ Jα ; aω, bω Γ −β J (u) + ℓ Jα ; aω, bω . (C.32) (r)ρa(s) e −πiρa(r)ρa(s) ρa(û 0 −mω)+ωa(ξ)+(ra−1)ν R B m . (C.39)The analysis of C ± m is analogous to the one for B m and it gives the following: (r)α(s) e πiα(r)α(s)ν R × C ± m . (C.40) There is a small caveat: the IR superconformal R-symmetry must be visible in the UV, i.e. it should not be accidental. The two recent papers[9,10] also investigate the entropy of BPS black holes in AdS 5 . 3 Notice that the superconformal index, up to a change of variables reviewed in Section 3.1, is a singlevalued function of the fugacities, while the partition function is not[24]. In particular, let us stress that the condition w · [û] = [û] in the definition of M BAE could be relaxed with no harm: in that case, we would simply include more poles in the sum, whose residues however combine to zero. The integers p, q appearing in this appendix should not be confused with the complex angular fugacities appearing in the rest of the paper. Indeed,ρ I is guaranteed to be a weight (and in particular a root) only if the spin j I is integer. It is easy to prove that Λ roots,J and Λ roots,J are disjoint. Suppose, on the contrary, that there exists some common elementβ J + ℓ Jα = −β J + k Jα for some ℓ J , k J . This would imply thatβ J = (k J − l J )α/2, but since (β J ,α) = 0, thenβ J = 0. Since the only roots proportional toα are −α andα itself, we have reached a contradiction. Since generic vectors r, s (C.33) have integer components, the condition A ij ∈ 4Z implies that also A ij r i s j ∈ 4Z for any choice of r, s. Contracting with the vectors, we obtain(C.45) Therefore, the condition A ij ∈ 4Z requires that the number of su(2)α representations with j aI ∈ 2Z + 1 2 be even, and this guarantees that φ = 0. Once again, in an anomaly-free theory φ ∈ Z. Indeed, labelling the chiral multiplets by a, their su(2)α representations by I and dubbing their spin j aI , the only non. integer contribuOnce again, in an anomaly-free theory φ ∈ Z. Indeed, labelling the chiral multiplets by a, their su(2)α representations by I and dubbing their spin j aI , the only non-integer contribu- Counting chiral primaries in N =1, d=4 superconformal field theories. C Romelsberger, 10.1016/j.nuclphysb.2006.03.037arXiv:hep-th/0510060Nucl. Phys. 747hep-thC. Romelsberger, "Counting chiral primaries in N =1, d=4 superconformal field theories," Nucl. Phys. B747 (2006) 329-353, arXiv:hep-th/0510060 [hep-th]. An index for 4 dimensional super conformal theories. J Kinney, J M Maldacena, S Minwalla, S Raju, 10.1007/s00220-007-0258-7arXiv:hep-th/0510251Commun. Math. Phys. 275hep-thJ. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, "An index for 4 dimensional super conformal theories," Commun. Math. Phys. 275 (2007) 209-254, arXiv:hep-th/0510251 [hep-th]. Indices for Superconformal Field Theories in 3,5 and 6 Dimensions. J Bhattacharya, S Bhattacharyya, S Minwalla, S Raju, 10.1088/1126-6708/2008/02/064arXiv:0801.1435JHEP. 0264hep-thJ. Bhattacharya, S. Bhattacharyya, S. Minwalla, and S. Raju, "Indices for Superconformal Field Theories in 3,5 and 6 Dimensions," JHEP 02 (2008) 064, arXiv:0801.1435 [hep-th]. Localization techniques in quantum field theories. V Pestun, 10.1088/1751-8121/aa63c1arXiv:1608.02952J. Phys. 50440301hep-thV. Pestun et al., "Localization techniques in quantum field theories," J. Phys. A50 (2017) 440301, arXiv:1608.02952 [hep-th]. Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N =1 Dual Theories. F A Dolan, H Osborn, 10.1016/j.nuclphysb.2009.01.028arXiv:0801.4947Nucl. Phys. 818hep-thF. A. Dolan and H. Osborn, "Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N =1 Dual Theories," Nucl. Phys. B818 (2009) 137-178, arXiv:0801.4947 [hep-th]. N =1 supersymmetric indices and the four-dimensional A-model. C Closset, H Kim, B Willett, 10.1007/JHEP08(2017)090arXiv:1707.05774JHEP. 0890hep-thC. Closset, H. Kim, and B. Willett, "N =1 supersymmetric indices and the four-dimensional A-model," JHEP 08 (2017) 090, arXiv:1707.05774 [hep-th]. Seifert fibering operators in 3d N =2 theories. C Closset, H Kim, B Willett, 10.1007/JHEP11(2018)004arXiv:1807.02328JHEP. 114hep-thC. Closset, H. Kim, and B. Willett, "Seifert fibering operators in 3d N =2 theories," JHEP 11 (2018) 004, arXiv:1807.02328 [hep-th]. F Benini, P Milan, arXiv:1812.09613Black holes in 4d N =4 Super-Yang-Mills. hep-thF. Benini and P. Milan, "Black holes in 4d N =4 Super-Yang-Mills," arXiv:1812.09613 [hep-th]. Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes. A Cabo-Bizet, D Cassani, D Martelli, S Murthy, arXiv:1810.11442hep-thA. Cabo-Bizet, D. Cassani, D. Martelli, and S. Murthy, "Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes," arXiv:1810.11442 [hep-th]. S Choi, J Kim, S Kim, J Nahmgoong, arXiv:1810.12067Large AdS black holes from QFT. hep-thS. Choi, J. Kim, S. Kim, and J. Nahmgoong, "Large AdS black holes from QFT," arXiv:1810.12067 [hep-th]. A topologically twisted index for three-dimensional supersymmetric theories. F Benini, A Zaffaroni, 10.1007/JHEP07(2015)127arXiv:1504.03698JHEP. 07127hep-thF. Benini and A. Zaffaroni, "A topologically twisted index for three-dimensional supersymmetric theories," JHEP 07 (2015) 127, arXiv:1504.03698 [hep-th]. Black hole microstates in AdS 4 from supersymmetric localization. F Benini, K Hristov, A Zaffaroni, 10.1007/JHEP05(2016)054arXiv:1511.04085JHEP. 0554hep-thF. Benini, K. Hristov, and A. Zaffaroni, "Black hole microstates in AdS 4 from supersymmetric localization," JHEP 05 (2016) 054, arXiv:1511.04085 [hep-th]. Supersymmetric partition functions on Riemann surfaces. F Benini, A Zaffaroni, 10.1090/pspum/096arXiv:1605.06120Proc. Symp. Pure Math. 96hep-thF. Benini and A. Zaffaroni, "Supersymmetric partition functions on Riemann surfaces," Proc. Symp. Pure Math. 96 (2017) 13-46, arXiv:1605.06120 [hep-th]. Exact microstate counting for dyonic black holes in AdS 4. F Benini, K Hristov, A Zaffaroni, 10.1016/j.physletb.2017.05.076arXiv:1608.07294Phys. Lett. 771hep-thF. Benini, K. Hristov, and A. Zaffaroni, "Exact microstate counting for dyonic black holes in AdS 4 ," Phys. Lett. B771 (2017) 462-466, arXiv:1608.07294 [hep-th]. Supersymmetric AdS 5 black holes. J B Gutowski, H S Reall, 10.1088/1126-6708/2004/02/006arXiv:hep-th/0401042JHEP. 026hep-thJ. B. Gutowski and H. S. Reall, "Supersymmetric AdS 5 black holes," JHEP 02 (2004) 006, arXiv:hep-th/0401042 [hep-th]. General supersymmetric AdS 5 black holes. J B Gutowski, H S Reall, 10.1088/1126-6708/2004/04/048arXiv:hep-th/0401129JHEP. 0448hep-thJ. B. Gutowski and H. S. Reall, "General supersymmetric AdS 5 black holes," JHEP 04 (2004) 048, arXiv:hep-th/0401129 [hep-th]. General non-extremal rotating black holes in minimal five-dimensional gauged supergravity. Z W Chong, M Cvetic, H Lu, C N Pope, 10.1103/PhysRevLett.95.161301arXiv:hep-th/0506029Phys. Rev. Lett. 95161301hep-thZ. W. Chong, M. Cvetic, H. Lu, and C. N. Pope, "General non-extremal rotating black holes in minimal five-dimensional gauged supergravity," Phys. Rev. Lett. 95 (2005) 161301, arXiv:hep-th/0506029 [hep-th]. Supersymmetric multi-charge AdS 5 black holes. H K Kunduri, J Lucietti, H S Reall, 10.1088/1126-6708/2006/04/036arXiv:hep-th/0601156JHEP. 0436hep-thH. K. Kunduri, J. Lucietti, and H. S. Reall, "Supersymmetric multi-charge AdS 5 black holes," JHEP 04 (2006) 036, arXiv:hep-th/0601156 [hep-th]. Partition Functions of N =(2, 2) Gauge Theories on S 2 and Vortices. F Benini, S Cremonesi, 10.1007/s00220-014-2112-zarXiv:1206.2356Commun. Math. Phys. 334hep-thF. Benini and S. Cremonesi, "Partition Functions of N =(2, 2) Gauge Theories on S 2 and Vortices," Commun. Math. Phys. 334 (2015) 1483-1527, arXiv:1206.2356 [hep-th]. Higgs branch localization in three dimensions. F Benini, W Peelaers, 10.1007/JHEP05(2014)030arXiv:1312.6078JHEP. 0530hep-thF. Benini and W. Peelaers, "Higgs branch localization in three dimensions," JHEP 05 (2014) 030, arXiv:1312.6078 [hep-th]. Comments on twisted indices in 3d supersymmetric gauge theories. C Closset, H Kim, 10.1007/JHEP08(2016)059arXiv:1605.06531JHEP. 0859hep-thC. Closset and H. Kim, "Comments on twisted indices in 3d supersymmetric gauge theories," JHEP 08 (2016) 059, arXiv:1605.06531 [hep-th]. Supersymmetric partition functions and the three-dimensional A-twist. C Closset, H Kim, B Willett, 10.1007/JHEP03(2017)074arXiv:1701.03171JHEP. 0374hep-thC. Closset, H. Kim, and B. Willett, "Supersymmetric partition functions and the three-dimensional A-twist," JHEP 03 (2017) 074, arXiv:1701.03171 [hep-th]. The Casimir Energy in Curved Space and its Supersymmetric Counterpart. B Assel, D Cassani, L Di Pietro, Z Komargodski, J Lorenzen, D Martelli, 10.1007/JHEP07(2015)043arXiv:1503.05537JHEP. 0743hep-thB. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen, and D. Martelli, "The Casimir Energy in Curved Space and its Supersymmetric Counterpart," JHEP 07 (2015) 043, arXiv:1503.05537 [hep-th]. Supersymmetric Casimir Energy and the Anomaly Polynomial. N Bobev, M Bullimore, H.-C Kim, 10.1007/JHEP09(2015)142arXiv:1507.08553JHEP. 09142hep-thN. Bobev, M. Bullimore, and H.-C. Kim, "Supersymmetric Casimir Energy and the Anomaly Polynomial," JHEP 09 (2015) 142, arXiv:1507.08553 [hep-th]. Bethe/Gauge correspondence on curved spaces. N A Nekrasov, S L Shatashvili, 10.1007/JHEP01(2015)100arXiv:1405.6046JHEP. 01100hep-thN. A. Nekrasov and S. L. Shatashvili, "Bethe/Gauge correspondence on curved spaces," JHEP 01 (2015) 100, arXiv:1405.6046 [hep-th]. Constraints on Supersymmetry Breaking. E Witten, 10.1016/0550-3213(82)90071-2Nucl. Phys. 202253E. Witten, "Constraints on Supersymmetry Breaking," Nucl. Phys. B202 (1982) 253. The Hagedorn transition, deconfinement and N =4 SYM theory. B Sundborg, 10.1016/S0550-3213(00)00044-4arXiv:hep-th/9908001Nucl. Phys. 573hep-thB. Sundborg, "The Hagedorn transition, deconfinement and N =4 SYM theory," Nucl. Phys. B573 (2000) 349-363, arXiv:hep-th/9908001 [hep-th]. The Hagedorn/deconfinement phase transition in weakly coupled large N gauge theories. O Aharony, J Marsano, S Minwalla, K Papadodimas, M Van Raamsdonk, 10.4310/ATMP.2004.v8.n4.a1arXiv:hep-th/0310285Adv. Theor. Math. Phys. 8hep-thO. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, "The Hagedorn/deconfinement phase transition in weakly coupled large N gauge theories," Adv. Theor. Math. Phys. 8 (2004) 603-696, arXiv:hep-th/0310285 [hep-th]. S Benvenuti, B Feng, A Hanany, Y.-H He, 10.1088/1126-6708/2007/11/050arXiv:hep-th/0608050Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics. 50hep-thS. Benvenuti, B. Feng, A. Hanany, and Y.-H. He, "Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics," JHEP 11 (2007) 050, arXiv:hep-th/0608050 [hep-th]. Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. V Pestun, 10.1007/s00220-012-1485-0arXiv:0712.2824Commun. Math. Phys. 313hep-thV. Pestun, "Localization of gauge theory on a four-sphere and supersymmetric Wilson loops," Commun. Math. Phys. 313 (2012) 71-129, arXiv:0712.2824 [hep-th]. The N =1 Chiral Multiplet on T 2 × S 2 and Supersymmetric Localization. C Closset, I Shamir, 10.1007/JHEP03(2014)040arXiv:1311.2430JHEP. 0340hep-thC. Closset and I. Shamir, "The N =1 Chiral Multiplet on T 2 × S 2 and Supersymmetric Localization," JHEP 03 (2014) 040, arXiv:1311.2430 [hep-th]. Localization on Hopf surfaces. B Assel, D Cassani, D Martelli, 10.1007/JHEP08(2014)123arXiv:1405.5144JHEP. 08123hep-thB. Assel, D. Cassani, and D. Martelli, "Localization on Hopf surfaces," JHEP 08 (2014) 123, arXiv:1405.5144 [hep-th]. The elliptic gamma function and sl(3, Z) ⋉ Z 3. G Felder, A Varchenko, 10.1006/aima.2000.1951arXiv:math/9907061Adv. Math. 15644G. Felder and A. Varchenko, "The elliptic gamma function and sl(3, Z) ⋉ Z 3 ," Adv. Math. 156 (2000) 44, arXiv:math/9907061. An SU(2) Anomaly. E Witten, 10.1016/0370-2693(82)90728-6Phys. Lett. 117E. Witten, "An SU(2) Anomaly," Phys. Lett. B117 (1982) 324-328. Nonperturbative Anomalies in Higher Dimensions. S Elitzur, V P Nair, 10.1016/0550-3213(84)90024-5Nucl. Phys. 243205S. Elitzur and V. P. Nair, "Nonperturbative Anomalies in Higher Dimensions," Nucl. Phys. B243 (1984) 205. Global Gauge Anomalies for Simple Lie Algebras. H Zhang, S Okubo, Y Tosa, 10.1103/PhysRevD.37.2946Phys. Rev. 372946H. Zhang, S. Okubo, and Y. Tosa, "Global Gauge Anomalies for Simple Lie Algebras," Phys. Rev. D37 (1988) 2946.
[]
[ "A search for directional violations of the Lorentz invariance through the study of a possible anisotropy of particle lifetimes", "A search for directional violations of the Lorentz invariance through the study of a possible anisotropy of particle lifetimes" ]
[ "Il ", "Cimento ", "\nLIP/IST\nINFN and INAF Trieste\nLisboaItaly, Portugal (\n" ]
[ "LIP/IST\nINFN and INAF Trieste\nLisboaItaly, Portugal (" ]
[]
From the study of a sample of about 62.3 million well reconstructed K 0 S → π + π − decays recorded by the KLOE detector at the DAΦNE accelerator in Frascati, the lifetimes of K 0 S mesons parallel and antiparallel to the direction of motion of the Earth with respect to the Cosmic Microwave Background (CMB) reference frame have been studied. No difference has been found, and a limit on a possible asymmetry of the lifetime with respect to the CMB has been set at 95% C.L.: |A|CMB = |τ+CMB − τ−CMB|/(τ+CMB + τ−CMB) < 0.98 × 10 −3 . This is presently the best experimental limit on such quantity, and it is smaller of the speed, expressed in natural units, of the Solar System with respect to the CMB (V /c = 1.23 × 10 −3 ). The present limit might constrain possible Lorentz-violating anisotropical theories. PACS 03.30.+p -Special relativity. PACS 04.60.-m -Quantum gravity. PACS 95.85.Pw -Gamma rays astronomical observations.
10.1393/ncc/i2011-10894-9
[ "https://arxiv.org/pdf/1011.3720v2.pdf" ]
118,833,307
1011.3720
c50f98f21a16a3b9841c43e962cfc0f4b7e5a795
A search for directional violations of the Lorentz invariance through the study of a possible anisotropy of particle lifetimes 20 Jan 2011 Il Cimento LIP/IST INFN and INAF Trieste LisboaItaly, Portugal ( A search for directional violations of the Lorentz invariance through the study of a possible anisotropy of particle lifetimes 20 Jan 20111 From the study of a sample of about 62.3 million well reconstructed K 0 S → π + π − decays recorded by the KLOE detector at the DAΦNE accelerator in Frascati, the lifetimes of K 0 S mesons parallel and antiparallel to the direction of motion of the Earth with respect to the Cosmic Microwave Background (CMB) reference frame have been studied. No difference has been found, and a limit on a possible asymmetry of the lifetime with respect to the CMB has been set at 95% C.L.: |A|CMB = |τ+CMB − τ−CMB|/(τ+CMB + τ−CMB) < 0.98 × 10 −3 . This is presently the best experimental limit on such quantity, and it is smaller of the speed, expressed in natural units, of the Solar System with respect to the CMB (V /c = 1.23 × 10 −3 ). The present limit might constrain possible Lorentz-violating anisotropical theories. PACS 03.30.+p -Special relativity. PACS 04.60.-m -Quantum gravity. PACS 95.85.Pw -Gamma rays astronomical observations. Possible violations of the Lorentz invariance have been recently suggested to explain anomalies in the propagation of cosmic rays and the transparency [1] of the Universe to gamma rays [2,3,4]. Violations of the Lorentz invariance might in particular imply the existence of a preferred reference frame; this could introduce a globally anisotropical mechanics [5]. A global anisotropy might explain a possible different behavior of photon propagation for different high energy gamma sources [6,7], and it has been recently suggested as a possible explanation of physical observations [8]. Different frameworks have been proposed [9,10] to host a global anisotropy; widely used are the theories of Robertson [11], and Mansouri and Sexl [12], together generally called the RMS-theories, and the so-called "Standard Model Extension" [13]. Many experiments measuring particle lifetimes [15] give evidence for a time dilation in accordance with the Lorentz transformations, up to the present experimental accuracy. ( * ) On leave of absence from Università di Udine, Via delle Scienze 208, Udine, Italy. c Società Italiana di Fisica Such experimental verifications do not rule out however the possibility that measured particle lifetimes depend on their direction of motion: present isotropy tests are not very accurate [14]. Accurate isotropy tests have instead been performed about the speed of light, and they did not give firm indications of anisotropies [15,16]. In special relativity, spatial isotropy is a crucial issue in the synchronization procedure of two distant clocks; Einstein stressed however [17] that the fact that light travels at equal speeds along the opposite directions of a particular path is "neither a supposition nor a hypothesis about the physical nature of light, but a stipulation" that can be freely made so as to arrive at a definition of simultaneity. The so-called conventionalist thesis proposed by Reichenbach [18] states that quantities as the one-way speed of light are inherently conventional, and that to recognize this aspect is to recognize a profound feature of nature. Only proper time has "objective status in special relativity" [19]; this is because one-way velocity's value is nothing about the pattern of coincidences of events at a given space locations, but it refers to the comparison of remote events, and so is inevitably conventional. Thus a test of isotropy of particle lifetimes is independent of a test of the isotropy of the speed of light. The Cosmic Microwave Background (CMB) dipole anisotropy [20], interpreted as a Doppler effect, indicates the motion of the Local Group in the direction ( , b) CMB = (264 • , 48 • ) in galactic coordinates, with a speed of V = (369 ± 1) km/s [21], i.e., V /c = (1.231 ± 0.003) × 10 −3 .(1) The CMB is a unique rest frame: even if this fact does not imply by itself any anisotropy of the physical laws (although at a small level QED should became anisotropical due to interactions with a nontrivial vacuum), the existence of such a natural rest frame provides a rational framework for the interpretation of any asymmetry that might possibly be discovered. The old idea of an absolute "aether" is exploited, with the only difference that the preferred frame is now identified with one in which the cosmic background radiation is locally isotropic. In fact, there is only one frame with this property, being all other frames experiencing the dipole anisotropy, and therefore distinguishable. Collider detectors with 4π acceptance can be used as a probe for detecting global asymmetries in the Universe [14]: the Earth's rotation provides in the different seasons and hours of the day different orientations of the symmetry axes with respect to arbitrary directions, and this fact entails a strong reduction of detector effects ( fig. 1). This note reports on a test of the isotropy of K 0 S lifetime, which has been done [22,23] by comparing the lifetimes of K 0 S measured by KLOE parallel and antiparallel with respect to the direction of motion of the particles with respect to the CMB system. Selected data on K 0 S decays into charged pion pairs have been used from the data collected by KLOE in 2004 and 2005. After a severe quality selection, a total sample of about 62.3 million well reconstructed decays has been used. The K 0 S momentum( 1 ) has been transformed from local-KLOE into galactic coordinates [22]. We retained only events inside a cone with an opening angle of 30 • ( 2 ) parallel (up) and antiparallel (down) ( 1 ) DAΦNE operates at an energy slightly above the φ mass; the K 0 S have a peak momentum of about 0.11 GeV/c. ( 2 ) Monte Carlo studies [14] demonstrated that this criterion maximizes the sensitivity to asymmetries. The relation between the measured asymmetry in a cone and the asymmetry with to the direction of motion with respect to the CMB, and in each cone we measured the K 0 S lifetime, τ, by a fit to the proper time distribution as in [23]. We define the asymmetry A cone as A cone = τ up − τ down τ up + τ down , finding [23] A cone, CMB = (−0.13 ± 0.40) × 10 −3 , consistent with zero; the error is purely statistical, since systematics largely cancel due to the definition of the asymmetry (this fact has been explicitly verified in a subsample corresponding to about 1/4 of the total statistics in [23]). As a cross check, the asymmetry has been measured in two directions perpendicular to the direction of motion with respect to the CMB, and found again to be consistent with zero( 3 ). Using eq. 2, such a result on the asymmetry with respect to a cone translates in an respect to a fixed a direction u is, in case of uniform population (which is a good approximation for the data set): A u 1.072Acone .(2) ( 3 ) The so-called "absolute" direction, defined in [24]: ( , b) abs = (277.6 • , −34.5 • ) in galactic coordinates, has been also studied; a value A cone, abs = (−0.1 ± 0.3) × 10 −3 , again consistent with 0, has been found. upper limit at 95% CL referred to the direction of motion with respect to the CMB: |A| CMB = |τ +CM B − τ −CM B | τ +CM B + τ −CM B < 0.98 × 10 −3 (95% CL) .(3) This result sets limits to non-relativistic theories, and to possible anisotropical interactions of neutral kaons with the matter in the universe, in an unexplored domain, improving by one order of magnitude the previous results ( [14]). The present upper limit on the asymmetry is smaller than the relative velocity (eq. 1), in natural units, of the Solar System with respect to the CMB. Fig. 1 . 1-Collider detectors with 4π acceptance can be used as a probe for detecting global asymmetries: the Earth's rotation provides in the different seasons and hours of the day different orientations of the symmetry axes with respect to arbitrary directions, thus reducing detector effects. . Science. 3201752MAGIC Collaboration, Science, 320 (2008) 1752 . Coleman S Glashow, S L , Phys. Lett. B. 405249Coleman S. & Glashow S.L., Phys. Lett. B, 405 (1997) 249 . T Kifune, Astrophys. J. 51821Kifune T., Astrophys. J., 518 (1999) L21 . W Kluzniak, Astropart. Phys. 1117Kluzniak W., Astropart. Phys., 1 (1999) 117 Real and conventional anisotropy, generalized Lorentz transformations and physical effects. Udinethe University of UdinePhD Thesis atSee for example Pin M., "Real and conventional anisotropy, generalized Lorentz transformations and physical effects" (Udine 2005), PhD Thesis at the University of Udine (see http://www.infn.it) . J Ellis, MAGIC CollaborationPhys. Lett. B. 668253MAGIC Collaboration & Ellis J. et al., Phys. Lett. B, 668 (2008) 253 . Nature. 462331Fermi Collaboration, Nature, 462 (2009) 331 Question Isotropy. See For Example Ralston, J P , arXiv:1011.2240See for example Ralston J.P., "Question Isotropy", arXiv:1011.2240; Constant change: are there no universal laws?. M Brooks, New Scientist. Brooks M., "Constant change: are there no universal laws?", New Scientist, 25 October 2010 Modern Tests of Lorentz Invariance. D Mattingly, Living Rev. Relativity. 85Mattingly D., "Modern Tests of Lorentz Invariance", Living Rev. Relativity, 8 (2005) 5 -URL (cited on Nov. 12, 2010): http://www.livingreviews.org/lrr-2005-5 . C Lämmerzahl, Lect. Notes Phys. 702349Lämmerzahl C., Lect. Notes Phys., 702 (2006) 349 . H P Robertson, Rev. Mod. Phys. 21378Robertson H.P., Rev. Mod. Phys., 21 (1949) 378 . R Mansouri, R U Sexl, Gen. Rel. Grav. 8497ibid. 515; ibid. 809Mansouri R. & Sexl R.U., Gen. Rel. Grav., 8 (1977) 497; ibid. 515; ibid. 809 . D Colladay, V A Kostelecky, Phys. Rev. D. 58116002Colladay D., Kostelecky V.A., Phys. Rev. D, 58 (1998) 116002 Search for Intrinsic Anisotropies of Time Dilation. De Angelis, A , 92/01/AAUniversity of UdineReportRevDe Angelis A., "Search for Intrinsic Anisotropies of Time Dilation", University of Udine Report 92/01/AA, January 1992 (Rev. June 1992); Search for Intrinsic Anisotropies of Time Dilation. A De Angelis, B Smalska, CERN DELPHI 1996 141 PHYS 640GenevaDe Angelis A. & Smalska B., "Search for Intrinsic Anisotropies of Time Dilation", CERN DELPHI 1996 141 PHYS 640 (Geneva 1996) For a review of the tests of time dilation and of the isotropy of the speed of light see Zhang Y.Z., "Special Relativity and Its Experimental Foundations. World ScientificSingaporeand [9For a review of the tests of time dilation and of the isotropy of the speed of light see Zhang Y.Z., "Special Relativity and Its Experimental Foundations" (World Scientific, Singapore 1997), and [9]; A Review of One-Way and Two-Way Experiments to Test the Isotropy of the Speed of Light. Farid Ahmed Md, arXiv:1011.1318Farid Ahmed Md. et al., "A Review of One-Way and Two-Way Experiments to Test the Isotropy of the Speed of Light", arXiv:1011.1318 Relativity: The Special and the General Theory. A Einstein, New YorkEinstein A., "Relativity: The Special and the General Theory" (New York 1920) H Reichenbach, The Philosophy of Space and Time. New YorkDoverReichenbach H., "The Philosophy of Space and Time" (Dover, New York 1957) Foundations of Space-Time Theories, Relativistic Physics and Philosophy of Science. M Friedman, Princeton University PressPrincetonFriedman M., "Foundations of Space-Time Theories, Relativistic Physics and Philosophy of Science" (Princeton University Press, Princeton 1983) . Astrophys. J. 4191COBE Collaboration, Astrophys. J., 419 (1993) 1 . K Nakamura, Particle Data GroupJ. Phys. G. 75021and references thereinNakamura K. et al. (Particle Data Group), J. Phys. G, 37 (2010) 075021 and references therein Searching for Intrinsic Anisotropies of the Universe through the Study of Kaon Lifetimes. De Maria, M , Udinethe University of UdinePhD Thesis atDe Maria M., "Searching for Intrinsic Anisotropies of the Universe through the Study of Kaon Lifetimes" (Udine 2010), PhD Thesis at the University of Udine (see http://www.infn.it) Precision Measurement of KS meson lifetime with the KLOE detector. arXiv:1011.2668Eur. Phys. J. C. hep-ex. in pressKLOE Collaboration, "Precision Measurement of KS meson lifetime with the KLOE detector", arXiv:1011.2668 [hep-ex] (Eur. Phys. J. C, in press) . R T Cahill, Apeiron. 53Cahill R.T., Apeiron, 11 (2004) 53
[]
[ "STABILITY OF CAPILLARY HYPERSURFACES IN A EUCLIDEAN BALL", "STABILITY OF CAPILLARY HYPERSURFACES IN A EUCLIDEAN BALL" ]
[ "Haizhong Li ", "Changwei Xiong " ]
[]
[]
We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.2010 Mathematics Subject Classification. 53A10, 49Q10.
10.2140/pjm.2018.297.131
[ "https://arxiv.org/pdf/1408.2086v1.pdf" ]
119,698,575
1408.2086
bda0e9c5ccb2040d047f9d58d4b0d5e92091b33e
STABILITY OF CAPILLARY HYPERSURFACES IN A EUCLIDEAN BALL Haizhong Li Changwei Xiong STABILITY OF CAPILLARY HYPERSURFACES IN A EUCLIDEAN BALL We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.2010 Mathematics Subject Classification. 53A10, 49Q10. Introduction Capillarity is an important physical phenomena, which occurs when two different materials contact and do not mix. Given a container B with an incompressible liquid drop T in it, the interface of the liquid and the air is a capillary surface M . In absence of gravity, the interface M is of constant mean curvature and the contact angle of M to the boundary ∂B is constant. One should compare this setting with soap bubble (resp. soap film), where the surface has no boundary (resp. fixed boundary) and constant mean curvature. The literature for the study of capillarity is extensive and we refer to the book of Finn [5], where the treatment of the theory is mainly in the nonparametric case and in the more general situation of presence of gravity. Also we mention [6] for a more recent survey about this topic. In this paper we are concerned with the special case that the container B is a unit Euclidean ball and no gravity is involved. We study the (weak) stability for capillary hypersurfaces. This problem has been discussed by Ros and Souam [13], where they dealt with surface case and obtained some topological and geometrical restrictions. For the hypersurface case with free boundary (the contact angle is π/2), Ros and Vergasta also proved some interesting results in [14]. Applying the same argument as in the proof of Proposition 1.1 in [13], we know that the hyperplanes and the spherical caps in a unit Euclidean ball are capillarily stable. Recently, Marinov [12] proved that, for surface case, all other known examples are unstable. We generalize Marinov's result to the hypersurface case. In fact we prove the following theorem. Theorem 1. Let x : M n → R n+1 be an immersed capillary hypersurface in a unit Euclidean ball B n+1 and Ω the wetted part of the boundary of the ball. Denote by T the generalized body enclosed by x(M ) and Ω. If the mass center of T is at the origin, the capillary hypersurface M is unstable. Here since we assume M is immersed, x(M ) may have self-intersections. Thus we need to consider the generalized body T . When M is embedded, T is understood in the common sense. See Remark 1 below for more explanation. For n = 2, our Theorem 1 reduces to Marinov's result in [12]. We also note that his argument relies on the conformal coordinates, which can not be generalized to the higher dimensional case. Applying Theorem 1 to Delaunay hypersurfaces, we get the following Corollary 2. Recall that Delaunay hypersurfaces are the ones of revolution with constant mean curvature. By Proposition 4.3 in [11], Delaunay hypersurfaces are classified as: an unduloid, cylinder, nodoid, sphere, catenoid, or a hyperplane. To guarantee the portion of a Delaunay hypersurface in a Euclidean ball is also capillary, it shall have some symmetry. See Section 2 below for more details. And in that case, we call it Delaunay capillary hypersurface. From Theorem 1 we have Corollary 2. The only stable Delaunay capillary hypersurface M n in a unit Euclidean ball B n+1 is a totally geodesic hyperplane or a spherical cap. Our approach for proving Theorem 1 is as follows. In higher dimensional case, we find that we can construct a conformal killing vector field Y [ξ] for any fixed ξ ∈ S n from the natural conformal transformation family on B n+1 . Using the normal part Y [ξ], N as the test function we can define a symmetric quadratic form Q(ξ 1 , ξ 2 ). By summing Q over (n + 1) coordinate directions we find Q has at least one negative eigenvalue. This summation technique can be compared with J. Simons' work [16]. At last, under the hypothesis of Theorem 1 we can derive the instability of the hypersurface. Our argument indicates that this conformal field is very important and we can use it to conclude that the mass center of minimal submanifolds with free boundary in a unit Euclidean ball is at the origin (See Proposition 11). We refer the readers to [7][8][9] for the very recent work on the minimal submanifolds with free boundary. At last, as an application of our argument, we give a new proof of the classical result due to Barbosa and do Carmo [1], which states that the only closed stable immersed hypersurface of constant mean curvature in R n+1 is the round sphere. An outline of this paper is as follows. In Section 2 after fixing some notations and definitions, we prove the stability of hyperplanes and spherical caps. Then we construct the crucial conformal vector field. We also review some known facts about the Delaunay hypersurfaces. In Section 3 we give the proof of Theorem 1. In last section, we discuss some applications of our method. {T i } m i=1 , choose one and fix it. In the proof we will see that only the property ∂T = x(M )∪Ω is needed. And if there is no confusion, we will write M (resp. ∂M ) for x(M ) (resp. x(∂M )) for simplicity. Let N be the unit normal of M pointing inwards to T andN the unit outward normal of ∂B n+1 . Denote by ν andν the conormals of ∂M in M and Ω, respectively. Let D (resp. ∇) be the connection of R n+1 (resp. M ). Then the second fundamental form of M in R n+1 is given by σ(X 1 , X 2 ) = D X 1 X 2 , N for ∀ X 1 , X 2 ∈ T M . When taking an orthonormal basis {e i } n i=1 on T M , we also denote by h ij the components σ(e i , e j ). So the mean cur- vature H of M is H = 1 n n i=1 h ii . And the second fundamental form of ∂B in R n+1 is given by Π(Y 1 , Y 2 ) = D Y 1 Y 2 , −N for ∀ Y 1 , Y 2 ∈ T (∂B) . At last let θ ∈ (0, π) be the angle between ν andν. See Figure 1 for an illustration. Following [13], we discuss the variation of M . Definition 2.1. An admissible variation of x : M n → R n+1 is a differen- tiable map X : (−ε, ε) × M → R n+1 so that X t : M n → R n+1 , t ∈ (−ε, ε) given by X t (p) = X(t, p), p ∈ M is an immersion satisfying X t (int M ) ⊂ int B and X t (∂M ) ⊂ ∂B for all t, and X 0 = x. Now for given θ ∈ (0, π), we define a energy functional E(t) = |M (t)| − cos θ|Ω(t)|,(1) where | · | denotes the area function. And the volume functional can be defined as V (t) = [0,t]×M X * dv, where dv is the standard volume element of R n+1 . Under these constraints, we have Definition 2.2. An immersed hypersurface x : M n → R n+1 is called capil- lary if E (0) = 0 for any admissible volume-preserving variation of x. Note that we have the following formulas, E (0) = −n M Hf da + ∂M Y, ν − cos θν ds,(2)V (0) = − M f da,(3) where Y is the variational vector field Y (p) = ∂X ∂t (p)| t=0 , f its normal component f = Y, N , and da and ds are the corresponding area elements. From these formulas we see that M is capillary if and only if it has constant mean curvature and makes constant contact angle θ with ∂B. Furthermore, one can compute the second derivative at t = 0 with respect to an admissible volume-preserving variation to get (see e.g. the appendix of [13]) E (0) = − M ∆f + (|σ| 2 + Ric(N ))f f da + ∂M ( ∂f ∂ν − qf )f ds, (4) where f ∈ F := {f ∈ H 1 (M ), M f da = 0}, Ric(N ) is the Ricci curvature of the ambient space and q = 1 sin θ Π(ν,ν) + cot θσ(ν, ν).(5) In our setting, Ric(N ) = 0 and Π(ν,ν) = 1. Definition 2.3. A capillary hypersurface M is called (weakly) stable if E (0) ≥ 0 for all f ∈ F. In the sequel we will denote by ∂ 2 E(φ) the quantity E (0) for a given function φ. 2.2. Stable examples of capillary hypersurfaces. First we prove the stability of totally geodesic capillary hypersurfaces and spherical caps. The proof is similar to that of Proposition 1.1 in [13]. We include it for completeness. Proposition 3. Let B n+1 ⊂ R n+1 be a unit Euclidean ball. Then totally geodesic capillary hypersurfaces and spherical caps are stable. Proof. First assume M is a totally geodesic capillary hypersurface, i.e., an n-dimensional ball B n (R) with radius R in B n+1 . Then the contact angle θ satisfies sin θ = R. By the definition of stability, we have to prove M |∇f | 2 da ≥ 1 R ∂M f 2 ds, for ∀f ∈ F.(6) Consider now the (n + 1)-dimensional ball B of radius R having M as an equatorial totally geodesic hypersurface. Then by [2], M is area minimizing for partitioning problem in B . Thus M is stable in B , which is equivalent to the inequality (6). Next assume M is a spherical cap in B n+1 with R the radius of the sphere containing M and θ the contact angle. Consider the n-dimensional hyperplane P containing ∂M . Then M is a capillary hypersurface in a halfspace with a contact angle θ . By [10], M is stable in the halfspace, which means M (|∇f | 2 − n R 2 f 2 )da ≥ cot θ R ∂M f 2 ds, for ∀f ∈ F.(7) Elementary calculation leads to 1 sin θ + cot θ R = cot θ R .(8) Now (7) and (8) together yield the stability of M in B n+1 . 2.3. Conformal transformations on the Euclidean ball. Now we construct a conformal vector field. Fix a vector a ∈ B n+1 . Then (see e.g. Section 3.8 in [15]) ϕ a (x) = (1 − |a| 2 )x − (1 − 2 a, x + |x| 2 )a 1 − 2 a, x + |a| 2 |x| 2(9) defines a map from B n+1 to B n+1 and from S n to S n , since we have 1 − |ϕ a (x)| 2 = (1 − |a| 2 )(1 − |x| 2 ) 1 − 2 a, x + |a| 2 |x| 2 . Moreover ϕ a is conformal. In fact, by a direct calculation we can check that |dϕ a | 2 = 1 − |a| 2 1 − 2 a, x + |a| 2 |x| 2 2 |dx| 2 . Note that ϕ a (a) = 0, ϕ a (0) = −a, ϕ a fixes two points ± a |a| and ϕ 0 is an identity. Next fix ξ ∈ S n . Let a = tξ with −1 < t < 1. Then f t (x) = ϕ tξ (x) = (1 − t 2 )x − (1 − 2t ξ, x + |x| 2 )tξ 1 − 2t ξ, x + t 2 |x| 2(10) is a family of conformal transformations with parameter t. Thus f t determines a conformal vector field Y [ξ] as follows. Y [ξ] = d dt t=0 f t (x) = −(1 + |x| 2 )ξ + 2 ξ, x x.(11) Note that Y [ξ] is tangential along the sphere S n , since for ∀x ∈ S n , Y [ξ], x = −(1 + |x| 2 ) ξ, x + 2 ξ, x |x| 2 = 0. 2.4. Delaunay hypersurfaces in Euclidean space. In this subsection, following [11] we review some facts about Delaunay hypersurfaces, which is rotational and of constant mean curvature H. These hypersurfaces are the models we are concerned with in Theorem 1. Let M n ⊂ R n+1 be a hypersurface which is invariant under the action of the orthogonal group O(n) fixing the x 1 -axis. Assume M is generated by a curve Γ contained in the x 1 x 2 -plane. Then it suffices to determine the curve Γ. Parametrize the curve Γ = (x 1 , x 2 ) by arc-length s. Denote by α the angle between the tangent to Γ and the positive x 1 -direction and choose the normal vector N = (sin α, − cos α). Then (x 1 , x 2 ; α) satisfies the following system of ordinary differential equations      (x 1 ) = cos α, (x 2 ) = sin α, α = −nH + (n − 1) cos α x 2 . The first integral of this system is given by (1) If F H > 0 then Γ is a periodic graph over the x 1 -axis. It generates a periodic embedded unduloid, or a cylinder. (2) If F H < 0 then Γ is a locally convex curve and M is a nodoid, which has self-intersections. From this proposition, it is easy to see if M n is the portion of an unduloid, cylinder, nodoid or a catenoid in a unit Euclidean ball B n+1 with revolution axis x 1 and moreover M is symmetric with respect to the hyperplane {x 1 = 0}, then M is a capillary hypersurface in B n+1 . In that case we call them Delaunay capillary hypersurfaces in B n+1 . Furthermore the generalized body T enclosed by M and the wetted part of the sphere has the mass center at the origin. So Theorem 1 is applicable. (x 2 ) n−1 cos α − H(x 2 ) n = F, Instability of capillary hypersurfaces With the preparations above, we can define a "test function" φ[ξ] = Y [ξ], N = −(1 + |x| 2 )ξ + 2 ξ, x x, N .(12) We mention that we will also use the following expression of φ[ξ] φ[ξ] = ξ, −(1 + |x| 2 )N + 2 x, N x .(13) Recall the second variational formula ∂ 2 E(φ) = − M Lφ · φda + ∂M (φ ν − qφ)φds,(14) where L = ∆ + |σ| 2 and q = 1 sin θ + cot θσ(ν, ν). Now we can prove the following lemmas. Lemma 5. ν is a principal direction for σ along ∂M . In particular, D ν N = −σ(ν, ν)ν. Proof. It suffices to prove that, for ∀X ∈ T p (∂M ), σ(ν, X) = 0. In fact, we have σ(ν, X) = D X ν, N = D X (cos θν + sin θN ), − sin θν + cos θN = D Xν ,N = −II(ν, X) = 0, where we used θ is constant,ν andN are unit vectors, and ∂B is totally umbilical. Thus we complete the proof of Lemma 5. Lemma 6. Along ∂M , there holds φ ν − qφ = 0.(15) Proof. First from (13) and Lemma 5 we have φ ν = ξ, −(1 + |x| 2 )N + 2 x, N x ν = ξ, −2 x, ν N + (1 + |x| 2 )σ(ν, ν)ν − 2 x, σ(ν, ν)ν x + 2 x, N ν = 2 ξ, − x, ν N + σ(ν, ν)(ν − x, ν x) + x, N ν , where in the third line we used |x| = 1 along ∂M . Next noticing that x =N = cos θN + sin θν, we get φ ν = 2 ξ, − sin θN + σ(ν, ν)(ν − sin θ(cos θN + sin θν)) + cos θν = 2 ξ, (σ(ν, ν) cos θ + 1)(cos θν − sin θN ) . On the other hand, there holds qφ = ( 1 sin θ + cot θσ(ν, ν)) ξ, −(1 + |x| 2 )N + 2 x, N x = ( 1 sin θ + cot θσ(ν, ν))2 ξ, −N + cos θ(cos θN + sin θν) = ( 1 sin θ + cot θσ(ν, ν))2 ξ, − sin 2 θN + cos θ sin θν = (1 + cos θσ(ν, ν))2 ξ, − sin θN + cos θν , where again in the second line we used |x| = 1 along ∂M . Hence we obtain φ ν − qφ = 0. The next lemma, which indicates the geometric meaning of Lemma 6, may have its own interest. Thus we also include it here. Lemma 7. Under the flow f t , there holds d dt t=0 θ(t) = φ ν − qφ.(16) In particular, since f t is conformal (angle preserving), φ ν − qφ = 0. Proof. Following [13], we denote by a "prime" the convariant derivative D dt t=0 . Also by the appendix of [13], we have ν = ( ∂φ ∂ν + σ(Y 0 , ν))N + φS 0 (ν) − φσ(ν, ν)ν − S 1 (Y 1 ) + cot θ∇φ, ν = −Π(Y,ν)N − S 2 (Y 1 ) + 1 sin θ∇ φ, where∇ denotes the gradient on ∂M , Y 0 (resp. Y 1 ) the tangent part of the variational vector field Y of M (resp. to ∂M ), S 0 the shape operator of M in R n+1 with respect to N , and S 1 (resp. S 2 ) the shape operator of ∂M in M (resp. ∂B) with respect to ν (resp.ν). Note that cos θ(t) = ν,ν , which implies − sin θ d dt t=0 θ(t) = ν ,ν + ν,ν . Taking into account thatν = − sin θN + cos θν, N = cos θN + sin θν, we have − sin θ d dt t=0 θ(t) = ( ∂φ ∂ν + σ(Y 0 , ν))N + φS 0 (ν) − φσ(ν, ν)ν, − sin θN + cos θν + ν, −Π(Y,ν)(cos θN + sin θν) = − sin θ( ∂φ ∂ν + σ(Y 0 , ν)) − sin θΠ(Y,ν), or d dt t=0 θ(t) = ∂φ ∂ν + σ(Y 0 , ν) + Π(Y,ν).(17) Again from the appendix of [13], there hold Y 0 = Y 1 − cot θφν, Y = Y 1 − 1 sin θ φν, σ(Y 1 , ν) + Π(Y 1 ,ν) = 0. Now plugging these equalities into (17), the lemma follows immediately. Lemma 8. Lφ = −2n ξ, N + Hx . Proof. The proof is a direct calculation using moving frame method. This method is very powerful in differential geometry. Take an orthonormal basis {e i , i = 1, · · · , n; e n+1 = N }. Then we have the structure equations: (see e.g. [4]) dx = n i=1 ω i e i , de i = n j=1 ω ij e j + n j=1 h ij ω j e n+1 , de n+1 = − n i,j=1 h ij ω i e j , where ω i is the dual forms and ω ij the connection forms. Thus there holds ∆φ = ∆ ξ, −(1 + |x| 2 )N + 2 x, N x = ξ, −∆((1 + |x| 2 )N ) + 2∆( x, N x) = ξ, −(∆|x| 2 · N + 2 n i=1 (|x| 2 ) ,i N ,i + (1 + |x| 2 )∆N ) + 2(∆ x, N · x + 2 n i=1 x, N ,i x ,i + x, N ∆x) .(19) Note that ∆|x| 2 = 2 x, ∆x + 2|∇x| 2 = 2nH x, N + 2n, n i=1 (|x| 2 ) ,i N ,i = −2 n i,j=1 x, e i h ij e j . And using Codazzi equation n i=1 h ij,i = n i=1 h ii,j = nH ,j = 0 we have ∆N = n i=1 N ,ii = n i,j=1 (−h ij e j ) ,i = − n i,j=1 h ij h ij N = −|σ| 2 N. Moreover we can get ∆ x, N = ∆x, N + 2 n i=1 x ,i , N ,i + x, ∆N = nHN, N + 2 n i,j=1 e i , −h ij e j + x, −|σ| 2 N = −nH − |σ| 2 x, N , n i=1 x, N ,i x ,i = n i,j=1 x, −h ij e j e i = n i,j=1 −h ij x, e j e i . Now substituting all these terms into (19) gives rise to ∆φ = ξ, −((2nH x, N + 2n) · N − 4 n i,j=1 x, e i h ij e j − (1 + |x| 2 )|σ| 2 N ) + 2((−nH − |σ| 2 x, N ) · x − 2 n i,j=1 h ij x, e j e i + x, N nHN ) = ξ, (−2n + (1 + |x| 2 )|σ| 2 )N − 2(nH + |σ| 2 x, N )x = ξ, −2n(N + Hx) − |σ| 2 φ. Therefore, Lφ = ∆φ + |σ| 2 φ = −2n ξ, N + Hx . Thus we obtain ∂ 2 E(φ) = −2n M ξ, N + Hx · ξ, (1 + |x| 2 )N − 2 x, N x da. To analyze ∂ 2 E(φ), we define a quadratic form Q(ξ 1 , ξ 2 ) = −2n M ξ 1 , N + Hx · ξ 2 , (1 + |x| 2 )N − 2 x, N x da,(20) for ∀ ξ 1 , ξ 2 ∈ S n . Denote by {∂ A } n+1 A=1 the standard coordinate vectors in R n+1 . Then we have the following lemma. Lemma 9. Q has the following properties. (1) Q is symmetric. Proof. (1) First we prove Q is symmetric. Note that in fact Q is defined as Q(ξ 1 , ξ 2 ) = − M L(φ[ξ 1 ]) · φ[ξ 2 ]da. Then Green's formula implies Q(ξ 1 , ξ 2 ) = − M φ[ξ 1 ] · L(φ[ξ 2 ])da + ∂M (φ[ξ 1 ](φ[ξ 2 ]) ν − (φ[ξ 1 ]) ν φ[ξ 2 ]) ds. But Lemma 6 yields (φ[ξ i ]) ν = qφ[ξ i ], i = 1, 2. So the boundary term vanishes and then Q(ξ 1 , ξ 2 ) = Q(ξ 2 , ξ 1 ). (2) Next we calculate trQ. trQ = n+1 A=1 Q(∂ A , ∂ A ) = −2n M n+1 A=1 ∂ A , N + Hx · ∂ A , (1 + |x| 2 )N − 2 x, N x da = −2n M N + Hx, (1 + |x| 2 )N − 2 x, N x da = −2n M H x, N (1 − |x| 2 ) + 1 + |x| 2 − 2 x, N 2 da ≤ −2n M (H x, N + 1)(1 − |x| 2 )da. Also we have ∆|x| 2 = 2n(H x, N + 1). Consequently, trQ ≤ − M ∆|x| 2 · (1 − |x| 2 )da = M ∇|x| 2 · ∇(1 − |x| 2 )da − ∂M ∂|x| 2 ∂ν (1 − |x| 2 )ds = − M |∇(|x| 2 )| 2 da ≤ 0, where we have used |x| = 1 on ∂M to remove the boundary term. And it is easy to see trQ = 0 if and only if |x| = const. So we complete the proof. Thus Q has at least one negative eigenvalue. But on the other hand, div R n+1 Y [ξ] = n+1 A=1 D ∂ A Y [ξ], ∂ A = 2(n + 1) ξ, x ,(21) which by integration implies M φda = M Y [ξ], N da = − T div R n+1 Y [ξ]dv + Ω Y [ξ],N da = −2(n + 1) T ξ, x dv.(22) So generally M φda = 0. That means φ[ξ] is not a test function. However, under the hypothesis of Theorem 1 that the mass center of T is at the origin, we have M φda = −2(n + 1) T ξ, x dv = 0 for ∀ ξ ∈ S n . So if we choose ξ as an eigenvector corresponding to the negative eigenvalue of Q, we have ∂ 2 E(φ[ξ]) = Q(ξ, ξ) < 0, which implies that M is unstable. This completes the proof of Theorem 1. Other applications and question In this section we shall give several applications of the above argument and propose a conjecture on the topic. 4.1. Another criteria for instability. The following proposition is an immediate result. Proposition 10. If the quadratic form Q has two negative eigenvalues, then M is unstable. Proof. Assume Q is diagonalized such that ξ 1 and ξ 2 are the eigenvectors corresponding to the two negative eigenvalues. Then for real numbers c 1 and c 2 with c 2 1 + c 2 2 = 0, Q(c 1 ξ 1 + c 2 ξ 2 , c 1 ξ 1 + c 2 ξ 2 ) = c 2 1 Q(ξ 1 , ξ 1 ) + c 2 2 Q(ξ 2 , ξ 2 ) < 0.(23) On the other hand, M φ[c 1 ξ 1 + c 2 ξ 2 ]da = −2(n + 1) T c 1 ξ 1 + c 2 ξ 2 , x dv = −2(n + 1)(c 1 T ξ 1 , x dv + c 2 T ξ 2 , x dv). So we can always choose proper c 1 and c 2 with c 2 1 + c 2 2 = 0 such that M φ[c 1 ξ 1 + c 2 ξ 2 ]da = 0. Then using φ[c 1 ξ 1 + c 2 ξ 2 ] as a test function, from (23) we know M is unstable. The significance of the above proposition is as follows. For a given concrete capillary hypersurface M in B n+1 , the quadratic form Q is computable in principle. Then if Q has two negative eigenvalues, we can assert its instability. Also from this proposition we know that for hyperplanes and spherical caps Q has exactly one negative eigenvalue. 4.2. The mass center of minimal submanifolds with free boundary. By free boundary we mean that M intersects ∂B n+1 orthogonally, that is, ν = x along ∂M . By analyzing the vector field Y [ξ], we have the following proposition. Proposition 11. The mass center of a minimal submanifold M k with free boundary in a Euclidean ball is at the origin. Proof. Along M k choose the orthonormal basis {e i , i = 1, · · · , k; e α , α = k + 1, · · · , n + 1} such that {e i , i = 1, · · · , k} ⊂ T M . Then we have This proposition shows that minimal submanifolds with free boundary have some symmetry. Comparing with it, we mention two other properties of M k : div M Y [ξ] T = div M (Y [ξ] − (1) The mass center of the boundary ∂M is at the origin (a simple argument). (2) The volume of M has a lower bound |M k | ≥ |B k |, where B k is a k-dimensional unit ball ( [3,7,14] There are some remarks on this conjecture. (1) For n ≥ 2, H = 0 and θ = π 2 , M must be totally geodesic [14]. (2) For n = 2 and θ = π 2 , M is a totally geodesic disk, a spherical cap or a surface of genus 1 with embedded boundary having at most two connected components [14]. (3) For n = 2 and H = 0, M is a totally geodesic disk or a surface of genus 1 with at most three connected boundary components [13]. Figure 1 . 1A Notations and definitions. Let x : M n → R n+1 be an orientable immersed hypersurface in the unit Euclidean ball B n+1 ⊂ R n+1 with x(int M ) ⊂ B n+1 and x(∂M ) ⊂ ∂B n+1 . Suppose Ω ⊂ ∂B n+1 such that ∂Ω = x(∂M ). And denote by T ⊂ B n+1 the part of ball satisfying ∂T = x(M ) ∪ Ω. Remark 1 . 1If x(M ) has self-intersections, T may be viewed as the finite union of some domains T i , i = 1, · · · , m, i.e. T = ∪ m i=1 T i . Here T i may intersect with each other. If there are not one choices for where the constant F is called the force of the curve Γ and it together with H will determine the curve as follows.(See Proposition 4.3 in [11]) Proposition 4. The curve Γ and the hypersurface M generated by Γ have the following several possible types. ( 3 ) 3If F = 0 and H = 0 then M is a sphere. (4) If H = 0 and F = 0 we obtain a catenary which generates an embedded catenoid M with F > 0 if the normal points down and F < 0 if the normal points up. ( 5 ) 5If H = 0 and F = 0 then Γ is a straight line orthogonal to the x 1 -axis which generates a hyperplane.(6) If M touches the x 1 -axis, then it must be a sphere or a hyperplane.(7)The curve Γ is determined, up to translation along the x 1 -axis, by the pair (H, F ). ∂ A , ∂ A ) ≤ 0 with equality if and only if |x| = const on M. Y [ξ], e α e α ) = 2k ξ, x + Y [ξ], k H = 2k ξ, x . have used the fact Y [ξ] is tangential to ∂B n+1 . ). 4.3. Stable immersed closed CMC hypersurfaces in R n+1 . At last we give a new proof of a theorem by Barbosa and do Carmo.Theorem 12 ( [1]). The only stable immersed closed hypersurface of constant mean curvature in R n+1 is the round sphere.Proof. By translation, assume the mass center of generalized body T enclosed by M is at the origin. So M φ[ξ]da = 0 for all ξ ∈ S n . If M is the round sphere, we are done. Otherwise |x| = const. So by Lemma 9 the quadratic form Q has a negative eigenvalue. Choosing ξ as an eigenvector corresponding to the negative eigenvalue, we have∂ 2 E(φ[ξ]) = − Lφ[ξ] · φ[ξ]da = Q(ξ, ξ) < 0,which shows that M is unstable.4.4.An open question. Since all the examples, i.e. the Delaunay type capillary hypersurfaces is known to be stable or unstable, we propose a conjecture as follows.Conjecture 13. The only stable capillary hypersurface M n (n ≥ 3) in a unit Euclidean ball B n+1 is a totally geodesic hyperplane or a spherical cap.M Stability of hypersurfaces with constant mean curvature. J L Barbosa, M Carmo, Math. Z. 1853J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean cur- vature, Math. Z. 185 (1984), no. 3, 339-353. Zerlegung konvexer Körper durch minimale Trennflächen. J Bokowski, E SpernerJr, J. Reine Angew. Math. 311J. Bokowski and E. Sperner Jr., Zerlegung konvexer Körper durch minimale Tren- nflächen, J. Reine Angew. Math. 311/312 (1979), 80-100. A sharp bound for the area of minimal surfaces in the unit ball. S Brendle, Geom. Funct. Anal. 223S. Brendle, A sharp bound for the area of minimal surfaces in the unit ball, Geom. Funct. Anal. 22 (2012), no. 3, 621-626. r-minimal submanifolds in space forms. L Cao, H Li, Ann. Global Anal. Geom. 324L. Cao and H. Li, r-minimal submanifolds in space forms, Ann. Global Anal. Geom. 32 (2007), no. 4, 311-341. Equilibrium Capillary Surfaces. R Finn, Grundlehren der Mathematischen Wissenschaften. New YorkSpringer-Verlag284R. Finn, Equilibrium Capillary Surfaces, Grundlehren der Mathematischen Wis- senschaften, vol. 284, Springer-Verlag, New York, 1986. Capillary surface interfaces. R Finn, Notices Amer. Math. Soc. 467R. Finn, Capillary surface interfaces, Notices Amer. Math. Soc. 46 (1999), no. 7, 770-781. The first Steklov eigenvalue, conformal geometry, and minimal surfaces. A Fraser, R Schoen, Adv. Math. 226A. Fraser and R. Schoen, The first Steklov eigenvalue, conformal geometry, and min- imal surfaces, Adv. Math. 226 (2011), 4011-4030. A Fraser, R Schoen, arXiv:1209.3789Sharp eigenvalue bounds and minimal surfaces in the ball. A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, arXiv: 1209.3789. A Fraser, R Schoen, arXiv:1304.0851Minimal surfaces and eigenvalue problems. A. Fraser and R. Schoen, Minimal surfaces and eigenvalue problems, arXiv: 1304.0851. Existence and regularity for the problem of a pendent liquid drop. E Gonzalez, U Massari, I Tamanini, Pacific J. Math. 882E. Gonzalez, U. Massari, and I. Tamanini, Existence and regularity for the problem of a pendent liquid drop, Pacific J. Math. 88 (1980), no. 2, 399-420. Proof of the double bubble conjecture. M Hutchings, F Morgan, M Ritoré, A Ros, Ann. of Math. 2M. Hutchings, F. Morgan, M. Ritoré, and A. Ros, Proof of the double bubble conjec- ture, Ann. of Math. (2) 155 (2002), no. 2, 459-489. Stability analysis of capillary surfaces with planar or spherical boundary in the absence of gravity. P I Marinov, The University of Toledo (Ohio)Ph.D. thesisP. I. Marinov, Stability analysis of capillary surfaces with planar or spherical boundary in the absence of gravity, Ph.D. thesis, The University of Toledo (Ohio), 2010. On stability of capillary surfaces in a ball. A Ros, R Souam, Pacific J. Math. 1782A. Ros and R. Souam, On stability of capillary surfaces in a ball, Pacific J. Math. 178 (1997), no. 2, 345-361. Stability for hypersurfaces of constant mean curvature with free boundary. A Ros, E Vergasta, Geom. Dedicata. 561A. Ros and E. Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), no. 1, 19-33. R Schoen, S.-T Yau, Lectures on Differential Geometry. International Press1R. Schoen and S.-T. Yau, Lectures on Differential Geometry, Conf. Proc. Lecture Notes Geom. Topol., vol. 1, International Press, 1994. Minimal varieties in riemannian manifolds. J Simons, Ann. of Math. 2J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968) 62-105.
[]
[ "The short-run impact of COVID-19 on the activity in the insurance industry in the Republic of North Macedonia", "The short-run impact of COVID-19 on the activity in the insurance industry in the Republic of North Macedonia" ]
[ "Viktor Stojkoski [email protected] \nFaculty of Economics\nUniversity Ss. Cyril and Methodius in Skopje and Macedonian Academy of Sciences and Arts\n\n", "Petar Jolakoski [email protected] \nFaculty of Economics, University Ss. Cyril and Methodius in Skopje\nFaculty of Economics, University Ss. Cyril and Methodius in Skopje\n\n", "Igor Ivanovski [email protected] ", "\nIntroduction\n\n" ]
[ "Faculty of Economics\nUniversity Ss. Cyril and Methodius in Skopje and Macedonian Academy of Sciences and Arts\n", "Faculty of Economics, University Ss. Cyril and Methodius in Skopje\nFaculty of Economics, University Ss. Cyril and Methodius in Skopje\n", "Introduction\n" ]
[]
The COVID-19 pandemic had a significant impact on the social and economic actions of every individual. As a consequence, it is expected this impact to transpose into the nature and methods of insuring risky ventures, and thus drastically change the business models of the insurance industry both on short and long run. Despite the abundance of predictions and potential implications, the literature lacks investigations which targets the short-run economic impact of the COVID-19 pandemic on the insurance industry. This paper contributes towards this literature, by performing a first of a kind analysis based on data of the activities in one developing country and insurance market -North Macedonia. By utilizing a seasonal autoregressive model and data on the gross claims paid (GCP) and gross written premiums (GWP) in 11 insurance classes, we evaluate the overall and class-specific impact of the pandemic on the insurance activities in the country. We find that, during the first half of 2020, the activity in GCP and GWP shrank by more than 10% to what was expected. The total loss in the industry amounted to approximately 8.2 million EUR which was, however, much less than the volume of reserves that the Insurance Supervision Agency made available as funds for the companies to deal with the potential crisis. In addition, the pandemic induced changes in the insurance activity structure -the share of motor vehicles class in the total industry activity fell at the expense of the property classes. Altogether, our results suggest that the insurance industry in North Macedonia was well prepared to tackle the consequences of the pandemic crisis. Nevertheless, the presence of automatic stabilizers had a major influence on weakening the overall impact of the pandemic.
10.1111/rmir.12187
[ "https://arxiv.org/pdf/2011.10826v1.pdf" ]
227,126,568
2011.10826
3fdf7ff2cd34921456392e6b1354d1fffb4b4079
The short-run impact of COVID-19 on the activity in the insurance industry in the Republic of North Macedonia 21 Nov 2020 November 24, 2020 Viktor Stojkoski [email protected] Faculty of Economics University Ss. Cyril and Methodius in Skopje and Macedonian Academy of Sciences and Arts Petar Jolakoski [email protected] Faculty of Economics, University Ss. Cyril and Methodius in Skopje Faculty of Economics, University Ss. Cyril and Methodius in Skopje Igor Ivanovski [email protected] Introduction The short-run impact of COVID-19 on the activity in the insurance industry in the Republic of North Macedonia 21 Nov 2020 November 24, 2020COVID-19insurance industrySARIMANorth Macedonia JEL codes: C22G22 The COVID-19 pandemic had a significant impact on the social and economic actions of every individual. As a consequence, it is expected this impact to transpose into the nature and methods of insuring risky ventures, and thus drastically change the business models of the insurance industry both on short and long run. Despite the abundance of predictions and potential implications, the literature lacks investigations which targets the short-run economic impact of the COVID-19 pandemic on the insurance industry. This paper contributes towards this literature, by performing a first of a kind analysis based on data of the activities in one developing country and insurance market -North Macedonia. By utilizing a seasonal autoregressive model and data on the gross claims paid (GCP) and gross written premiums (GWP) in 11 insurance classes, we evaluate the overall and class-specific impact of the pandemic on the insurance activities in the country. We find that, during the first half of 2020, the activity in GCP and GWP shrank by more than 10% to what was expected. The total loss in the industry amounted to approximately 8.2 million EUR which was, however, much less than the volume of reserves that the Insurance Supervision Agency made available as funds for the companies to deal with the potential crisis. In addition, the pandemic induced changes in the insurance activity structure -the share of motor vehicles class in the total industry activity fell at the expense of the property classes. Altogether, our results suggest that the insurance industry in North Macedonia was well prepared to tackle the consequences of the pandemic crisis. Nevertheless, the presence of automatic stabilizers had a major influence on weakening the overall impact of the pandemic. Introduction The COVID-19 outbreak led to an unprecedented abrupt economic shock to many developing countries, among which was the Republic of North Macedonia. In order to reduce the impact of the disease spread, the government of North Macedonia implemented social distancing restrictions such as closure of schools, airports, borders, restaurants and shopping malls. In the most severe cases there were even lockdowns -the citizens of certain municipalities were prohibited from leaving their homes. This subsequently led to a major economic downturn: stock markets plummeted, inter-national trade slowed down, businesses went bankrupt and people were left unemployed. The resulting pandemic and the government actions taken in response, notably altered the social and economic activities undertaken by the population (Stojkoski et al., 2020). There is no doubt this will lead to a drastic change in the way economies manage and distribute their risks. As such, it is expected to have a significant impact on the insurance industry, whose goal is to assure people's activities by developing and supplying products and services for absorption and transferring of risks. The impact of the crisis should translate in the structure of insurers' claims, assets and business flows. Even though, many of the pandemic's consequences will take time to materialize, and the ultimate effects will depend on the severity of the economic crisis, with preliminary studies suggesting that the insurance industry has remained strong, with insurers generally maintaining their solvency position. As elaborated in a report by Insurance-Europe (2020): "Insurers across Europe have maintained business continuity, including their ability to continue to serve customers effectively. Since the outset of the pandemic, many insurers have taken a very broad range of actions to help their clients. On a case-by-case basis, they have offered, where possible, temporary fee deferrals, fee waivers or even partial refunds of premium payments. Moreover, insurers have launched a broad range of voluntary goodwill actions to support citizens and businesses..." The general global losses in the industry are expected to amount to an estimated 84 billion EUR. Nevertheless, the magnitude of the impact will not be the same in every economy. This is a result of the large heterogeneity in the impact of the pandemic in the economies, as well as the differences in the measures introduced by the governments. In addition, it is also because of to the national features of insurance, that is, the extent to which insurers cover certain losses is dependent on national regulatory and supervisory regimes, besides being dependent on individual company circumstances. Our contribution Definitely, the impact of the pandemic should be studied from a country-specific perspective. To the best of our knowledge, such analyses are yet in their infancy, with so far being made only several contributions, among which (Richter and Wilson, 2020). In this aspect, here we aim to introduce a coherent statistical procedure of evaluating the short-run impact of the COVID-19 crisis on the activity in the insurance industry, and specifically in the Republic of North Macedonia. North Macedonia is a small Southeastern European developing economy. Such economies are characterized with developing insurance industries and unstable and yet upward trending dynamics of the gross written premiums (GWP) and gross claims paid (GCP) due to the previous lower development, increased liberalization and competition of the market and the constant restructuring of the activities within the insurance classes. GWP are the total revenue from a contract expected to be received by an insurer before deductions for reinsurance or ceding commissions, whereas GCP are all demands made by the insured, for payment of the benefits provided by the insurance contract or for coverage of an incurred loss. Hence, by construction, these two observables are enough to quantify the activity on both the demand and supply side of the insurance industry, i.e., the former represents the demand side, and the latter, implicitly describes the supply side of the industry. Our paper answers whether, indeed, GWP and GCP decreased (or increased) more than what was expected and to what extent. The paper also clarifies the magnitude of the effect -what were the total losses in the industry? It also addresses the class-specific impact. Precisely, it answers how the gross insurance premiums and gross claims payments in certain classes decreased more than others, and how they affected the overall structure of the insurance industry activities. To answer these questions, one would have to compare the expected number and volume of claims and premiums to the one realized in the first and second quarter of 2020. While the realized claims and premiums are easily obtainable from the reports of the national Insurance Supervision Agency (ISA), calculating the expected amount might represent an exhausting task. In particular, the expected amount of GWP and GCP is premised on statistical predictions that are made via distinct mathematical models. The choice of model ultimately determines the computational cost and the predictive power -models that are more computationally expensive also have greater predictive power and vice versa. To obtain reliable and inexpensive predictions for the expected value one usually has to make a trade-off and chooses the model which has the best cost-benefit characteristics. As an example, here we consider the Seasonal Auto-regressive Integrated Moving Average model (SARIMA). The advantage of modeling through SARIMA is that it specifies that the output variable depends linearly on its own previous values and on a stochastic term, thus using the best fit recurrence relation to predict future values (Makridakis and Hibon, 1997). In addition, it is able to capture the possible seasonal pattern which has been observed in the dynamics of GCP and GWP (Ulyah, Mardianto et al., 2019). By utilizing quarterly data, spanning from the second quarter of 2012 up until the last quarter of 2019, we fit a separate SARIMA model for the realized gross insurance premiums and claims for 11 insurance classes. We use the results to construct out of sample forecast for the first and second quarter of 2020 and take the difference of our predictions with the realized values as our measure for the impact of COVID-19. By performing an analysis of our predictions with respect to the realized values, we report that the activity in GCP and GWP decreased by more than 10 percentages during the first half of 2020, compared to what was expected. These values are much larger when compared with the activity in the same quarter in the previous year, when the changes were between 0 and 4.5 percentages. In nominal terms, our predictions suggest the nominal effect of the crisis is around 8.2 million EUR. In comparison, the total non-life and life GWP activity in the first half of 2020 amounted to 172 million EUR. Therefore, on the first sight it might appear that the pandemic had an enormous impact on the activity in the insurance industry. Nevertheless, in contrast to the actions taken by the ISA in order to reduce the impact of the crisis, which were in a value of around 30 million EUR, it seems that the impact was much smaller than what was initially estimated 1 . Besides the overall impact, the COVID-19 pandemic induced a drastic change in the structure of GCP. In particular, we provide evidence that during the first two quarters of 2020 the share of the Property, other and Property, fire and nat. forces classes increased, while at the same time the share of Motor vehicles (Casco) and MTPL classes decreased. We argue that the changes in the structure were a consequence of an automatic stabilizer effect in the risky social and economic undertakings of the population due to the introduced social distancing measures, even though the MTPL is mandatory and administratively organized. These conclusions, which can not be inferred from a simple descriptive analysis based solely on comparison with past data, may be more consistent with the "perceived" economic situation in some countries than otherwise suggested. The rest of the paper is organized as follows. In Section 2 we conduct a thorough review of the literature describing the factors which determine the insurance claims and premiums and discuss their relation with the dynamics of the COVID-19 pandemic. Next, in Section 3 we describe in detail the SARIMA model and the data used to perform the analysis. Section 4 is constituted of three parts. In the first part we provide a descriptive analysis for the dynamics of the gross insurance claims and premiums in the Republic of North Macedonia during the pandemic. We continue by presenting our ARIMA results and quantifying the impact of the COVID-19 crisis. In the last part of this section, we present the implications created by our results. Finally, in Section 5 we summarize our findings and offer directions for future research. 1 The description of the measures introduced by ISA can be found at http://aso.mk/en/new-set-of-measures-for-insurance-undertakings-to-facilitate-work-in-emergency-conditions/ Literature Review The primary contribution of this paper is empirical. It provides new information on the dynamics of the two main insurance activities in North Macedonia following the COVID-19 outbreak. It also enables identifying the differential impact on insurance classes. As such, it joins the growing literature on financial markets impact of an epidemic, focusing mainly on the risky undertakings and the insurance reactions of firms to an epidemic, see for example (Acharya and Steffen, 2020;McAleer, 2020;Richter and Wilson, 2020). The general consensus of this literature is that epidemics can obstruct social and economic welfare by changing expectations about how the economy will function and by deterring investment and tourism. In many situations, the immediate costs of an epidemic are apparent, while the long-term costs are unclear. In any case, it is apparent that epidemics are extremely costly. As noted in Bloom, Canning et al. (2004), preventing epidemics requires overcoming a range of obstacles, as does responding to an epidemic once it begins. Our contribution also joins the growing literature on how the COVID-19 outbreak specifically contributed to the changes in the activity of the insurance industry Mansour et al. (2020); Babuna et al. (2020); Acs and Karpman (2020); Richter and Wilson (2020). Various approaches have been implemented to investigate the impact of COVID-19 on the insurance industry. For instance, Mansour et al. (2020) used the patient perspective to propose improvements in the coverage of the health insurance class. Babuna et al. (2020) conducted interviews with representatives from the insurance industry in Ghana and found out that there is a trend of decreasing profits but increasing claims. Richter and Wilson (2020) developed a scenario analysis in which they evaluate and summarize the lessons learned from the pandemic crisis by baselining actual developments against a reasonable, pre-COVID-19 scenario. Their results support the hypothesis that financial market developments dominate claims losses due to the demographics of pandemics and other factors. Nevertheless, this literature is still in its infancy and, we believe that, our paper can be a stepping stone towards a more data-driven understanding of the COVID-19 impact on the industry. From a methodological perspective the paper contributes to the time-series properties of the insurance activities dynamics, Clinebell, Kahl and Stevens (1994); Thomann (2013) Diao and Wang (2019). This paper presents a way to use a reliable existing data source that is published on a quarterly basis, the ISA activity reports, to study short-run insurance dynamics. This would allow to easily expand the analysis done here in the future. We also hope that this work will stimulate similar work in other countries with similar data sources. Materials and methods Data To empirically evaluate the impact of the COVID-19 pandemic on the insurance industry in the Republic of North Macedonia, we use quarterly data for the gross insurance premiums and claims per insurance class. We focus on the period from the second quarter of 2012, up until the last quarter of 2019. This is the optimal period which allows us to utilize the largest amount of data: including longer periods will result in a smaller longitudinal dataset. During this period, there are a total of 11 insurance classes for which there is no missing data and are included in our analysis. Ten of them belong to the non-life insurance type: Accidents; Health; Motor vehicles (casco); Cargo, Property; fire and nat. forces; Property, other; MTPL (total); General liability; Financial losses and Tourists assistance; whereas one of them belongs to the life insurance type -the Life assurance. The life assurance class incorporates insurances with respect to life, death insurance, mixed life insurance, rent insurance and unit-linked life insurance. From the analysis we exclude the classes for which there are gaps in the reported data and for which the activity is negligible (i.e. most of the time there were either no claims paid or written premiums). Overall, these are a total of 11 classes of insurance. In total the excluded classes account for less than 1% in the activity of GCP and GWP in 2020, and therefore we believe that their exclusion will not significantly impact the interpretation of our findings. The data for the gross insurance premiums and claims were collected from ISA's preliminary reports. Altogether we end up with 33 observations for each class: 31 of which span the estimation period, while the remaining two quarters represent the basis for quantifying the current impact of the crisis. All data was collected on 28th September 2020 from https://aso.mk/en/preliminary-data/. The cleared and processed data used in our analysis is available at github.com/pero-jolak/insurance-activity-mkd. Table 1 gives the summary statistics for each of the insurance classes. During the studied period the MTPL class was the one with most gross claims paid followed by the Motor vehicles (casco) class and the Accident class, whereas the Financial losses class has the lowest number of gross claims paid. Average gross written premiums is also highest for the MTPL (total) class, followed by the Property, other class. Cargo and Financial losses are the classes with the smallest average gross written premiums. 2020, whereas the red lines emphasize the volumes in 2020. For each class and type of activity, we observe that there is a significant drop in the first quarter of 2020, followed by either a stationary or a slight upward trend. Altogether, this indicates that the COVID-19 pandemic had a negative impact on the activity in the insurance industry in North Macedonia. Econometric model The simplest, yet effective model for studying seasonal changes is the seasonal autoregressive integrated moving average model (SARIMA). Under this model, a stationary transformation of the data under study, let's name it x i,t , where i is a nomenclature for the insurance classes and t is the period of observation, is dependent on its past values, there is a seasonal (periodic) effect and it is determined by the amount of previous shocks in the class 2 . Formally, the mathematical specification of an econometric model described solely by an autoregressive and moving average terms is x i,t = µ + φ 1 x i,t−1 + θ 1 ǫ i,t−1 + ... + φ p y i,t−p + θ q ǫ i,t−q + ǫ i,t ,(3.1) where the φî are the parameters of the marginal effect of the t −î-th period on the current value of the dependent variable, and θî are the respective marginal effects of the previous errors (shocks) ǫ i,t−î . The AR term (p) of these models indicates that the evolving variable of interest is modeled as a function of its own prior values. In other words, we have a linear regression setting where the model predicts the next data point by looking at previous data points. The MA (for "moving average") term (q) of the model represents the regression error as a linear combination of the lagged Figure 1: Dynamics of GCP. Note: On the x-axis is time, measured in quarters, whereas on the y-axis is the value of the gross claims paid in 000 MKD (61.5 MKD = 1 EUR). The red lines correspond to values in 2020, whereas the black lines are values before 2020. error terms. Finally, I or "integrated" refers to the order of differencing that renders the initial time-series data stationary. In order to capture the multiplicative effect of seasonal cycles in the data we introduce the SARIMA model. This model is often represented as SARIMA (p, d, q) × (P, D, Q) s where the lowercase letters refer to the specification of simple ARIMA model, while the uppercase letters refer to the specification of the seasonal component. Finally, the subscript s indicates the periodicity of the seasons (in our case 4, for quarterly data). Generically, the data process is written as: φ p (L)φ P (L s )x i,t = A(t) + θ q (L)θ Q (L s )ǫ t ,(3.2) where φ p (L) andφ p (L s ) are respectively, the non-seasonal and seasonal autoregressive lag polynomials, and A(t) is the intercept. Moreover, the Θ q (L) andΘ Q (L s ) are respectively, the non-seasonal Note: On the x-axis is time, measured in quarters, whereas on the y-axis is the value of the gross written premiums in 000 MKD (61.5 MKD = 1 EUR). The red lines correspond to values in 2020, whereas the black lines are values before 2020. and seasonal moving average lag polynomial. To present the model construction of SARIMA, let us assume that we have selected a SARIMA (2, 1, 0) × (1, 1, 0) 4 model. After a careful manipulation of the non-seasonal and seasonal lag polynomials we arrive at the following model representation: x i,t = µ + φ 1 x i,t−1 + φ 2 x i,t−2 +φ 1 x i,t−4 − φ 1φ1 x i,t−5 − φ 2φ1 x i,t−6 + ǫ t . (3.3) Note that the stationary transformation of the variable x i,t represents taking first difference (d = 1) and then taking the fourth difference (D = 4). The final model effectively represents an ARIMA model with autoregressive multiplicative effects by taking combinations of the underlying seasonal and non-seasonal parameters. Model selection and diagnostics The optimal fit of a particular SARIMA model can be done easily by performing a numerical optimization. The choice of the seasonal, autoregressive and moving average parameters, however, is a far more important task. This is because, in general, it is them which determine the performance of the model. A standard way for choosing models is by using an information criterion estimator. A such estimator evaluates the relationship between the goodness of fit of the model and its complexity in terms of the number of explanatory variables. Here, we opt to use the Akaike information criterion (AIC), AIC is founded in information theory, and its inference is done by comparing a given a set of candidate models for the data. Always, the preferred model is the one with the minimum AIC value. The inference is done by rewarding goodness of fit, but also including a penalty that is an increasing function of the number of estimated parameters. The penalty discourages overfitting, which is desired because increasing the number of parameters in the model almost always improves the goodness of the fit. The implementation of AIC allows us to choose a total of 22 distinct SARIMA models, each being able to adequately predict the future GCP and GWP values of a particular insurance class. In order to produce consistent predictions, besides providing an adequate fit, the model should also satisfy two assumptions: i) homoscedasticity and ii) absence of autocorrelation between the residuals. To investigate whether our models satisfy these assumptions we conduct two statistical tests. For the first test, we estimate the ARCH LM statistic of each model. The ARCH LM test is the standard approach for detecting whether the model satisfies autoregressive conditional homoscedasticity (Engle, 1982). With the second test, we infer the Ljung-Box statistic (Ljung and Box, 1978). This statistic shows whether there is any autocorrelation between the residuals of every model. Under the null hypothesis, both tests assume that the assumptions are satisfied, and thus the models can be used for predictive purposes. Results Descriptive analysis We begin the analysis by performing a simple comparative analysis for the dynamics of GCP and GWP in the two quarters of 2020. For this purpose, we conduct two distinct evaluations. In particular, first we investigate the differences between the realized values in the first and second quarter of 2020 with the with the ones observed in the previous quarter (last quarter of 2019 and first quarter of 2020, respectively). The quantity which we formally estimate is the single period growth rate of the value of GCP and GWP. Formally, this is given as r i,t−1 = 100 × y i,t − y i,t−1 y i,t−1 , (4.1) where y i (t) is the value of either GCP or GWP of class i in time t, measured in thousands of MKD (61.5 MKD = 1 EUR). This comparison allows us to examine the trend patterns in the time-series and whether they drastically changed during the pandemic. Second, we examine the seasonal patterns during 2020 by comparing the realized values in the two quarters of 2020 with the corresponding quarters of 2019, i.e., r i,t−4 = 100 × y i,t − y i,t−4 y i (t − 4) . (4.2) Table 2 gives the results for the first quarter of 2020. They reveal that the, for the descriptive dynamics of GCP, in the first quarter there was a gain of 0.11 percentages compared to the same value in the same quarter of the previous year (column base p.y.). Out of the classes, the largest increase was in the Financial losses which exhibited a growth of more than 50000 percentages, followed by the Property, fire and nat. forces class (92.74%), whereas the GCP in the Cargo class decreased the most during this period (−94.69%). Interestingly, when compared to the previous quarter (column p.q.), only the Financial losses category exhibited, growth whereas every other class had a significant decline in the GCP. The total decline in the activity was −76.35 percentages. We believe that this is majorly a consequence of the seasonal patterns present in the GCP dynamics. Identical conclusions hold for the changes in GWP during the first quarter of 2020. Concretely, the total change in GWP compared to the same values in the first quarter of 2019 is 4.42%. The Property, other class showed the largest growth during this period (32.51%), followed by the Financial losses class (23.73%), and the Tourists assistance class was the one with the smallest growth (−14.76%). Every class showed decline in GWP when compared to the previous quarter. Table 2 complements the previous table by While these results offer a simple depiction for the activity in the insurance industry during the initial COVID-19 pandemic crisis, they fail to highlight the differences between the expected and realized activity in the insurance industry due to three reasons. In particular, the health crisis began at the end of the first quarter, thus the total of the crisis in this period can be only partially evaluate by simply studying the relative changes. Second, even though there is an evident seasonal pattern in the dynamics of most classes, a large amount of them also display a general trend of growth in both GCP and GWP, i.e., the observed changes by the descriptive analysis may be simply a result of this trend. Finally, there is no doubt that the COVID-19 pandemic is reshaping and redefining the social activities within an economy. In this aspect, it is obvious that an analysis that does not capture the expected dynamics and is instead strictly rooted in naive predictions will fail to be useful as a coherent tool for developing economic policies. In spite of all of its shortcomings and due to their simple interpretation, the findings from the analysis in this part can be still used as a baseline, though with a dose of caution, for identifying the implications created by the SARIMA analysis performed in the following part. Measuring the COVID-19 pandemic impact As described in Section 3, we use the SARIMA model to measure the COVID-19 crisis impact on the activity in the insurance industry. This is done by fitting a separate model to each class and type of activity, based on the data from the 2nd quarter of 2012 up until the last quarter of 2019. Appendix A provides tables with detailed information regarding the choice and performance of the models in fitting our studied data. Each model satisfies the homoscedasticity and no autocorrelation assumptions. We utilize the models to predict the activity in the 1st and 2nd quarter of 2020 and use the expected value of the prediction y p i (t) as our assessment for the expected behavior of the activity in the insurance industry. Subsequently, we use this expected value to estimate the relative percentage error, defined as r p i (t) = 100 × y p i (t) − y i (t) y i (t) . (4.3) This is our measure for the COVID-19 pandemic impact. Table 4 displays the results for each class divided per quarter and per type of activity. The total relative error in the GCP activity was 5.29% in the first quarter of 2020 and 11.69%. The positive sign implies that, in fact, during the first half of 2020 the GCPs decreased more than expected. The decrease was of similar magnitude in both quarters. Nevertheless, one has to keep in mind that the COVID-19 pandemic began at the end of the first quarter, and hence, only a part of the differences should be attributed to it. In order to investigate the drivers of the differences between the expected and observed GCP activity we inspect the class specific behavior. In this case, in the first quarter we observe large disparities in the behavior: there were 4 classes which experienced activity greater than expected GCP and 7 classes whose GCP activity decreased. Remark that a negative sign in the GCP relative error implies that the expected GCP were smaller than the realized ones. The class whose expected activity decreased mostly than expected was Cargo, whereas the Financial losses was the one with the largest increase in the activity. In the second quarter the same classes show increase in the activity that is bigger than what was expected, and the Financial losses class was again the one with the largest increase. In this quarter, it is also the Cargo class which experienced the largest discrepancies in the GCP activity. Regarding the GWP activity, the total relative error in the first quarter is 8.12% and in the second quarter it is 11.32%. In the case of GWP, the positive value of the relative error indicates that there was less activity than what was expected. The magnitude of the relative error is significant, therefore suggesting that the COVID-19 pandemic had a large impact on the reduction in the overall activity of GWP in the economy of North Macedonia. When we look at the intra-class errors, we find that even much larger discrepancies. Specifically, in the first quarter it was the Financial losses class whose activity decreased significantly, whereas in the second quarter it is the Tourists assistance class which decreased mostly. On the other hand, it was the Property, other class and the Financial losses class whose GWP was much more than expected in the first, and second quarter respectively. In Appendix B we show the confidence intervals of our predictions. They appear tight, and thus we can infer that our results are robust. To summarize, these findings allow us to conclude that, overall, the COVID-19 pandemic might have had a serious impact on the activities in the insurance industry in North Macedonia. In absolute terms, there were 0.8 million EUR less paid in GCP during the first quarter of 2020 and 3.4 million EUR less in the second quarter of 2020. In the same fashion, there were a total of 3.3 million EUR less spent in purchases of GWP in the first quarter of 2020, and 9.2 million EUR less in the second quarter of 2020. Altogether, this implies that the loss in the insurance industry can be quantified as a total of 8.2 million EUR. In comparison, ISA produced a set of measures in which it included a release 30 million EUR in liquid assets whose main purpose is to strengthen the support towards the companies in the segment of managing the negative consequences of the COVID-19 crisis. On the one hand, might suggest that the potential impact of the crisis was less than it was actually estimated by ISA. On the other hand, it implies that the insurance industry in North Macedonia was well prepared for the emergence of a such crisis. Implications While the COVID-19 crisis had a major impact on the overall activity in the insurance industry, it also induced a significant impact in the class-specific behavior. An exact interpretation of our results requires a detailed investigation of the structure of GCP and GWP in 2020. As a means to provide a such analysis, we compare the expected share of each insurance class in the total expected activity of GCP and GWP with respect to the realized one. The results are presented in Figure 3 Evidently, there were several major changes in the structure of the GCP. During the first half of 2020, it was the MTPL and Motor vehicles classes whose share decreased the most. In particular, we discover that in the first quarter of 2020 the share of the these two classes in the total GCP was around 2 percentage points less. At the same time, it was the Life assurance class whose share increased most, by 9 percentage points more than expected in the first quarter, by 3 percentage points. In the second quarter, we observe an even larger decrease in the MTPL (total) class in the first quarter (6 percentage points), and a large increase in the Property, other class (4 percentage points). In general, it is predicated that the insurance activity in a developing economy, such as the one of North Macedonia, follows an easily predictable pattern (Arena, 2008). Hence, it seems that during the 2020 pandemic crisis there was a drastic change in the structure of the GCP. But why was this the case? The pandemic forced drastic changes in the social and economic activities of the whole population. In particular, there were the government imposed social distancing measures such as movement and travel restrictions, people were instructed to work from home and closure of public places. Moreover, most people were cautious and further decreased the social activities by self-imposing additional distancing measures. In this aspect, it is obvious that vehicle insurance is usually purchased for travels, and this activity was significantly reduced especially during the second quarter of 2020. At Health and MTPL classes display a decrease. As a result, we can conjecture that the magnitude of the impact of the COVID-19 pandemic was reduced simply because of automatic stabilizer effects that offset the fluctuations in the insurance activity through their normal operation without additional timely authorization by the government or policymakers. Conclusion We investigated the impact of the COVID-19 pandemic crisis on the activity in the insurance industry in North Macedonia. By including quarterly data on gross claims paid and gross written premiums for 11 classes of insurance, we introduced predictions for the expected behavior of the activity in GCP and GWP in the absence of the pandemic. We measured the impact of the crisis as the difference between our predictions and the realized values in the first and second quarter of 2020. Similarly to previous works dealing with the dynamics of the insurance activity within an economy, we described the dynamics of the activities of each class as a seasonal autoregressive model. This resulted in a total of 22 SARIMA models, each predicting the gross claims paid of the gross premiums purchased of a particular class. Our analysis showed that, the GCP activity decreased by 5.29 percentages in the first quarter and by 11.69 percentages in the second quarter, compared to what was expected. At the same time, the GWP activity fell by 8.12% in the first quarter and by 11.32% in the second quarter. In comparison to the behavior with the same quarter in the previous year, we observed notable differences. This led us to believe that the COVID-19 pandemic had a significant impact on the insurance activity in North Macedonia. Importantly, we estimated that in nominal value the loss in the 11 studied insurance classes amounted to approximately 8.2 million EUR. This is less than one third of the reserves which ISA made available to the insurance companies in order to increase their liquidity during the crisis. Thus, we argued that the insurance industry was ready to intervene in case of an even larger crisis. Next, we investigated the structural changes in the activity of the insurance industry during the crisis. We found that the pandemic induced drastic changes in the structure of GCPs and GWPs. In particular, the share of the Property classes in the total activity increased, at the expense of the shares of the vehicle insurance classes. We argued that this was a result of the automatic stabilizers inducing changes due to the introduced social distancing measures. In this aspect, it remains an unresolved question why the automatic stabilizers had a such big effect on the impact of the crisis. We believe that the answer to this question lies in the insurance protection gap in the country (Richter and Wilson, 2020). Formally, as defined in Schanz (2018), the insurance protection gap is the difference between the amount of insurance that is economically beneficial and the amount of coverage actually purchased. It is believed that in developing and emerging countries, such as North Macedonia, this gap is largest because combined insurance premiums still fall significantly short of these countries' and regions' share in global GDP. A larger protection gap makes the structure of the industry more susceptible to shocks in case of extreme events. In North Macedonia, the exact magnitude of the protection gap is still unknown. A more detailed analysis is certainly needed to uncover the effect of the protection gap on the activity in the insurance industry in North Macedonia, especially in the times of the COVID-19 pandemic and its aftermath. This, however, requires a highly accurate and disaggregated data for the affordability of the insurance policies, their quality and the cultural and social character of the insurer behavior. In regard to this, building an explanatory model for the underlying protection gap would bring novel insights about the relations between the insurance classes in the GCP and GWP and, importantly, about the policies required for fast and stable recovery of the insurance industry from the ongoing crisis. This is a subject of our current work and it aims to offer new recommendations for promoting growth and development of the insurance market. Notes: The lower and upper columns show the 95% bounds of the confidence intervals, the expected column is our prediction, the real column is the observed value of the class specific activity and the diff column calculates the differences between the realized and expected value. The values are in in 000 MKD (61.5 MKD = 1 EUR). Notes: The lower and upper columns show the 95% bounds of the confidence intervals, the expected column is our prediction, the real column is the observed value of the class specific activity and the diff column calculates the differences between the realized and expected value. The values are in in 000 MKD (61.5 MKD = 1 EUR). Notes: The lower and upper columns show the 95% bounds of the confidence intervals, the expected column is our prediction, the real column is the observed value of tha class specific activity and the diff column calculates the differences between the realized and expected value. The values are in in 000 MKD (61.5 MKD = 1 EUR). Notes: The lower and upper columns show the 95% bounds of the confidence intervals, the expected column is our prediction, the real column is the observed value of tha class specific activity and the diff column calculates the differences between the realized and expected value. The values are in in 000 MKD (61.5 MKD = 1 EUR). ; Arena (2008); Ulyah, Mardianto et al. (2019); Kumar et al. (2020); Cummins and Griepentrog (1985); Mohammadi and Rich (2013); Andrews et al. (2013). This literature suggests that GWP and GCP are usually very complex and exhibits seasonal patterns Pesantez-Narvaez, Guillen and Alcañiz (2019); Choi and Varian (2009). Moreover, usually such research on GWP and GCP relies on large scale class specific data, which cannot be integrated together Hong and Martin (2017); Yang, Qian and Zou (2018); Guelman (2012); Figure 1 1displays the dynamics of the values for the gross claims paid for each class. Every class, except Health and financial losses exhibits a seasonal pattern, thus suggesting the appropriateness of using a seasonal model. The Health and financial losses classes, on the other hand have an upward trend pattern with seasonal adjustments. Hence, under a suitable stationary transformation a seasonal econometric model can be also implemented to them. Similarly, Figure 2 displays the dynamics of the values for the gross written premiums for each class. The same dynamical pattern for each class as in the gross claims values appear again, hence a seasonal model can be also implemented for this insurance activity. In the figures, the black lines display the dynamics before Figure 2 : 2Dynamics of GWP. displaying the changes in the activity in the insurance industry in the second quarter of 2020. During this quarter there are more drastic changes. In total, the GCP declined by −6.89% compared to its value in the same quarter in 2019. As in the results for the first quarter, the Financial losses were the class with the largest increase, and the Cargo class had the largest decrease. Differently, from the first quarter of 2019, though, in this case every class exhibited growth in GCP when compared to the previous quarter, and the overall growth in the value of the GCPs was 90.36%. When examining the descriptive dynamics of GWP we observe the nearly same behavior -the overall decline in GWPs compared to the same quarter in the previous year was −2.51 percentages and there was an increase of almost 100 percentages compared the first quarter of 2020. Figure 3 : 3the same time, the social distancing measures increased the risky ventures underdone in the home of the insured person. Hence, there has been an increase in GCP share of the Property classes.Similar changes appear in the structure of GWP. That is, we observe extreme increase in the amount of premiums paid in the Property, other class in the second quarter of 2020, whereas the Expected and realized share of the classes in the total insurance activity . Table 1 : 1Summary StatisticsNote: The data is in 000 MKD. 61.5 MKD = 1 EUR. Source: Own calculations using data from ISA.GCP GWP Table 2 : 2Percentage changes in the activity in the insurance industry in the 1st quarter of 2020 Source: Own calculations using data from ISA.GCP GWP Insurance class base p.y. base p.q. base p.y. base p.q. MTPL (total) -2.47 -76.38 0.16 -79.67 Financial losses 52183.33 509.13 23.73 -88.16 Property, fire and nat.forces 92.74 -88.05 9.16 -66.68 Property, other 3.91 -79.18 32.51 -70.68 Cargo -94.69 -98.67 5.79 -68.22 Motor vehicles (casco) -16.69 -78.8 -2.51 -76.33 Accident 1.51 -74.26 5.97 -66.16 General liability -58.04 -82.77 -3.55 -69.67 Tourists assistance -11.83 -77.87 -14.76 -84.92 Health 36.66 -68.93 13.48 -40.00 Life assurance 26.85 -67.55 -1.66 -78.23 TOTAL 0.11 -76.35 4.42 -75.54 Table 3 : 3Percentage changes in the activity in the insurance industry in the 2nd quarter of 2020 Source: Own calculations using data from ISA.GCP GWP Insurance class base p.y. base p.q. base p.y. base p.q. MTPL (total) -15.16 76.76 -10.91 103.11 Financial losses 1503.28 118.23 254.69 474.03 Property, fire and nat.forces 11.82 325.19 20.65 94.47 Property, other 24.54 145.36 20.33 133.64 Cargo -68.68 971.43 7.67 102.95 Motor vehicles (casco) -13.75 102.67 -4.96 103.85 Accident -12.96 73.30 -3.35 68.53 General liability -58.51 24.95 -4.73 80.16 Tourists assistance -4.26 76.44 -59.2 20.99 Health 57.10 116.18 21.99 48.48 Life assurance 19.69 75.95 -8.78 96.18 TOTAL -6.89 90.36 -2.51 98.74 Table 4 : 4Relative error in predicted changes in the activity in the insurance industry in 2020.GCP GWP Insurance Class 1Q 2Q 1Q 2Q MTPL (total) 9.09 24.90 5.56 18.46 Financial losses -97.74 -95.42 164.36 -31.70 Property, fire and nat.forces -29.11 -7.39 16.18 4.16 Property, other -33.20 -37.86 -26.95 -22.07 Cargo 6136.45 358.77 -3.61 -6.03 Motor vehicles (casco) 24.76 20.19 6.56 8.31 Accident 3.64 17.41 -4.43 3.50 General liability 318.55 202.35 14.91 14.56 Tourists assistance 23.19 10.96 21.68 155.80 Health 22.84 20.07 103.90 66.44 Life assurance -22.27 -19.51 12.10 18.65 TOTAL 5.29 11.69 8.12 11.32 Source: Own calculations using data from ISA. Table A1 : A1Selected GCP models and diagnostics.Table A2: Selected GWP models and diagnostics. Property, fire and nat.forces SARIMA (1, 0, 0)x(0, 1, 0, 4) -85.39Insurance class Model AIC ARCH-LM (p-value) Ljung-Box (p-value) MTPL (total) SARIMA (1, 0, 0)x(2, 1, 0, 4) -100.29 0.62 0.04 Financial losses SARIMA (1, 1, 3)x(0, 0, 0, 0) 130.38 0.67 0.13 Property, fire and nat.forces SARIMA (3, 1, 3)x(0, 0, 0, 0) 54.06 0.24 0.69 Property, other SARIMA (1, 0, 0)x(2, 1, 0, 4) 2.93 0.99 0.27 Cargo SARIMA (1, 1, 4)x(0, 0, 0, 0) 95.06 0.44 0.31 Motor vehicles (casco) SARIMA (1, 0, 0)x(0, 1, 0, 4) -99.72 0.34 0.96 Accident SARIMA (1, 0, 0)x(0, 1, 0, 4) -80.25 0.85 1.00 General liability SARIMA (1, 0, 0)x(0, 0, 0, 0) 84.81 0.76 0.86 Tourists assistance SARIMA (1, 0, 0)x(2, 1, 0, 4) -25.68 0.78 0.38 Health SARIMA (3, 1, 3)x(0, 0, 0, 0) 48.67 0.04 0.94 Life assurance SARIMA (1, 0, 0)x(2, 1, 0, 4) -19.31 0.85 0.84 Insurance class Model AIC ARCH-LM (p-value) Ljung-Box (p-value) MTPL (total) SARIMA (0, 1, 2)x(2, 1, 0, 4) -180.22 0.56 0.05 Financial losses SARIMA (3, 1, 3)x(0, 0, 0, 0) 49.26 0.39 0.05 0.84 0.38 Property, other SARIMA (0, 1, 2)x(2, 1, 0, 4) -20.02 0.48 0.91 Cargo SARIMA (1, 0, 0)x(0, 1, 0, 4) -43.35 0.33 0.52 Motor vehicles (casco) SARIMA (1, 0, 0)x(0, 1, 0, 4) -125.19 0.54 0.89 Accident SARIMA (2, 0, 0)x(1, 1, 0, 4) -105.34 0.93 0.20 General liability SARIMA (0, 1, 2)x(2, 1, 0, 4) -102.45 0.19 0.96 Tourists assistance SARIMA (0, 1, 2)x(2, 1, 0, 4) -78.39 0.83 0.98 Health SARIMA (1, 1, 1)x(0, 0, 0, 0) 67.20 0.43 0.70 Life assurance SARIMA (0, 1, 2)x(2, 1, 0, 4) -125.70 0.51 0.58 A SARIMA model Table A1 and A1A2 show the 22 SARIMA models for each class and the two types of activities that had the best performance under the AIC criterion.B Confidence intervals of the predicted valuesTables B1, B2 B3 and B4 show the upper and lower confidence interval bounds for the predicted values of our models. In general, the total upper and lower values of the intervals do not significantly differ from the predicted value, and thus we conclude that our results offer a consistent depiction of the expected activity in the insurance industry in case of no pandemic. Table B1 : B1Prediction and realized values for GCP in the first quarter of 2020.Insurance class 95% CI (lower) Expected 95% CI (upper) Real Difference MTPL (total) 488,683 530,810 576,569 486,558 -44,252 Financial losses 3 71 1,743 3,137 3,066 Property, fire and nat.forces 6,165 14,807 35,562 20,887 6,080 Property, other 27,730 42,928 66,454 64,263 21,335 Cargo 506 3,492 24,093 56 -3,436 Motor vehicles (casco) 131,294 143,517 156,878 115,033 -28,484 Accident 99,434 112,320 126,876 108,374 -3,946 General liability 3,957 21,459 116,362 5,127 -16,332 Tourists assistance 13,901 18,310 24,118 14,863 -3,447 Health 12,576 28,068 62,642 22,849 -5,219 Life assurance 62,633 84,995 115,343 109,341 24,346 TOTAL 846,882 1,000,777 1,306,640 950,488 -50,289 Table B2 : B2Prediction and realized values for GCP in the second quarter of 2020.Insurance class 95% CI (lower) Expected 95% CI (upper) Real Difference MTPL (total) 975,619 1,074,204 1,182,752 860,049 -214,155 Financial losses 8 313 11,592 6,846 6,533 Property, fire and nat.forces 33,074 82,249 204,537 88,810 6,561 Property, other 53,978 97,979 177,848 157,677 59,698 Cargo 196 2,753 38,566 600 -2,153 Motor vehicles (casco) 248,111 280,212 316,467 233,135 -47,077 Accident 193,140 220,511 251,761 187,812 -32,699 General liability 3,294 19,369 113,889 6,406 -12,963 Tourists assistance 21,050 29,098 40,222 26,224 -2,874 Health 19,258 59,309 182,651 49,394 -9,915 Life assurance 99,852 154,856 240,161 192,388 37,532 TOTAL 1,647,580 2,020,853 2,760,448 1,809,341 -211,512 Table B3 : B3Prediction and realized values for GWP in the first quarter of 2020.Insurance class 95% CI (lower) Expected 95% CI (upper) Real Difference MTPL (total) 960,985 982,417 1,004,326 930,651 -51,766 Financial losses 9,348 21,033 47,323 7,956 -13,077 Property, fire and nat.forces 260,102 290,976 325,515 250,442 -40,534 Property, other 163,220 218,617 292,816 299,261 80,644 Cargo 20,034 24,988 31,167 25,924 936 Motor vehicles (casco) 199,289 211,408 224,264 198,391 -13,017 Accident 208,045 224,523 242,307 234,936 10,413 General liability 73,219 79,105 85,464 68,839 -10,266 Tourists assistance 36,144 40,508 45,399 33,291 -7,217 Health 67,385 231,079 792,431 113,329 -117,750 Life assurance 371,055 391,308 412,667 349,066 -42,242 TOTAL 2,368,825 2,715,962 3,503,679 2,512,086 -203,876 Table B4 : B4Prediction and realized values for GWP in the second quarter of 2020.Insurance class 95% CI (lower) Expected 95% CI (upper) Real Difference MTPL (total) 2,184,572 2,239,191 2,295,176 1,890,229 -348,962 Financial losses 9,610 31,192 101,242 45,670 14,478 Property, fire and nat.forces 434,212 507,319 592,736 487,044 -20,275 Property, other 395,523 544,882 750,643 699,201 154,319 Cargo 38,217 49,442 63,964 52,614 3,172 Motor vehicles (casco) 406,710 438,030 471,762 404,421 -33,609 Accident 373,338 409,792 449,804 395,928 -13,864 General liability 131,396 142,078 153,629 124,022 -18,056 Tourists assistance 89,650 103,032 118,412 40,278 -62,754 Health 68,963 280,083 1,137,509 168,275 -111,808 Life assurance 764,619 812,527 863,437 684,787 -127,740 TOTAL 4,896,810 5,557,569 6,998,314 4,992,469 -565,100 In the case of GCPs and GWPs, the suitable stationary transformation is given by the logarithm of the reported value Stress tests" for banks as liquidity insurers in a time of COVID. Viral Acharya, Sascha Steffen, VoxEU. org. Acharya, Viral, and Sascha Steffen. 2020. "Stress tests" for banks as liquidity insurers in a time of COVID." VoxEU. org, March, 22. Employment, Income, and Unemployment Insurance during the COVID-19 Pandemic. Gregory Acs, Michael Karpman, Urban InstituteAcs, Gregory, and Michael Karpman. 2020. "Employment, Income, and Unemployment In- surance during the COVID-19 Pandemic." Urban Institute. Building ARIMA and ARIMAX models for predicting long-term disability benefit application rates in the public/private sectors. Bruce H Andrews, D Matthew, Robert Dean, Caroline Swain, Cole, Society of Actuaries. Andrews, Bruce H, Matthew D Dean, Robert Swain, and Caroline Cole. 2013. "Building ARIMA and ARIMAX models for predicting long-term disability benefit application rates in the public/private sectors." Society of Actuaries, 1-54. Does insurance market activity promote economic growth? A cross-country study for industrialized and developing countries. Marco Arena, Journal of risk and Insurance. 754Arena, Marco. 2008. "Does insurance market activity promote economic growth? A cross-country study for industrialized and developing countries." Journal of risk and Insurance, 75(4): 921-946. The Impact of COVID-19 on the Insurance Industry. Pius Babuna, Xiaohua Yang, Amatus Gyilbag, Doris Abra Awudi, David Ngmenbelle, Dehui Bian, International journal of environmental research and public health. 17165766Babuna, Pius, Xiaohua Yang, Amatus Gyilbag, Doris Abra Awudi, David Ngmenbelle, and Dehui Bian. 2020. "The Impact of COVID-19 on the Insurance Industry." International journal of environmental research and public health, 17(16): 5766. Interactions Between Global Change and Human Health (Scripta Varia. David E Bloom, David Canning, 106Epidemics and economicsBloom, David E, David Canning, et al. 2004. "Epidemics and economics." Interactions Between Global Change and Human Health (Scripta Varia, 106: 304-331. Predicting initial claims for unemployment benefits. Hyunyoung Choi, Hal Varian, Google Inc1Choi, Hyunyoung, and Hal Varian. 2009. "Predicting initial claims for unemployment bene- fits." Google Inc, 1: 1-5. Time-Series Properties of the Equity Risk Premium. John M Clinebell, Jerry L Douglas R Kahl, Stevens, Journal of Financial Research. 171Clinebell, John M, Douglas R Kahl, and Jerry L Stevens. 1994. "Time-Series Properties of the Equity Risk Premium." Journal of Financial Research, 17(1): 105-116. Forecasting automobile insurance paid claim costs using econometric and ARIMA models. J Cummins, Gary L David, Griepentrog, International Journal of Forecasting. 13Cummins, J David, and Gary L Griepentrog. 1985. "Forecasting automobile insurance paid claim costs using econometric and ARIMA models." International Journal of Forecasting, 1(3): 203-215. Research on Premium Income Prediction Based on LSTM Neural Network. Li Diao, Ning Wang, Advances in Social Sciences Research Journal. 611Diao, Li, and Ning Wang. 2019. "Research on Premium Income Prediction Based on LSTM Neural Network." Advances in Social Sciences Research Journal, 6(11): 256-260. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Robert F Engle, Econometrica: Journal of the Econometric Society. Engle, Robert F. 1982. "Autoregressive conditional heteroscedasticity with estimates of the vari- ance of United Kingdom inflation." Econometrica: Journal of the Econometric Society, 987-1007. Gradient boosting trees for auto insurance loss cost modeling and prediction. Leo Guelman, Expert Systems with Applications. 393Guelman, Leo. 2012. "Gradient boosting trees for auto insurance loss cost modeling and predic- tion." Expert Systems with Applications, 39(3): 3659-3667. A flexible Bayesian nonparametric model for predicting future insurance claims. Liang Hong, Ryan Martin, North American Actuarial Journal. 212Hong, Liang, and Ryan Martin. 2017. "A flexible Bayesian nonparametric model for predicting future insurance claims." North American Actuarial Journal, 21(2): 228-241. Response to IAIS consultation on the impact of COVID-19. Insurance-Europe, Insurance Europe Position PapersInsurance-Europe. 2020. "Response to IAIS consultation on the impact of COVID-19." Insurance Europe Position Papers. Forecasting motor insurance claim amount using ARIMA model. V Kumar, Dipak Selva, Kumar Satpathi, Ptv Praveen Kumar, AIP Publishing LLC2246Kumar, V Selva, Dipak Kumar Satpathi, PTV Praveen Kumar, and VV Haragopal. 2020. "Forecasting motor insurance claim amount using ARIMA model." Vol. 2246, 020005, AIP Publishing LLC. On a measure of lack of fit in time series models. Greta M Ljung, George Ep Box, Biometrika. 652Ljung, Greta M, and George EP Box. 1978. "On a measure of lack of fit in time series models." Biometrika, 65(2): 297-303. ARMA models and the Box-Jenkins methodology. Spyros Makridakis, Michele Hibon, Journal of Forecasting. 163Makridakis, Spyros, and Michele Hibon. 1997. "ARMA models and the Box-Jenkins method- ology." Journal of Forecasting, 16(3): 147-163. Outpatient Parenteral Antimicrobial Therapy in the Time of COVID-19: The Urgent Need for Better Insurance Coverage. Omar Mansour, Sara Keller, Morgan Katz, Jennifer L Townsend, Oxford University Press US7Mansour, Omar, Sara Keller, Morgan Katz, and Jennifer L Townsend. 2020. "Outpa- tient Parenteral Antimicrobial Therapy in the Time of COVID-19: The Urgent Need for Better Insurance Coverage." Vol. 7, ofaa287, Oxford University Press US. Prevention is better than the cure: Risk management of COVID-19. Michael Mcaleer, McAleer, Michael. 2020. "Prevention is better than the cure: Risk management of COVID-19." Dynamics of unemployment insurance claims: an application of ARIMA-GARCH models. Hassan Mohammadi, Daniel P Rich, Atlantic Economic Journal. 414Mohammadi, Hassan, and Daniel P Rich. 2013. "Dynamics of unemployment insurance claims: an application of ARIMA-GARCH models." Atlantic Economic Journal, 41(4): 413-425. Predicting motor insurance claims using telematics data-XGBoost versus logistic regression. Jessica Pesantez-Narvaez, Montserrat Guillen, Manuela Alcañiz, Risks. 7270Pesantez-Narvaez, Jessica, Montserrat Guillen, and Manuela Alcañiz. 2019. "Predict- ing motor insurance claims using telematics data-XGBoost versus logistic regression." Risks, 7(2): 70. Covid-19: implications for insurer risk management and the insurability of pandemic risk. Andreas Richter, Thomas C Wilson, The Geneva risk and insurance review. 45Richter, Andreas, and Thomas C Wilson. 2020. "Covid-19: implications for insurer risk management and the insurability of pandemic risk." The Geneva risk and insurance review, 45(2): 171-199. Understanding and addressing global insurance protection gaps. Kai-Uwe Schanz, The Geneva Association. Schanz, Kai-Uwe. 2018. "Understanding and addressing global insurance protection gaps." The Geneva Association. The socio-economic determinants of the coronavirus disease (COVID-19) pandemic. Viktor Stojkoski, Zoran Utkovski, Petar Jolakoski, Dragan Tevdovski, Ljupco Kocarev, arXiv:2004.07947arXiv preprintStojkoski, Viktor, Zoran Utkovski, Petar Jolakoski, Dragan Tevdovski, and Ljupco Kocarev. 2020. "The socio-economic determinants of the coronavirus disease (COVID-19) pan- demic." arXiv preprint arXiv:2004.07947. The impact of catastrophes on insurer stock volatility. Christian Thomann, Journal of Risk and Insurance. 801Thomann, Christian. 2013. "The impact of catastrophes on insurer stock volatility." Journal of Risk and Insurance, 80(1): 65-94. Comparing the Performance of Seasonal ARI-MAX Model and Nonparametric Regression Model in Predicting Claim Reserve of Education Insurance. S M Ulyah, Mardianto, IOP Publishing139712074Ulyah, SM, MFF Mardianto, et al. 2019. "Comparing the Performance of Seasonal ARI- MAX Model and Nonparametric Regression Model in Predicting Claim Reserve of Education Insurance." Vol. 1397, 012074, IOP Publishing. Insurance premium prediction via gradient tree-boosted Tweedie compound Poisson models. Yi Yang, Wei Qian, Hui Zou, Journal of Business & Economic Statistics. 363Yang, Yi, Wei Qian, and Hui Zou. 2018. "Insurance premium prediction via gradient tree-boosted Tweedie compound Poisson models." Journal of Business & Economic Statistics, 36(3): 456-470.
[]
[ "No need for dark matter: resolved kinematics of the ultra-diffuse galaxy AGC 114905", "No need for dark matter: resolved kinematics of the ultra-diffuse galaxy AGC 114905" ]
[ "Pavel E Mancera Piña \nKapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands\n", "Filippo 2★ \nASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991 PDDwingelooThe Netherlands\n", "Fraternali \nKapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands\n", "Tom Oosterloo \nKapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands\n\nASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991 PDDwingelooThe Netherlands\n", "Elizabeth A K Adams \nKapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands\n\nASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991 PDDwingelooThe Netherlands\n", "Kyle A Oman \nInstitute for Computational Cosmology\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUnited Kingdom\n", "Lukas Leisman \nDepartment of Physics and Astronomy\nValparaiso University\n1610 Campus Drive East46383ValparaisoINUSA\n\nDepartment of Astronomy\nUniversity of Illinois\n1002 W. Green St61801UrbanaILUSA\n" ]
[ "Kapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands", "ASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991 PDDwingelooThe Netherlands", "Kapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands", "Kapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands", "ASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991 PDDwingelooThe Netherlands", "Kapteyn Astronomical Institute\nUniversity of Groningen\nLandleven 129747 ADGroningenThe Netherlands", "ASTRON\nNetherlands Institute for Radio Astronomy\nOude Hoogeveensedijk 47991 PDDwingelooThe Netherlands", "Institute for Computational Cosmology\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUnited Kingdom", "Department of Physics and Astronomy\nValparaiso University\n1610 Campus Drive East46383ValparaisoINUSA", "Department of Astronomy\nUniversity of Illinois\n1002 W. Green St61801UrbanaILUSA" ]
[ "MNRAS" ]
We present new H interferometric observations of the gas-rich ultra-diffuse galaxy AGC 114905, which previous work, based on low-resolution data, identified as an outlier of the baryonic Tully-Fisher relation. The new observations, at a spatial resolution ∼ 2.5 times higher than before, reveal a regular H disc rotating at about 23 km s −1 . Our kinematic parameters, recovered with a robust 3D kinematic modelling fitting technique, show that the flat part of the rotation curve is reached. Intriguingly, the rotation curve can be explained almost entirely by the baryonic mass distribution alone. We show that a standard cold dark matter halo that follows the concentration-halo mass relation fails to reproduce the amplitude of the rotation curve by a large margin. Only a halo with an extremely (and arguably unfeasible) low concentration reaches agreement with the data. We also find that the rotation curve of AGC 114905 deviates strongly from the predictions of Modified Newtonian dynamics. The inclination of the galaxy, which is measured independently from our modelling, remains the largest uncertainty in our analysis, but the associated errors are not large enough to reconcile the galaxy with the expectations of cold dark matter or Modified Newtonian dynamics.
10.1093/mnras/stab3491
[ "https://arxiv.org/pdf/2112.00017v3.pdf" ]
244,773,372
2112.00017
c3a45875223aaeadb9ebb24e5927e223084c7b2f
No need for dark matter: resolved kinematics of the ultra-diffuse galaxy AGC 114905 2015 Pavel E Mancera Piña Kapteyn Astronomical Institute University of Groningen Landleven 129747 ADGroningenThe Netherlands Filippo 2★ ASTRON Netherlands Institute for Radio Astronomy Oude Hoogeveensedijk 47991 PDDwingelooThe Netherlands Fraternali Kapteyn Astronomical Institute University of Groningen Landleven 129747 ADGroningenThe Netherlands Tom Oosterloo Kapteyn Astronomical Institute University of Groningen Landleven 129747 ADGroningenThe Netherlands ASTRON Netherlands Institute for Radio Astronomy Oude Hoogeveensedijk 47991 PDDwingelooThe Netherlands Elizabeth A K Adams Kapteyn Astronomical Institute University of Groningen Landleven 129747 ADGroningenThe Netherlands ASTRON Netherlands Institute for Radio Astronomy Oude Hoogeveensedijk 47991 PDDwingelooThe Netherlands Kyle A Oman Institute for Computational Cosmology Department of Physics Durham University South RoadDH1 3LEDurhamUnited Kingdom Lukas Leisman Department of Physics and Astronomy Valparaiso University 1610 Campus Drive East46383ValparaisoINUSA Department of Astronomy University of Illinois 1002 W. Green St61801UrbanaILUSA No need for dark matter: resolved kinematics of the ultra-diffuse galaxy AGC 114905 MNRAS 0002015Accepted XXX. Received YYY; in original form ZZZPreprint 17 February 2022 Compiled using MNRAS L A T E X style file v3.0galaxies: dwarfs -galaxies: irregular -galaxies: kinematics and dynamics -cosmology: dark matter We present new H interferometric observations of the gas-rich ultra-diffuse galaxy AGC 114905, which previous work, based on low-resolution data, identified as an outlier of the baryonic Tully-Fisher relation. The new observations, at a spatial resolution ∼ 2.5 times higher than before, reveal a regular H disc rotating at about 23 km s −1 . Our kinematic parameters, recovered with a robust 3D kinematic modelling fitting technique, show that the flat part of the rotation curve is reached. Intriguingly, the rotation curve can be explained almost entirely by the baryonic mass distribution alone. We show that a standard cold dark matter halo that follows the concentration-halo mass relation fails to reproduce the amplitude of the rotation curve by a large margin. Only a halo with an extremely (and arguably unfeasible) low concentration reaches agreement with the data. We also find that the rotation curve of AGC 114905 deviates strongly from the predictions of Modified Newtonian dynamics. The inclination of the galaxy, which is measured independently from our modelling, remains the largest uncertainty in our analysis, but the associated errors are not large enough to reconcile the galaxy with the expectations of cold dark matter or Modified Newtonian dynamics. INTRODUCTION The properties, origin, and formation mechanisms of ultra-diffuse galaxies (UDGs) have been widely discussed in the last years. UDGs (van Dokkum et al. 2015) are low surface brightness galaxies (e.g. Impey et al. 1988, see also discussion in Conselice 2018) with an extended light distribution. At fixed stellar mass or luminosity, UDGs have significantly larger effective radii than the 'classical' dwarf galaxy population (e.g. van Dokkum et al. 2015;Mihos et al. 2015;Mancera Piña et al. 2019a;Chamba et al. 2020). UDGs are mostly found by number in massive galaxy clusters, but they are also present in galaxy groups, in the field, and even in voids (e.g. van der Burg et al. 2016;Leisman et al. 2017;Román & Trujillo 2017;Mancera Piña et al. 2019a;Román et al. 2019; Barbosa et al. 2020;Karunakaran et al. 2020). The ubiquity of UDGs across different environments indicates that even if some of them form due to environmental processes, this is not the case for all UDGs, and they can also form due to their own internal processes. The population of UDGs is likely a mixed bag of galaxies with similar sizes and surface brightness, but perhaps multiple formation mechanisms. The above idea seems confirmed by a number of results from semi-analytic models and hydrodynamics simulations that produce UDG-like simulated galaxies based on different physical processes. ★ e-mail: [email protected] On the one hand, different authors report that classical dwarf galaxies can become UDGs (i.e. increase their size and likely decrease their surface brightness) due to cluster pre-processing phenomena such as tidal striping and tidal heating (Carleton et al. 2019;Tremmel et al. 2020;Sales et al. 2020). On the other hand, it has also been suggested that internal processes can explain the optical properties of UDGs. Amorisco & Loeb (2016) proposed a model where UDGs form due to a higher than average dark-matter angular momentum, which then gives rise to an extended stellar effective radius (see also Rong et al. 2017). Here it is worth mentioning that even if UDGs inhabit normal dark matter haloes, they could still have a higher than average retained fraction of stellar specific angular momentum (Posti et al. 2018;Mancera Piña et al. 2020). Another idea is that UDGs are dwarfs that became larger due to feedback-driven outflows, which change the dark matter and baryonic potential and allow the stars to migrate to more external orbits, increasing their effective radius (Di Cintio et al. 2017;Chan et al. 2018). It has also been argued that the expansion of the stellar orbits can be the result of massive mergers at early ( > 1) epochs (Wright et al. 2021) or the by-product of very efficient globular cluster feedback (Trujillo-Gomez et al. 2021). While the above models and simulations seem to produce simulated galaxies that match a number of properties of real UDGs, surprising observations of two different populations of UDGs have been more challenging to reproduce. First, it has been found that two gas-poor UDGs (DF-2 and DF-4) near (at least in projection) the galaxy NGC 1052, contain significantly less dark matter than expected based on stellar and globular cluster kinematics (e.g. van Dokkum et al. 2018Danieli et al. 2019;Emsellem et al. 2019). While caveats regarding the distance and accuracy of the kinematic tracers used to study these UDGs have been raised (e.g. Laporte et al. 2019;Trujillo et al. 2019), DF-2 and DF-4 have motivated multiple studies aiming to explain the existence of dark-matter free galaxies. The main ideas to explain their existence involve dark matter removing mechanisms such as high-velocity collisions and tidal stripping, or a tidal dwarf origin (Haslbauer et al. 2019;Silk 2019;Montes et al. 2020;Shin et al. 2020;Doppel et al. 2021;Jackson et al. 2021). The second set of puzzling observations, still very much lacking a quantitative explanation, is related to the H kinematics of some gas-rich UDGs (sometimes also called H -bearing UDGs). Using unresolved ALFALFA data (see Haynes et al. 2018), Leisman et al. (2017, see also Jones et al. 2018Karunakaran et al. 2020) first found that gas-rich UDGs have narrow global H profiles for their gas mass. Then, Mancera Piña et al. (2019bPiña et al. ( , 2020) studied a set of six of those gas-rich UDGs with low-resolution (two independent resolution elements per galaxy side) interferometric H observations. Using a state-of-the-art kinematic modelling fitting technique ( 3D Barolo, Di Teodoro & Fraternali 2015) to overcome beam smearing effects, they recovered the circular speeds of their galaxies, unveiling two intriguing features. First, that having a baryonic mass a factor 10 − 100 larger than galaxies with similar circular speed, H -rich UDGs shift off the baryonic Tully-Fisher relation (BTFR, McGaugh et al. 2000), with the offset from the BTFR correlating with the UDGs optical disc scale lengths. And second, that their dynamical mass within the extent of the H disc is about the same as their baryonic mass, meaning that the galaxies have very low dark matter fractions within scales as large as 10 kpc. These features suggest that gas-rich UDGs have atypical non-luminous mass distributions, making them a promising population to test dark matter theories. It is also important to stress that these gas-rich UDGs are selected to be fairly isolated (Leisman et al. 2017), and they lie at distances of several tens of Mpc where Hubble flow distances are robust, negating some of the main concerns raised for DF-2 and DF-4. Given all this, it is important to further characterise the properties of these UDGs, which are apparently gas-rich but also dark-matter poor (at least within the observed radii). One way to do this is by studying their H rotation curves, as H provides arguably the best kinematic tracer for disc galaxies, both massive and dwarfs (e.g. Begeman 1987;de Blok et al. 2008;Iorio et al. 2017). In this work, we present and analyse new, high-resolution interferometric observations of one of these peculiar gas-rich UDGs, AGC 114905. As we will show, the galaxy seems to pose a challenge to the currently favoured galaxy formation framework. This paper is organised in the following way. In Section 2, we describe the main properties of AGC 114905 and we present the data used in this work. In Section 3 we show the kinematic modelling of the galaxy, and in Section 4 the resulting mass models. We discuss our results in Section 5, to then present our conclusions and summary in Section 6. DATA AND PROPERTIES OF AGC 114905 AGC 114905 (01:25:18.60, +07:21:41.11, J2000) lies at a (Hubbleflow) distance = 76 ± 5 Mpc (Leisman et al. 2017). The stellar distribution of AGC 114905, consists of an exponential disc with a disc scale length d = 1.79 ± 0.04 kpc. The left panel of Fig. 1 shows its −band stellar image. The galaxy has a relatively blue colour, ( − ) = 0.3 ± 0.1 (Mancera Piña et al. 2020; Gault et al. 2021). We estimate the stellar mass ( * ) of the galaxy using the mass-tolight-colour relation from Du et al. (2020), which has been recently calibrated using a large sample of low surface brightness galaxies. We obtain * = (1.3 ± 0.3) × 10 8 ; this is slightly smaller than the value used in Mancera Piña et al. (2019bPiña et al. ( , 2020, owing to the different mass-to-light-colour calibrations. We gathered H data of AGC 114905 at different resolutions, obtained with the Karl G. Jansky Very Large Array. Specifically, we collected data from the D-, C-and B-array configurations. Details on the D and C configuration observations (PI: Leisman, ID 17A-210) can be found in Leisman et al. (2017) and Gault et al. (2021). The new B-array observations (PI: Mancera Piña, ID 20A-095) were obtained between July and October 2020. 40 hours were observed (about 34 hours on-source) and combined with the existing C-and D-array observations. The data reduction was done with the software Miriad (Sault et al. 1995) following standard procedures, using a robust weighting of 0.75 to make the final data cube, which results in a cleaned beam of size 7.88 arcsec × 6.36 arcsec. After Hanningsmoothing, our final cube has a rms noise per channel of about 0.26 mJy/beam and a spectral resolution of 3.4 km s −1 . The peak H column density is 8.4×10 20 atoms cm −2 , and the noise level is 4.1×10 19 cm −2 . The integral flux of the total H map is 0.73±0.07 Jy km s −1 , close to the value of 0.78 Jy km s −1 used in Mancera Piña et al. (2019b), although lower than the 0.96 ± 0.04 Jy km s −1 reported by Leisman et al. (2017) from unresolved ALFALFA observations. At the distance of AGC 114905, our integral flux yields HI = (9.7 ± 1.4) × 10 8 . 1 We combine HI and * to obtain the baryonic mass bar = 1.33 HI + * = (1.4 ± 0.2) × 10 9 . The factor 1.33 accounts for the presence of helium, and we neglect any contribution of molecular gas, whose mass is expected to be negligible compared to HI (e.g. Hunter et al. 2019;Wang et al. 2020). The galaxy gas fraction gas = gas / bar ≈ 0.9, ensures that bar is robust against possible systematics related to * , since the main uncertainty in gas comes from the distance, which is well constrained. The total H map of the galaxy is shown in the middle panel of Fig. 1, and also on the left panel with the H contours overlaid on top of the stellar emission. It is clear that the gas extends well beyond the optical emission, despite UDGs being optically extended. There is also some degree of misalignment between the optical and H position angles (see also Gault et al. 2021), although the optical morphology is likely affected by bright, patchy star formation regions. The right panel of Fig. 1 shows the surface mass density profiles of the stellar and gaseous discs of AGC 114905. The stellar profile comes from converting our optical surface brightness profiles into mass density using a mass-to-light ratio in the −band of 0.47 (Du et al. 2020). The gas profile is obtained using the gipsy (van der Hulst et al. 1992) task , and converted to mass density using the conversion factor 1 pc −2 = 1.25 × 10 20 atoms cm −2 . Once this conversion is applied, we multiply by the factor 1.33 to account for the presence of helium. KINEMATICS In order to obtain reliable kinematic information (rotation velocity and velocity dispersion) for our galaxy, we use the software 3D Barolo (Di Teodoro & Fraternali 2015). As extensively explained in Di Teodoro & Fraternali (2015), Di Teodoro et al. (2016), and Iorio et al. (2017), 3D Barolo 2 builds 3D realisations of tilted-ring models of a galaxy data cube, which are then convolved with the beam of the observations and compared channel by channel with the real data. This allows for a robust recovery of the rotation curve and gas velocity dispersion, since the method largely mitigates the effects of beam smearing (Bosma 1978;Begeman 1987;Swaters 1999;Di Teodoro & Fraternali 2015). Before delving into the details and results of our modelling, we will briefly discuss the observed kinematics as well as the derivation of two important geometrical parameters: the position angle of the galaxy and its inclination. Velocity field and geometrical parameters The velocity field (1 st moment map) of AGC 114905, shown in the top panel of Fig. 2, has the clear kinematic pattern of a regularly rotating disc. The position angle of the velocity field is estimated by trial and error as the angle that maximizes the amplitude of the major-axis position-velocity (PV) diagram (e.g. Mancera Piña et al. 2020). We find a position angle of 89 • , as shown in Fig. 2 with a line on top of the velocity field. Our value is similar to the 84 • reported in Mancera Piña et al. (2019bPiña et al. ( , 2020 derived from the less resolved data. If we fit the position angle during our kinematic modelling (see below) we find values between 85 − 92 • depending on our initial estimates. The middle (bottom) panel of Fig. 2 shows, in blue background and black contours, the PV diagram along the major (minor) axis of AGC 114905. The major-axis PV shows the typical pattern of a rotating disc and seems to reach a flat velocity in the outer regions. While differences of a few degrees in the position angle do not significantly affect the final rotation curve, the inclination of the galaxy is more critical, as small changes at low inclinations can severely affect the value of the deprojected rotation curve. Undoubtedly, the inclination is the main uncertainty in our kinematic modelling and results, and we pay special attention to it. Traditionally, the inclination of high-resolution data can be obtained during the kinematic fitting using the velocity field (e.g. de ). However, 2 https://editeodoro.github.io/Bbarolo/ this method is not particularly robust as it depends on the shape of the rotation curve: for solid-body rotation the iso-contours on the velocity field are parallel, nullifying the power to measure the inclination. Given the above, and following Mancera Piña et al. (2020), we decide to estimate the inclination with an approach that is independent of the kinematics, relying only on the H map of the galaxy. Our method works as follows. We use 3D Barolo to build azimuthal models of the galaxy at different inclinations, with these inclinations being drawn from a flat prior distribution between 10 • − 80 • and sampled using a Markov chain Monte Carlo (MCMC) routine (based on the Python package emcee, see Foreman-Mackey et al. 2013). Each model is convolved with the beam of the observations, and its total intensity map is built. We then compare these model intensity maps with the real data, with our MCMC routine minimizing the absolute residuals between model and observed intensity maps. We have tested this method extensively using artificial data cubes matching our resolution and signal-to-noise (S/N), finding it robust and reliable (see also Fraternali et al. 2021). In the end, we find an inclination of 32 ± 3 • for AGC 114905, which we adopt as our fiducial value; the posterior distribution is shown in Fig B1 in Appendix. B. As an extra check, we also estimated the inclination in two other ways 3 . First, using the method described in Mancera Piña et al. (2020), which is equivalent to our method described above but independent of the MCMC sampling. We find an inclination of 34 ± 5 • , although the quoted uncertainty is just an expected mean value rather than a well defined statistical uncertainty. Similarly, Mancera Piña et al. (2019bPiña et al. ( , 2020 found an inclination of 33 ± 5 • from their lowerresolution data. Second, we derive kinematic-dependent inclinations. We use both 3D Barolo (fitting the whole data cube) and the gipsy task (fitting the velocity field). Depending on the exact initial value, mask, and ring separation, both methods find inclinations between 30 • − 37 • . It is reassuring that despite not being our favoured approaches to measure the inclination, we find these different values consistent with the results of our preferred method. Overall, it 3 The optical inclination, derived from the optical axis ratio, is around 45 • . We do not use this inclination in our analysis as it is not clear if the optical data follows the H emission (see also e.g. Gault et al. 2021;Kado-Fong et al. 2021), but we provide the value for completeness. Clearly, this inclination would lower the value of the circular speed, strengthening our results. is important to highlight that we do not find any evidence favouring inclinations lower than 30 • . Finally, it is worth mentioning that while deriving the inclination we assume that the H resides on a razor-thin disc, a significant thickness of the disc would imply a higher inclination than what we have derived due to projection effects (Iorio 2018). From this point of view 32±3 • gives a lower limit on the inclination of AGC 114905 (see also Sec 5.4). On the other hand, if the gas disc is non-axisymmetric, but instead has some intrinsic elongation, we could be overestimating its inclination. While some simulations suggest this is possible 4 , in 4 Marasco et al. (2018) have reported that about half of the massive dwarf galaxies (60 < max < 120 km s −1 , with max the maximum rotation velocity) in the APOSTLE simulations (Sawala et al. 2015;Fattahi et al. 2016) inhabit dark matter halos with intrinsic axis ratios / < 0.8; if the disc of what follows we assume that the observed HI total intensity map and velocity gradient correspond to an inclined axisymmetric disc galaxy with gas moving in circular orbits. Kinematic modelling With the position angle and inclination determined, we proceeded to perform our kinematic modelling with 3D Barolo, leaving as free parameters the systemic velocity ( sys ), the rotation velocity ( rot ), and the velocity dispersion ( HI ). We fit an azimuthal model and we use a ring separation of 6 arcsec. This represents a minor oversampling of less than 10 percent with respect to the size of the beam along the major axis of the galaxy (6.5 arcsec), allowing us to trace the rotation curve of the galaxy with five, basically uncorrelated, resolution elements per galaxy side. We also check that the rotation velocities obtained using four or five rings (see below) are well consistent with each other. We first perform an iteration where sys is a free parameter. The best sys turns out to be 5435 km s −1 , which matches the centre of a Gaussian fit to the global H profile. For our final model we keep this sys fixed and we only fit rot and HI . The final model faithfully reproduces the observations. This can be seen in the middle and bottom panels of Fig. 2, where we compare the PV diagrams of the best-fitting model (red) and data (black). There are some low-S/N features at low velocities ( 5 km s −1 ) not reproduced, but 3D Barolo closely mimics the overall kinematics of the galaxy. This can also be seen in Appendix A, where we show representative channel maps of AGC 114905 and of our best-fitting model. The resulting rotation curve, uncorrected for inclination, is shown as yellow points overlaid on top of the major-axis PV diagram of Fig. 2, and it is clear that it reaches its flat part well before our outermost radius. To take into account pressure-supported motions, we apply the asymmetric drift correction to our rotation curve (see Iorio et al. 2017), ending up with the circular speed c . The correction is found to be very small, contributing less than 2 km s −1 at all radii. In Fig. 3, we explicitly show the circular speed profile of AGC 114905, as well as its velocity dispersion profile. The uncertainties in c include the uncertainties in the inclination, by means of a Monte Carlo sampling approach as detailed in Mancera Piña et al. (2020). The flat part of the circular speed profile has a velocity of 23 km s −1 . This, together with the bar of the galaxy, confirms its position as an outlier of the BTFR. In Fig. 3 we also include for comparison the values for c and HI obtained in Mancera Piña et al. (2019bPiña et al. ( , 2020 at lower resolution, showing the good agreement between them and our new determinations. This is important not only for AGC 114905, but for all the UDGs in Mancera Piña et al. (2019bPiña et al. ( , 2020, as it is a direct validation of the lower-resolution results presented previously. Local and global disc stability With a median value of ∼ 5 km s −1 , the velocity dispersion HI of AGC 114905 is slightly below the average value in rotation-supported dwarfs (∼ 8 km s −1 , e.g. Iorio et al. 2017) although consistent within the uncertainties. The low values of c and HI of our UDG imply a relatively low value of the Toomre parameter gas = HI /( Σ gas ), with the epicycle frequency (Toomre 1964). The gas profile shows a slight decrease with radius, with a median (mean) value of 0.95 AGC 114905 has a similar intrinsic / (i.e. it is an elongated disc instead of an inclined circular disc) its inclination could be as low as 10 • . (1.6), after applying a small correction to account for thickness, see Romeo 1994;Romeo & Falstad 2013). The uncertainties are relatively large (typically a factor 2 − 3), but these values of gas suggest that the galaxy could be susceptible to local instabilities (see Romeo & Falstad 2013 and references therein for a detailed discussion on the interpretation of gas ). While these local instabilities may lead to fragmentation and subsequent star formation, observations suggest this is not always the case (Hunter et al. 1998;Leroy et al. 2008;Elmegreen & Hunter 2015). The value of gas for AGC 114905 is lower on average but consistent within 2 with the median values of LITTLE THINGS dwarf galaxies ). Finally, it should be noted that a more detailed calculation that takes into account the gas disc flaring (e.g. Elmegreen & Hunter 2015;Bacchini et al. 2020) would increase the value of gas , especially in the outer parts. While gas is in principle only related to local instabilities, we can also investigate the global disc stability of our UDG. The ordered kinematics seen in Fig. 2 and the isolation (see Mancera Piña et al. 2020) of the galaxy strongly suggest an equilibrium state. We further tested this by allowing 3D Barolo to fit radial motions overlaid on the rotation, but we did not find evidence of such radial motions as their amplitude is always consistent with zero within the uncertainties. We also computed the global stability parameter 2 = 2 /(4 Σ gas ) (Toomre 1981), finding a median of 1.2 and with 2 being smaller than 1 (0.9) only at the outermost radius, suggesting the system is stable against bar instabilities ( 2 1 is the instability condition often used for dwarf galaxies, see e.g. Mihos et al. 1997;Hidalgo-Gámez 2004). Overall, these investigations show that it is reasonable to assume that the cold gas in AGC 114905 is in closed orbits tracing its gravitational potential and allowing us to build mass models based on the derived rotation curve. MASS MODELLING A baryon-dominated rotation curve AGC 114905 has a baryonic mass much higher than other dwarf galaxies with similar circular speeds Mancera Piña et al. 2020). It is therefore interesting to see if AGC 114905, like most dwarfs, is dominated by dark matter at all radii. Prior to obtaining any mass model, we can compare the circular speed profile of the galaxy with the circular speed profile of the baryonic distribution ( bar ), which is simply the sum in quadrature of the contributions of the stellar and gas discs, this is 2 bar = 2 * + 2 gas . We derive c, * and c,gas using the software (Iorio 2018). 5 takes as input the mass density profile of a given component, fitted with an appropriate function (see below), computes its gravitational potential via numerical integration, and returns the associated circular speed. In the case of the stellar disc, we use an exponential profile with * = 1.3 × 10 8 and an exponential disc scale length d = 1.79 kpc; this profile can be compared with the data in Fig. 1. We assume a sech 2 profile along the vertical direction, and a constant thickness d = 0.196 0.633 d ≈ 280 pc, as found in low-inclination star forming galaxies (Bershady et al. 2010). For the gas component (H plus helium), we fit the density profile with a profile of the form Σ gas = Σ 0,gas − / 1 (1 + / 2 ) ,(1) where Σ 0,gas is the gas central surface density, is the cylindrical radius, and 1 , 2 , and are the fitting parameters (equal to 3.2 /pc 2 , 1.1 kpc, 16.5 kpc, and 18, respectively). This profile provides a good fit to the observations, as seen in Fig. 1. For the vertical structure of the gaseous disc we assume a Gaussian profile and a constant vertical scale-height d = 250 pc. It is worth mentioning that the results we show below do not depend significantly on the assumed thickness of the stellar or gaseous discs. Fig. 4 shows the contribution of * , gas , and bar to the total c of AGC 114905. Remarkably, bar provides a reasonable description of c at all radii. This implies that as opposed to classical dwarf galaxies (e.g. Iorio et al. 2017;Read et al. 2017), the dynamics of AGC 114905, at least within the observed radii extending to about 10 kpc, are baryon-dominated rather than dark-matter dominated. This was already postulated in Mancera Piña et al. (2019b), but it is now confirmed with a well traced rotation curve. Fitting cold dark matter halos In our current framework of galaxy formation, we expect every galaxy to be embedded in a cold dark matter (CDM) halo. Because of this, it is relevant to investigate whether or not physically motivated CDM haloes can be consistent with our rotation curve, even if Fig. 4 suggests the absence of a dynamically significant halo in AGC 114905. We aim to find a dark matter halo whose circular speed DM meets 2 c = 2 * + 2 gas + 2 DM . Dark matter haloes are often described with the so-called NFW profile (Navarro et al. 1997), whose density as a function of the spherical radius in cylindrical coordinates ( = √︁ 2 + 2 ) is given by NFW ( ) = 4 s ( / s ) (1 + / s ) 2 ,(3) where s is a 'scale radius' and s is the density at s . We will denote the corresponding mass profile as NFW ( ): NFW (< ) = 200 ln(1 + 200 ) − 200 1 + 200 ln 1 + s − s 1 + s −1 .(4) The parameter 200 is defined as the mass within a sphere with radius 200 within which the average density is 200 times the critical density of the universe, while the concentration 200 is defined as 200 = 200 / s . While NFW haloes provide good descriptions for massive galaxies, this is not the case for dwarf galaxies (see Bullock & Boylan-Kolchin 2017). Therefore, for our UDG, we assume that the dark matter halo is described by a ' NFW' profile (Read et al. 2016a), which is an extension of the classical NFW profile that has the flexibility to develop -or not-a core. In Section 5.2 we discuss other halo profiles. The NFW profile has been found to fit very well rotation curves of dwarf galaxies, both real and simulated (Read et al. 2016a,b). The density profile of the NFW halo can be written as coreNFW ( ) = NFW ( ) + −1 (1 − 2 ) 4 2 c NFW ( ) .(5) Here, NFW and NFW are the above NFW parameters, while is a function (defined as = tanh( / c )) that generates a core of size c . In principle, c can be a fitting parameter, but as discussed in detail by Read et al. (2016aRead et al. ( ,b, 2017, fixing it to c = 2.94 d 6 is in good agreement with simulations and observations where c is fitted as free parameter. Importantly, the factor 2.94 cannot be significantly larger as there is not enough energy from supernovae to create cores of size much larger than 2.94 d (see also e.g. Benítez-Llambay et al. 2019;Lazar et al. 2020;Trujillo-Gomez et al. 2021). The degree of transformation from cusp to core is described by the parameter , with = 0 defining a cuspy NFW profile and = 1 a completely cored profile. The parameter is defined as = tanh( SF / dyn ), with = 0.04 a fixed parameter, SF the time whilst the galaxy has been forming stars (set to 14 Gyr), and dyn the NFW dynamical time at the scale radius s , which can be expressed in terms of NFW , s , 6 In principle, c = 1.75 e , with e the half-light radius. For an exponential profile ( e = 1.678 d ) this becomes c = 2.94 d . and (the Newtonian gravitational constant) as dyn = 2 √︄ 3 s NFW (< s ) .(6) The dark matter profile has then the same two free parameters as a NFW halo: the mass of the halo ( 200 ) and its concentration ( 200 ). N-body cosmological simulations find a strong correlation between 200 and 200 (e.g. Dutton & Macciò 2014;Ludlow et al. 2014), so in practice one can even fit NFW-like profiles with one single parameter. While the the other parameters of the halo ( , s , c ), are not considered free parameters, they also change on each step of the MCMC, as they depend on 200 and 200 as described above. To find the best-fitting CDM halo we use a MCMC routine (also based on emcee) that minimises the residuals of Eq. 2 using a standard exp(−0.5 2 ) function as likelihood, with 2 given by 2 = ∑︁ ( c − c,mod ) 2 2 c ,(7) where c and c,mod are the observed and model circular speed profiles, respectively, and c is the 3D Barolo uncertainty in the kinematic modelling, which we assume to be Gaussian (see Di Teodoro 2015). As we discuss next, the inclination is a free parameter in our MCMC, and thus c itself does not include the contribution from the inclination uncertainty; these error bars are, therefore, smaller than those shown in Figs. 3 and 4. The MCMC explores the ( 200 , 200 ) parameter space and retrieves the best-fitting combination. In addition to 200 and 200 , we include the distance and the inclination as nuisance parameters. In practice, we impose a Gaussian prior on centered at 76 Mpc and with a standard deviation of 5 Mpc, exploring the 2 range 66 ≤ /Mpc ≤ 86. Similarly, for , we impose a Gaussian prior centered at 32 • with a standard deviation of 3 • , within 26 • ≤ ≤ 38 • ; in Section 5 we also discuss the case where the priors for and are wider. It is worth pointing out that a change in introduces a change in the conversion factor between arcsecond and kpc, thus modifying our sampling of the rotation curve. Additionally, it affects the value of d , which in turn changes c and the thickness of the stellar disc. On the other hand, affects the overall normalization of the rotation curve and of the gas circular speed profile. Having established this, we explore different scenarios, which differ by our chosen priors on 200 −3 , = (76 ± 5) Mpc, and = 33 ± 3 • . While the resulting fit is close to the data (given that bar ≈ c and DM is subdominant), the value of 200 ∼ 10 8 is too low to be plausible in a CDM cosmology. Given bar = 1.4×10 9 , the very minimum expected 200 (assuming the galaxy has a baryon fraction as high as the cosmological average: c,bar 0.16, see Cimatti et al. 2019) is about 0.9 × 10 10 . It is clear that the MCMC routine finds a low mass since c bar , but the resulting halo does not seem to have a physical justification. Taking the above into consideration, we decided to impose a lower boundary to the prior of 200 such that the minimum halo would produce bar / 200 0.16 (i.e. the cosmological baryon fraction). With this, the prior for 200 becomes 10 ≤ log( 200 / ) ≤ 12. We stress that the lower limit on the prior corresponds to the minimum expected value of 200 . In theory one expects the galaxy to have a significantly larger 200 . For example, the ΛCDM stellar-to-halo mass relation from Posti et al. (2020) In what follows, we refer to these two scenarios as Case 1 and Case 2, respectively. The posterior distributions of both Case 1 and Case 2 are shown in Appendix B. Somewhat unsurprisingly, for both cases the MCMC finds log( 200 / ) 10, with posterior distributions that simply try to go to the lower bound. In Case 1, (imposing the Gaussian prior on the 200 − 200 relation), we find = (67 ± 1) Mpc and = 26.1 •+0.2 −0.1 , with posterior distributions also trying to go towards their lower bounds (see Fig. B2). The concentration, on the other hand, is well constrained, and we find 200 = 11.7 ± 0.3. The other parameters of the NFW profile are = 0.7, c = 4.6 kpc, and s = 3.8 kpc. For Case 2, while and are well constrained following their priors ( = 73 ± 4 Mpc, = 29 • ± 2 • ), the posterior distribution of Fig. 5 shows the two resulting mass models. Case 1, on the left panel, is in clear disagreement with the data as it significantly overestimates c , even when the distance and inclination go to their lowest allowed values. Case 2, on the right panel, lies closer to the data but presents other problems, as we discuss in the next Section. DISCUSSION Having presented our main results, we now discuss their implications. Provided our rotation curve for AGC 114905 faithfully traces its circular speed, the fact that it is baryon-dominated out to the outermost observed radius (Fig. 4) implies two possible scenarios is a ΛCDM context: that our UDG lacks a significant amount of dark matter across all radii (even beyond the range probed by our data), or that it has a peculiar dark matter halo with little mass within 10 kpc (e.g. right panel in Fig. 5). AGC 114905 compared to 'dark-matter free' galaxies Since van Dokkum et al. (2018) and van Dokkum et al. (2019) postulated that DF-2 and DF-4 have very low or no dark matter content, different mechanisms to create such peculiar galaxies have been proposed. One of the leading ideas is that high-velocity (∼ 300 km s −1 ) collisions between gas-rich dwarf galaxies can create dark-matter free (or almost dark-matter free, DM ∼ 10 5 ) galaxies (Silk 2019;Shin et al. 2020). Importantly, those types of galaxies are expected to form in dense environments and to have a baryonic mass dominated by stars rather than cold gas. Another mechanism proposed to explain the existence of DF-2 and DF-4 are tidal interactions with massive neighbouring galaxies that strip the dark matter away (Jackson et al. 2021;Doppel et al. 2021); Montes et al. (2020) claim that in fact DF-4 currently shows signs of such interactions. While the above scenarios can manage to produce dark-matter poor, UDG-like galaxies that show some degree of similarity with DF-2 and DF-4, it is important to bear in mind that gas-rich UDGs are rather different objects. Not only are they gas-dominated ( gas 0.9 for AGC 114905), but they are also isolated (by selection, see Leisman et al. 2017). In the specific case of AGC 114905, the nearest galaxy within a recession velocity of 500 km s −1 with confirmed (optical or HI) redshift, is the faint dwarf AGC 114806 at a projected distance of 2.1 Mpc and with a systemic velocity within a few km s −1 . Using data from the Sloan Digital Sky Survey (Alam et al. 2015), we also looked for possible unconfirmed massive companions of AGC 114905. We explored the area within 45 arcmin of AGC 114905, corresponding to a circle of radius 1 Mpc at the distance of AGC 114905, querying for galaxies with e ≥ 1 kpc and with color ( − ) ≤ 1 mag. In this region, there are only seven galaxies with * 10 9 (assuming a distance of 76 Mpc and the mass-to-light-colour relation from Du et al. 2020) with unknown distance. All of them resemble confirmed background red galaxies, and the closest in projection lies at 700 kpc. All this evidence, together with the lack of tidal features in the optical and H morphology of our UDG, suggests that it is truly isolated. An idea that could reconcile a tidal origin with the current isolation of AGC 114905 is that it is an old tidal dwarf galaxy (TDG, e.g., Duc et al. 2014), since TDGs are expected to have a low dark matter content and low rotation velocities (e.g. Hunter et al. 2000;Lelli et al. 2015). If the interaction that originated the TDG happened at high redshift ( ∼ 4 − 6) and the galaxy had an escape velocity of ∼ 400 km s −1 , the parent galaxy would lie today at distances about ∼ 5 Mpc from AGC 114905. While this scenario is impossible to test in practice, the population of known old TDGs in the nearby universe both in observations and simulations are found at much closer distances and recessional velocities from their parent galaxies than what AGC 114905 (and the similar gas-rich UDGs from Mancera Piña et al. 2019bPiña et al. , 2020 is from any massive galaxy (Hunter et al. 2000;Kaviraj et al. 2012;Duc et al. 2014;Haslbauer et al. 2019). Overall, while is difficult to give a final answer, it seems unlikely that the small (if any) amount of dark matter in AGC 114905 can be attributed to the above mentioned mechanisms perhaps valid for DF-2 and DF-4. Recently, Trujillo-Gomez et al. (2021) proposed a semi-empirical model where strong feedback from globular clusters can produce UDGs with dark matter cores as large as 10 − 30 kpc. However, the model does not include a detailed treatment of the gas component which is the dominant mass budget of gas-rich UDGs, and a thorough comparison with our data is not yet possible to carry out. It would be instructive to obtain information about the kinematics of AGC 114905 beyond 10 kpc, where the contribution of stars and gas becomes smaller and would produce a declining rotation curve. Instead, if a flat rotation curve were to be found, it would suggest the presence of dark matter. In the next Section, motivated by our results in Fig. 5, we discuss which type of CDM haloes are in agreement or disagreement with our observations. The 200 of a CDM halo for AGC 114905 is too low It follows from Drawing conclusions from those galaxies might be less straightforward: the rotation curve of AGC 242019 does not seem to reach its flat part, while the data of UGC 2165 have low resolution and its rotation curve (apparently rising as solid-body) is significantly oversampled. Still, it is interesting that similarly low values of 200 are reported. It is important to highlight that while low surface brightness galaxies have been historically found to inhabit low-concentration haloes (e.g. McGaugh et al. 2003), the concentrations of those haloes are still usually in broad to good agreement with ΛCDM cosmology (Macciò et al. 2007), while the concentration of AGC 114905 is rejected at a high significance level. Given the volume explored by Leisman et al. (2017) when building the parent sample of AGC 114905 (∼ 10 6 Mpc 3 , see Haynes et al. 2018;Jones et al. 2018), finding a single galaxy with the properties of AGC 114905 should be practically impossible in a CDM Universe. This result becomes even stronger considering the rest of the sample studied in Mancera Piña et al. (2019bPiña et al. ( , 2020 possibly shows similar properties, even if slightly less extreme as AGC 114905 presents the largest offset from the BTFR. In this context, it is important to bear in mind that gas-rich UDGs as a whole population have significantly narrower velocity widths (a proxy for their rotation velocities) than galaxies of similar mass (Leisman et al. 2017;Jones et al. 2018). We also note here that the implausibility of the CDM halo needed in AGC 114905 is not just related with the cusp-core problem (Bullock & Boylan-Kolchin 2017), since by fitting a NFW profile we do not force the halo to be cored or cuspy per se. It is also clear that the scales at which dark matter is deficient in AGC 114905 (10 kpc) are larger than any realistic core size for dwarf galaxies in both observations and simulations (e.g. Read et al. 2016aRead et al. , 2017Lazar et al. 2020). To further explore this, we performed a run of our MCMC routine where c is kept as a free parameter. In practice, we use a flat prior exploring the range 0 ≤ c /kpc ≤ 44. The maximum value of 44 kpc is chosen because it is the value of 200 given 200 = 10 10 . Additionally, we impose a minimum value on the prior of 200 , log( 200 / ) = 10, as well as the Gaussian prior on the 200 − 200 relation. The MCMC routine finds the parameters log( 200 / ) ≈ 10 10 , 200 ≈ 12, ≈ 71 Mpc, ≈ 27 • , and c ≈ 41 kpc, with the and c posterior distributions simply going to their minimum and maximum allowed values, respectively. While the fit is just slightly worse than Case 2 (right panel of Fig. 5) it seems non-physical. Expressing the core radius as c = e implies ∼ 15. As discussed by Read et al. (2017), there is not enough supernovae energy in galaxies to drive > 2.75, and = 1.75 fits real and simulated galaxies well. Even if other energy sources (e.g. Cimatti et al. 2019) can affect the distribution of dark matter in galaxies, it seems unlikely that they would contribute much more than supernovae, as required to achieve c ≈ 200 . Finally, it is worth clarifying that the problem of fitting a CDM halo to AGC 114905 is not restricted to specific functional forms such as the NFW profile. In addition to NFW, we also try with the Einasto halo, which allows the Einasto profile to develop a core, and which has been found by Lazar et al. (2020) to successfully reproduce the cored dark matter profile of a variety of galaxies in the FIRE-2 simulations . For this profile we also impose a minimum log( 200 / ) = 10, but the only way to find agreement with our data is again if the size of the core is as large as 200 . Overall, the existence of galaxies like AGC 114905 seems to pose a major challenge for CDM haloes. An interesting line of research is to explore whether or not the current issues can be mitigated by invoking a different type of dark matter (e.g. Kaplinghat et al. 2020;Yang et al. 2020). AGC 114905 challenging MOND Modified Newtonian dynamics (MOND, Milgrom 1983;Sanders & McGaugh 2002) is an alternative approach to dark matter theories which aims to explain dark matter physics by invoking a modification to Newtonian dynamics. One of the major achievements of MOND is how well it predicts rotation curves of galaxies (e.g. Sanders & McGaugh 2002;Famaey & McGaugh 2012 and references therein). Rather than a fit, MOND makes a direct prediction of the shape of the rotation curve given the stellar and gas mass distributions. The only other parameter is 0 1.2 × 10 −8 cm s −2 , postulated to be a universal constant. The fact that some isolated UDGs are off the BTFR already posed a challenge to MOND, which predicts a tight (zero intrinsic scatter) BTFR with slope of 4 for isolated galaxies. Yet, some doubts about this may exist, as it could be argued that the circular speeds reported in Mancera Piña et al. (2019bPiña et al. ( , 2020 were not tracing the flat part of the rotation curve or were too affected by the resolution; here we have shown that this is not the case for AGC 114905. Following Gentile (2008), in the absence of an 'external' gravitational field of a neighbouring massive galaxy, the MOND rotation curve can be written as We stress that the above formula is applicable to our UDG given the lack of massive galaxies in its vicinity. In order to test this prediction, we performed another MCMC fit with and as free parameters (which affect both c and MOND ), following the same Gaussian priors as for our Case 1 and Case 2 above. Even with the posterior distributions for and going to their lower bounds ( ≈ 66 Mpc and ≈ 26 • , see Fig. B4 in Appendix B), the MOND prediction markedly overestimates the circular speed of our UDG (consistent with the offset from the BTFR), as we show in Fig. 6. There may be also some tension with the shape of the rotation curve, which is not predicted to be flat as in the observations. Therefore, our UDG seemingly presents a challenge to MOND, which can only be reconciled by invoking a much lower inclination, as we discuss in Section 5.4. The effects of a lower inclination As described in Section 3, we measure the inclination of AGC 114905 to be 32 • ± 3 • , using a well tested method that relies exclusively on the total H map and is independent of the kinematics of the galaxy and our posterior kinematic modelling. While we argue that our inclination is robust, our results are certainly dependent on it. In particular, if AGC 114905 had a much lower inclination, the amplitude of its rotation curve would be significantly larger, having more room for dark matter within the observed radii and potentially alleviating some of the tensions presented in this paper (see for instance the case of IC 1613 in Oman et al. 2016). Given this, it is interesting to quantify by how much the inclination Similarly, an inclination of = 10.8 ± 0.3 • would be needed in order to find agreement between the MOND prediction and the c profile of AGC 114905, at least on average, since the shape predicted by MOND seems to also differ from our rotation curve. Note, however, that a radially varying inclination could potentially alleviate this tension between the rotation curves shapes. The above values for the inclination are about 20 • degrees off the inclination we determined in Section 3. This is a discrepancy a factor 6-7 larger than the nominal uncertainty estimate of our measurement (see Fig. B1), although inclinations below ∼ 25 • become increasingly difficult to constrain as ellipses with lower inclinations all have very similar shapes. We can also inspect visually if inclinations as low as 11 • − 15 • can be consistent with the data. In Fig. 7 axisymmetric discs (but see Section 3.1). The model at 32 • does an overall good job at following the H contour, while the contour for the model at 11 • appears inconsistent with it, being significantly more elongated along the minor axis. This is also shown in Fig. A1 where we compare the channel maps of AGC 114905 with the channel maps of our best-fitting model and those of a model with an inclination of 11 • . Given all of the above, we find it unlikely that we are severely overestimating the inclination of our UDG, although this remains the largest source of uncertainty in our analysis. Something else to consider is that there are other gas-rich UDGs showing a similar set of properties, all at different inclinations (Mancera Piña et al. 2019bPiña et al. , 2020, see also Sengupta et al. 2019;Shi et al. 2021, and the spatially unresolved data from e.g. Leisman et al. 2017;Karunakaran et al. 2020). This means that the inclinations of all of them would need to be overestimated by a large margin. Still, it is desirable to repeat our analysis with a gas-rich UDG at a similar resolution as we have now for AGC 114905, but at higher inclination, and we aim to do this in the near future. CONCLUSIONS We obtained new H interferometric observations of the gas-rich ultra-diffuse galaxy (UDG) AGC 114905 using the Karl G. Jansky Very Large Array in its B-, C-and D-configurations. The new data, tracing the H emission up to 10 kpc from the galaxy centre, have a spatial resolution a factor about 2.5 higher than previous data, and confirm that AGC 114905 has a regularly rotating gas disc. We performed 3D kinematic modelling of the data cube using 3D Barolo, which allowed us to recover the intrinsic rotation curve and velocity dispersion profile of the galaxy. AGC 114905 has a regular rotation curve that reaches a flat part with a circular speed (after a minor correction for asymmetric drift) of about 23 km s −1 . This result confirms that this UDG lies off the baryonic Tully-Fisher relation, as suggested by Mancera Piña et al. (2019bPiña et al. ( , 2020 with low-resolution data. The observed circular speed profile of our UDG can be explained almost entirely by the contribution of the baryons alone, with little room for dark matter within our observed outermost radius ( out ≈ 10 kpc). Moreover, we found that the circular speed profile cannot be reproduced by standard cold dark matter (CDM) halos: the only possibility to find a good fit to the data is if the concentration of the halo is as low as ∼ 0.3, completely off CDM expectations. We tested whether the rotation of our UDG is instead reproduced within the MOND framework, but we find that there is a significant mismatch on the normalization and shape of the MOND rotation curve with respect to our observations. The geometry of the system (assumed to be an inclined axisymmetric disc) is the main source of uncertainty in our results. The inclination of AGC 114905 (32 ± 3 • ), which we measure from its total H map independently of its kinematics, is a significant caveat, but a number of independent pieces of evidence suggest that it cannot be overestimated to the extent of significantly changing the above results. Efforts to observe another gas-rich UDG at a similar spatial resolution but at higher inclination are under way. Finally, it is important to consider that we have confirmed for one UDG the robustness of the results obtained by Mancera Piña et al. (2019bPiña et al. ( , 2020 at low resolution. The fact that the six UDGs (and see also e.g. Leisman et al. 2017;Shi et al. 2021) at different inclinations show the same behavior argues in favor of them being really exotic and suggests that our results are not the byproduct of systematic uncertainties. We have strengthened and clarified previous results on the nature and startling dynamics of gas-rich UDGs. Yet, their origin and precise evolutionary pathways remain largely a mystery. The present work has also shown that gas-rich UDGs are a promising population to study dark matter, as they can potentially provide telltale clues to understand its nature. Fig. A1 shows representative channel maps of the data cube of AGC 114905. The observed emission is shown in grey background and dark blue contours (open contours for negative values). The green cross shows the centre of the galaxy and the velocity of each channel map is given on the bottom right corner of each panel. In red, we show the contours for the best-fitting 3D Barolo azimuthal tilted-ring model; while low S/N features are not fully reproduced, the model captures well the overall kinematics of the galaxy, as also shown in Fig. 2 with the PV diagrams. We also overlay in light blue the contours for a model with a fixed inclination of 11 • (as needed to match CDM and MOND expectations, see Section 5.4). A close inspection shows that the model at 11 • has an excess of flux along the minor axis in the channels close to the systemic velocity; the model at 32 • does a better job in this regard (however, this comparison is better appreciated in Fig. 7). Moreover, the model at 32 • matches in a better way the spectral extent of the observations. APPENDIX A: CHANNEL MAPS APPENDIX B: MCMC POSTERIOR DISTRIBUTIONS In this appendix we provide the main posterior distributions obtained with our MCMC analyses as described in the main text. This paper has been typeset from a T E X/L A T E X file prepared by the author. Dec. Data Figure 1 . 1Left: Stellar image of AGC 114905 with the total H contours overlaid. The contours are at 1, 2, 4×10 20 atoms cm −2 , the noise level is 4.1×10 19 atoms cm −2 . Middle: Total H intensity map; contours as in the previous panel. The grey ellipse shows the beam of our data. Right: Stellar (orange) and gas (blue, includes helium correction) surface mass density profiles of AGC 114905. The dashed black lines on top show the fits to the distributions used to obtain the stellar and gas circular speeds (see Section 4). Figure 2 . 2Top: Observed velocity field (same physical scale as the total H map inFig. 1); the grey ellipse shows the beam of the observations, the grey line the kinematic major axis, and the black cross the kinematic centre. Middle (Bottom): Major (minor)-axis PV diagram; data are shown in blue background and black contours (grey for negative values), and the best-fitting 3D Barolo azimuthal model in red contours. The yellow points show the recovered rotation velocities. Figure 3 . 3Circular speed (grey circles) and velocity dispersion (blue circles) profiles of AGC 114905, as obtained with our kinematic modelling. Squares show previous results obtained at a lower spatial resolution. Figure 4 . 4Circular speed profile of AGC 114905 (red points) compared to the contribution expected from stars (orange line), gas (blue line), and baryons (stars plus gas, magenta line). and 200 . In a very first attempt, we use the flat priors 6 ≤ log( 200 / ) ≤ 12 and 0.1 ≤ 200 ≤ 30. However, 200 remains completely unconstrained as its posterior distribution is flat over all the explored range. Upon imposing the 200 − 200 relation of Dutton & Macciò (2014) as a Gaussian prior on 200 , we find log( 200 / ) = 7.6 +0.7 −1.0 , 200 = 21 +5 200 we explore two scenarios: one where we impose again the 200 − 200 relation of Dutton & Macciò (2014) as a Gaussian prior in the MCMC routine, and one where 200 has a flat wide prior 0.1 ≤ 200 ≤ 30. 200 goes to its lower bound, 200 = 0.3 +0.3 −0.2 (see Fig. B3). The other parameters of the NFW profile are log( 200 / ) = 10.2, = 0.03 (i.e. NFW ≈ NFW) driven by a high dyn (Eq. 6), c = 5.1 kpc, and s = 166 kpc, driven by the extremely low 200 ( s = 200 / 200 ). Figure 5 . 5Fig. 5that it does not seem possible to fit the circular speed profile of AGC 114905 with a CDM-motivated 200 . As mentioned above, if the 200 − 200 relation is imposed (Case 1, left panel ofFig. 5), it fails by a large margin at reproducing the amplitude of the circular speed profile. Case 2 (right panel ofFig. 5), fitting a free Mass models of AGC 114905. Case 1 and Case 2 are shown on the left and right panels, respectively. In both panels the red points show the c profile of AGC 114905, while the dashed magenta lines represent the c expected from the baryons (stars plus gas). The dark matter haloes are shown with black lines, and the red lines give the total contribution of baryons and dark matter together. Case 1, which follows the CDM 200 − 200 relation, is inconsistent with the observations. Case 2 fits the data better, but it has a 200 too low for CDM. Note also that the assumed distance and inclination are different between both panels. Because the assumed distance is different on each panel, the sampling of the rotation curve along the horizontal axes is also different. In a similar way, the normalization of the rotation curves differ from each other due to the different inclinations. See the text for details. simulations 7 (e.g.Dutton & Macciò 2014;Ludlow et al. 2014) and it might even be non-physical:McGaugh et al. (2003) argue that CDM haloes with 200 < 2 are not produced in any sensible cosmology.Sengupta et al. (2019) andShi et al. (2021) also suggested that the gas-rich UDGs AGC 242019 and UGC 2165, respectively, have a 200 around 2. 7 Assuming that the scatter of the 200 − 200 relation measured at high masses ( log( 200 ) = 0.11 dex, see Dutton & Macciò 2014) is applicable also at 200 10 10 , then 200 of AGC 114905 is about 15 below the expected value, although this number could be reduced if the 200 − 200 or its scatter depart from Gaussian (Kong, D. et al. in prep.). Figure 6 . 6MOND prediction (green line) of the circular speed profile of AGC 114905 (red points). The baryonic circular speed profile is shown with a magenta line. of our UDG would need to decrease to make it consistent with the CDM (and MOND) expectation. For this exploration we assume a NFW profile with c = 2.94 d . The 'minimum' expectation is such that the galaxy has log( 200 / ) = 10 and a 200 in agreement (within some scatter) with the 200 − 200 relation. We run again our MCMC routine, but this time we use wider Gaussian priors for and : 50 ≤ /Mpc ≤ 100 and 5 • ≤ < 85 • , with centre and standard deviations as in Case 1. The resulting parameters are log( 200 / ) = 10.03 +0.06 −0.03 , 200 = 12.1 ± 0.3, = (72.7 ± 5) Mpc, and = (15 ± 1) • . The low inclination brings up the circular speed of AGC 114905 to velocities around 45 km s −1 ,which are consistent with a NFW halo similar to the halo in Case 1 (the other parameters are = 0.73, c = 5 kpc, and s = 3.8 kpc). With an inclination of 15 • the galaxy would be still quite puzzling, as it would have a baryon fraction about 70 per cent the value of the cosmological average (90 per cent if = 76 Mpc), as opposed to most dwarf galaxies that have low baryon fractions of a few per cent (McGaugh et al. 2010; Read et al. 2017). Additionally, AGC 114905 would still lie off the BTFR. If one instead imposes log( 200 / ) = 10.6 (which gives a baryon fraction of about 20 per cent) assuming the ΛCDM stellar-to-halo mass relation from Posti et al. (2020), a corresponding 200 = 11.5 (following the 200 − 200 relation), and = 76 Mpc, the needed inclination is 10.4 ± 0.3 • . Figure 7 . 7we show the outer contour of the H map of AGC 114905 overlaid on the −band optical image, and we compare it with the equivalent contours of two 3D Barolo azimuthal models (convolved with the observed beam) of AGC 114905 at different inclinations. The models are razor-thin Comparison between the outer contours (S/N = 3) of the H map of AGC 114905 (white) and two azimuthal models at different inclinations. While the model at 32 • (solid black line) provides a good fit to the data, the model at 10 • (dashed blue) is significantly more elongated than the data along the minor axis. The background shows the optical image of AGC 114905. Figure A1 .Figure B1 .Figure B4 . A1B1B4Representative channel maps of AGC 114905. The emission of the galaxy is shown in grey background and dark blue contours (open contours for negative values). The green crosses show the centre of the galaxy, and we indicate the velocity corresponding to each channel map on the bottom right corner. The contours for the best-fitting 3D Barolo azimuthal tilted-ring model are shown in red, while the contours for a model at 11 • are shown in light blue. Contours are at -2, 2, 4 times the rms noise per channel. MCMC posterior distribution of the inclination of AGC 114905. The central value, shown in blue, is the median of the distribution, while the uncertainties represent the difference between the median and the 16 th and 84 th percentiles (dashed black lines). See Section 3.1 for details. MCMC posterior distribution for the MOND model. Lines are as in Fig. B1. See Section 5.3 for details. Pavel E. ManceraPiña et al. The ALFALFA flux would instead imply HI ≈ 1.3 × 10 9 , only strengthening the results shown below. MNRAS 000, 1-12(2015) https://gitlab.com/iogiul/galpynamics/ MNRAS 000, 1-12(2015) , does a better job and it is consistent with the circular speed profile within the uncertainties. However, 200 is too low and completely off the expected 200 − 200 relation that emerges from cosmological MNRAS 000, 1-12(2015) ACKNOWLEDGEMENTSWe would like to thank Federico Lelli and Justin Read for useful comments on this manuscript, as well as Alessandro Romeo for useful discussions on disc stability. We appreciate the feedback from an anonymous referee. P.E.M.P. and F.F. are supported by the Netherlands Research School for Astronomy (Nederlandse Onderzoekschool voor Astronomie, NOVA), Phase-5 research programme Network 1, Project 10.1.5.6. E.A.K.A. is supported by the WISE research programme, which is financed by the Netherlands Organization for Scientific Research (NWO). K.A.O. acknowledges support by the European Research Council (ERC) through Advanced Investigator grant to C.S. Frenk, DMIDAS (GA 786910).This work is based on observations made with the Karl G. Jansky Very Large Array (VLA). The VLA is a facility of the National Radio Astronomy Observatory (NRAO). NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.We have used extensively SIMBAD and ADS services, as well the Python packages NumPy(Oliphant 2007), Matplotlib(Hunter 2007), SciPy(Virtanen et al. 2020), Astropy (Astropy Collaboration et al. 2018), spectral-cube(Ginsburg et al. 2019), and corner (Foreman-Mackey 2016), for which we are thankful.DATA AVAILABILITYThe datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. . S Alam, 10.1088/0067-0049/219/1/12ApJS. 21912Alam S., et al., 2015, ApJS, 219, 12 . N C Amorisco, A Loeb, 10.1093/mnrasl/slw055MNRAS. 45951Amorisco N. C., Loeb A., 2016, MNRAS, 459, L51 . 10.3847/1538-3881/aabc4fAJ. 156123Astropy Collaboration et al., 2018, AJ, 156, 123 . C Bacchini, F Fraternali, G Pezzulli, A Marasco, 10.1051/0004-6361/202038962A&A. 644125Bacchini C., Fraternali F., Pezzulli G., Marasco A., 2020, A&A, 644, A125 . C E Barbosa, 10.3847/1538-4365/ab7660ApJS. 24746Barbosa C. E., et al., 2020, ApJS, 247, 46 . K G Begeman, C S Frenk, A D Ludlow, J F Navarro, 10.1093/mnras/stz1890MNRAS. 4882387Kapteyn Astronomical Institute, University of Groningen Benítez-Llambay A.PhD thesisBegeman K. G., 1987, PhD thesis, Kapteyn Astronomical Institute, University of Groningen Benítez-Llambay A., Frenk C. S., Ludlow A. D., Navarro J. F., 2019, MNRAS, 488, 2387 . M A Bershady, M A W Verheijen, K B Westfall, D R Andersen, R A Swaters, T Martinsson, 10.1088/0004-637X/716/1/234ApJ. 716234Bershady M. A., Verheijen M. A. W., Westfall K. B., Andersen D. R., Swaters R. A., Martinsson T., 2010, ApJ, 716, 234 . A Bosma, 10.1146/annurev-astro-091916-055313ARA&A. Bullock J. S., Boylan-Kolchin M55343PhD thesisBosma A., 1978, PhD thesis, - Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343 . T Carleton, R Errani, M Cooper, M Kaplinghat, J Peñarrubia, Y Guo, 10.1093/mnras/stz383MNRAS. 485382Carleton T., Errani R., Cooper M., Kaplinghat M., Peñarrubia J., Guo Y., 2019, MNRAS, 485, 382 . N Chamba, I Trujillo, J H Knapen, 10.1051/0004-6361/201936821A&A. 6333Chamba N., Trujillo I., Knapen J. H., 2020, A&A, 633, L3 . T K Chan, D Kereš, A Wetzel, P F Hopkins, C A Faucher-Giguère, K El-Badry, S Garrison-Kimmel, M Boylan-Kolchin, 10.1093/mnras/sty1153MNRAS. 478906Chan T. K., Kereš D., Wetzel A., Hopkins P. F., Faucher-Giguère C. A., El-Badry K., Garrison-Kimmel S., Boylan-Kolchin M., 2018, MNRAS, 478, 906 A Cimatti, F Fraternali, C Nipoti, Introduction to Galaxy Formation and Evolution: From Primordial Gas to Present-Day Galaxies. Cambridge University PressCimatti A., Fraternali F., Nipoti C., 2019, Introduction to Galaxy Formation and Evolution: From Primordial Gas to Present-Day Galaxies. Cambridge University Press . C J Conselice, 10.3847/2515-5172/aab7f6Research Notes of the American Astronomical Society. 243Conselice C. J., 2018, Research Notes of the American Astronomical Society, 2, 43 . S Danieli, P Van Dokkum, C Conroy, R Abraham, A J Romanowsky, 10.3847/2041-8213/ab0e8cApJ. 87412Danieli S., van Dokkum P., Conroy C., Abraham R., Romanowsky A. J., 2019, ApJ, 874, L12 . Di Cintio, A Brook, C B Dutton, A A Macciò, A V Obreja, A Dekel, A , 10.1093/mnrasl/slw210MNRAS. 4661Di Cintio A., Brook C. B., Dutton A. A., Macciò A. V., Obreja A., Dekel A., 2017, MNRAS, 466, L1 . Di Teodoro, E M , 10.1093/mnras/stv1213MNRAS. Bologna Di Teodoro E. M., Fraternali F.4513021University ofPhD thesisDi Teodoro E. M., 2015, PhD thesis, University of Bologna Di Teodoro E. M., Fraternali F., 2015, MNRAS, 451, 3021 . Di Teodoro, E M Fraternali, F Miller, S H , 10.1051/0004-6361/201628315A&A. 59477Di Teodoro E. M., Fraternali F., Miller S. H., 2016, A&A, 594, A77 . J E Doppel, L V Sales, J F Navarro, M G Abadi, E W Peng, E Toloba, F Ramos-Almendares, 10.1093/mnras/staa3915MNRAS. 5021661Doppel J. E., Sales L. V., Navarro J. F., Abadi M. G., Peng E. W., Toloba E., Ramos-Almendares F., 2021, MNRAS, 502, 1661 . W Du, C Cheng, Z Zheng, H Wu, 10.3847/1538-3881/ab6efbAJ. 159138Du W., Cheng C., Zheng Z., Wu H., 2020, AJ, 159, 138 . P.-A Duc, S Paudel, R M Mcdermid, J.-C Cuillandre, P Serra, F Bournaud, M Cappellari, E Emsellem, 10.1093/mnras/stu330MNRAS. 4401458Duc P.-A., Paudel S., McDermid R. M., Cuillandre J.-C., Serra P., Bournaud F., Cappellari M., Emsellem E., 2014, MNRAS, 440, 1458 . A A Dutton, A V Macciò, 10.1093/mnras/stu742MNRAS. 4413359Dutton A. A., Macciò A. V., 2014, MNRAS, 441, 3359 . B G Elmegreen, D A Hunter, 10.1088/0004-637X/805/2/145ApJ. 805145Elmegreen B. G., Hunter D. A., 2015, ApJ, 805, 145 . E Emsellem, 10.1051/0004-6361/201834909A&A. 62576Emsellem E., et al., 2019, A&A, 625, A76 . B Famaey, S S Mcgaugh, 10.12942/lrr-2012-10Living Reviews in Relativity. 1510Famaey B., McGaugh S. S., 2012, Living Reviews in Relativity, 15, 10 . A Fattahi, 10.1093/mnras/stv2970MNRAS. 457844Fattahi A., et al., 2016, MNRAS, 457, 844 . D Foreman-Mackey, 10.21105/joss.00024The Journal of Open Source Software. 124Foreman-Mackey D., 2016, The Journal of Open Source Software, 1, 24 . D Foreman-Mackey, D W Hogg, D Lang, J Goodman, 10.1086/670067PASP. 125306Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306 . F Fraternali, A Karim, B Magnelli, C Gómez-Guijarro, E F Jiménez-Andrade, A C Posses, 10.1051/0004-6361/202039807A&A. 647194Fraternali F., Karim A., Magnelli B., Gómez-Guijarro C., Jiménez-Andrade E. F., Posses A. C., 2021, A&A, 647, A194 . L Gault, 10.3847/1538-4357/abd79dApJ. 90919Gault L., et al., 2021, ApJ, 909, 19 . G Gentile, 10.1086/590048ApJ. 6841018Gentile G., 2008, ApJ, 684, 1018 . A Ginsburg, 10.5281/zenodo.2573901radio-astro-tools/spectral-cube: v0.4.4Ginsburg A., et al., 2019, radio-astro-tools/spectral-cube: v0.4.4, doi:10.5281/zenodo.2573901 . M Haslbauer, J Dabringhausen, P Kroupa, B Javanmardi, I Banik, 10.1051/0004-6361/201833771A&A. 62647Haslbauer M., Dabringhausen J., Kroupa P., Javanmardi B., Banik I., 2019, A&A, 626, A47 . M P Haynes, 10.3847/1538-4357/aac956ApJ. 86149Haynes M. P., et al., 2018, ApJ, 861, 49 . A M Hidalgo-Gámez, Rev. Mex. Astron. Astrofis. 4037Hidalgo-Gámez A. M., 2004, Rev. Mex. Astron. Astrofis., 40, 37 . P F Hopkins, 10.1093/mnras/sty1690MNRAS. 480800Hopkins P. F., et al., 2018, MNRAS, 480, 800 . J D Hunter, 10.1109/MCSE.2007.55Computing in Science & Engineering. 990Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 . D A Hunter, B G Elmegreen, A L Baker, 10.1086/305158ApJ. 493595Hunter D. A., Elmegreen B. G., Baker A. L., 1998, ApJ, 493, 595 . D A Hunter, S D Hunsberger, E W Roye, 10.1086/309542ApJ. 542137Hunter D. A., Hunsberger S. D., Roye E. W., 2000, ApJ, 542, 137 . D A Hunter, B G Elmegreen, C L Berger, 10.3847/1538-3881/ab1e54AJ. 157241Hunter D. A., Elmegreen B. G., Berger C. L., 2019, AJ, 157, 241 . C Impey, G Bothun, D Malin, 10.1086/166500ApJ. 330634Impey C., Bothun G., Malin D., 1988, ApJ, 330, 634 . G Iorio, G Iorio, F Fraternali, C Nipoti, Di Teodoro, E Read, J I Battaglia, G , 10.1093/mnras/stw3285MNRAS. 4664159PhD thesisIorio G., 2018, PhD thesis, alma, http://amsdottorato.unibo.it/ 8449/ Iorio G., Fraternali F., Nipoti C., Di Teodoro E., Read J. I., Battaglia G., 2017, MNRAS, 466, 4159 . R A Jackson, 10.1093/mnras/stab093MNRAS. 5021785Jackson R. A., et al., 2021, MNRAS, 502, 1785 . M G Jones, E Papastergis, V Pandya, L Leisman, A J Romanowsky, L Y A Yung, R S Somerville, E A K Adams, 10.1051/0004-6361/201732409A&A. 61421Jones M. G., Papastergis E., Pandya V., Leisman L., Romanowsky A. J., Yung L. Y. A., Somerville R. S., Adams E. A. K., 2018, A&A, 614, A21 . E Kado-Fong, arXiv:2106.05288arXiv e-printsKado-Fong E., et al., 2021, arXiv e-prints, p. arXiv:2106.05288 . M Kaplinghat, T Ren, H.-B Yu, 10.1088/1475-7516/2020/06/027J. Cosmology Astropart. Phys. 202027Kaplinghat M., Ren T., Yu H.-B., 2020, J. Cosmology Astropart. Phys., 2020, 027 . A Karunakaran, K Spekkens, D Zaritsky, R L Donnerstein, J Kadowaki, A Dey, 10.3847/1538-4357/abb464ApJ. 90239Karunakaran A., Spekkens K., Zaritsky D., Donnerstein R. L., Kadowaki J., Dey A., 2020, ApJ, 902, 39 . S Kaviraj, D Darg, C Lintott, K Schawinski, J Silk, 10.1111/j.1365-2966.2011.19673.xMNRAS. 41970Kaviraj S., Darg D., Lintott C., Schawinski K., Silk J., 2012, MNRAS, 419, 70 . C F P Laporte, A Agnello, J F Navarro, 10.1093/mnras/sty2891MNRAS. 484245Laporte C. F. P., Agnello A., Navarro J. F., 2019, MNRAS, 484, 245 . A Lazar, 10.1093/mnras/staa2101MNRAS. 4972393Lazar A., et al., 2020, MNRAS, 497, 2393 . L Leisman, 10.3847/1538-4357/aa7575ApJ. 842133Leisman L., et al., 2017, ApJ, 842, 133 . F Lelli, 10.1051/0004-6361/201526613A&A. 584113Lelli F., et al., 2015, A&A, 584, A113 . A K Leroy, F Walter, E Brinks, F Bigiel, W J G De Blok, B Madore, M D Thornley, 10.1088/0004-6256/136/6/2782AJ. 1362782Leroy A. K., Walter F., Brinks E., Bigiel F., de Blok W. J. G., Madore B., Thornley M. D., 2008, AJ, 136, 2782 . A D Ludlow, J F Navarro, R E Angulo, M Boylan-Kolchin, V Springel, C Frenk, S D M White, 10.1093/mnras/stu483MNRAS. 441378Ludlow A. D., Navarro J. F., Angulo R. E., Boylan-Kolchin M., Springel V., Frenk C., White S. D. M., 2014, MNRAS, 441, 378 . A V Macciò, A A Dutton, F C Van Den Bosch, B Moore, D Potter, J Stadel, 10.1111/j.1365-2966.2007.11720.xMNRAS. 37855Macciò A. V., Dutton A. A., van den Bosch F. C., Moore B., Potter D., Stadel J., 2007, MNRAS, 378, 55 . Mancera Piña, P E Aguerri, J A L Peletier, R F Venhola, A Trager, S , Choque Challapa, N , 10.1093/mnras/stz238MNRAS. 4851036Mancera Piña P. E., Aguerri J. A. L., Peletier R. F., Venhola A., Trager S., Choque Challapa N., 2019a, MNRAS, 485, 1036 . Mancera Piña, P E , 10.3847/2041-8213/ab40c7ApJ. 88333Mancera Piña P. E., et al., 2019b, ApJ, 883, L33 . Mancera Piña, P E , 10.1093/mnras/staa1256MNRAS. 4953636Mancera Piña P. E., et al., 2020, MNRAS, 495, 3636 . A Marasco, K A Oman, J F Navarro, C S Frenk, T Oosterloo, 10.1093/mnras/sty354MNRAS. 4762168Marasco A., Oman K. A., Navarro J. F., Frenk C. S., Oosterloo T., 2018, MNRAS, 476, 2168 . S S Mcgaugh, J M Schombert, G D Bothun, W J G De Blok, 10.1086/312628ApJ. 53399McGaugh S. S., Schombert J. M., Bothun G. D., de Blok W. J. G., 2000, ApJ, 533, L99 . S S Mcgaugh, M K Barker, W J G De Blok, 10.1086/345806ApJ. 584566McGaugh S. S., Barker M. K., de Blok W. J. G., 2003, ApJ, 584, 566 . S S Mcgaugh, J M Schombert, W J G De Blok, M J Zagursky, 10.1088/2041-8205/708/1/L14ApJ. 70814McGaugh S. S., Schombert J. M., de Blok W. J. G., Zagursky M. J., 2010, ApJ, 708, L14 . J C Mihos, S S Mcgaugh, W J G De Blok, 10.1086/310528ApJ. 47779Mihos J. C., McGaugh S. S., de Blok W. J. G., 1997, ApJ, 477, L79 . J C Mihos, 10.1088/2041-8205/809/2/L21ApJ. 80921Mihos J. C., et al., 2015, ApJ, 809, L21 . M Milgrom, 10.1086/161130ApJ. 270365Milgrom M., 1983, ApJ, 270, 365 . M Montes, R Infante-Sainz, A Madrigal-Aguado, J Román, M Monelli, A S Borlaff, I Trujillo, 10.3847/1538-4357/abc340ApJ. 904114Montes M., Infante-Sainz R., Madrigal-Aguado A., Román J., Monelli M., Borlaff A. S., Trujillo I., 2020, ApJ, 904, 114 . J F Navarro, C S Frenk, S D M White, 10.1086/304888ApJ. 490493Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493 . T E Oliphant, Computing in Science Engineering. 910Oliphant T. E., 2007, Computing in Science Engineering, 9, 10 . K A Oman, J F Navarro, L V Sales, A Fattahi, C S Frenk, T Sawala, M Schaller, S D M White, 10.1093/mnras/stw1251MNRAS. 4603610Oman K. A., Navarro J. F., Sales L. V., Fattahi A., Frenk C. S., Sawala T., Schaller M., White S. D. M., 2016, MNRAS, 460, 3610 . L Posti, G Pezzulli, F Fraternali, Di Teodoro, E M , 10.1093/mnras/stx3168MNRAS. 475232Posti L., Pezzulli G., Fraternali F., Di Teodoro E. M., 2018, MNRAS, 475, 232 . L Posti, B Famaey, G Pezzulli, F Fraternali, R Ibata, A Marasco, 10.1051/0004-6361/202038474A&A. 64476Posti L., Famaey B., Pezzulli G., Fraternali F., Ibata R., Marasco A., 2020, A&A, 644, A76 . J I Read, O Agertz, M L M Collins, 10.1093/mnras/stw713MNRAS. 4592573Read J. I., Agertz O., Collins M. L. M., 2016a, MNRAS, 459, 2573 . J I Read, G Iorio, O Agertz, F Fraternali, 10.1093/mnras/stw1876MNRAS. 4623628Read J. I., Iorio G., Agertz O., Fraternali F., 2016b, MNRAS, 462, 3628 . J I Read, G Iorio, O Agertz, F Fraternali, 10.1093/mnras/stx147MNRAS. 467Read J. I., Iorio G., Agertz O., Fraternali F., 2017, MNRAS, 467, 2019 . J Román, I Trujillo, 10.1093/mnras/stx694MNRAS. 4684039Román J., Trujillo I., 2017, MNRAS, 468, 4039 . J Román, M A Beasley, T Ruiz-Lara, D Valls-Gabaud, 10.1093/mnras/stz835MNRAS. 486823Román J., Beasley M. A., Ruiz-Lara T., Valls-Gabaud D., 2019, MNRAS, 486, 823 . A B Romeo, A&A. 286799Romeo A. B., 1994, A&A, 286, 799 . A B Romeo, N Falstad, 10.1093/mnras/stt809MNRAS. 4331389Romeo A. B., Falstad N., 2013, MNRAS, 433, 1389 . Y Rong, Q Guo, L Gao, S Liao, L Xie, T H Puzia, S Sun, J Pan, 10.1093/mnras/stx1440MNRAS. 4704231Rong Y., Guo Q., Gao L., Liao S., Xie L., Puzia T. H., Sun S., Pan J., 2017, MNRAS, 470, 4231 . L V Sales, J F Navarro, L Peñafiel, E W Peng, S Lim, L Hernquist, 10.1093/mnras/staa854MNRAS. 4941848Sales L. V., Navarro J. F., Peñafiel L., Peng E. W., Lim S., Hernquist L., 2020, MNRAS, 494, 1848 . R H Sanders, S S Mcgaugh, 10.1146/annurev.astro.40.060401.093923ARA&A. 40263Sanders R. H., McGaugh S. S., 2002, ARA&A, 40, 263 R J Sault, P J Teuben, M C H Wright, arXiv:astro-ph/0612759Astronomical Data Analysis Software and Systems IV. p. Shaw R. A., Payne H. E., Hayes J. J. E.77433Astronomical Society of the Pacific Conference SeriesSault R. J., Teuben P. J., Wright M. C. H., 1995, in Shaw R. A., Payne H. E., Hayes J. J. E., eds, Astronomical Society of the Pacific Conference Series Vol. 77, Astronomical Data Analysis Software and Systems IV. p. 433 (arXiv:astro-ph/0612759) . T Sawala, 10.1093/mnras/stu2753MNRAS. 4482941Sawala T., et al., 2015, MNRAS, 448, 2941 . C Sengupta, T C Scott, A Chung, O I Wong, 10.1093/mnras/stz1884MNRAS. 4883222Sengupta C., Scott T. C., Chung A., Wong O. I., 2019, MNRAS, 488, 3222 . Y Shi, Z.-Y Zhang, J Wang, J Chen, Q Gu, X Yu, S Li, 10.3847/1538-4357/abd777ApJ. 90920Shi Y., Zhang Z.-Y., Wang J., Chen J., Gu Q., Yu X., Li S., 2021, ApJ, 909, 20 . E.-J Shin, M Jung, G Kwon, J.-H Kim, J Lee, Y Jo, B K Oh, 10.3847/1538-4357/aba434ApJ. 89925Shin E.-j., Jung M., Kwon G., Kim J.-h., Lee J., Jo Y., Oh B. K., 2020, ApJ, 899, 25 . J Silk, 10.1093/mnrasl/slz090MNRAS. 48824Silk J., 2019, MNRAS, 488, L24 . R A Swaters, 10.1086/147861ApJ. 1391217Kapteyn Astronomical Institute, University of Groningen Toomre A.PhD thesisSwaters R. A., 1999, PhD thesis, Kapteyn Astronomical Institute, University of Groningen Toomre A., 1964, ApJ, 139, 1217 A Toomre, Structure and Evolution of Normal Galaxies. Fall S. M., Lynden-Bell D.Toomre A., 1981, in Fall S. M., Lynden-Bell D., eds, Structure and Evolution of Normal Galaxies. pp 111-136 . M Tremmel, A C Wright, A M Brooks, F Munshi, D Nagai, T R Quinn, 10.1093/mnras/staa2015MNRAS. 4972786Tremmel M., Wright A. C., Brooks A. M., Munshi F., Nagai D., Quinn T. R., 2020, MNRAS, 497, 2786 . S Trujillo-Gomez, J M D Kruijssen, M Reina-Campos, arXiv:2103.08610Trujillo-Gomez S., Kruijssen J. M. D., Reina-Campos M., 2021, arXiv e- prints, p. arXiv:2103.08610 . I Trujillo, 10.1093/mnras/stz771MNRAS. 4861192Trujillo I., et al., 2019, MNRAS, 486, 1192 . P Virtanen, 10.1038/s41592-019-0686-2Nature Methods. 17261Virtanen P., et al., 2020, Nature Methods, 17, 261 . J Wang, K Yang, Z.-Y Zhang, M Fang, Y Shi, S Liu, J Li, F Li, 10.1093/mnrasl/slaa150MNRAS. 49926Wang J., Yang K., Zhang Z.-Y., Fang M., Shi Y., Liu S., Li J., Li F., 2020, MNRAS, 499, L26 . A C Wright, M Tremmel, A M Brooks, F Munshi, D Nagai, R S Sharma, T R Quinn, 10.1093/mnras/stab081MNRAS. 5025370Wright A. C., Tremmel M., Brooks A. M., Munshi F., Nagai D., Sharma R. S., Quinn T. R., 2021, MNRAS, 502, 5370 . D Yang, H.-B Yu, H An, 10.1103/PhysRevLett.125.111105Phys. Rev. Lett. 125111105Yang D., Yu H.-B., An H., 2020, Phys. Rev. Lett., 125, 111105 . W J G De Blok, F Walter, E Brinks, C Trachternach, S H Oh, R C J Kennicutt, 10.1088/0004-6256/136/6/2648AJ. 1362648de Blok W. J. G., Walter F., Brinks E., Trachternach C., Oh S. H., Kennicutt R. C. J., 2008, AJ, 136, 2648 . P G Van Dokkum, R Abraham, A Merritt, J Zhang, M Geha, C Conroy, 10.1088/2041-8205/798/2/L45ApJ. 79845van Dokkum P. G., Abraham R., Merritt A., Zhang J., Geha M., Conroy C., 2015, ApJ, 798, L45 . P Van Dokkum, 10.1038/nature25767Nature. 555629van Dokkum P., et al., 2018, Nature, 555, 629 . P Van Dokkum, S Danieli, R Abraham, C Conroy, A J Romanowsky, 10.3847/2041-8213/ab0d92ApJ. 8745van Dokkum P., Danieli S., Abraham R., Conroy C., Romanowsky A. J., 2019, ApJ, 874, L5 R F J Van Der Burg, A Muzzin, H Hoekstra, J M Van Der Hulst, J P Terlouw, K G Begeman, W Zwitser, P R Roelfsema, 10.1051/0004-6361/201628222Astronomical Data Analysis Software and Systems I. p. 131 log(M200/M. Worrall D. M., Biemesderfer C., Barnes J.59010Astronomical Society of the Pacific Conference Seriesvan der Burg R. F. J., Muzzin A., Hoekstra H., 2016, A&A, 590, A20 van der Hulst J. M., Terlouw J. P., Begeman K. G., Zwitser W., Roelfsema P. R., 1992, in Worrall D. M., Biemesderfer C., Barnes J., eds, Astronomical Society of the Pacific Conference Series Vol. 25, Astronomical Data Analysis Software and Systems I. p. 131 log(M200/M ) = 10
[]
[ "On Channel Reciprocity in Reconfigurable Intelligent Surface Assisted Wireless Network", "On Channel Reciprocity in Reconfigurable Intelligent Surface Assisted Wireless Network" ]
[ "Wankai Tang ", "Xiangyu Chen ", "JunMing Zheng Chen ", "Yan Dai ", "Yu Han ", "Shi Jin ", "Qiang Cheng ", "Geoffrey Ye Li ", "Tie Jun Cui " ]
[]
[]
Channel reciprocity greatly facilitates downlink precoding in time-division duplexing (TDD) multipleinput multiple-output (MIMO) communications without the need for channel state information (CSI) feedback. Recently, reconfigurable intelligent surfaces (RISs) emerge as a promising technology to enhance the performance of future wireless networks. However, since the artificial electromagnetic characteristics of RISs do not strictly follow the normal laws of nature, it brings up a question: does the channel reciprocity hold in RIS-assisted TDD wireless networks? After briefly reviewing the reciprocity theorem, in this article, we show that there still exists channel reciprocity for RIS-assisted wireless networks satisfying certain conditions. We also experimentally demonstrate the reciprocity at the sub-6GHz and the millimeter-wave frequency bands by using two fabricated RISs. Furthermore, we introduce several RIS-assisted approaches to realizing nonreciprocal channels. Finally, potential opportunities brought by reciprocal/nonreciprocal RISs and future research directions are outlined.
10.1109/mwc.001.2100136
[ "https://arxiv.org/pdf/2103.03753v2.pdf" ]
232,135,147
2103.03753
77cbee7d3c8e954e7d1be7f8f3be8d3707dc2dd6
On Channel Reciprocity in Reconfigurable Intelligent Surface Assisted Wireless Network Wankai Tang Xiangyu Chen JunMing Zheng Chen Yan Dai Yu Han Shi Jin Qiang Cheng Geoffrey Ye Li Tie Jun Cui On Channel Reciprocity in Reconfigurable Intelligent Surface Assisted Wireless Network 1 Channel reciprocity greatly facilitates downlink precoding in time-division duplexing (TDD) multipleinput multiple-output (MIMO) communications without the need for channel state information (CSI) feedback. Recently, reconfigurable intelligent surfaces (RISs) emerge as a promising technology to enhance the performance of future wireless networks. However, since the artificial electromagnetic characteristics of RISs do not strictly follow the normal laws of nature, it brings up a question: does the channel reciprocity hold in RIS-assisted TDD wireless networks? After briefly reviewing the reciprocity theorem, in this article, we show that there still exists channel reciprocity for RIS-assisted wireless networks satisfying certain conditions. We also experimentally demonstrate the reciprocity at the sub-6GHz and the millimeter-wave frequency bands by using two fabricated RISs. Furthermore, we introduce several RIS-assisted approaches to realizing nonreciprocal channels. Finally, potential opportunities brought by reciprocal/nonreciprocal RISs and future research directions are outlined. The rest of this article is organized as follows. In Section II, we give a brief review on the reciprocity theorem for traditional wireless channels and unveil the key factors that determine whether the RIS-assisted wireless channel is reciprocal. We then present our experimental results to demonstrate the channel reciprocity of common RIS-assisted wireless networks in Section III. Then, in Section IV, we introduce several approaches to realizing nonreciprocal RISs that break channel reciprocity. We then discuss potential opportunities and future research directions of reciprocal/nonreciprocal RIS-assisted wireless networks in Section V before we conclude the article in Section VI. II. ON CHANNEL RECIPROCITY OF RIS-ASSISTED WIRELESS CHANNELS Electromagnetic wave is the fundamental basis and physical carrier of modern wireless communications, which makes it possible to realize convenient radio access and communication without the need for wire connections. Electromagnetic wave has many useful physical properties, such as propagating at the speed of light and providing various degrees of freedom. These useful properties endow modern wireless communication systems with advantages, including high mobility, low transmission latency, and superior network capacity. In this article, we focus on the important property in electromagnetics known as reciprocity, with an emphasis on whether the channel reciprocity holds in RIS-assisted wireless networks. Electromagnetic reciprocity refers to the phenomenon that the electromagnetic field generated at the observation point by a source remains the same when the source and observation point swap their positions. This phenomenon comes from the symmetry of Maxwell's equations with respect to time. In particular, as derived from Maxwell's equations for time-invariant and linear media, the most common representation of reciprocity in electromagnetics is the Rayleigh-Carson reciprocity theorem as follows [7] V A J A · E B dV A = V B J B · E A dV B ,(1) where volume V i contains source i with current density J i that creates field E i . Equation 1 indicates that the reaction of field E B on source A is the same as that of field E A on source B, i.e., the interaction between any pair of electromagnetic sources is reciprocal. In particular, Rayleigh-Carson reciprocity theorem unveils that the transmission between a pair of antennas is reciprocal for opposite propagation directions. This property is widely exploited in the field of wireless communications and is known as channel reciprocity. Without loss of generality, a simple yet typical RIS-assisted TDD wireless communication system is considered in the rest of this article to elaborate on whether the channel reciprocity holds in RIS-assisted TDD wireless networks. As shown in Fig. 1, the considered system is comprised of a single-antenna access point, a single-antenna user, and an RIS configured by an RIS controller. The wireless channel between the access point and the user can be expressed as h = h RIS + h D ,(2) where h RIS and h D denote the cascaded channel of the RIS-assisted link and the effective channel of the direct link without the reflection of the RIS, respectively. h RIS is composed of three cascaded parts, namely the channel between AP and the RIS, the reflection coefficient matrix of the RIS, and the channel between the RIS and the user. The reflection coefficient matrix of the RIS is often modeled as a diagonal matrix, whose elements are often phase-adjustable. Therefore, the RIS can flexibly regulate the channel h RIS [4]- [6]. It is worth noting that the direct link between the access point and the user does not necessarily refer to free-space transmission. It is obvious that the wireless channel of the direct link h D is reciprocal because it is a common TDD channel. In other words, the property of the cascaded channel h RIS that is provided by the reflection of the RIS determines whether the entire wireless channel of an RIS-assisted wireless network is reciprocal. In order to better understand whether h RIS is reciprocal, let us first revisit the prerequisite of the Rayleigh-Carson reciprocity theorem, which is written as follows [7] ε = ε T µ = µ T ,(3) where ε and µ denote the permittivity and permeability tensors, respectively. The tensor identity given in (3) In general, when the control signals applied to the unit cells remain unchanged, commonly designed and fabricated RISs inherently obey the reciprocity theorem. This is because the materials that constitute RISs, such as metal patches, dielectric layers, and electronic components, often conform to the conditions given in (3). Nevertheless, there are several special approaches that break (3), thereby breaking channel reciprocity in RIS-assisted TDD wireless networks, which will be also discussed subsequently. III. EXPERIMENTAL MEASUREMENTS ON TWO TYPICAL RISS In this section, we present the experimental measurements at the sub-6 GHz and millimeterwave bands to validate whether the channel reciprocity holds in common RIS-assisted TDD wireless communication systems by using two typical fabricated RISs. A. Two Typical RISs For ease of exposition, the two RISs in Fig. 2 are referred to as RIS 1 and RIS 2, respectively. As shown in Fig. 2(a), RIS 1 belongs to the varactor-diode-based type and works in the sub-6 GHz frequency band. Every two columns of RIS 1 share the same control signal and form a super-column, whose reflection phase can be continuously manipulated as the control signal varies between 0 V and 21 V. Meanwhile, as shown in Fig. 2(b), RIS 2 belongs to the PINdiode-based type and works in the millimeter-wave frequency band. Every seven columns of RIS 2 share the same control signal and form a super-column, whose reflection phase is only 1-bit programmable. More details of these two RISs can be found in [8] and [9], respectively. B. Experiment Setup We have built an experiment setup, as illustrated in Fig. 3, in order to practically explore whether the reciprocity of RIS-assisted wireless channels holds. In Fig. 3, d 1 , d 2 , θ 1 , and θ 2 denote the distance between antenna 1 and the center of the RIS, the distance between antenna 2 and the center of the RIS, the elevation angle from the center of the RIS to antenna 1, and the elevation angle from the center of the RIS to antenna 2, respectively. More specifically, θ 1 is the angle between path d 1 and the normal of the RIS, and θ 2 is the angle between path d 2 and the normal of the RIS. The configuration of the RIS is determined by the control signals from the RIS controller. During the measurements, an RF signal generator provides the transmitted signal with power P t and frequency f , and an RF signal analyzer displays the information of the received signal. As far as the uplink is concerned, antenna 2 is connected to the RF signal generator and antenna 1 is connected to the RF signal analyzer. In this setup, a part of the wireless signal transmitted from antenna 2 is reflected by the RIS and then received by antenna 1, and another part is reflected by the conventional scatterers (e.g., ground and walls) and then received by antenna 1 as well. The wireless channels corresponding to the former and the latter are referred to as h RIS and h D as given in equation (2), respectively. As far as the downlink is concerned, antenna 1 is connected to the RF signal generator and C. Measurement Results The varactor-diode-based RIS 1 is first employed as the RIS in Fig. 3. Fig. 4(a). More importantly, from the experiment, the trajectory of the received signal obtained in the downlink mode is in quite good agreement with that obtained in the uplink mode, which indicates that the uplink and downlink wireless channels are reciprocal. For example, as illustrated in Fig. 4(a), the received signals in the uplink and downlink modes coincide well with each other when the control voltage is 0 V, which demonstrates channel reciprocity. The same experiment is repeated by considering d 1 = 1.5 m, θ 1 = 50 • , d 2 = 1 m, and θ 2 = 30 • . Fig. 4(b) illustrates the measured received signal in the complex plane. Once again, we observe that the trajectories of the received signals obtained in the downlink mode and in the uplink mode agree well with each other, which proves that the uplink and downlink wireless channels are reciprocal when RIS 1 is employed for RIS-assisted transmission. We should indicate that the trajectories of the received signals in Fig. 4(a) and in Fig. 4(b) are different since their corresponding measurement setups, such as d 1 , θ 1 , d 2 , and θ 2 , are different. The above-mentioned measurements are carried out under the condition that the control signals of RIS 1 vary dynamically from 0 V to 21 V. In the following, we present the experimental results when the employed RIS is configured by static coding patterns. As illustrated in Fig. 2(a), identical coding pattern and gradient coding pattern are applied to RIS 1 in the reciprocity measurements, respectively. Among many possible coding patterns to realize coded RIS 1, identical coding and gradient coding are just two examples that we use to explore the channel reciprocity when RIS 1 is utilized in the experiment. The corresponding measurement results are reported in the first two large rows of Table I. We observe that the powers (amplitudes) and phases of the received signals in the uplink and downlink are extremely consistent, which unveils that the channel reciprocity still holds. For example, when the measurement setup is The PIN-diode-based RIS 2 operating in the millimeter-wave frequency band is then employed as the RIS in Fig. 3. Limited by equipment conditions, we measure and compare the uplink and downlink received signal powers when RIS 2 is configured by identical coding pattern and stripe coding pattern, respectively, as shown in Fig. 2(b). The last two large rows of Table I report the corresponding measurement results. Once again, the received signal powers (amplitudes) in the uplink and downlink agree well with each other, which indicates channel reciprocity when RIS 2 is utilized for RIS-assisted wireless transmission. d 1 =1.5 m, θ 1 =30 • , d 2 =0.5 m, θ 2 =0 • It is worth noting that although the wireless channel of a realistic TDD wireless network is non-stationary, an important prerequisite for channel reciprocity is that the duration of a certain uplink time slot and its corresponding downlink time slot should be shorter than the channel coherence time. Therefore, even though the measurements are conducted in a stationary indoor environment, the experimental results reported above are sufficient to validate that the channel reciprocity holds in RIS-assisted TDD wireless communication systems, as long as the employed RISs are commonly designed and fabricated, and conform to the tensor identity given in (3). In addition, Table I suggests that an appropriate RIS coding pattern can significantly increase the received signal power. Moreover, due to the channel reciprocity, the improvements of the received signal powers in the uplink and downlink are always the same. IV. BREAKING CHANNEL RECIPROCITY Although channel reciprocity has been often relied upon to design efficient wireless communication protocols, there are some potential scenarios where channel reciprocity is not desired, such as wireless power transfer and secure wireless communication, which can be facilitated by breaking the tensor identity in (3). In this section, we will introduce some approaches to breaking channel reciprocity in RIS-assisted wireless communication systems, including using active nonreciprocal circuits, performing time-varying controls, and employing nonlinearities and structural asymmetries [11]. A. Nonreciprocal RISs Enabled by Active Nonreciprocal Circuits By utilizing active nonreciprocal circuits, e.g., microwave amplifiers and isolators, in the structure designs of unit cells, nonreciprocal RISs can be achieved with good integrability. For example, an RIS integrated with microwave amplifiers is able to bring about a distinct nonreciprocity between the uplink and downlink wireless channels [12]. As depicted in Fig. 5(a), in the downlink, wireless signals are first captured by the metal patches on RIS's top layer and transformed into circuit signals. After passing through amplifiers and via holes, the circuit signals are then radiated into space again by the metal patches on RIS's bottom layer. In the uplink, however, due to the nonreciprocity of integrated amplifiers, the signals captured by the metal patches on RIS's bottom layer are isolated. In addition, it is worth noting that the varactordiodes and PIN-diodes, which are often employed in RISs, will not directly cause nonreciprocity, because they are usually only unidirectional for direct current (DC) signals. B. Nonreciprocal RISs Enabled by Time-varying Controls Since the electromagnetic characteristics of RISs are real-time programmable, a straightforward approach to breaking channel reciprocity is to apply different coding patterns to RISs in the uplink and downlink time slots. Moreover, the space-time reconfigurability of RISs further enriches the manifestations of time-varying enabled nonreciprocity. For example, by designing a sophisticated space-time coding matrix applied to an RIS, the wireless signal impinging at an angle θ 1 and frequency f 1 can be anomalously reflected at an angle θ 2 and frequency f 2 in the uplink, as illustrated in Fig. 5(b). In the downlink, however, an incident wireless signal at an angle θ 2 and frequency f 2 is reflected at an angle θ 3 = θ 1 and frequency f 3 = f 1 [13]. Different space-time coding matrix designs can realize different nonreciprocal features. This example demonstrates that RISs have a great potential to achieve programmable channel nonreciprocity. C. Nonreciprocal RISs Enabled by Nonlinearities and Structural Asymmetries Another approach to breaking reciprocity is based on nonlinearities combined with structural asymmetries. For example, the nonreciprocal RIS in [14] uses a nonlinear resonator in the asymmetric dielectric structure of each unit cell, as shown in Fig. 5(c). Due to the spatial asymmetry of the two-layer dielectric structure, the amplitudes of the electric fields exciting the nonlinear resonator are different for the uplink and downlink channels, resulting in nonreciprocal transmission properties. As depicted in Fig. 5(c), the proposed RIS allows wave propagation in the downlink channel, and blocks wave propagation in the uplink channel. Compared with the aforementioned approaches, the resulting RIS needs neither active power supplies nor timevarying control signals to achieve nonreciprocity. Moreover, the uplink transmission coefficient is even related to the power of the incident wireless signals [14]. V. OPPORTUNITIES AND FUTURE RESEARCH DIRECTIONS We have shown that common RISs are reciprocal because they often inherently conform to the reciprocity theorem. Meanwhile, there are several available approaches to breaking the symmetry of material tensors in order to realize nonreciprocal RISs. In the following, we present some potential opportunities and future research directions on reciprocal/nonreciprocal RIS-assisted wireless network designs. A. Signal Amplification Function In existing research works on RIS-assisted wireless networks, RISs are considered to be nearly-passive, i. e., they have no signal amplification functions. The amplitude component of each unit cell of an RIS is thus modeled as less than or equal to 1. However, as mentioned in the previous section, RISs may contain microwave amplifiers, which can provide considerable gains on wireless signals during the wave-matter interaction. For example, a practical transmission gain of about 13 dB is achieved at 5.55 GHz in [12]. Therefore, it is necessary to develop original signal models for RISs with amplification functions by comprehensively considering the reciprocity, performance, hardware cost, power consumption, and introduced noise. In addition, the RIS reported in [12] cannot yet regulate the phase component, and further studies are needed to realize simultaneous phase adjustment and amplitude amplification functions. B. Switchable Transmission State The dramatic increase in wireless devices leads to complex electromagnetic environments, which will cause severe data flow congestion and electromagnetic pollution. Fortunately, nonreciprocal RISs have demonstrated superior capability in controlling transmission state, which may fundamentally solve these problems. The RIS presented in [12] has four switchable transmission states, including bidirectional transmission, forward transmission, backward transmission, and no transmission. Moreover, the building block proposed in [14] realizes nonreciprocal transmission states. These RIS designs make it possible to directly control data flows from the electromagnetic level while the application cases and analytical models still need further studies. C. Angle-Dependent Model The design methods of RISs are often based on the resonant structures, which inherently makes the reflection/transmission coefficients of the unit cells of RISs angle-dependent [15]. For example, the phase shift of the unit cells of RIS 1 in Fig. 2(a) is highly sensitive to the angle of incidence [8]. Meanwhile, the measurement results in this article prove that RIS 1 is reciprocal. Therefore, angle-dependent regulation and electromagnetic reciprocity coexist for reciprocal RISs, which unveils that the reflection/transmission coefficient of each unit cell is reciprocal for the exchange of the angle of incidence and the angle of observation. In other words, the reflection/transmission coefficient is not only sensitive to the angle of incidence but also sensitive to the angle of observation. Based on the preliminary angle-dependent phase shifter model proposed in [15], which only considers the dependence on the angle of incidence, a more tractable and reliable angle-dependent model for reciprocal RISs is urgently needed in order to optimize and predict the performance of RIS-assisted wireless networks. D. Prototyping and Measurement Prototyping and measurement works are the basis of theoretical modeling, link budget analysis, function extension, performance prediction, and configuration optimization of RIS-assisted wireless networks, and can help reveal the practical performance limits offered by RISs. At present, studies on the prototyping and measurement of RISs are still in the initial stage and the concept of smart radio environments has not yet been fully demonstrated in practice. It is desired for researchers in academia and industry to jointly explore and promote related works. In addition, currently reported prototyping and measurement works are mostly based on common reciprocal RISs. Whether RISs with signal amplification function and programmable reciprocity can provide performance leaps in RIS-assisted wireless networks needs extensive experiments and measurements to validate [11]- [14]. VI. CONCLUSION In this article, we have unveiled that the uplink and downlink wireless channels of an RISassisted TDD wireless network can be either reciprocal or nonreciprocal, depending on the specific design of the employed RIS. In particular, two typical RISs were utilized in the measurements to validate that the channel reciprocity often holds for commonly designed and fabricated RISs. In addition, we provided an overview of three available approaches to designing nonreciprocal RISs and thus breaking channel reciprocity in RIS-assisted TDD wireless networks. Furthermore, we outlined several future research directions that can facilitate the full potential of reciprocal/nonreciprocal RIS-assisted wireless networks. They include exploring RISs with signal amplification functions, utilizing RISs with switchable transmission states, developing angle-dependent models for RISs, and conducting prototyping and measurement works with both reciprocal and nonreciprocal RISs. of direct link: ℎ D Cascaded channel of RIS-assisted link: ℎ RIS Uplink: User to AP Downlink: AP to User Fig. 1. A simple yet typical RIS-assisted TDD wireless communication system. are expected to be transformed into artificially customized ones [6]. On the other hand, natural wireless channels in time-division duplexing (TDD) wireless communication networks have the inherent and fundamental property of channel reciprocity. Under a same operating frequency, the access point (AP) can acquire the channel state information (CSI) of the downlink based on the channel estimation of the uplink after some necessary calibrations. This mechanism minimizes the overhead requirements for CSI feedback, and thus facilitates the realization of accurate precoding and beamforming in the downlink of TDD wireless networks. However, it is not clear whether the channel reciprocity continues to work in scenarios where RISs are deployed. This article aims to provide insights on this topic both theoretically and experimentally. is used in the process of deriving the Rayleigh-Carson reciprocity theorem from Maxwell's equations. Therefore, this prerequisite unveils that materials and transmission media, for which these conditions expressed in (3) are satisfied, are reciprocal, otherwise they are nonreciprocal. The material tensors of the transmission media and scatterers, such as air, ground, and walls, in the conventional wireless communication systems conform to (3), i.e., the characteristics of the transmission media and scatterers are symmetric for the uplink and the downlink, which is the basic reason and principle for the channel reciprocity in the current TDD wireless networks. As for the RIS-assisted link shown in Fig. 1, whether the RIS follows conditions in (3) determines whether the cascaded wireless channel h RIS is reciprocal. Fig. 2 2illustrates the pictures and parameters of the two typical RISs utilized in the measurements. The word "typical" here lies in the fact that the two RISs are commonly designed and fabricated by using standard printed circuit board (PCB) technologies. In other words, no special nonreciprocal materials and designs are utilized in these two RISs. In addition, they do cover the most widely used types of RISs, namely varactor-diode-based RISs and positive-intrinsicnegative (PIN)-diode-based RISs. Furthermore, the operating frequencies of these two RISs are in the sub-6 GHz and millimeter-wave frequency bands, respectively. Therefore, the two RISs are sufficiently representative to demonstrate whether the uplink and downlink channels of common RIS-assisted wireless communication systems are reciprocal. Fig. 3 . 3Experiment setup for measuring whether the uplink and downlink channels of an RIS-assisted wireless communication system are reciprocal. Fig. 4 . 4Received signals in the complex plane when a control voltage that varies linearly from 0 V to 21 V is applied to all the unit cells of RIS 1. (a) d1 = 1.5 m, θ1 = 30 • , d2 = 0.5 m, and θ2 = 0 • . (b) d1 = 1.5 m, θ1 = 50 • , d2 = 1 m, and θ2 = 30 • .antenna 2 is connected to the RF signal analyzer. Therefore, to switch the experiment setup from the uplink mode to the downlink mode can be simply realized by swapping the RF connections of the RF signal generator and the RF signal analyzer, as shown inFig.3. In consequence, whether the channel reciprocity holds can be proved by observing whether the power (amplitude) and the phase of the received signals in the uplink mode and the downlink mode are consistent with each other. The measurement setup is P t = 0 dBm, f = 4.25 GHz, d 1 = 1.5 m, θ 1 = 30 • , d 2 = 0.5 m, and θ 2 = 0 • . By letting the RIS controller apply a control voltage that varies linearly from 0 V to 21 V to all the unit cells of RIS 1 (i.e., let control signals V 1 , V 2 , V 3 , and V 4 shown inFig. 2(a) share a same varying voltage), the corresponding received signal in the complex plane is illustrated inFig. 4(a). We observe that the received signal moves in the complex plane and forms a trajectory as the control voltage varies from 0 V to 21 V. More specifically, the trajectory of the received signal is actually generated by the sum of h D and h RIS , i.e., h D + h RIS given in(2) [10]. In particular, since the measurement is conducted in a stationary indoor environment, the wireless channel of the direct link h D is fixed once the measurement setup is determined. Meanwhile, the and gradient coding pattern is applied, the differences between the received powers and the received phases of the uplink and downlink are only 0.1 dB (-42.6+42.7=0.1) and 1 degree (71−70=1), respectively. These negligible differences are caused by the slight disturbance in swapping the RF connections during the measurements. Fig. 5 . 5Approaches to breaking channel reciprocity in RIS-assisted TDD wireless networks. (a) Using active nonreciprocal circuits. (b) Performing time-varying controls. (c) Employing nonlinearities and structural asymmetries. Uplink: Antenna 2 Tx and Antenna 1 Rx Downlink: Antenna 1 Tx and Antenna 2 RxAntenna 1 Antenna 2 RF signal generator RF signal analyzer RIS RIS controller RF signal generator RF signal analyzer Zoom in Swap the RF connections to observe whether the wireless channel is reciprocal TABLE I IMEASUREMENT RESULTS ON THE CHANNEL RECIPROCITY BY USING TWO FABRICATED RISS.wireless channel of the RIS-assisted link h RIS is continuously adjusted by the varying control voltage, which leads to the circular trajectory of the received signal shown inRIS Measurement setup RIS coding pattern Received signal power (Uplink/Downlink, dBm) Received signal phase (Uplink/Downlink, degree) RIS 1 Pt = 0 dBm, f = 4.25 GHz, d1=1.5 m, θ1=30 • , d2=0.5 m, θ2=0 • Identical -45.5 / -45.5 53 / 52 Gradient -42.6 / -42.7 71 / 70 RIS 1 Pt = 0 dBm, f = 4.25 GHz, d1=1.5 m, θ1=50 • , d2=1 m, θ2=30 • Identical -44.9 / -44.9 -76 / -77 Gradient -53.6 / -53.7 -139 / -140 RIS 2 Pt = 0 dBm, f = 27 GHz, d1=1 m, θ1=35 • , d2=0.5 m, θ2=0 • Identical -58.0 / -57.8 NA Stripe -49.4 / -49.2 NA RIS 2 Pt = 0 dBm, f = 27 GHz, d1=1 m, θ1=5 • , d2=0.5 m, θ2=0 • Identical -49.3 / -49.2 NA Stripe -58.2 / -58.2 NA A Speculative Study on 6G. F Tariq, M R A Khandaker, K.-K Wong, M Imran, M Bennis, M Debbah, IEEE Wireless Commun. 274F. Tariq, M. R. A. Khandaker, K.-K. Wong, M. Imran, M. Bennis, and M. Debbah, "A Speculative Study on 6G," IEEE Wireless Commun., vol. 27, no. 4, Aug. 2020, pp. 118-125. Towards Smart and Reconfigurable Environment: Intelligent Reflecting Surface Aided Wireless Network. Q Wu, R Zhang, IEEE Commun. Mag. 581Q. Wu and R. Zhang, "Towards Smart and Reconfigurable Environment: Intelligent Reflecting Surface Aided Wireless Network," IEEE Commun. Mag., vol. 58, no. 1, Jan. 2020, pp. 106-112. Wireless Communications with Programmable Metasurface: Transceiver Design and Experimental Results. W Tang, X Li, J Y Dai, S Jin, Y Zeng, Q Cheng, T J Cui, China Commun. 165W. Tang, X. Li, J. Y. Dai, S. Jin, Y. Zeng, Q. Cheng, and T. J. Cui, "Wireless Communications with Programmable Metasurface: Transceiver Design and Experimental Results," China Commun., vol. 16, no. 5, May 2019, pp. 46-61. Large Intelligent Surface-Assisted Wireless Communication Exploiting Statistical CSI. Y Han, W Tang, S Jin, C.-K Wen, X Ma, IEEE Trans. Veh. Technol. 688Y. Han, W. Tang, S. Jin, C.-K. Wen, and X. Ma, "Large Intelligent Surface-Assisted Wireless Communication Exploiting Statistical CSI," IEEE Trans. Veh. Technol., vol. 68, no. 8, Aug. 2019, pp. 8238-8242. Smart Radio Environments Empowered by Reconfigurable Intelligent Surfaces: How It Works, State of Research, and the Road Ahead. M Di Renzo, A Zappone, M Debbah, M.-S Alouini, C Yuen, J De Rosny, S Tretyakov, IEEE J. Sel. Areas Commun. 3811M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, "Smart Radio Environments Empowered by Reconfigurable Intelligent Surfaces: How It Works, State of Research, and the Road Ahead," IEEE J. Sel. Areas Commun., vol. 38, no. 11, Nov. 2020, pp. 2450-2525. Reconfigurable Intelligent Surfaces for Energy Efficiency in Wireless Communication. C Huang, A Zappone, G C Alexandropoulos, M Debbah, C Yuen, IEEE Trans. Wireless Commun. 188C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, "Reconfigurable Intelligent Surfaces for Energy Efficiency in Wireless Communication," IEEE Trans. Wireless Commun., vol. 18, no. 8, Aug. 2019, pp. 4157-4170. Tutorial on Electromagnetic Nonreciprocity and Its Origins. V S Asadchy, M S Mirmoosa, A Díaz-Rubio, S Fan, S A Tretyakov, Proceedings of the IEEE. 10810V. S. Asadchy, M. S. Mirmoosa, A. Díaz-Rubio, S. Fan, and S. A. Tretyakov, "Tutorial on Electromagnetic Nonreciprocity and Its Origins," Proceedings of the IEEE, vol. 108, no. 10, Oct. 2020, pp. 1684-1727. Wireless Communications with Reconfigurable Intelligent Surface: Path Loss Modeling and Experimental Measurement. W Tang, M Z Chen, X Chen, J Y Dai, Y Han, M Di Renzo, Y Zeng, S Jin, Q Cheng, T J Cui, IEEE Trans. Wireless Commun. 201W. Tang, M. Z. Chen, X. Chen, J. Y. Dai, Y. Han, M. Di Renzo, Y. Zeng, S. Jin, Q. Cheng, and T. J. Cui, "Wireless Communications with Reconfigurable Intelligent Surface: Path Loss Modeling and Experimental Measurement," IEEE Trans. Wireless Commun., vol. 20, no. 1, Jan. 2021, pp. 421-439. Path Loss Modeling and Measurements for Reconfigurable Intelligent Surfaces in the Millimeter-Wave Frequency Band. W Tang, X Chen, M Z Chen, J Y Dai, Y Han, M Di Renzo, S Jin, Q Cheng, T J Cui, arXiv:2101.086072021W. Tang, X. Chen, M. Z. Chen, J. Y. Dai, Y. Han, M. Di Renzo, S. Jin, Q. Cheng, and T. J. Cui, "Path Loss Modeling and Measurements for Reconfigurable Intelligent Surfaces in the Millimeter-Wave Frequency Band," 2021, arXiv:2101.08607. [Online]. Available: https://arxiv.org/abs/2101.08607 RFocus: Beamforming Using Thousands of Passive Antennas. V Arun, H Balakrishnan, 17th USENIX Symposium on Networked Systems Design and Implementation (NSDI'20). V. Arun and H. Balakrishnan, "RFocus: Beamforming Using Thousands of Passive Antennas," 17th USENIX Symposium on Networked Systems Design and Implementation (NSDI'20), Feb. 2020, pp. 1047-1061. Microwave Nonreciprocity. A Kord, D L Sounas, A Alù, Proceedings of the IEEE. 10810A. Kord, D. L. Sounas, and A. Alù, "Microwave Nonreciprocity," Proceedings of the IEEE, vol. 108, no. 10, Oct. 2020, pp. 1728-1758. Q Ma, L Chen, H B Jing, Q R Hong, H Y Cui, Y Liu, L Li, T J Cui, Controllable and Programmable Nonreciprocity Based on Detachable Digital Coding Metasurface. 7Q. Ma, L. Chen, H. B. Jing, Q. R. Hong, H. Y. Cui, Y. Liu, L. Li, and T. J. Cui, "Controllable and Programmable Nonreciprocity Based on Detachable Digital Coding Metasurface," Adv. Optical Mater., vol. 7, no. 24, Dec. 2019, pp. 1-7. Breaking Reciprocity with Space-Time-Coding Digital Metasurfaces. L Zhang, X Q Chen, R W Shao, J Y Dai, Q Cheng, G Castaldi, V Galdi, T J Cui, Adv. Mater. 3141L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, "Breaking Reciprocity with Space-Time-Coding Digital Metasurfaces," Adv. Mater., vol. 31, no. 41, Oct. 2019, pp. 1-10. All-Passive Nonreciprocal Metastructure. A M Mahmoud, A R Davoyan, N Engheta, Nat. Commun. 68359A. M. Mahmoud, A. R. Davoyan, and N. Engheta, "All-Passive Nonreciprocal Metastructure," Nat. Commun., vol. 6, Sept. 2015, Art. no. 8359. Angle-Dependent Phase Shifter Model for Reconfigurable Intelligent Surfaces: Does the Angle-Reciprocity Hold?. W Chen, L Bai, W Tang, S Jin, W X Jiang, T J Cui, IEEE Commun. Lett. 249W. Chen, L. Bai, W. Tang, S. Jin, W. X. Jiang, and T. J. Cui, "Angle-Dependent Phase Shifter Model for Reconfigurable Intelligent Surfaces: Does the Angle-Reciprocity Hold?" IEEE Commun. Lett., vol. 24, no. 9, Sept. 2020, pp. 2060-2064.
[]
[ "The ZEUS Collaboration M5S 1A7 a", "The ZEUS Collaboration M5S 1A7 a" ]
[ "S Chekanov ", "M Derrick ", "S Magill ", "S Miglioranzi \nUniversity College London\nUK\n", "B Musgrave ", "D Nicholass \nUniversity College London\nUK\n", "J Repond ", "R Yoshida ", "M C K Mattingly ", "M Jechow ", "N Pavel ", "A G Yagües Molina ", "S Antonelli ", "P Antonioli ", "G Bari ", "M Basile ", "L Bellagamba ", "M Bindi ", "D Boscherini ", "A Bruni ", "G Bruni ", "L Cifarelli ", "F Cindolo ", "A Contin ", "M Corradi \nUniversity of Hamburg\nHamburgGermany, Germany\n\nUniversity of Liverpool\nUK\n", "S De Pasquale ", "G Iacobucci ", "A Margotti ", "R Nania ", "A Polini ", "L Rinaldi ", "G Sartorelli ", "A Zichichi ", "D Bartsch ", "I Brock ", "S Goers ", "H Hartmann ", "E Hilger ", "P Irrgang ", "H.-P Jakob ", "M Jüngst ", "O M Kind ", "E Paul ", "R Renner ", "U Samson ", "V Schönberg ", "R Shehzadi ", "M Wlasenko ", "N H Brook ", "G P Heath ", "J D Morris ", "T Namsoo ", "H H Wills \nInstitute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan\n\nDepartamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q\n\nUniversidad Autónoma de Madrid\nMadridSpain\n\nDipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly\n", "M Capua ", "S Fazio ", "A Mastroberardino ", "M Schioppa ", "G Susinno ", "E Tassi ", "J Y Kim ", "K J Ma ", "Z A Ibrahim ", "B Kamaluddin ", "W A T Wan Abdullah ", "Jabatan Fizik ", "Universiti Malaya ", "Kuala Lumpur ", "Malaysia ", "Y Ning ", "Z Ren ", "F Sciulli ", "J Chwastowski ", "A Eskreys ", "J Figiel ", "A Galas ", "M Gil ", "K Olkiewicz ", "P Stopa ", "L Zaw-Iejski ", "L Adamczyk ", "T Bo ", "I Grabowska-Bo ", "D Ld ", "J Kisielewska ", "M Lukasik ", "L Przybycień ", "Suszycki ", "I A Kotański ", "W S Lomiński ", "V Adler ", "U Behrens ", "I Bloch ", "C Blohm ", "A Bonato ", "K Borras ", "N Coppola ", "A Dos-Sanov ", "J Fourletova ", "A Geiser ", "D Gladkov ", "P Göttlicher ", "I Gregor ", "T Haas ", "W Hain ", "C Horn ", "B Kahle ", "U Klein ", "U Kötz ", "H Kowalski ", "E Lobodzinska ", "B Löhr ", "R Mankel ", "I.-A Melzer-Pellmann ", "A Montanari ", "D Notz ", "A E Nuncio-Quiroz ", "I Rubinsky ", "R San-Tamarta ", "U Schneekloth ", "A Spiridonov \nInstitut of Theoretical and Experimental Physics\nMoscowRussia\n", "H Stadie ", "D Szuba \nalso at INP\nCracowPoland\n", "J Szuba \non leave of absence from FPACS, AGH-UST\nCracowPoland\n", "T Theedt ", "G Wolf ", "K Wrona ", "C Youngman ", "W Zeuner ", "W Lohmann ", "S Schlenstedt ", "G Barbagli ", "E Gallo ", "P G Pelfer \nPhysikalisches Institut\nUniversität Bonn\nBonnGermany\n\nChonnam National University\nKwangjuSouth Korea\n\nFaculty of Exact Sciences\nSchool of Physics\nRutherford Appleton Laboratory, Chilton\nTel-Aviv University\nDidcot, Tel-AvivOxonUnited Kingdom, Israel\n\nUniversità del Piemonte Orientale\nNovaraItaly\n\nDepartment of Physics\nPhysics and Astronomy Department\nUniversity of Toronto\nTorontoOntarioCanada\n\nUniversity College London\nLondonUnited King\n\nInstitute of Experimental Physics, Warsaw\nInstitute for Nuclear Studies, Warsaw\nDepartment of Particle Physics, Weizmann Institute\nWarsaw University\nRehovotPoland, Poland, Israel\n", "E A Bamberger ", "D Dobur ", "F Karstens ", "N N Vlasov ", "I Br \nFaculty of Physics and Applied Computer Science\nAGH-University of Science and Tech-nology\nCracowPoland\n", "Freiburg I Br \nFaculty of Physics and Applied Computer Science\nAGH-University of Science and Tech-nology\nCracowPoland\n", "Germany \nPhysics Laboratory\nUniversity of Bristol\nBristolUnited Kingdom\n\nPhysics Department and INFN\nCalabria University\nCosenzaItaly\n\nHigh Energy Nuclear Physics Group\nImperial College London\nLondonUnited King-dom\n", "P J Bussey ", "A T Doyle ", "W Dunne ", "J Ferrando ", "D H Saxon ", "I O Skillicorn \nInstitute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan\n\nDepartamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q\n\nUniversidad Autónoma de Madrid\nMadridSpain\n\nDipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly\n", "I Gialas ", "T Gosau ", "U Holm ", "R Klanner ", "E Lohrmann ", "H Salehi ", "P Schleper ", "T Schörner-Sadenius ", "J Sztuk ", "K Wichmann ", "K Wick ", "C Foudas ", "C Fry ", "K R Long ", "A D Tapper ", "M Kataoka ", "T Matsumoto ", "K Nagano ", "K Tokushuku ", "S Yamada ", "Y Yamazaki ", "A N Barakbaev ", "E G Boos ", "N S Pokrovskiy ", "B O Zhautykov ", "D Son ", "South Korea \nThe Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences\nNevis Laboratories\nColumbia University\nIrvington on Hudson10027o, CracowNew YorkPoland\n", "Ii J De Favereau ", "K Piotrzkowski ", "F Barreiro ", "C Glasman ", "M Jimenez ", "L Labarga ", "J Del Peso ", "E Ron ", "M Soares ", "J Terrón ", "M Zambrana ", "F Corriveau ", "C Liu ", "R Walsh ", "C Zhou ", "T Tsurugai ", "A Antonov ", "B A Dolgoshein ", "V Sosnovtsev ", "A Stifutkin ", "S Suchkov ", "R K Dementiev ", "P F Ermolov ", "L K Gladilin ", "I I Katkov ", "L A Khein ", "I A Korzhavina ", "V A Kuzmin ", "B B Levchenko ", "O Yu Lukina ", "A S Proskuryakov ", "L M Shcheglova ", "D S Zotkin ", "S A Zotkin ", "I Abt ", "C Büttner ", "A Caldwell ", "D Kollar ", "W B Schmidke ", "J Sutiak ", "G Grigorescu ", "A Keramidas ", "E Koffeman ", "P Kooijman ", "A Pellegrino ", "H Tiecke ", "M Vázquez ", "L Wiggers ", "N Brümmer ", "B Bylsma ", "L S Durkin ", "A Lee ", "T Y Ling ", "P D Allfrey ", "M A Bell ", "A M Cooper-Sarkar ", "A Cottrell ", "R C E Devenish ", "B Foster ", "K Korcsak-Gorzo ", "S Patel ", "V Roberfroid ", "A Robertson ", "P B Straub ", "C Uribe-Estrada ", "R Walczak ", "P Bellan ", "A Bertolin ", "R Brugnera ", "R Carlin ", "R Ciesielski ", "F Dal Corso ", "S Dusini ", "A Garfagnini ", "S Limentani ", "A Longhin ", "L Stanco ", "M Turcato \nPhysikalisches Institut\nUniversität Bonn\nBonnGermany\n\nChonnam National University\nKwangjuSouth Korea\n\nFaculty of Exact Sciences\nSchool of Physics\nRutherford Appleton Laboratory, Chilton\nTel-Aviv University\nDidcot, Tel-AvivOxonUnited Kingdom, Israel\n\nUniversità del Piemonte Orientale\nNovaraItaly\n\nDepartment of Physics\nPhysics and Astronomy Department\nUniversity of Toronto\nTorontoOntarioCanada\n\nUniversity College London\nLondonUnited King\n\nInstitute of Experimental Physics, Warsaw\nInstitute for Nuclear Studies, Warsaw\nDepartment of Particle Physics, Weizmann Institute\nWarsaw University\nRehovotPoland, Poland, Israel\n", "B Y Oh ", "A Raval ", "J Ukleja ", "J J Whitmore ", "J E Cole ", "J C Hart \nInstitute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan\n\nDepartamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q\n\nUniversidad Autónoma de Madrid\nMadridSpain\n\nDipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly\n", "H Abramowicz ", "A Gabareen ", "R Ingbir ", "S Kananov ", "A Levy Raymond ", "Beverly Sackler ", "M Kuze ", "R Hori ", "S Kagawa ", "N Okazaki ", "S Shimizu ", "T Tawara ", "R Hamatsu ", "H Kaji ", "S Kitamura ", "O Ota ", "Y D Ri ", "M I Ferrero ", "V Monaco ", "R Sacchi ", "A Solano ", "M Arneodo ", "M Ruspa ", "S Fourletov ", "J F Martin ", "S K Boutle ", "J M Butterworth ", "C Gwenlan ", "R Hall-Wilton ", "T W Jones ", "J H Loizides ", "M R Sutton ", "C Targett-Adams ", "M Wing \nInstitute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan\n\nDepartamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q\n\nUniversidad Autónoma de Madrid\nMadridSpain\n\nDipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly\n", "Dom M B Brzozowska ", "J Ciborowski ", "G Grzelak ", "P Kulinski ", "P Lużniak \nLódź University\nPoland\n", "J Malka \nLódź University\nPoland\n", "R J Nowak ", "J M Pawlak ", "T Tymieniecka ", "A Ukleja ", "A F Żarnecki ", "M Adamus ", "P Plucinski ", "Y Eisenberg ", "I Giller ", "D Hochman ", "U Karshon ", "M Rosin ", "E Brownson ", "T Danielson ", "A Everett ", "D Kçira ", "D D Reeder ", "P Ryan ", "A A Savin ", "W H Smith ", "H Wolfe ", "S Bhadra ", "C D Catterall ", "Y Cui ", "G Hartner ", "S Menary ", "U Noor ", "J Standage ", "J Whyte ", "\nArgonne National Laboratory\n60439-4815ArgonneIllinoisUSA\n", "\nInstitut für Physik der Humboldt\nAndrews University\n49104-0380Berrien SpringsMichiganUSA\n", "\nUniversity and INFN Bologna\nUniversität zu Berlin\nBerlin, BolognaGermany, Italy\n", "\nDepartment of Physics\nDeutsches Elektronen-Synchrotron DESY\nDeutsches Elektronen-Synchrotron DESY\nUniversity and INFN\nFakultät für Physik\nJagellonian University\nCracow, Hamburg, Zeuthen, FlorencePoland, Germany, Germany, Italy\n", "\nDepartment of Physics and Astronomy\nDepartment of Engineering in Management and Finance\nUniversität Freiburg\nUniversity of Glasgow\nGlasgowUnited King\n", "\nUniv. of Aegean\nGreece\n", "\nInstitute of Exp. Physics\nHamburg University\nHamburgGermany\n", "\nDepartment of Physics\nMcGill University\nH3A 2T8MontréalQuébecCanada\n", "\nFaculty of General Education\nMeiji Gakuin University\nYokohamaJapan\n", "\nMoscow Engineering Physics Institute\nMoscowRussia\n", "\nInstitute of Nuclear Physics\nMoscow State University\nMoscowRussia\n", "\nMax-Planck-Institut für Physik\nNIKHEF and University of Amsterdam\nMünchen, AmsterdamGermany, Netherlands\n", "\nPhysics Department\nOhio State University\n43210ColumbusOhio\n", "\nDepartment of Physics\nUniversity of Oxford\nOxfordUnited Kingdom\n", "\nIga Polytechnic University\nSagamiharaJapan\n", "\nIII G. D'Agostini, G. Marini, A. Nigro Dipartimento di Fisica, Università 'La Sapienza' and INFN\nRomeItaly\n", "\nDepartment of Physics\nTokyo Institute of Technology\nTokyoJapan\n", "\nDepartment of Physics\nUniversity of Tokyo\nTokyoJapan\n", "\nDepartment of Physics\nTokyo Metropolitan University\nTokyoJapan\n", "\nUniversità di Torino and INFN\nTorinoItaly\n", "\nDepartment of Physics\nUniversity of Wisconsin\n53706MadisonWisconsinUSA\n", "\nDepartment of Physics\nYork University\nM3J 1P3 a IV 1 supported by DESYOntarioCanada, Germany\n" ]
[ "University College London\nUK", "University College London\nUK", "University of Hamburg\nHamburgGermany, Germany", "University of Liverpool\nUK", "Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan", "Departamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q", "Universidad Autónoma de Madrid\nMadridSpain", "Dipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly", "Institut of Theoretical and Experimental Physics\nMoscowRussia", "also at INP\nCracowPoland", "on leave of absence from FPACS, AGH-UST\nCracowPoland", "Physikalisches Institut\nUniversität Bonn\nBonnGermany", "Chonnam National University\nKwangjuSouth Korea", "Faculty of Exact Sciences\nSchool of Physics\nRutherford Appleton Laboratory, Chilton\nTel-Aviv University\nDidcot, Tel-AvivOxonUnited Kingdom, Israel", "Università del Piemonte Orientale\nNovaraItaly", "Department of Physics\nPhysics and Astronomy Department\nUniversity of Toronto\nTorontoOntarioCanada", "University College London\nLondonUnited King", "Institute of Experimental Physics, Warsaw\nInstitute for Nuclear Studies, Warsaw\nDepartment of Particle Physics, Weizmann Institute\nWarsaw University\nRehovotPoland, Poland, Israel", "Faculty of Physics and Applied Computer Science\nAGH-University of Science and Tech-nology\nCracowPoland", "Faculty of Physics and Applied Computer Science\nAGH-University of Science and Tech-nology\nCracowPoland", "Physics Laboratory\nUniversity of Bristol\nBristolUnited Kingdom", "Physics Department and INFN\nCalabria University\nCosenzaItaly", "High Energy Nuclear Physics Group\nImperial College London\nLondonUnited King-dom", "Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan", "Departamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q", "Universidad Autónoma de Madrid\nMadridSpain", "Dipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly", "The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences\nNevis Laboratories\nColumbia University\nIrvington on Hudson10027o, CracowNew YorkPoland", "Physikalisches Institut\nUniversität Bonn\nBonnGermany", "Chonnam National University\nKwangjuSouth Korea", "Faculty of Exact Sciences\nSchool of Physics\nRutherford Appleton Laboratory, Chilton\nTel-Aviv University\nDidcot, Tel-AvivOxonUnited Kingdom, Israel", "Università del Piemonte Orientale\nNovaraItaly", "Department of Physics\nPhysics and Astronomy Department\nUniversity of Toronto\nTorontoOntarioCanada", "University College London\nLondonUnited King", "Institute of Experimental Physics, Warsaw\nInstitute for Nuclear Studies, Warsaw\nDepartment of Particle Physics, Weizmann Institute\nWarsaw University\nRehovotPoland, Poland, Israel", "Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan", "Departamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q", "Universidad Autónoma de Madrid\nMadridSpain", "Dipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly", "Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan\nCenter for High Energy Physics\nInstitut de Physique Nucléaire\nKyungpook National University\nAlmaty, DaeguKazakhstan", "Departamento de Física Teórica\nUniversité Catholique de Louvain\nLouvain-la-NeuveBel-gium q", "Universidad Autónoma de Madrid\nMadridSpain", "Dipartimento di Fisica dell' Università and INFN\nDepartment of Physics\nPennsylvania State University\n16802Padova, University ParkPennsylvaniaItaly", "Lódź University\nPoland", "Lódź University\nPoland", "Argonne National Laboratory\n60439-4815ArgonneIllinoisUSA", "Institut für Physik der Humboldt\nAndrews University\n49104-0380Berrien SpringsMichiganUSA", "University and INFN Bologna\nUniversität zu Berlin\nBerlin, BolognaGermany, Italy", "Department of Physics\nDeutsches Elektronen-Synchrotron DESY\nDeutsches Elektronen-Synchrotron DESY\nUniversity and INFN\nFakultät für Physik\nJagellonian University\nCracow, Hamburg, Zeuthen, FlorencePoland, Germany, Germany, Italy", "Department of Physics and Astronomy\nDepartment of Engineering in Management and Finance\nUniversität Freiburg\nUniversity of Glasgow\nGlasgowUnited King", "Univ. of Aegean\nGreece", "Institute of Exp. Physics\nHamburg University\nHamburgGermany", "Department of Physics\nMcGill University\nH3A 2T8MontréalQuébecCanada", "Faculty of General Education\nMeiji Gakuin University\nYokohamaJapan", "Moscow Engineering Physics Institute\nMoscowRussia", "Institute of Nuclear Physics\nMoscow State University\nMoscowRussia", "Max-Planck-Institut für Physik\nNIKHEF and University of Amsterdam\nMünchen, AmsterdamGermany, Netherlands", "Physics Department\nOhio State University\n43210ColumbusOhio", "Department of Physics\nUniversity of Oxford\nOxfordUnited Kingdom", "Iga Polytechnic University\nSagamiharaJapan", "III G. D'Agostini, G. Marini, A. Nigro Dipartimento di Fisica, Università 'La Sapienza' and INFN\nRomeItaly", "Department of Physics\nTokyo Institute of Technology\nTokyoJapan", "Department of Physics\nUniversity of Tokyo\nTokyoJapan", "Department of Physics\nTokyo Metropolitan University\nTokyoJapan", "Università di Torino and INFN\nTorinoItaly", "Department of Physics\nUniversity of Wisconsin\n53706MadisonWisconsinUSA", "Department of Physics\nYork University\nM3J 1P3 a IV 1 supported by DESYOntarioCanada, Germany" ]
[]
The production of D * ± (2010) mesons in e ± p scattering in the range of exchanged photon virtuality 0.05 < Q 2 < 0.7 GeV 2 has been measured with the ZEUS detector at HERA using an integrated luminosity of 82 pb −1 . The decay channels D * + → D 0 π + with D 0 → K − π + and corresponding antiparticle decay were used to identify D * mesons and the ZEUS beampipe calorimeter was used to identify the scattered electron. Differential D * cross sections as functions of Q 2 , inelasticity, y, transverse momentum of the D * meson, p T (D * ), and pseudorapidity of the D * meson, η(D * ), have been measured in the kinematic region 0.02 < y < 0.85, 1.5 < p T (D * ) < 9.0 GeV and |η(D * )| < 1.5. The measured differential cross sections are in agreement with two different NLO QCD calculations. The cross sections are also compared to previous ZEUS measurements in the photoproduction and DIS regimes.
10.1016/j.physletb.2007.04.003
[ "https://export.arxiv.org/pdf/hep-ex/0702034v2.pdf" ]
119,101,654
hep-ex/0702034
90aea9284c53259a6c36043f3608aa61f0ba114b
The ZEUS Collaboration M5S 1A7 a arXiv:hep-ex/0702034v2 10 Apr 2007 13th February 2007 (2005-2008) S Chekanov M Derrick S Magill S Miglioranzi University College London UK B Musgrave D Nicholass University College London UK J Repond R Yoshida M C K Mattingly M Jechow N Pavel A G Yagües Molina S Antonelli P Antonioli G Bari M Basile L Bellagamba M Bindi D Boscherini A Bruni G Bruni L Cifarelli F Cindolo A Contin M Corradi University of Hamburg HamburgGermany, Germany University of Liverpool UK S De Pasquale G Iacobucci A Margotti R Nania A Polini L Rinaldi G Sartorelli A Zichichi D Bartsch I Brock S Goers H Hartmann E Hilger P Irrgang H.-P Jakob M Jüngst O M Kind E Paul R Renner U Samson V Schönberg R Shehzadi M Wlasenko N H Brook G P Heath J D Morris T Namsoo H H Wills Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan Center for High Energy Physics Institut de Physique Nucléaire Kyungpook National University Almaty, DaeguKazakhstan Departamento de Física Teórica Université Catholique de Louvain Louvain-la-NeuveBel-gium q Universidad Autónoma de Madrid MadridSpain Dipartimento di Fisica dell' Università and INFN Department of Physics Pennsylvania State University 16802Padova, University ParkPennsylvaniaItaly M Capua S Fazio A Mastroberardino M Schioppa G Susinno E Tassi J Y Kim K J Ma Z A Ibrahim B Kamaluddin W A T Wan Abdullah Jabatan Fizik Universiti Malaya Kuala Lumpur Malaysia Y Ning Z Ren F Sciulli J Chwastowski A Eskreys J Figiel A Galas M Gil K Olkiewicz P Stopa L Zaw-Iejski L Adamczyk T Bo I Grabowska-Bo D Ld J Kisielewska M Lukasik L Przybycień Suszycki I A Kotański W S Lomiński V Adler U Behrens I Bloch C Blohm A Bonato K Borras N Coppola A Dos-Sanov J Fourletova A Geiser D Gladkov P Göttlicher I Gregor T Haas W Hain C Horn B Kahle U Klein U Kötz H Kowalski E Lobodzinska B Löhr R Mankel I.-A Melzer-Pellmann A Montanari D Notz A E Nuncio-Quiroz I Rubinsky R San-Tamarta U Schneekloth A Spiridonov Institut of Theoretical and Experimental Physics MoscowRussia H Stadie D Szuba also at INP CracowPoland J Szuba on leave of absence from FPACS, AGH-UST CracowPoland T Theedt G Wolf K Wrona C Youngman W Zeuner W Lohmann S Schlenstedt G Barbagli E Gallo P G Pelfer Physikalisches Institut Universität Bonn BonnGermany Chonnam National University KwangjuSouth Korea Faculty of Exact Sciences School of Physics Rutherford Appleton Laboratory, Chilton Tel-Aviv University Didcot, Tel-AvivOxonUnited Kingdom, Israel Università del Piemonte Orientale NovaraItaly Department of Physics Physics and Astronomy Department University of Toronto TorontoOntarioCanada University College London LondonUnited King Institute of Experimental Physics, Warsaw Institute for Nuclear Studies, Warsaw Department of Particle Physics, Weizmann Institute Warsaw University RehovotPoland, Poland, Israel E A Bamberger D Dobur F Karstens N N Vlasov I Br Faculty of Physics and Applied Computer Science AGH-University of Science and Tech-nology CracowPoland Freiburg I Br Faculty of Physics and Applied Computer Science AGH-University of Science and Tech-nology CracowPoland Germany Physics Laboratory University of Bristol BristolUnited Kingdom Physics Department and INFN Calabria University CosenzaItaly High Energy Nuclear Physics Group Imperial College London LondonUnited King-dom P J Bussey A T Doyle W Dunne J Ferrando D H Saxon I O Skillicorn Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan Center for High Energy Physics Institut de Physique Nucléaire Kyungpook National University Almaty, DaeguKazakhstan Departamento de Física Teórica Université Catholique de Louvain Louvain-la-NeuveBel-gium q Universidad Autónoma de Madrid MadridSpain Dipartimento di Fisica dell' Università and INFN Department of Physics Pennsylvania State University 16802Padova, University ParkPennsylvaniaItaly I Gialas T Gosau U Holm R Klanner E Lohrmann H Salehi P Schleper T Schörner-Sadenius J Sztuk K Wichmann K Wick C Foudas C Fry K R Long A D Tapper M Kataoka T Matsumoto K Nagano K Tokushuku S Yamada Y Yamazaki A N Barakbaev E G Boos N S Pokrovskiy B O Zhautykov D Son South Korea The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences Nevis Laboratories Columbia University Irvington on Hudson10027o, CracowNew YorkPoland Ii J De Favereau K Piotrzkowski F Barreiro C Glasman M Jimenez L Labarga J Del Peso E Ron M Soares J Terrón M Zambrana F Corriveau C Liu R Walsh C Zhou T Tsurugai A Antonov B A Dolgoshein V Sosnovtsev A Stifutkin S Suchkov R K Dementiev P F Ermolov L K Gladilin I I Katkov L A Khein I A Korzhavina V A Kuzmin B B Levchenko O Yu Lukina A S Proskuryakov L M Shcheglova D S Zotkin S A Zotkin I Abt C Büttner A Caldwell D Kollar W B Schmidke J Sutiak G Grigorescu A Keramidas E Koffeman P Kooijman A Pellegrino H Tiecke M Vázquez L Wiggers N Brümmer B Bylsma L S Durkin A Lee T Y Ling P D Allfrey M A Bell A M Cooper-Sarkar A Cottrell R C E Devenish B Foster K Korcsak-Gorzo S Patel V Roberfroid A Robertson P B Straub C Uribe-Estrada R Walczak P Bellan A Bertolin R Brugnera R Carlin R Ciesielski F Dal Corso S Dusini A Garfagnini S Limentani A Longhin L Stanco M Turcato Physikalisches Institut Universität Bonn BonnGermany Chonnam National University KwangjuSouth Korea Faculty of Exact Sciences School of Physics Rutherford Appleton Laboratory, Chilton Tel-Aviv University Didcot, Tel-AvivOxonUnited Kingdom, Israel Università del Piemonte Orientale NovaraItaly Department of Physics Physics and Astronomy Department University of Toronto TorontoOntarioCanada University College London LondonUnited King Institute of Experimental Physics, Warsaw Institute for Nuclear Studies, Warsaw Department of Particle Physics, Weizmann Institute Warsaw University RehovotPoland, Poland, Israel B Y Oh A Raval J Ukleja J J Whitmore J E Cole J C Hart Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan Center for High Energy Physics Institut de Physique Nucléaire Kyungpook National University Almaty, DaeguKazakhstan Departamento de Física Teórica Université Catholique de Louvain Louvain-la-NeuveBel-gium q Universidad Autónoma de Madrid MadridSpain Dipartimento di Fisica dell' Università and INFN Department of Physics Pennsylvania State University 16802Padova, University ParkPennsylvaniaItaly H Abramowicz A Gabareen R Ingbir S Kananov A Levy Raymond Beverly Sackler M Kuze R Hori S Kagawa N Okazaki S Shimizu T Tawara R Hamatsu H Kaji S Kitamura O Ota Y D Ri M I Ferrero V Monaco R Sacchi A Solano M Arneodo M Ruspa S Fourletov J F Martin S K Boutle J M Butterworth C Gwenlan R Hall-Wilton T W Jones J H Loizides M R Sutton C Targett-Adams M Wing Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan Center for High Energy Physics Institut de Physique Nucléaire Kyungpook National University Almaty, DaeguKazakhstan Departamento de Física Teórica Université Catholique de Louvain Louvain-la-NeuveBel-gium q Universidad Autónoma de Madrid MadridSpain Dipartimento di Fisica dell' Università and INFN Department of Physics Pennsylvania State University 16802Padova, University ParkPennsylvaniaItaly Dom M B Brzozowska J Ciborowski G Grzelak P Kulinski P Lużniak Lódź University Poland J Malka Lódź University Poland R J Nowak J M Pawlak T Tymieniecka A Ukleja A F Żarnecki M Adamus P Plucinski Y Eisenberg I Giller D Hochman U Karshon M Rosin E Brownson T Danielson A Everett D Kçira D D Reeder P Ryan A A Savin W H Smith H Wolfe S Bhadra C D Catterall Y Cui G Hartner S Menary U Noor J Standage J Whyte Argonne National Laboratory 60439-4815ArgonneIllinoisUSA Institut für Physik der Humboldt Andrews University 49104-0380Berrien SpringsMichiganUSA University and INFN Bologna Universität zu Berlin Berlin, BolognaGermany, Italy Department of Physics Deutsches Elektronen-Synchrotron DESY Deutsches Elektronen-Synchrotron DESY University and INFN Fakultät für Physik Jagellonian University Cracow, Hamburg, Zeuthen, FlorencePoland, Germany, Germany, Italy Department of Physics and Astronomy Department of Engineering in Management and Finance Universität Freiburg University of Glasgow GlasgowUnited King Univ. of Aegean Greece Institute of Exp. Physics Hamburg University HamburgGermany Department of Physics McGill University H3A 2T8MontréalQuébecCanada Faculty of General Education Meiji Gakuin University YokohamaJapan Moscow Engineering Physics Institute MoscowRussia Institute of Nuclear Physics Moscow State University MoscowRussia Max-Planck-Institut für Physik NIKHEF and University of Amsterdam München, AmsterdamGermany, Netherlands Physics Department Ohio State University 43210ColumbusOhio Department of Physics University of Oxford OxfordUnited Kingdom Iga Polytechnic University SagamiharaJapan III G. D'Agostini, G. Marini, A. Nigro Dipartimento di Fisica, Università 'La Sapienza' and INFN RomeItaly Department of Physics Tokyo Institute of Technology TokyoJapan Department of Physics University of Tokyo TokyoJapan Department of Physics Tokyo Metropolitan University TokyoJapan Università di Torino and INFN TorinoItaly Department of Physics University of Wisconsin 53706MadisonWisconsinUSA Department of Physics York University M3J 1P3 a IV 1 supported by DESYOntarioCanada, Germany The ZEUS Collaboration M5S 1A7 a arXiv:hep-ex/0702034v2 10 Apr 2007 13th February 2007 (2005-2008)Measurement of D * ± meson production in e ± p scattering at low Q 2 ZEUS Collaboration Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan f 10 now at DESY group FEB, 24 also at Max Planck Institute, Munich, Germany, Alexander von Humboldt Research Award 25 now at KEK, Tsukuba, Japan 26 now at Nagoya University, Japan 27 Department of Radiological Science 28 PPARC Advanced fellow 29 also at Lódź University, Poland The production of D * ± (2010) mesons in e ± p scattering in the range of exchanged photon virtuality 0.05 < Q 2 < 0.7 GeV 2 has been measured with the ZEUS detector at HERA using an integrated luminosity of 82 pb −1 . The decay channels D * + → D 0 π + with D 0 → K − π + and corresponding antiparticle decay were used to identify D * mesons and the ZEUS beampipe calorimeter was used to identify the scattered electron. Differential D * cross sections as functions of Q 2 , inelasticity, y, transverse momentum of the D * meson, p T (D * ), and pseudorapidity of the D * meson, η(D * ), have been measured in the kinematic region 0.02 < y < 0.85, 1.5 < p T (D * ) < 9.0 GeV and |η(D * )| < 1.5. The measured differential cross sections are in agreement with two different NLO QCD calculations. The cross sections are also compared to previous ZEUS measurements in the photoproduction and DIS regimes. 1 supported by DESY, Germany 2 also affiliated with University College London, UK 3 also at University of Hamburg, Germany, Alexander von Humboldt Fellow 4 self-employed 5 now at Siemens, Lindau, Germany Introduction The production of charm quarks at HERA has been studied both in deep inelastic scattering (DIS) [1][2][3][4][5] and photoproduction [6][7][8][9][10]. In general, reasonable agreement is seen with next-to-leading-order (NLO) QCD predictions. This paper presents measurements of the D * cross section in the range 0.05 < Q 2 < 0.7 GeV 2 . The beampipe calorimeter of ZEUS [11,12] was used for the measurement of the scattered lepton, which allows the first measurements of the transition region between photoproduction (photon virtuality, Q 2 ∼ 0 GeV 2 ) and DIS (Q 2 > 1 GeV 2 ). The cross sections are compared to the predictions of two different NLO QCD calculations, one designed for DIS, the other for the photoproduction region. This paper investigates whether the calculations remain valid in this transition region. Experimental set-up This analysis was performed with data taken from 1998 to 2000, when HERA collided electrons or positrons 1 with energy E e = 27.5 GeV with protons of energy E p = 920 GeV. The combined data sample has an integrated luminosity of L = 81.9 ± 1.8 pb −1 . A detailed description of the ZEUS detector can be found elsewhere [13]. A brief outline of the components that are most relevant for this analysis is given below. Charged particles are tracked in the central tracking detector (CTD) [14], which operates in a magnetic field of 1.43 T provided by a thin superconducting coil. The CTD consists of 72 cylindrical drift chamber layers, organized in nine superlayers covering the polarangle 2 region 15 • < θ < 164 • . The transverse-momentum resolution for full-length tracks is σ(p T )/p T = 0.0058p T ⊕ 0.0065 ⊕ 0.0014/p T , with p T in GeV. The high-resolution uranium-scintillator calorimeter (CAL) [15] consists of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each part is subdivided transversely into towers and longitudinally into one electromagnetic section and either one (in RCAL) or two (in BCAL and FCAL) hadronic sections. The smallest subdivision of the calorimeter is called a cell. The CAL energy resolutions, as measured under test-beam conditions, are σ(E)/E = 0.18/ √ E for electrons and σ(E)/E = 0.35/ √ E for hadrons, with E in GeV. 1 Hereafter, both electrons and positrons are referred to as electrons. 2 The ZEUS coordinate system is a right-handed Cartesian system, with the Z axis pointing in the proton beam direction, referred to as the "forward direction", and the X axis pointing left towards the centre of HERA. The coordinate origin is at the nominal interaction point. The scattered electron was detected in the beampipe calorimeter (BPC). The BPC allowed the detection of low-Q 2 events, where the electron is scattered through a small angle. The BPC was used in previous measurements of the proton structure function, F 2 , at low Q 2 [11,12]. It originally consisted of two tungsten-scintillator sampling calorimeters with the front faces located at Z = −293.7 cm, the centre at Y = 0.0 cm, and the inner edge of the active area at X = ±4.4 cm, as close as possible to the electron-beam trajectory. At the end of 1997 one of the two BPC calorimeters was removed; hence, for the analysis in this paper, only the calorimeter located on the +X side of the beampipe was utilised. It had an active area of 12.0 × 12.8 cm 2 in X × Y and a depth of 24 radiation lengths. The relative energy resolution as determined in test-beam measurements with 1 -6 GeV electrons was ∆E/E = 17%/ E ( GeV). The luminosity was measured from the rate of the bremsstrahlung process ep → eγp, where the photon was measured in a lead-scintillator calorimeter [16] placed in the HERA tunnel at Z=-107m. A three-level trigger system was used to select events online [13,17]. At all three levels, the event was required to contain a scattered electron candidate in the BPC. Additionally, at the third level, a reconstructed D * candidate was required for the event to be kept for further analysis. The efficiency of the online D * reconstruction, determined relative to an inclusive DIS trigger, was above 95% [5]. Kinematic reconstruction and event selection Deep inelastic electron-proton scattering, ep → eX, can be described in terms of two kinematic variables, chosen here to be y and Q 2 , where y is the inelasticity. They are defined as Q 2 = −q 2 = −(k − k ′ ) 2 and y = Q 2 /(2P · q) , where k and P are the fourmomenta of the incoming electron and proton, respectively, and k ′ is the four-momentum of the scattered electron. The inelasticity, which is the fractional energy transferred to the proton in its rest frame, is related to the Bjorken scaling variable x and Q 2 by Q 2 = sxy, where s = 4E e E p is the square of the electron-proton centre-of-mass energy of 318 GeV. The values of y and Q 2 were calculated using the measured electron scattering angle and the energy deposited in the BPC as detailed in a previous analysis [11], which also describes the method used for the energy calibration of the BPC. A time dependent re-calibration of the energy response was necessary [18], as radiation damage of the scintillator resulted in a degradation of about 10% by the end of the 2000 running period. A series of cuts was applied to reject background. The events were required to have a primary vertex within 50 cm in Z of the nominal interaction point. The electron candidates in the BPC were required to have E BPC > 4 GeV, as the trigger efficiency is low below this energy. The electron impact point on the face of the BPC was required to be more than 0.7 cm from the inner edge to ensure good shower containment. Photoproduction events were efficiently rejected by requiring the events to have 35 < E − P Z < 65 GeV, where E − P Z = i (E − P Z ) i is summed over all CAL deposits, including the scattered electron candidate in the BPC. Finally, events with an additional well-reconstructed electron candidate in the CAL with energy greater than 5 GeV were rejected to reduce background from DIS events with Q 2 > 1 GeV 2 . The measured kinematic region in y and Q 2 was restricted to the range of high acceptance, 0.02 < y < 0.85, 0.05 < Q 2 < 0.7 GeV 2 . With these cuts, the reconstructed invariant mass of the hadronic system, W , lies between 50 and 300 GeV, with a mean of 190 GeV. Selection of D * candidates The D * mesons were identified using the decay channel D * + → D 0 π + s with the subsequent decay D 0 → K − π + and the corresponding antiparticle decay chain, where π + s refers to a low-momentum ("slow") pion accompanying the D 0 . Charged tracks measured by the CTD and assigned to the primary event vertex 3 were selected. The transverse momentum was required to be greater than 0.12 GeV. The p T cut was raised to 0.25 GeV for a data subsample corresponding to (16.9 ± 0.4) pb −1 , for which the low-momentum track-reconstruction efficiency was lower due to the operating conditions of the CTD [19]. Each track was required to reach at least the third superlayer of the CTD. These restrictions ensured that the track acceptance was high and the momentum resolution was good. Tracks in the CTD with opposite charges and transverse momenta p T > 0.45 GeV were combined in pairs to form D 0 candidates. The tracks were alternately assigned the kaon and the pion mass and the invariant mass of the pair, M Kπ , was determined. Each additional track, with charge opposite to that of the kaon track, was assigned the pion mass and combined with the D 0 -meson candidate to form a D * candidate. The D * mesons were selected in the kinematic region 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5. The ∆M distribution for events with an electron reconstructed in the BPC is shown in Fig. 1. To extract the number of D * mesons, the ∆M distribution was fit using an unbinned likelihood method, with a Gaussian to describe the signal and a threshold function to describe the combinatorial background. A first estimate of the background was given by D * candidates with wrong-sign combinations, in which both tracks forming the D 0 candidates have the same charge and the third track has the opposite charge. These are shown as the shaded region in Fig. 1. The number of D * mesons obtained from the fit was N(D * ) = 253 ± 25. Acceptance corrections and systematic uncertainties The acceptances were calculated using the Herwig 6.1 [20] and Rapgap 2.08 [21] Monte Carlo (MC) models. Both models simulate charm and beauty production and include contributions from both direct and resolved photoproduction. In direct photoproduction the photon participates as a point-like particle in the hard scattering process, while in resolved photoproduction a parton in the photon scatters on a parton in the proton. The generated events were passed through a full simulation of the detector, using Geant 3.13 [22] and then processed and selected with the same programs as used for the data. The CTEQ5L [23] parton density function (PDF) was used for the proton and GRV-LO [24] was used for the photon. The charm-quark mass was set to 1.5 GeV. The Herwig predictions are in good agreement with the data distributions for both the scattered lepton and hadronic variables and so this Monte Carlo was used to correct the data for detector effects. For the kinematic region of the measurement 0.05 < Q 2 < 0.7 GeV 2 , 0.02 < y < 0.85, 1.5 < p T (D * ) < 9 GeV, and |η(D * )| < 1.5 the acceptance was (1.11 ± 0.03)%. This includes the geometrical acceptance of the BPC, which was about 9%, and the reconstruction efficiency for the D * decay chain. The Rapgap MC gives a similarly good representation of the data and was used to estimate part of the systematic uncertainties, as described below. The differential cross section for a given observable Y was determined using dσ dY = N A · L · B · ∆Y , where N is the number of D * events in a bin of size ∆Y , A is the acceptance (which takes into account migrations and efficiencies for that bin) and L is the integrated luminosity. The product, B, of the appropriate branching ratios for the D * and D 0 decays was set to (2.57 ± 0.05)% [25]. The systematic uncertainties of the measured cross sections were determined by changing in turn the selection cuts or the analysis procedure within their uncertainties and repeating the extraction of the cross sections [26]. The major experimental sources of systematic uncertainty were (the variation of the total cross section is given in parentheses): the BPC alignment ( +2.5 −3.1 %) and energy scale ( +0.4 −1.2 %); the uncertainty in the CTD momentum scale ( +0.2 −1.5 %) and the CAL energy scale (±1%); the p T (D * )/E θ>10 • T cut ( +3.0 −1.7 %) and the D * signal extraction ( +0.1 −1.5 %). The uncertainty due to the MC model ( +9.5 −4.8 %) was determined by using Rapgap to evaluate the acceptance correction rather than Herwig, as well as by varying the fraction of resolved and direct photoproduction processes in the simulation. All the above errors were added in quadrature separately for the positive and negative variations to determine the overall systematic uncertainty. The overall normalisation has additional uncertainties of 2.2% due to the luminosity measurement and 2.0% due to knowledge of branching ratios. These are included in the error quoted for the total cross section but not in the systematic uncertainties of the differential cross sections. Theoretical predictions Two different calculations were used to evaluate the theoretical expectation for charm production. The HVQDIS program [27] implements an NLO calculation of charm production in DIS. At low Q 2 , the hadron-like structure of the photon, not included in HVQDIS, is needed to regularise the NLO calculation. Therefore predictions from this program are expected to lose accuracy in the limit Q 2 → 0. The ZEUS measurements of D * production in DIS for Q 2 > 1.5 GeV 2 are in good agreement with the HVQDIS prediction [5]. The FMNR program [28] implements an NLO calculation of charm photoproduction which includes the hadron-like component of the photon. Electroproduction cross sections can be obtained with FMNR using the Weizsäcker-Williams approximation [29] and are therefore expected to be reliable only at low Q 2 , where this approximation is valid. The FMNR predictions are in reasonable agreement with ZEUS measurements of D * photoproduction [7], considering the theoretical uncertainties. It is therefore interesting to see whether these calculations are able to reproduce the data in the transition region between photoproduction and DIS. The following parameters were used in the calculations for both programs. They were chosen to be the same as in a previous publication [5]. A variant of the ZEUS-S NLO QCD global fit [30] to structure-function data was used as the parameterisation of the proton PDFs. This fit was repeated in the fixed-flavour-number scheme, FFNS, in which the PDF has three active quark flavours in the proton, and Λ (3) QCD is set to 0.363 GeV. The mass of the charm quark was set to 1.35 GeV. The renormalisation and factorisation scales were set to µ R = µ F = Q 2 + 4m 2 c in HVQDIS, while for FMNR they were set to the usual choice of µ R = µ F = p 2 T + m 2 c , where p 2 T is the average transverse momentum squared of the charm quarks. The charm fragmentation to a D * is carried out using the Peterson function [31]. The hadronisation fraction, f (c → D * ), was taken to be 0.238 [32] and the Peterson parameter, ǫ, was set to 0.035 [33]. The parameters used here for the FMNR calculation are different from those used in a previous photoproduction analysis [7] (which used m c = 1.5 GeV) leading to a 20% larger predicted photoproduction cross section. For the FMNR calculation the electroproduction cross section, σ ep , was obtained from the photoproduction cross section, σ γp (W ), using σ ep = ymax y min dy Φ(y, Q 2 min , Q 2 max )σ γp ( √ ys), where Φ(y, Q 2 min , Q 2 max ) = α em 2π (1 + (1 − y) 2 ) y ln Q 2 max Q 2 min − 2m e y 1 Q 2 min − 1 Q 2 max (1) is the photon flux and y min , y max , Q 2 min , Q 2 max define the measurement range in y and Q 2 . The NLO QCD predictions for D * production are affected by systematic uncertainties, which were also evaluated as in a previous ZEUS paper [5] 4 . The sources of systematic uncertainties on the total cross section are: charm quark mass ( +15 −13 % for HVQDIS, +16 −14 % for FMNR); renormalisation and factorisation scale ( +1 −13 % for HVQDIS, +23 −10 % for FMNR); ZEUS PDF (±5%); fragmentation ( +10 −6 %). For both programs, the systematic uncertainties were added in quadrature and are displayed as a band in the figures. Theoretical calculations of the total charm cross section in this Q 2 range can not be compared to the present data since D * are only measured in a limited p T and η range. Cross section measurements The total cross section for 0.05 < Q 2 < 0.7 GeV 2 , 0.02 < y < 0.85, 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5 is: σ(ep → eD * X) = 10.1 ± 1.0(stat.) +1.1 −0.8 (syst.) ± 0.20(BR) nb, where the first uncertainty is statistical, the second from systematic effects (including the luminosity uncertainty) and the third from the uncertainties in the branching ratios. The prediction from the HVQDIS program is 8.6 +1.9 −1.8 nb, in agreement with the data, while the prediction from FMNR is 8.9 +2.4 −1.4 nb 5 , also in good agreement. The measured differential D * cross sections as a function of Q 2 , y, p T (D * ) and η(D * ) for the data are shown in Fig. 2 and given in Table 1. The predictions of the NLO calculations, including their uncertainties, are shown as bands. The measured differential cross sections are well described over the full measured kinematic region by both calculations. This analysis was also compared to previous ZEUS measurements of D * production in DIS [5] made in the kinematic region 1.5 < Q 2 < 1000 GeV 2 , 0.02 < y < 0.7, 1.5 < p T (D * ) < 15 GeV and |η(D * )| < 1.5. In order to directly compare with the results presented there, the cross sections were recalculated in the modified kinematic region 0.02 < y < 0.7. No correction was made for the different upper cut on p T (D * ), as the size of the effect is ≈ 1%. For this modified kinematic region, the differential cross section as a function of Q 2 is presented in Fig. 3 and given in Table 2. The systematic errors were assumed to be the same as those in the full y range. Figure 3 also shows the previous ZEUS measurement and the HVQDIS prediction. The combination of both measurements shows that the slope of dσ/dQ 2 changes with Q 2 ; at high Q 2 the slope is steeper than at low Q 2 . The NLO calculation describes the measured data well over the full Q 2 range. The D * electroproduction cross sections were converted to γp cross sections, σ γp , in the range 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5 (measured in the laboratory frame) using the photon flux from Eq. 1. The cross sections are given for W = 160 GeV, which corresponds to y = 0.25, close to the mean y of the measured cross sections. The W dependence of σ γp was evaluated from the data. The uncertainty of this procedure was estimated to be 10%. A comparison of the charm photoproduction cross section [7], this measurement and the DIS cross sections [5] is shown in Fig. 4. The numbers are tabulated in Table 3. The photoproduction point was corrected for the different kinematic range and centre-of-mass energy used here using the FMNR program. As can be seen, the present measurements are consistent with the photoproduction cross section. A fit using a function of the form σ(Q 2 ) = SM 2 /(Q 2 + M 2 ), where S is the photoproduction cross section at Q 2 = 0 and M 2 is the scale at which the γp cross section changes from the photoproduction value to the DIS 1/Q 2 behaviour, gives a good description of the data over the whole Q 2 range with S = 823 ± 63 nb and M 2 = 13 ± 2 GeV 2 . The value of M 2 found here for charm production is close to 4m 2 c [34] and significantly larger than that found for inclusive data M 2 0 = 0.52 ± 0.05 GeV 2 [12]. Conclusions Charm production has been measured as a function of Q 2 , y, p T (D * ) and η(D * ) in the kinematic region 0.05 < Q 2 < 0.7 GeV 2 , 0.02 < y < 0.85, 1.5 < p T (D * ) < 9.0 GeV and |η(D * )| < 1.5. These measurements extend the previous ZEUS measurements in DIS to lower Q 2 . The measured differential cross sections are well described by two different NLO QCD calculations: one (FMNR) is designed for the photoproduction region; while the other (HVQDIS) is designed for DIS. Both calculations predict similar cross sections in the intermediate Q 2 region measured here, which agree well with the measurements. The measurements, converted to γp cross sections, also agree well with the D * photoproduction data. Table 1: Measured differential cross sections as a function of Q 2 , y, p T (D * ) and η(D * ) for 0.05 < Q 2 < 0.7 GeV 2 , 0.02 < y < 0.85, 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5. The statistical and systematic uncertainties are shown separately. The normalisation uncertainties from the luminosity measurement and the branching ratios are not included in the systematic uncertainties. Table 2: Measured differential cross sections as a function of Q 2 for 0.02 < y < 0.7, 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5. The systematic uncertainties are assumed to be the same as those for the kinematic range 0.02 < y < 0.85. Table 3: γp cross sections for D * production in the range 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5 as a function of Q 2 for W = 160 GeV . The values at Q 2 ≈ 0 and for Q 2 > 2.7 GeV 2 are obtained from previous photoproduction [7] and DIS measurements [5] in the range 1. The γp cross section for D * ± production in the range 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5 as a function of Q 2 from this paper (BPC), compared with previous results on D * production in DIS [5] and photoproduction [7] for 1.5 < p T (D * ) < 15 GeV and |η(D * )| < 1.5. The data are represented by points. The inner error bars are statistical while the open error bars are the sum of statistical and systematic uncertainties added in quadrature. The photoproduction point is drawn at Q 2 = 0.003 GeV 2 for convenience. The curve shows a fit to the data described in the text. Q 2 bin dσ/dQ 2 ∆ stat ∆ syst ( GeV 2 ) ( nb/ GeV 2 ) 0Q 2 bin dσ/dQ 2 ∆ stat ( GeV 2 ) ( nb/ GeV 2 ) 0Q 2 σ γp ∆ stat ∆ syst ( GeV 2 ) ( nb) ∼ 0 729 ±46 +110 − 6 retired 7 supported by Chonnam National University in 2005 8 supported by a scholarship of the World Laboratory Björn Wiik Research Project 9 supported by the research grant no. 1 P03B 04529 (2005-2008) 10 now at DESY group FEB, Hamburg, Germany 11 now at University of Liverpool, UK 12 also at Institut of Theoretical and Experimental Physics, Moscow, Russia 13 also at INP, Cracow, Poland 14 on leave of absence from FPACS, AGH-UST, Cracow, Poland 15 partly supported by Moscow State University, Russia 16 also affiliated with DESY 17 now at CERN, Geneva, Switzerland 18 also at University of Tokyo, Japan 19 Ramón y Cajal Fellow 20 partly supported by Russian Foundation for Basic Research grant no. 05-02-39028-NSFC-a 21 EU Marie Curie Fellow 22 partially supported by Warsaw University, Poland 23 This material was based on work supported by the National Science Foundation, while working at the Foundation. 24 also at Max Planck Institute, Munich, Germany, Alexander von Humboldt Research Award 25 now at KEK, Tsukuba, Japan 26 now at Nagoya University, Japan 27 Department of Radiological Science 28 PPARC Advanced fellow 29 also at Lódź University, Poland 30 Lódź University, Poland 31 supported by the Polish Ministry for Education and Science grant no. 1 P03B 12629 32 supported by the Polish Ministry for Education and Science grant no. 1 P03B 14129 † deceased part by the MINERVA Gesellschaft für Forschung GmbH, the Israel Science Foundation (grant no. 293/02-11.2) and the U.S.-Israel Binational Science Foundation d supported by the German-Israeli Foundation and the Israel Science Foundation e supported by the Italian National Institute for Nuclear Physics (INFN) f supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and its grants for Scientific Research g supported by the Korean Ministry of Education A mass window for the signal region of the D 0 varying from 1.82 < M Kπ < 1.91 GeV to 1.79 < M Kπ < 1.94 GeV was used, reflecting the dependence of the CTD resolution on p T (D * ). The signal region for the reconstructed mass difference ∆M = (M Kππs − M Kπ ) was 0.1435 < ∆M < 0.1475 GeV. The requirement of p T (D * )/E θ>10 • T > 0.1 was also applied, where E θ>10 • T is the transverse energy outside a cone of θ = 10 • defined with respect to the proton direction. This cut rejects background without significantly affecting the signal. 5 < p T (D * ) < 15 GeV and |η(D * )| < 1.5. Figure 1 :Figure 2 :Figure 3 : 123The distribution of the mass difference, ∆M = M(Kππ s ) − M(Kπ), for D * ± candidates with a measured scattered electron in the BPC. The histogram shows the ∆M distribution for wrong charge combinations, normalised to the data in the region 0.151 < ∆M < 0.167. The normalisation factor is 1.07. The solid curve is the result of the fit described in the text. Differential D * production cross sections as a function of (a) Q 2 , (b) y,(c) p T (D * ) and (d) η(D * ) compared to the HVQDIS and FMNR NLO predictions. Data are represented by points. The inner error bars are the statistical errors of the measurement while the open error bars are the sum of statistical and systematic uncertainties added in quadrature. The shaded area indicates the theoretical uncertainties obtained by variation of the HVQDIS parameters. The dashed and dotted lines represent the central value of the FMNR calculation and its uncertainty, respectively. The D * production cross section as a function of Q 2 in the kinematic region 0.02 < y < 0.7, 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5 for this measurement (BPC) and previous results on D * production in DIS [5] (for 1.5 < p T (D * ) < 15 GeV ), compared to the HVQDIS NLO prediction. The data are represented by points. The inner error bars are statistical while the open error bars are the sum of statistical and systematic uncertainties added in quadrature. The shaded area indicates the theoretical uncertainties obtained by variations of the HVQDIS parameters. Figure 4 : 4Figure 4: The γp cross section for D * ± production in the range 1.5 < p T (D * ) < 9 GeV and |η(D * )| < 1.5 as a function of Q 2 from this paper (BPC), compared with previous results on D * production in DIS [5] and photoproduction [7] for 1.5 < p T (D * ) < 15 GeV and |η(D * )| < 1.5. The data are represented by points. The inner error bars are statistical while the open error bars are the sum of statistical and systematic uncertainties added in quadrature. The photoproduction point is drawn at Q 2 = 0.003 GeV 2 for convenience. The curve shows a fit to the data described in the text. and Korea Science and Engineering Foundation h supported by the Netherlands Foundation for Research on Matter (FOM) supported by the Polish State Committee for Scientific Research, grant no. 620/E-77/SPB/DESY/P-03/DZ 117/2003-2005 and grant no. 1P03B07427/2004-2006 partially supported by the German Federal Ministry for Education and Research (BMBF) supported by RF Presidential grant N 8122.2006.2 for the leading scientific schools and by the Russian Ministry of Education and Science through its grant Research on High Energy Physics l supported by the Spanish Ministry of Education and Science through funds provided by CICYT supported by the Particle Physics and Astronomy Research Council, UK supported by the US Department of Energy supported by the US National Science Foundation. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. supported by the Polish Ministry of Science and Higher Education supported by FNRS and its associated funds (IISN and FRIA) and by an Inter-University Attraction Poles Programme subsidised by the Belgian Federal Science Policy Office supported by the Malaysian Ministry of Science, Technology and Innovation/Akademi Sains Malaysia grant SAGA 66-02-03-0048i j k m n o p q r The resolution of such tracks is not good enough to separate primary and secondary vertices from c and b hadron decays. For the HVQDIS case, following[5], the minimum value for the scales was set to 2m c . The contribution from the hadron-like component of the photon is 9%. AcknowledgementsWe would like to thank B. Harris, E. Laenen and S. Frixione for helpful discussions on the application of QCD calculations in this intermediate regime. We thank the DESY Directorate for their strong support and encouragement. The remarkable achievements of the HERA machine group were essential for the successful completion of this work. The design, construction and installation of the ZEUS detector have been made possible by the effort of many people who are not listed as authors. . J Breitweg, ZEUS CollaborationPhys. Lett. B. 407402ZEUS Collaboration, J. Breitweg et al., Phys. Lett. B 407 (1997) 402. . C Adloff, H1 CollaborationNucl. Phys. B. 54521H1 Collaboration, C. Adloff et al., Nucl. Phys. B 545 (1999) 21. . J Breitweg, ZEUS CollaborationEur. Phys. J. C. 1235ZEUS Collaboration, J. Breitweg et al., Eur. Phys. J. C 12 (2000) 35. . C Adloff, H1 CollaborationPhys. Lett. B. 528199H1 Collaboration, C. Adloff et al., Phys. Lett. B 528 (2002) 199. . S Chekanov, ZEUS CollaborationPhys. Rev. D. 6912004ZEUS Collaboration, S. Chekanov et al., Phys. Rev. D 69 (2004) 012004. . S Aid, H1 CollaborationNucl. Phys. B. 47232H1 Collaboration, S. Aid et al., Nucl. Phys. B 472 (1996) 32. . J Breitweg, ZEUS CollaborationEur. Phys. J. C. 667ZEUS Collaboration, J. Breitweg et al., Eur. Phys. J. C 6 (1999) 67. . S Chekanov, ZEUS CollaborationNucl. Phys. B. 729492ZEUS Collaboration, S. Chekanov et al., Nucl. Phys. B 729 (2005) 492. . A Aktas, H1 CollaborationEur. Phys. J. C. 47597H1 Collaboration, A. Aktas et al., Eur. Phys. J. C 47 (2006) 597. . A Aktas, H1 Collaborationhep-ex/0608042Eur. Phys. J. C. Preprint. Submitted toH1 Collaboration, A. Aktas et al., Preprint hep-ex/0608042, 2006. Submitted to Eur. Phys. J. C. . J Breitweg, ZEUS CollaborationPhys. Lett. B. 407432ZEUS Collaboration, J. Breitweg et al., Phys. Lett. B 407 (1997) 432. . J Breitweg, ZEUS CollaborationPhys. Lett. B. 48753ZEUS Collaboration, J. Breitweg et al., Phys. Lett. B 487 (2000) 53. The ZEUS Detector. Status Report (unpublished), DESY. U. HolmZEUS Collaboration, U. Holm (ed.), The ZEUS Detector. Status Report (unpublished), DESY (1993), available on http://www-zeus.desy.de/bluebook/bluebook.html. . N Harnew, Nucl. Inst. Meth. A. 279290N. Harnew et al., Nucl. Inst. Meth. A 279 (1989) 290; . B Foster, Nucl. Phys. Proc. Suppl. B. 32181B. Foster et al., Nucl. Phys. Proc. Suppl. B 32 (1993) 181; . B Foster, Nucl. Inst. Meth. A. 338254B. Foster et al., Nucl. Inst. Meth. A 338 (1994) 254. . M Derrick, Nucl. Inst. Meth. A. 30977M. Derrick et al., Nucl. Inst. Meth. A 309 (1991) 77; . A Andresen, Nucl. Inst. Meth. A. 309101A. Andresen et al., Nucl. Inst. Meth. A 309 (1991) 101; . A Caldwell, Nucl. Inst. Meth. A. 321356A. Caldwell et al., Nucl. Inst. Meth. A 321 (1992) 356; . A Bernstein, Nucl. Inst. Meth. A. 33623A. Bernstein et al., Nucl. Inst. Meth. A 336 (1993) 23. . J Andruszków, DESYPreprint DESY-92-066J. Andruszków et al., Preprint DESY-92-066, DESY, 1992; . M Derrick, ZEUS CollaborationZ. Phys. C. 63391ZEUS Collaboration, M. Derrick et al., Z. Phys. C 63 (1994) 391; . J Andruszków, Acta Phys. Pol. B. 322025J. Andruszków et al., Acta Phys. Pol. B 32 (2001) 2025. W H Smith, K Tokushuku, L W Wiggers, Proc. Computing in High-Energy Physics. C. Verkerk and W. WojcikComputing in High-Energy PhysicsAnnecy, France, CERN222Also in preprint DESY 92-150BW.H. Smith, K. Tokushuku and L.W. Wiggers,. Proc. Computing in High-Energy Physics (CHEP), C. Verkerk and W. Wojcik (eds.), p. 222. Annecy, France, CERN (1992). Also in preprint DESY 92-150B, 1992. . J Tandler, BONN-IR-2003-06Bonn, GermanyUniversität BonnReportJ. Tandler. Ph.D. Thesis, Universität Bonn, Bonn, Germany, Report BONN-IR-2003-06 (unpublished), (2003), available on http://www-zeus.physik.uni-bonn.de/german/phd.html. . D S Bailey, R Hall-Wilton, Nucl. Inst. Meth. A. 51537D.S. Bailey and R. Hall-Wilton, Nucl. Inst. Meth. A 515 (2003) 37. . G Marchesini, hep-ph/991239617Preprint Cavendish-HEP-99G. Marchesini et al., Preprint Cavendish-HEP-99/17 (hep-ph/9912396), 1999; . G Marchesini, Comp. Phys. Comm. 67465G. Marchesini et al., Comp. Phys. Comm. 67 (1992) 465. . H Jung, Comp. Phys. Comm. 86147H. Jung, Comp. Phys. Comm. 86 (1995) 147. . R Brun, CERN-DD/EE/84-1CERN3Technical ReportR. Brun et al., geant3, Technical Report CERN-DD/EE/84-1, CERN, 1987. . H L Lai, CTEQ CollaborationEur. Phys. J. C. 12375CTEQ Collaboration, H.L. Lai et al., Eur. Phys. J. C 12 (2000) 375. . M Glück, E Reya, A Vogt, Phys. Rev. D. 46M. Glück, E. Reya and A. Vogt, Phys. Rev. D 46 (1992) 1973. . W.-M Particle Data Group, Yao, J. Phys G. 331Particle Data Group, W.-M. Yao et al., J. Phys G 33 (2006) 1. . P Irrgang, BONN-IR-2004-016Bonn, GermanyUniversität BonnReportP. Irrgang. Ph.D. Thesis, Universität Bonn, Bonn, Germany, Report BONN-IR-2004-016 (unpublished), (2004), available on http://www-zeus.physik.uni-bonn.de/german/phd.html. . B W Harris, J Smith, Phys. Rev. D. 572806B.W. Harris and J. Smith, Phys. Rev. D 57 (1998) 2806. . S Frixione, P Nason, G Ridolfi, Nucl. Phys. B. 454S. Frixione, P. Nason and G. Ridolfi, Nucl. Phys. B 454 (1995). . S Frixione, Phys. Lett. B. 319339S. Frixione et al., Phys. Lett. B 319 (1993) 339. . S Chekanov, ZEUS CollaborationPhys. Rev. D. 6712007ZEUS Collaboration, S. Chekanov et al., Phys. Rev. D 67 (2003) 012007. . C Peterson, Phys. Rev. D. 27105C. Peterson et al., Phys. Rev. D 27 (1983) 105. . L Gladilin, Preprint hep-ex/9912064L. Gladilin, Preprint hep-ex/9912064, 1999; . S Chekanov, ZEUS CollaborationEur. Phys. J. C. 44351ZEUS Collaboration, S. Chekanov et al., Eur. Phys. J. C 44 (2005) 351. . P Nason, C Oleari, Nucl. Phys. B. 565245P. Nason and C. Oleari, Nucl. Phys. B 565 (2000) 245. . V M Budnev, Phys. Rep. 15181V.M. Budnev et al., Phys. Rep. 15C (1974) 181.
[]
[ "SETS OF LENGTHS OF POWERS OF A VARIABLE", "SETS OF LENGTHS OF POWERS OF A VARIABLE" ]
[ "Richard Belshoff ", "Daniel Kline ", "Mark W Rogers " ]
[]
[]
A positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x n in R[x], where (R, m) is an Artinian local ring with m 2 = 0.Preliminaries.For the rest of this paper, unless otherwise specified, (R, m) is a commutative Artinian local ring identity, having unique maximal ideal m = 0 and residue field R = R/m; R[x] is the polynomial ring in the variable x with coefficients in R. The concept of an irreducible element is usually defined only for integral domains. For rings with zero-divisors, several different notions of irreducible have been proposed ([A1], [A2], [A3].) Our definition of irreducible will be the usual one, i.e., the one that is used when R is an integral domain. We begin by recalling this and a few other definitions and equivalences. Let f (x) = a 0 + a 1 x + a 2 x 2 + · · · + a d x d denote a polynomial of degree d in R[x].• The polynomial f (x) is a unit if a 0 is a unit and a i ∈ m for all i > 0. 2000
10.1216/rmj-2019-49-3-729
[ "https://arxiv.org/pdf/1607.02236v1.pdf" ]
119,155,301
1607.02236
afed1d90f88d1420e9c0b20f5abfd6687812da04
SETS OF LENGTHS OF POWERS OF A VARIABLE 8 Jul 2016 Richard Belshoff Daniel Kline Mark W Rogers SETS OF LENGTHS OF POWERS OF A VARIABLE 8 Jul 2016 A positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x n in R[x], where (R, m) is an Artinian local ring with m 2 = 0.Preliminaries.For the rest of this paper, unless otherwise specified, (R, m) is a commutative Artinian local ring identity, having unique maximal ideal m = 0 and residue field R = R/m; R[x] is the polynomial ring in the variable x with coefficients in R. The concept of an irreducible element is usually defined only for integral domains. For rings with zero-divisors, several different notions of irreducible have been proposed ([A1], [A2], [A3].) Our definition of irreducible will be the usual one, i.e., the one that is used when R is an integral domain. We begin by recalling this and a few other definitions and equivalences. Let f (x) = a 0 + a 1 x + a 2 x 2 + · · · + a d x d denote a polynomial of degree d in R[x].• The polynomial f (x) is a unit if a 0 is a unit and a i ∈ m for all i > 0. 2000 1. Introduction. In this paper we study the non-uniqueness of factorizations of x n in R[x], where (R, m) is a commutative Artinian local ring with identity, with the added restriction that m 2 = 0. For example, R could be the ring Z/p 2 Z = Z p 2 where p is prime. Example 1.1. Consider the following factorizations of x 6 in Z 9 [x]. (1) x 6 = x · x · x · x · x · x (2) x 6 = x · x · (x 2 + 3) · (x 2 − 3) (3) x 6 = (x 2 + 3) · (x 2 + 3) · (x 2 + 3) (4) x 6 = (x 3 + 3) · (x 3 − 3) The first factorization expresses x 6 in the usual way as a product of 6 irreducible polynomials; for this reason, we say that 6 is a length of x 6 , and if R were a unique factorization domain, this would be the only length of x 6 . However, the remaining factorizations show that 4, 3, and 2 are lengths of x 6 . As we will later see, these are all of the lengths of x 6 , and we write L(x 6 ) = {2, 3, 4, 6}. In general, the set of lengths of x n in Z p 2 [x] depends on whether p = 2 or p is an odd prime. For example, in Z 4 [x], L(x 6 ) = {2, 4, 6}. Our goal in this paper is the collection of results Proposition 4.6, Lemma 4.10, and Theorem 4.14, which completely determine L(x n ) over Artinian local rings that are not fields but for which the square of the maximal ideal is zero. The result depends on whether n is even or odd, and whether the cardinality of R is 4 or not (there are only two such rings with cardinality 4). For example, if n is an even integer and the cardinality of R is greater than 4, we show that L(x n ) = {2, 3, 4, 5, . . . , n − 2} ∪ {n}, as we saw above for n = 6. For a recent survey of sets of lengths, we refer the reader to the recent paper [G] by Alfred Geroldinger. • The polynomial f (x) is a zero divisor if each a i ∈ m. Note that any multiple of a zero divisor is a zero divisor. • The polynomial f (x) is regular if f (x) is not a zero divisor. Note that if a product is regular, so is each factor. • The nonunit polynomial f (x) is irreducible if f (x) = g(x)h(x) implies g(x) or h(x) is a unit. • The order of the polynomial f (x) (denoted ord(f )) is the least i such that a i = 0. • By f (x) we mean the image of f (x) in R[x]. The following proposition is proved by B. R. McDonald ([M], Theorem XIII.6) for finite rings. The result generalizes to the case where R is any Artinian local ring. We will need this result in Lemma 4.3. Proposition 2.1. Every regular polynomial f in R[x] is representable as f = uf * where u is a unit of R[x] and f * is a monic polynomial of R[x]. Also, deg(f * ) = deg(f ). The following simple corollary allows us to assume that irreducible factors of a monic polynomial are themselves monic, and thus nonconstant. We use this corollary implicitly throughout the paper. Corollary 2.2. If f is a monic polynomial in R[x] such that f = f 1 f 2 · · · f k for some polynomials f 1 , f 2 , . . . , f k then there are monic polynomials f * 1 , f * 2 , . . . , f * k such that f = f * 1 f * 2 · · · f * k . If each f i is irreducible, then each f * i is irreducible (and nonconstant). Proof. Since the product f 1 · · · f k is regular, so is each f i . By Proposition 2.1, each f i = u i f * i for some unit u i and some monic polynomial f * i . Since f = (u 1 · · · u k )f * 1 · · · f * k and f and f * 1 · · · f * k are both monic, the leading coefficient of the unit u 1 · · · u k is 1. The only unit with this property is 1, so f = f * 1 f * 2 · · · f * k . If each f i is irreducible, then so are the associates f * i ; they cannot be constant, since the only monic constant is 1, and units aren't considered irreducible. Generalized Eisenstein Polynomials. We begin by showing that while factorization in R[x] may be non-unique, it is at least possible. We remind the reader that (R, m) is an Artinian local ring with m = 0. Proof . Suppose f (x) = a d x d + · · · + a 1 x + a 0 is a zero divisor of R[x]. Then, for 0 ≤ k ≤ d, we have a k ∈ m, hence a k is nilpotent. Now 1 − f (x) = −a d x d − · · · − a 1 x + (1 − a 0 ) and 1 − a 0 is a unit. Therefore 1 − f (x) is a unit of R[x].Definition 3.2. A generalized Eisenstein polynomial (abbreviated GE polynomial ) is a non-constant monic polynomial f (x) = x d + f d−1 x d−1 + · · · + f 1 x + f 0 with the property that f i ∈ m for each i = 0, . . . , d − 1. Equivalently, f (x) is a GE polynomial if f (x) is non-constant, monic and f (x) = x d in R[x], where d = deg f . We note that x n is a GE polynomial for any positive integer n. Proof. Assume both f and g are GE polynomials; then f = x k and g = x ℓ where k and ℓ are the degrees of f and g. Thus x k+ℓ = f g = f g, showing that f g is a GE polynomial. Conversely, if f and g are monic of degrees k and ℓ respectively, then f g = f g = x k+ℓ since f g is a GE polynomial. Since R[x] is a UFD, it follows easily that f = x k and g = x ℓ . Therefore both f and g are GE polynomials. The next theorem is the reason for our terminology "generalized Eisenstein polynomial." Theorem 3.4. If f is a GE polynomial in R[x] whose constant term is in m \ m 2 , then f is irreducible. Proof. Suppose by way of contradiction that there are two polynomials g, h with f = gh. By Corollary 2.2, f = g * h * for some monic polynomials g * , h * . By Lemma 3.3, either g * and h * are GE polynomials, or one of them is constant. If one of them is constant then it is a unit, and the proof is complete. If both were nonconstant, then since they are GE polynomials, the product of their constant terms would be in m 2 , and this would contradict the assumption on the constant term of f . Corollary 3.5. Suppose m 2 = 0. If f is a GE polynomial in R[x] with degree at least two, then f is irreducible if and only if f has a nonzero constant term. Proof. If f is a GE polynomial with a nonzero constant term, then the constant term is in m \ m 2 since m 2 = 0. According to Theorem 3.4, f is irreducible. If the constant term of f is zero then f = x d + a d−1 x d−1 + · · · + a j x j = x(x d−1 + a d−1 x d−2 + · · · + a j x j−1 ) where d ≥ 2 and 1 ≤ j ≤ d. The factorization displayed above is a factorization into a product of two non-units, since a j ∈ m. Therefore, if the constant term of f is zero, then f is reducible. Remark 3.6. Let (R, m) be a finite local ring such that m 2 = 0 and let k = |m|. By Corollary 3.5, the number of irreducible GE polynomials of degree 2 in R[x] is exactly k(k − 1). We will use this remark later in Lemma 4.10. The central idea of the following proof for the case k = 2 was inspired by the computations done at the start of [FF]. Proposition 3.7. Suppose m 2 = 0. If k ≥ 2 and f 1 , f 2 , . . . , f k are GE polynomials in R[x] with deg(f i ) = d i and d 1 ≥ d 2 ≥ · · · ≥ d k then there is a GE polynomial h of degree d 1 such that f 1 f 2 · · · f k = hx d2+d3+···+d k . If, furthermore, f 1 is irreducible and d 1 > d 2 , then h is irreducible and ord( k i=1 f i ) = k i=2 d i . Proof. We use induction on k. Suppose k = 2. We have f 1 = x d1 +f 1 and f 2 = x d2 +f 2 wherẽ f 1 ,f 2 ∈ m[x] have degrees less than d 1 , d 2 , respectively. Therefore f 1 f 2 = (x d1 +f 1 )(x d2 +f 2 ) = x d1+d2 + x d1f 2 + x d2f 1 +f 1f2 . Since m 2 = 0 we havef 1f2 = 0, so f 1 f 2 = x d1+d2 + x d1f 2 + x d2f 1 = (x d1 + x d1−d2f 2 +f 1 )x d2 and the polynomial h = x d1 + x d1−d2f 2 +f 1 is a GE polynomial of degree d 1 . If, furthermore, f 1 is irreducible and d 1 > d 2 , then h is irreducible by Lemma 3.5, since h and f 1 have the same constant term. Finally, becausef 1 has a nonzero constant term, we have ord(f 1 f 2 ) = d 2 . Now suppose k ≥ 2 and assume f 1 · · · f k = x d2+···+d k h 1 where h 1 is a GE polynomial with deg(h 1 ) = d 1 , and if f 1 is irreducible with d 1 > d 2 , then h 1 is irreducible. Then k+1 i=1 f i = f 1 · · · f k f k+1 = (h 1 x d2+···+d k )f k+1 = x d2+···+d k (h 1 f k+1 ) = x d2+···+d k (hx d k+1 ) for some GE polynomial h of degree d 1 by the k = 2 case, and if f 1 is irreducible with d 1 > d 2 , then h is irreducible. Therefore k+1 i=1 f i = hx d2+···+d k +d k+1 . Furthermore, if f 1 is irreducible with d 1 > d 2 , then h is irreducible, and so it has a nonzero constant term. Therefore ord A regular polynomial of degree n cannot have length greater than n, according to the next lemma. In fact, after we establish the next three lemmas, we will be able to determine the set of lengths of x n for n ≤ 5. Proof. If f is a unit, then L(f ) = ∅ since irreducibles aren't units and a product of nonunits can't be a unit; now assume f is not a unit. Suppose k ∈ L(f ); then there are irreducible polynomials f 1 , . . . , f k in R[x] such that f = f 1 · · · f k . Each f i must be regular, since f is, and thus each f i has positive degree, since the only regular constants are units. For each i = 1, . . . , k, we have f i = u i f * i for some unit u i and some monic f * i in R[x], by Proposition 2.1; since f i is not a unit, f * i has positive degree. We have f = f 1 · · · f k = u 1 · · · u k f * 1 · · · f * k and thus k ≤ k+1 i=1 f i = d 2 + · · · + d k + d k+1 .k i=1 deg(f * i ) ≤ deg(u 1 · · · u k f * 1 · · · f * k ) = deg(f ) = n. The assumption that f is a regular polynomial is necessary in Lemma 4.3: If R = Z 4 then the constant polynomial 2 ∈ R[x] is irreducible. Hence for the polynomial f = 2x of degree 1 we have 2 ∈ L(f ). Lemma 4.4. Suppose m 2 = 0. If n is a positive integer then n − 1 ∈ L(x n ), and if n is odd then 2 ∈ L(x n ). Proof. We prove the second part first. Suppose, to get a contradiction, that 2 ∈ L(x n ) for some odd positive integer n. By Lemma 4.3 we must have n ≥ 3; thus there are irreducible nonconstant monic polynomials f and g such that x n = f g. By Lemma 3.3, both f and g are GE polynomials. Since n is odd, deg(f ) = deg(g), so without loss of generality we assume deg(f ) > deg(g). By Proposition 3.7 there is an irreducible GE polynomial h such that x n = f g = hx deg(g) , so n = ord(x n ) = ord(f g) = deg(g) = n − deg(f ), contradicting f nonconstant. This shows 2 ∈ L(x n ) if n is odd. Now we prove the first part. Since x 2 is reducible, 1 ∈ L(x 2 ). We have just shown 2 ∈ L(x 3 ). Suppose, to get a contradiction, n − 1 ∈ L(x n ) for some integer n ≥ 4; then there are irreducible, nonconstant, monic GE polynomials f 1 , f 2 , . . . , f n−1 such x n = f 1 f 2 · · · f n−1 . Since deg(f 1 f 2 · · · f n−1 ) = n, exactly one f i has degree 2 and the rest are linear. Without loss of generality, assume polynomials f 2 through f n−1 are linear and f 1 has degree two. By Proposition 3.7, f 2 · · · f n−1 = hx n−3 where h is a linear GE polynomial. Thus x n = f 1 hx n−3 , which implies x 3 = f 1 h. This is a contradiction, since if x 3 = f 1 h then 2 ∈ L(x 3 ). Therefore n − 1 ∈ L(x n ) Lemma 4.5. Suppose m 2 = 0. Let q be an irreducible GE polynomial in R[x] . For any integer n ≥ 2, (1) If n is even, then {2, 4, 6, . . . , n − 2, n} ⊆ L(q n ). (2) If n is odd, then {3, 5, 7, . . . , n − 2, n} ⊆ L(q n ). Proof. Suppose n is even. Since q is irreducible, n ∈ L(q n ). Let k be any even integer such that 2 ≤ k < n; we will find a factorization of q n with length k. Let m be any nonzero element of the maximal ideal m and consider the factorization − m) = q n . Hence we have a product of k irreducible factors equal to q n for any even k such that 2 ≤ k < n. Therefore, {2, 4, 6, . . . , n − 2, n} ⊆ L(q n ). If n is odd, the proof follows the same argument and factorization as above, except this time n and k are both odd integers. L(x) = {1} L(x 2 ) = {2} L(x 3 ) = {3} L(x 4 ) = {2, 4} L(x 5 ) = {3,5} Proof. This follows directly from Lemmas 4. 3, 4.4, and 4.5. We now proceed to find the set of lengths of x 6 . By It remains to determine if 3 ∈ L(x 6 ); this depends on whether |R| > 4 or |R| = 4 as we will see in Lemma 4.10 below. We first establish some general results about local rings of cardinality 4. Proposition 4.7. Let (R, m) be any local ring. The following are equivalent. (1) char(R/m) = 2 (2) 2 ∈ m If m = 0 but m 2 = 0, then (1) and (2) Proof. Since R = R × ∪ m, the disjoint union of the units R × and the maximal ideal m, it suffices to show that R has exactly two units. Suppose m = {0, t}. If the only unit of R is 1, then R is a ring with three elements and is thus isomorphic to Z 3 , contradicting |m| = 2. Therefore there exists a unit u = 1 in R; we show u = t + 1. Since ut ∈ m, either ut = 0 or ut = t. The first case is impossible since t = 0. In the second case t(u − 1) = 0 and hence u − 1 ∈ m. This implies u − 1 = t so u = t + 1. This shows that R = {0, 1, t, t + 1}, a ring with four elements. Remark 4.9. It is known ([M,Exercise I.4,p.4]) that if R is any ring with four elements, then R must be isomorphic to one of the following: Z 4 , F 4 , Z 2 × Z 2 , or F 2 [t]/(t 2 ). Of these, the only ones that are local rings and are not fields are Z 4 and F 2 [t]/(t 2 ). Note that both of these have the equivalent properties (1), (2), (3) of Proposition 4.7. Also note that if R = Z 4 or R = F 2 [t]/(t 2 ) there are exactly two irreducible GE polynomials of degree 2 in R[x]. (See Remark 3.6.) We will need this fact in the next proof. In the next lemma, we find the set of lengths of x 6 ; it will also be used as the base for an induction in the proposition to follow. If, on the other hand, char(R/m) = 2, then by Proposition 4.7, there exists a nonzero element a ∈ m with 2a = 0. Now set b = a and c = −2a. Then a + b + c = 0 and all three elements are nonzero. Now by Corollary 3.5, each of the polynomials x 2 + a, x 2 + b, and x 2 + c is an irreducible GE polynomial. Since a + b + c = 0 and m 2 = 0, the factorization (x 2 + a)(x 2 + b)(x 2 + c) = x 6 shows that 3 ∈ L(x 6 ). Proof of (b): Suppose |R| = 4 and 3 ∈ L(x 6 ). Then there exists three irreducible, monic, nonconstant GE polynomials f 1 , f 2 , f 3 whose product is x 6 . Without loss of generality we have the following three cases for (deg(f 1 ), deg(f 2 ), deg(f 3 )): (4, 1, 1), (3,2,1), and (2, 2, 2). For the first two cases, deg(f 1 ) is greater than deg(f 2 ) and deg(f 3 ), so by Proposition 3.7, 6 = ord(x 6 ) = ord(f 1 f 2 f 3 ) = deg(f 2 ) + deg(f 3 ) < 6, which is a contradiction. For the last case, since, as noted in Remark 3.6, there are exactly two irreducible GE polynomials of degree 2 in R[x], at least two f i are the same, say f 1 = f 2 , and thus, since m 2 = 0, f 1 f 2 = x 4 . But since f 1 f 2 f 3 = x 6 , we have f 3 = x 2 , a contradiction since f 3 is irreducible. So 3 ∈ L(x 6 ). Proposition 3 . 1 . 31Every polynomial of positive degree in R[x] that is not a unit can be factored into a product of irreducible polynomials. Lemma 3 . 3 . 33Let f and g be monic, nonconstant polynomials in R[x]. Then f g is a GE polynomial if and only if both f and g are GE polynomials. This completes the proof by induction. 4. Sets of Lengths of x n . We begin with the definition of the set of lengths of an element. Definition 4 . 1 . 41Let R be a commutative Artinian local ring with identity and let f ∈ R[x]. We say that a positive integer n is a length of f if f factors into a product of n irreducible polynomials in R[x]. We define the set L(f ) = {n | n is a length of f } to be the set of lengths of f . Remark 4.2. To say that 1 ∈ L(f ) means precisely that f is irreducible, and in this case L(f ) = {1}. If R is a unique factorization domain, then L(f ) is a singleton for any polynomial f ∈ R[x]. Of course n ∈ L(x n ), and if R is a UFD then L(x n ) = {n}. Lemma 4 . 3 . 43If f is a regular polynomial in R[x] of degree n, then L(f ) ⊆ {1, 2, . . . , n}. − m is also irreducible. Multiplying both sides of equation (4.1) by q k−2 yields q k Proposition 4 . 6 . 46Suppose m 2 = 0. In R[x] we have Lemmas 4.3, 4.4, and 4.5 we have {2, 4, 6} ⊆ L(x 6 ) ⊆ {2, 3, 4, 6}. We have char(R/m) = 2 if and only if1 +1 =2 =0 in R/m if and only if 2 ∈ m. Now assume m = 0 but m 2 = 0. For(2)implies(3), given any m ∈ m, we have 2m ∈ m 2 = 0. Conversely, if 2m = 0 then 2 is not a unit (since m = 0), so 2 ∈ m.Proposition 4.8. Let (R, m) be any local ring. If |m| = 2, then |R| = 4. Lemma 4. 10 . 10Suppose m 2 = 0. If |R| > 4 then L(x 6 ) = {2, 3, 4, 6}; if |R| = 4 then L(x 6 ) = {2, 4, 6}. Proof. By the remarks after Proposition 4.6, it is enough to show that (a) if |R| > 4 then 3 ∈ L(x 6 ), and (b) if |R| = 4 then 3 ∈ L(x 6 ). Proof of (a): We first show there exist three nonzero elements a, b, c in m satisfying a + b + c = 0. Suppose char(R/m) = 2. By Proposition 4.8, we know there are two distinct nonzero elements a and b in m. We must have a + b = 0, since otherwise a = −b = b by Proposition 4.7 (3), a contradiction. With c = −(a + b) we have a + b + c = 0 for three nonzero elements a, b, c. This shows that the ring R[x] has harmless zero-divisors using the terminology of Frei-Frisch [FF, Definition 2.3]. Now the result follows from [FF, Lemma 2.8]. Acknowledgment: This article is a generalization of part of Daniel Kline's Master's Thesis [K]under the direction of Mark Rogers. The thesis work was inspired by the paper [FF] of Sophie Frisch and Christopher Frei, and a private email exchange with Sophie Frisch. The authors wish to thank the referee for several helpful suggestions and corrections.Proposition 4.11. Suppose m 2 = 0. For all n ≥ 6, |R| > 4 if and only if n − 3 ∈ L(x n ).Proof. If |R| > 4 then by Lemma 4.10, 3 ∈ L(x 6 ), so there is a factorization of x 6 into three irreducible polynomials. Multiplying this factorization by x n−6 gives a factorization of x n of length n − 3. Therefore n − 3 ∈ L(x n ). Now assume |R| = 4. We show n − 3 ∈ L(x n ) for n ≥ 6 (equivalently, n ∈ L(x n+3 ) for n ≥ 3) by induction on n. By Lemma 4.10, 3 ∈ L(x 6 ). Now assume k ∈ L(x k+3 ) for some k ≥ 3. We show k + 1 ∈ L(x k+4 ).Suppose by way of contradiction that k + 1 ∈ L(x k+4 ); then there exist k + 1 irreducible, monic, nonconstant GE polynomials f 1 , f 2 , . . . , f k+1 , whose product is x k+4 . At least one f i must be linear,Furthermore, at least one f i must be non-linear. Without loss of generality, let f 1 be linear and f k+1 be non-linear. Then by Proposition 3.7 there exists an irreducible GE polynomial h such that f k+1 f 1 = hx. Therefore f 1 · · · f k = (f k+1 f 1 )f 2 · · · f k = (hx)f 2 · · · f k . We now have hf 2 · · · f k = x k+3 which implies k ∈ L(x k+3 ). This contradicts our assumption. Therefore n ∈ L(x n+3 ) for n ≥ 3, or equivalently, n − 3 ∈ L(x n ).The next two Lemmas do not depend on the cardinality of the local ring R. Proof. Let n ≥ 7. Let m be a nonzero element of the maximal ideal m, let ℓ be a positive integer, and consider the following three factorizations.By Corollary 3.5, each polynomial on the left side of the three factorizations is irreducible. Assume n ≥ 7. There are three cases:n ≡ 0 (mod 3): Set ℓ = n−6 3 ; then from equation (4.4), 3 ∈ L(x n ). n ≡ 1 (mod 3): Set ℓ = n−4 3 ; then from equation (4.3), 3 ∈ L(x n ). n ≡ 2 (mod 3): Set ℓ = n−2 3 ; then from equation (4.2), 3 ∈ L(x n ). Therefore for any integer n ≥ 7, we have 3 ∈ L(x n ).Lemma 4.13. Suppose m 2 = 0. For any integer n ≥ 7:(1) {3, 4, 5, . . . , n − 4} ∪ {n − 2, n} ⊆ L(x n ) if n is odd.(2) {2, 3, 4, 5, . . . , n − 4} ∪ {n − 2, n} ⊆ L(x n ) if n is even. then {3, 5, 7} ⊆ L(x 7 ) by Lemma 4.5. Thus we may assume n ≥ 9. 7Again by Lemma 4.5, {3, 5, 7, · · · , n − 2, n} ⊆ L(x n. so it remains to show that if kProof. Suppose n ≥ 7 and n is odd. If n = 7, then {3, 5, 7} ⊆ L(x 7 ) by Lemma 4.5. Thus we may assume n ≥ 9. Again by Lemma 4.5, {3, 5, 7, · · · , n − 2, n} ⊆ L(x n ), so it remains to show that if k −k+3 ); that is, there exists a factorization of x n−k+3 of length 3. Multiplying both sides of. ∈ , ∈ L(x n−k+3 ); that is, there exists a factorization of x n−k+3 of length 3. Multiplying both sides of Factorization in commutative rings with zero divisors. A1, D D Anderson, Silvia Valdes-Leon, Rocky Mountain J. Math. 26A1. Anderson, D. D. and Valdes-Leon, Silvia, Factorization in commutative rings with zero divisors, Rocky Mountain J. Math 26 (1996), 439-480. Factorization in commutative rings with zero divisors II. A2, D D Anderson, Silvia Valdes-Leon, Lecture Notes in Pure and Appl. Math. 189DekkerA2. Anderson, D. D. and Valdes-Leon, Silvia, Factorization in commutative rings with zero divisors II, Lecture Notes in Pure and Appl. Math, vol. 189, 197-219. Dekker, New York, 1997. Factorization in commutative rings with zero divisors III. Ahmet G A3. Aḡargün, D D Anderson, Silvia Valdes-Leon, Rocky Mountain J. Math. 31A3. Aḡargün, Ahmet G. and Anderson, D. D. and Valdes-Leon, Silvia, Factorization in commutative rings with zero divisors III, Rocky Mountain J. Math 31 (2001), 1-21. Non-unique factorization of polynomials over residue class rings of the integers. Ff, C Frei, S Frisch, Comm. Algebra. 394FF. Frei, C. and Frisch, S., Non-unique factorization of polynomials over residue class rings of the integers, Comm. Algebra 39 (2011), no. 4, 1482-1490. G Geroldinger, A , arXiv:1509.07462Sets of Lengths. math.GRG. Geroldinger, A., Sets of Lengths, arXiv:1509.07462 [math.GR] Sets of Lengths over Residue Class Rings of the Integers, Master's thesis. K Kline, D , Missouri State UniversityK. Kline, D., Sets of Lengths over Residue Class Rings of the Integers, Master's thesis, Missouri State University, 2011. M Mcdonald, Bernard , Finite Rings with Identity. New York28Marcel DekkerM. McDonald, Bernard, Finite Rings with Identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker, New York, 1974
[]
[ "Unambiguous probe of parity-mixing of Cooper pairs in noncentrosymmetric superconductors", "Unambiguous probe of parity-mixing of Cooper pairs in noncentrosymmetric superconductors" ]
[ "Satoshi Fujimoto \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n" ]
[ "Department of Physics\nKyoto University\n606-8502KyotoJapan" ]
[]
We propose an experimental scheme to detect unambiguously parity-mixing of Cooper pairs in noncentrosymmetric superconductors, which utilizes crossed Andreev reflection processes between two oppositely spin-polarized normal metal leads and a noncentrosymmetric superconductor. It is demonstrated that a non-local conductance exhibits a clear signature of parity breaking of Cooper pairs, and thus, can be a direct probe for the parity-mixing.PACS numbers:In noncentrosymmetric superconductors (NCSs), which lack inversion symmetry in their crystal structure, antisymmetric spin-orbit (SO) interactions give rise to various exotic effects on superconducting In particular, pairing states can not be classified according to parity, but, instead, the admixture of spin-singlet pairing and spin-triplet pairing generally occurs.7,10,11,22This most striking effect, however, unfortunately, has not been detected so far by experimental studies. The difficulty of detecting parity-mixing of Cooper pairs is partly due to the fact that conventional experimental approaches, which are utilized for the determination of parity of Cooper pairs in centrosymmetric superconductors, such as NMR Knight shift measurements, do not provide any useful information concerning parity-mixing. That is, in centrosymmetric superconductors, the change of the Knight shift below the transition temperature T c tells us whether the pairing state is spin-singlet or spin-triplet. By contrast, in NCS, if the SO interaction is much larger than the energy scale of the superconducting gap, the behavior of the Knight shift below T c is mainly governed by the strong SO interaction, and does not yield any information on pairing states.12,22,23,31,32Thus, a novel experimental approach is required for the detection of parity-mixing of Cooper pairs. There have been several proposals for this aim, which use, for instance, tunneling characteristics, 33,34,35,36 accidental gap-node structures, 37 and a fractional vortex scenario, 38 etc.
10.1103/physrevb.79.220506
[ "https://arxiv.org/pdf/0904.3582v2.pdf" ]
118,684,394
0904.3582
1fe647af3a2285512e4beea4f6648a3a4837f3d1
Unambiguous probe of parity-mixing of Cooper pairs in noncentrosymmetric superconductors 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28 Satoshi Fujimoto Department of Physics Kyoto University 606-8502KyotoJapan Unambiguous probe of parity-mixing of Cooper pairs in noncentrosymmetric superconductors 29301,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28(Dated: June 9, 2009) We propose an experimental scheme to detect unambiguously parity-mixing of Cooper pairs in noncentrosymmetric superconductors, which utilizes crossed Andreev reflection processes between two oppositely spin-polarized normal metal leads and a noncentrosymmetric superconductor. It is demonstrated that a non-local conductance exhibits a clear signature of parity breaking of Cooper pairs, and thus, can be a direct probe for the parity-mixing.PACS numbers:In noncentrosymmetric superconductors (NCSs), which lack inversion symmetry in their crystal structure, antisymmetric spin-orbit (SO) interactions give rise to various exotic effects on superconducting In particular, pairing states can not be classified according to parity, but, instead, the admixture of spin-singlet pairing and spin-triplet pairing generally occurs.7,10,11,22This most striking effect, however, unfortunately, has not been detected so far by experimental studies. The difficulty of detecting parity-mixing of Cooper pairs is partly due to the fact that conventional experimental approaches, which are utilized for the determination of parity of Cooper pairs in centrosymmetric superconductors, such as NMR Knight shift measurements, do not provide any useful information concerning parity-mixing. That is, in centrosymmetric superconductors, the change of the Knight shift below the transition temperature T c tells us whether the pairing state is spin-singlet or spin-triplet. By contrast, in NCS, if the SO interaction is much larger than the energy scale of the superconducting gap, the behavior of the Knight shift below T c is mainly governed by the strong SO interaction, and does not yield any information on pairing states.12,22,23,31,32Thus, a novel experimental approach is required for the detection of parity-mixing of Cooper pairs. There have been several proposals for this aim, which use, for instance, tunneling characteristics, 33,34,35,36 accidental gap-node structures, 37 and a fractional vortex scenario, 38 etc. In this paper, we propose another method to probe parity-mixing of Cooper pairs in an unambiguous way. In this scheme, we utilize crossed Andreev reflection (CARE) between two ferromagnetic normal metal leads and a NCS. CARE is a non-local reflection process; an electron injected from one lead to a superconductor is converted to a hole in another lead, when the distance between two leads is smaller than the coherence length ξ. 39,40,41,42,43,44 To illustrate the basic idea, we consider a setup consisting of two oppositely spin-polarized leads and a NCS with the Rashba SO interaction 45 H SO = αL(k) · σ with L(k) = (k y , −k x , 0), and α the SO coupling constant, σ = (σ x , σ y , σ z ) the Pauli matrices 1). In the NCS, the SO interaction splits the electron band into two parts, ε k± = ε k ±α|k|, each of which is the eigen state of spin chirality. We assume that the SO split is sufficiently larger than the superconducting gap, and that there are only intra-band Cooper pairs formed between electrons in the same band. For instance, in the ε k− band, an electron with momentum k and spin ↑ and an electron with momentum −k and spin ↓ form a Cooper pairing state |k ↑ | − k ↓ , while in the ε k+ band, an electron with momentum k and spin ↓ and an electron with momentum −k and spin ↑ form a Cooper pairing state |k ↓ | − k ↑ . Here, we have chosen the spin quantization axis parallel to L(k) = (k y , −k x , 0). When the density of states in each band is different from each other, the superposition between these two pairing state is impossible. Then, the pairing state in each band is the admixture of a spin-singlet state and a spin-triplet state; e.g. 7,10,11,22,23 The superconducting gap of the parity-mixed pairing state is generally given by ∆ σσ ′ (k) = ∆ s (k)i(σ y ) σσ ′ + d(k) · (σiσ y ) σσ ′ . We assume that the d-vector of the triplet pairing is constrained by the Rashba SO interaction; i.e. d(k) = |d(k)|(sin φ k , − cos φ k , 0) with φ k = tan −1 (k y /k x ). 7,10,11 Then, the superconducting gaps on the ε k+ -band and the ε k− -band are, respectively,∆ |k ↑ | − k ↓ = 1 2 (|k ↑ | − k ↓ − |k ↓ | − k ↑ ) + 1 2 (|k ↑ | − k ↓ + |k ↓ | − k ↑ ).+ (k) = [∆ s (k) + |d(k)|]t k and∆ − (k) = [∆ s (k) − |d(k)|]t * k , where t k is an oddparity phase factor given by t k = ie −iφ k . Actually, the structure of the d-vector is determined not only by the SO interaction, but also by the detail of the pairing interaction. 25,46 The generalization of our argument to the case with more complex structure of d-vector is straightforward. In the following, for simplicity, we focus on one-dimensional (1D) scattering problem where currents flow only along the x-axis which is perpendicular to the interface between the leads and the NCS (see FIG.1). A qualitative feature which is important for the detection of parity-mixing is not largely affected by this simplification. In fact, the degrees of freedom along the z-axis are irrelevant for our argument. Effects of titling alignment of the leads on the xy-plane will be discussed later. In this setup, we assume that two leads are oppositely spinpolarized with the spin-quantization axis parallel to the y-axis. An important observation here is that the paritybroken structure of the Cooper pair |k ↑ | − k ↓ (or |k ↓ | − k ↑ ) is directly related to the parity-mixing, as explained above. The parity breaking of Cooper pairs can be detected by CARE as asymmetric reflection processes; a process in which an injected electron with spin ↑ in the lead 1 is converted to a hole with spin ↓ in the lead 2 is not equivalent to a process in which an injected electron with spin ↓ in the lead 2 is converted to a hole with spin ↑ in the lead 1 because of broken inversion symmetry of the NCS. In the former process (denoted as the process A), the Andreev-reflected hole is associated with the superconducting gap ∆ − , while, in the latter process (denoted as the process B), the relevant superconducting gap is ∆ + . As mentioned above, in the parity-mixed pairing state, the amplitudes of these two gaps are different, which can be clearly observed as a characteristic biasvoltage-dependence of the non-local conductance. Thus, the parity-mixing of Cooper pairs can be detected directly without ambiguity. The non-local conductance, which characterizes the CARE, is given by G 12 (V 1 ) = dI 2 /dV 1 and G 21 (V 2 ) = dI 1 /dV 2 , where I 1(2) and V 1(2) are, respectively, a current and a bias voltage in the lead 1 (2). The non-local conductance is expressed in terms of the reflection probabilities A σ ij for the process that an injected electron with spin σ in the lead i is converted to a hole with spin −σ in the lead j; i.e. when the electron spin in each lead is fully polarized, G 12 (V 1 ) = G N A ↑ 12 /2, G 21 (V 2 ) = G N A ↓ 21 /2. (1) Here G N is the conductance in the normal state. To obtain the probabilities, we solve the Bogoliubov-de Gennes (BdG) equation for the CARE. In the representation where σ y is diagonal, the Hamiltonian is decoupled into two partsĤ + andĤ − , each of which corresponds to the pairing state in one of two SO split bands. The BdG equation for the 1D scattering problem depicted in FIG.1 isĤ ν Ψ ν = EΨ ν (2) H ν = ε(k) + ναk x + V (x) −iν∆ ν (k) iν∆ ν (k) −ε(k) − ναk x − V (x) ,(3)with ν = ±. Here ε(k) = − 1 2m ∇ 2 − µ with µ a chem- ical potential, andk x = −i∂ x . The gap functions are ∆ ± (k) = ∆ s (k) ± |d(k)|. We assume that a barrier at the interface between the lead 1 (2) and the NCS is given by a Dirac-type potential, V (x) = kF m Z 1(2) δ(x). Here Z 1 and Z 2 are dimensionless parameters for the strength of the barrier potentials, and k F is defined by k 2 F /2m = µ: i.e. the Fermi momentum in the case without the SO split. To simplify the analysis, we consider the case of an s + p wave pairing state, and neglect k-dependence of ∆ ± . The following argument can be easily extended to the case with more general pairing states such as a d + f wave state, a g+h wave state etc. After a straightforward calculation, 47 we obtain the probability A σ ij , A ↑ 12 (E) = ∆ 2 ν s 2 ν 4E 2 |γ ν (E)| 2 ,(4) with ν = − and s ν = 1 − νmα/k F . Here, for E < ∆ ν 4E 2 |γ ν (E)| 2 = (s ν E − (Z 1 − Z 2 ) ∆ 2 ν − E 2 ) 2 +( ∆ 2 ν − E 2 (s ν + 2Z 1 Z 2 ) + s ν (Z 1 − Z 2 )E) 2 ,(5) and for E > ∆ ν , 4E 2 |γ ν (E)| 2 = (s ν E + (s ν + 2Z 1 Z 2 ) E 2 − ∆ 2 ν )) 2 +(Z 1 − Z 2 ) 2 (s ν E + E 2 − ∆ 2 ν ) 2 .(6) The probability A ↓ 12 (E), is given by Eq.(4) with ν = +. One can obtain the probability A σ 21 by interchanging Z 1 and Z 2 in the expression of A σ 12 . In the derivation of A σ ij , we have used the approximation that the Fermi momentum for ε k± is k F ± ≈ k F ∓ mα, which is valid when the SO split is much smaller than the Fermi energy. In fact, within this approximation, the shift of the Fermi momentum due to the SO split does not change qualitatively the feature of the non-local conductance that is important for the detection of parity-mixing of Cooper pairs, as will be shown below. We, first, consider the case that the leads 1, 2 are fully spin-polarized in the opposite directions, and the nonlocal conductance is given by (1). Although the spin polarization in the leads induces exchange fields in the NCS region which may affect the amplitude of the superconducting gap in a nontrivial way, we neglect this effect because it may not change our argument qualitatively. Because of the parity-broken structure of Cooper pairs mentioned above, G 12 (eV ) and G 21 (eV ) exhibit asymmetric behaviors as functions of V even when Z 1 = Z 2 ; i.e. for G 12 (eV ) a peak structure appears at eV = ∆ − , while for G 21 (eV ) it appears at eV 2 = ∆ + . When there is the admixture of spin-singlet pairing and spin-triplet pairing, ∆ + = ∆ − holds. Thus, the parity-mixing can be detected unambiguously from the measurement of the nonlocal conductance. Also, we can derive the BCS gap magnitudes for the spin-singlet pairs and spin-triplet pairs from ∆ s = (∆ + + ∆ − )/2, and |d(k)| = (∆ + − ∆ − )/2. Typical behaviors of the non-local conductance as functions of bias voltages are shown in FIG.2(a). It is noted that even in the case of Z 1 = Z 2 , the origin of this asymmetric behavior of G 12 and G 21 can be clearly attributed to the result of parity-mixing, since the most important factor which yields the asymmetric behavior of the nonlocal conductance is the existence of two gaps ∆ + and ∆ − associated with, respectively, opposite spin chirality of the two SO split Fermi surfaces. We emphasize that the two different gap structure which appears in G 12 and G 21 shown in FIG.2(a) is obviously different from conventional multi-gap behaviors of centrosymmetric superconductors with multi-bands. From this point of view, the CARE experiment is more advantageous than the conventional Andreev reflection experiment 33,34,35,36 as a probe for parity-mixing, though its realization is still challenging with current nanotechnology. In the above argument, a crucial assumption for the gap function is that there are no inter-band Cooper pairs, or, if they exist, the gap amplitude for the inter-band pairs is negligibly small. This assumption is valid as long as the SO split of the Fermi surface is sufficiently smaller than the Fermi energy. An apparent drawback of the above scheme is that in the setup shown in FIG.1, one needs to know beforehand the spin structure on the Fermi surfaces of the NCS determined by the SO interaction to align the direction of the spin polarization in two leads properly. Generally, this task is not so easy, because the structure of the SO interaction for real NCSs is a quite complicated function of momentum k. 48,49,50 For the purpose of extending our scenario to such realistic cases with the non-Rashba SO interactions, we consider a setup a bit different from that depicted in FIG.1; in two normal metal leads, there is no spontaneous magnetization, but, instead, spin polarization is induced by an external weak magnetic field H which is smaller than H c1 . We neglect effects of the magnetic field on the NCS, since it does not change our argument qualitatively. A main effect of a sufficiently small magnetic field on the Andreev reflection processes is to raise an imbalance of spin population in two normal metal leads. Suppose the Rashba SO interaction for a while. We will discuss more general cases later. Then, when the magnetic field is parallel to the positive direction of the y-axis, the non-local conductance G H>0 12 is 2G H>0 12 G N = A ↑ 12 + C 0 A ↓ 12 ,(7) where C 0 = N ↓ /N ↑ with N ↑(↓) the total number of elec- in Eq. (7) is the reflection probability in the case without a magnetic field. In the derivation of (7), we have taken into account the Zeeman effect up to the lowest order in µ B H/µ, but neglected the change of the Fermi momentum due to the Zeeman shift in the leads. Thus, the suppression of the Andreev reflection due to the spin polarization is not included in Eq.(7) up to O(µ B H/µ). This approximation does not affect an important feature of the non-local conductance relevant to the detection of paritymixing for a sufficiently small H, as will be clarified later. In a similar way, the non-local conductance in the case with a magnetic field parallel to the negative direction of the y-axis is obtained as G H<0 12 /G N = C 0 A ↑ 12 + A ↓ 12 . From the difference between the conductance for H > 0 and that for H < 0, G H>0 12 −G H<0 12 = (1−C 0 )(A ↑ 12 −A ↓ 12 ) , we can clearly see whether the parity of Cooper pairs is broken (A ↑ 12 = A ↓ 12 ) or not (A ↑ 12 = A ↓ 12 ). We show a typical behavior of (G H>0 12 − G H<0 12 )/(1 − C 0 ) as a function of a bias voltage in FIG.2(b). This quantity exhibits distinct peak structures at eV = ∆ + and ∆ − as a signature of the parity-mixing, and thus, can be a useful probe for the admixture of spin-singlet pairs and spin-triplet pairs. It is noted that the peak height at eV = ∆ ± is not affected by the Zeeman shift of the Fermi momentum in the NM leads up to the first order in µ B H/µ, since, up to this order, the magnetic field H enters into the expression of A σ ij in the form of (µ B H/µ) |∆ 2 ± − E 2 |. Thus, the approximation used in the derivation of Eq.(7) and the equation for G H<0 12 is valid for our purpose. This scheme which uses field-induced spin polarization in the leads can be utilized for the detection of parity-mixing in the general case that the structure of the antisymmetric SO interaction, H SO = αL(k) · σ, is unknown. Even in this case, when a magnetic field is applied, one can observe the asymmetry between the non-local conductance for a magnetic field with a certain direction, G H>0 12 , and the non-local conductance for a field anti-parallel to it, G H<0 12 , quite generally except in the case with H ⊥ L(k). Thus, it is not difficult to find a direction of the magnetic field for which the asymmetric behavior of the non-local conductance is observed, and the conductance difference G H>0 12 − G H<0 12 is nonzero. Then, one can detect paritymixing of Cooper pairs. Finally, we comment on effects of inversion symmetry breaking caused by the interface between the NCS and the NM leads. Generally, inversion symmetry is broken at a surface. However, this extrinsic inversion symmetry breaking does not affect our proposal be-cause of the following reason. For the (100)-interface depicted in FIG.1, the SO interaction due to the interface is typically the Rashba-type with the Hamiltonian, H ′ = α ′ (k z σ y − k y σ z ). On the other hand, CARE processes for this geometry are dominated by electrons and holes with momentum parallel to the x-axis, i.e. k y = k z = 0. Thus, effects of the extrinsic Rashba interaction on the CARE are negligible. During the preparation of this manuscript, we have become aware of the paper by Wu and Samokhin (arXiv:0904.2397), in which the conductance for conventional Andreev reflection between a ferromagnetic metal and a NCS is calculated in a thorough way. It is important to generalize the current study to more realistic situations as considered by Wu and Samokhin for quantitative comparison between theory and experiments. In summary, it is proposed that a crossed Andreev reflection experiment can be utilized as an unambiguous probe for parity-mixing of Cooper pairs in NCSs. This work is supported by the Grant-in-Aids for Scientific Research from MEXT of Japan (Grant No.18540347, Grant No.19052003). FIG. 1 : 1Setup for crossed Andreev reflection between a NCS and two spatially-separated normal metal (NM) leads with opposite spin polarization and bias voltages V1 and V2, respectively. Short arrows represent the directions of electron spins. In the NCS side, the circular Fermi surface split by the Rashba SO interaction is depicted. Cooper pairs are formed within the same band. Black (gray) long arrows in the leads represent the flow of injected electrons and reflected holes for the process A (B).(FIG. behaviors of the non-local conductance as functions of a bias voltage eV . G12(eV ) (solid line) and G21(eV ) (broken line) for ∆− = 1.0 (energy unit), ∆+ = 2.0, Z1 = 0.3, Z2 = 0.4, mα/kF = 0.1. Unit of the conductance is GN /2. (b) (G H>0 12 − G H<0 12 )/(1 − C0) as a function of a bias voltage for the same parameters as (a). trons with up (down) spin in the lead 1, and A ↑(↓) 12 . E Bauer, G Hilscher, H Michor, Ch Paul, E W Scheidt, A Gribanov, Yu Seropegin, H Noel, M Sigrist, P Rogl, Phys. Rev. Lett. 9227003E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A. Gribanov, Yu. Seropegin, H. Noel, M. Sigrist, and P. Rogl, Phys. Rev. Lett 92, 027003 (2004). . T Akazawa, H Hidaka, H Kotegawa, T Kobayashi, T Fujiwara, E Yamamoto, Y Haga, R Settai, Y Ōnuki, J. Phys. Soc. Jpn. 733129T. Akazawa, H. Hidaka, H. Kotegawa, T. Kobayashi, T. Fujiwara, E. Yamamoto, Y. Haga, R. Settai, and Y.Ōnuki, J. Phys. Soc. Jpn. 73, 3129 (2004). . K Togano, P Badica, Y Nakamori, S Orimo, H Takeya, K Hirata, Phys. Rev. Lett. 93247004K. Togano, P. Badica, Y. Nakamori, S. Orimo, H. Takeya, and K. Hirata, Phys. Rev. Lett. 93, 247004 (2004). . N Kimura, K Ito, K Saitoh, Y Umeda, H Aoki, T Terashima, Phys. Rev. Lett. 95247004N. Kimura, K. Ito, K. Saitoh, Y. Umeda, H. Aoki, and T. Terashima, Phys. Rev. Lett. 95, 247004 (2005). . I Sugitani, Y Okuda, H Shishido, T Yamada, A Thamizhavel, E Yamamoto, T D Matsuda, Y Haga, T Takeuchi, R Settai, Y Ōnuki, J. Phys. Soc. Jpn. 7543703I. Sugitani, Y. Okuda, H. Shishido, T. Yamada, A. Thamizhavel, E. Yamamoto, T. D. Matsuda, Y. Haga, T. Takeuchi, R. Settai, and Y.Ōnuki, J. Phys. Soc. Jpn. 75, 043703 (2006). . G Amano, S Akutagawa, T Muranaka, Y Zenitani, J Akimitsu, J. Phys. Soc. Jpn. 73530G. Amano, S. Akutagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, J. Phys. Soc. Jpn. 73, 530 (2004). . V M Edelstein, Sov. Phys. JETP. 681244V. M. Edelstein, Sov. Phys. JETP 68, 1244 (1989). . V M Edelstein, Phys. Rev. Lett. 752004V. M. Edelstein, Phys. Rev. Lett. 75, 2004 (1995). . V M Edelstein, J. Phys. Condens. Matter. 8339V. M. Edelstein, J. Phys. Condens. Matter 8, 339 (1996). . L P Gor&apos;kov, E Rashba, Phys. Rev. Lett. 8737004L. P. Gor'kov and E. Rashba, Phys. Rev. Lett. 87, 037004 (2001). . P A Frigeri, D F Agterberg, A Koga, M Sigrist, Phys. Rev. Lett. 9297001P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004). . P A Frigeri, D F Agterberg, M Sigrist, New J. Phys. 6115P. A. Frigeri, D. F. Agterberg, and M. Sigrist, New J. Phys. 6, 115 (2004). . I A Sergienko, S H Curnoe, Phys. Rev. 70214510I. A. Sergienko and S. H. Curnoe, Phys. Rev. B70, 214510 (2004). . K Samokhin, Phys. Rev. 70104521K. Samokhin, Phys. Rev. B70, 104521 (2004). . K Samokhin, Phys. Rev. Lett. 9427004K. Samokhin, Phys. Rev. Lett.94 027004 (2005). . S K Yip, Phys. Rev. B. 65144508S. K. Yip, Phys. Rev. B 65, 144508 (2002). . S K Yip, J. Low Temp. Phys. 14067S. K. Yip, J. Low Temp. Phys. 140, 67 (2005). . R P Kaur, D F Agterberg, M Sigrist, Phys. Rev. Lett. 94137002R. P. Kaur, D. F. Agterberg, and M. Sigrist, Phys. Rev. Lett.94, 137002 (2005). . V P Mineev, Phys. Rev. 7112509V. P. Mineev, Phys. Rev. B71, 012509 (2005). . S Fujimoto, Phys. Rev. 7224515S. Fujimoto, Phys. Rev. B72, 024515 (2005). . N Hayashi, K Wakabayashi, P A Frigeri, M Sigrist, Phys. Rev. 7324504N. Hayashi, K. Wakabayashi, P.A. Frigeri, and M. Sigrist, Phys. Rev. B73, 092508 (2006); ibid 73, 024504 (2006). . S Fujimoto, J. Phys. Soc. Jpn. 7634712S. Fujimoto, J. Phys. Soc. Jpn. 76, 034712 (2007). . S Fujimoto, J. Phys. Soc. Jpn. 7651008S. Fujimoto, J. Phys. Soc. Jpn. 76, 051008 (2007). . S Fujimoto, Phys. Rev. 76184504S. Fujimoto, Phys. Rev. B76, 184504 (2007). . Y Yanase, M Sigrist, J. Phys. Soc. Jpn. 76124709Y. Yanase and M. Sigrist, J. Phys. Soc. Jpn. 76, 124709 (2007). . Y Tada, N Kawakami, S Fujimoto, Phys. Rev. Lett. 101267006Y. Tada, N. Kawakami, and S. Fujimoto, Phys. Rev. Lett.101, 267006 (2008). . C.-K Lu, S K Yip, Phys. Rev. 7754515C.-K. Lu and S. K. Yip, Phys. Rev. B77, 054515 (2008). . V P Mineev, K V Samokhin, Phys. Rev. 78144503V. P. Mineev and K. V. Samokhin, Phys. Rev. B78, 144503 (2008). . Y Tanaka, T Yokoyama, A V Balatsky, N Nagaosa, Phys. Rev. 7960505Y. Tanaka, T. Yokoyama, A. V. Balatsky, and N. Nagaosa, Phys. Rev. B79, 060505 (2009). . M Sato, S Fujimoto, Phys. Rev. 7994504M. Sato and S. Fujimoto, Phys. Rev. B79, 094504 (2009). . M Yogi, Y Kitaoka, S Hashimoto, T Yasuda, R Settai, T D Matsuda, Y Haga, Y Ōnuki, P Rogl, E Bauer, Phys. Rev. Lett. 9327003M. Yogi, Y. Kitaoka, S. Hashimoto, T. Yasuda, R. Settai, T. D. Matsuda, Y. Haga, Y.Ōnuki, P. Rogl, and E. Bauer, Phys. Rev. Lett. 93, 027003 (2004). . M Yogi, H Mukuda, Y Kitaoka, S Hashimot, T Yasuda, R Settai, T D Matsuda, Y Haga, Y Onuki, P Rogl, E Bauer, J. Phys. Soc. Jpn. 7513709M. Yogi, H. Mukuda, Y. Kitaoka, S. Hashimot, T. Yasuda, R. Settai, T. D. Matsuda, Y. Haga, Y. Onuki, P. Rogl, and E. Bauer, J. Phys. Soc. Jpn. 75, 013709 (2006). . C Iniotakis, N Hayashi, Y Sawa, T Yokoyama, U May, Y Tanaka, M Sigrist, Phys. Rev. 7612501C. Iniotakis, N. Hayashi, Y. Sawa, T. Yokoyama, U. May, Y. Tanaka, and M. Sigrist, Phys. Rev. B76, 012501 (2007). . J Linder, A Sudbø, Phys. Rev. 7654511J. Linder and A. Sudbø, Phys. Rev. B76, 054511 (2007). . T Yokoyama, Y Tanaka, J Inoue, Phys. Rev. 7435318T. Yokoyama, Y. Tanaka, and J. Inoue, Phys. Rev. B74, 035318 (2006). . K Børkje, A Sudbø, Phys. Rev. 7454506K. Børkje and A. Sudbø, Phys. Rev. B74, 054506 (2006). . I Eremin, J F Annett, Phys. Rev. 74184524I. Eremin and J. F. Annett, Phys. Rev. B74, 184524 (2006). . C Iniotakis, S Fujimoto, M Sigrist, J. Phys. Soc. Jpn. 7783701C. Iniotakis, S. Fujimoto, and M. Sigrist, J. Phys. Soc. Jpn. 77, 083701 (2008). . G Deutscher, D Feinberg, Appl. Phys. Lett. 76487G. Deutscher and D. Feinberg, Appl. Phys. Lett.76, 487 (2000). . J M Byers, M E Flatte, Phys. Rev. Lett. 74306J. M. Byers and M. E. Flatte, Phys. Rev. Lett. 74, 306 (1995). . D Beckmann, H B Weber, H V Löhneysen, Phys. Rev. Lett. 93197003D. Beckmann, H. B. Weber, and H. v. Löhneysen, Phys. Rev. Lett. 93, 197003 (2004). . S Russo, M Kroug, T M Klapwijk, A F Morpurgo, Phys. Rev. Lett. 9527002S. Russo, M. Kroug, T. M. Klapwijk, and A. F. Morpurgo, Phys. Rev. Lett. 95, 027002 (2005). . C Benjamin, Phys. Rev. 74180503C. Benjamin, Phys. Rev. B74, 180503 (2006). . P Cadden-Zimansky, V Chandrasekhar, Phys. Rev. Lett. 97237003P. Cadden-Zimansky and V. Chandrasekhar, Phys. Rev. Lett. 97, 237003 (2006). . E I Rashba, Sov, Phys. Solid State. 21109E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). . Y Tada, N Kawakami, S Fujimoto, New J. Phys. 955070Y. Tada, N. Kawakami, and S. Fujimoto, New J. Phys. 9 055070 (2009). . G E Blonder, M Tinkham, T M Klapwijk, Phys. Rev. 254515G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B25, 4515 (1982). . K V Samokhin, E S Zijlstra, S K Bose, Phys. Rev. 6994514K. V. Samokhin, E. S. Zijlstra, and S. K. Bose, Phys. Rev. B69, 094514 (2004). . S Hashimoto, T Yasuda, T Kubo, H Shishido, T Ueda, R Settai, T D Matsuda, Y Haga, H Harima, Y Onuki, J. Phys.: Condens. Matter. 16287S. Hashimoto, T. Yasuda, T. Kubo, H. Shishido, T. Ueda, R. Settai, T. D. Matsuda, Y. Haga, H. Harima, and Y. Onuki, J. Phys.: Condens. Matter 16, L287 (2004). . T Terashima, Y Takahide, T Matsumoto, S Uji, N Kimura, H Aoki, H Harima, Phys. Rev. B. 7654506T. Terashima, Y. Takahide, T. Matsumoto, S. Uji, N. Kimura, H. Aoki, and H. Harima, Phys. Rev. B 76, 054506 (2007)
[]
[ "A persistent quiet-Sun small-scale tornado III. Waves", "A persistent quiet-Sun small-scale tornado III. Waves" ]
[ "K Tziotziou \nInstitute for Astronomy\nAstrophysics, Space Applications and Remote Sensing\nNational Observatory of Athens\nGR-15236Pen-teliGreece\n", "G Tsiropoula \nInstitute for Astronomy\nAstrophysics, Space Applications and Remote Sensing\nNational Observatory of Athens\nGR-15236Pen-teliGreece\n", "I Kontogiannis \nLeibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 1614482PotsdamGermany\n" ]
[ "Institute for Astronomy\nAstrophysics, Space Applications and Remote Sensing\nNational Observatory of Athens\nGR-15236Pen-teliGreece", "Institute for Astronomy\nAstrophysics, Space Applications and Remote Sensing\nNational Observatory of Athens\nGR-15236Pen-teliGreece", "Leibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 1614482PotsdamGermany" ]
[]
Context. Vortex flows can foster a variety of wave modes. A recent oscillatory analysis of a persistent 1.7 h vortex flow with a significant substructure has suggested the existence of various types of waves within it. Aims. We investigate the nature and characteristics of waves within this quiet-Sun vortex flow, over the course of an uninterrupted 48-min observing time interval, in order to better understand its physics and dynamics. Methods. We used a cross-wavelet spectral analysis between pairs of Hα and Ca ii 8542 Å intensity time series at different wavelengths and, hence, atmospheric heights, acquired with the CRisp Imaging SpectroPolarimeter (CRISP) at the Swedish Solar Telescope (SST), as well as the derived Hα Doppler velocity and full width at half maximum (FWHM) time series. We constructed halftone frequencyphase difference plots and investigated the existence and propagation characteristics of different wave modes. Results. Our analysis suggests the existence of Alfvénic type waves within the vortex flow that propagate upwards with phase speeds of ∼20-30 km s −1 . The dominant wave mode seems to be the fast kink wave mode, however, our analysis also suggests the existence of localised Alfvénic torsional waves, which are related to the dynamics of individual chromospheric swirls that characterise the substructure of the vortex flow. The Hα V-I phase difference analysis seems to imply the existence of a standing wave pattern that is possibly arising from the interference of upwards propagating kink waves with downwards propagating ones that are reflected at the transition region or the corona. Moreover, the results provide further evidence that the central chromospheric swirl drives the dynamics of the vortex flow.Conclusions. This is the first exhaustive phase difference analysis within a vortex flow that explores the nature and dynamics of different wave modes within it. The question, however, of whether, and how, the dissipation of the derived wave modes occurs remains open, and given that such structures are ubiquitous on the solar surface, it's also important to investigate whether they might ultimately play a significant role in the energy budget of the upper layers of the solar atmosphere.
10.1051/0004-6361/202038951
[ "https://arxiv.org/pdf/2010.06327v1.pdf" ]
127,379,336
2010.06327
1755dd1710f1e5dc3b20860d32823f1573dcd05f
A persistent quiet-Sun small-scale tornado III. Waves October 14, 2020 K Tziotziou Institute for Astronomy Astrophysics, Space Applications and Remote Sensing National Observatory of Athens GR-15236Pen-teliGreece G Tsiropoula Institute for Astronomy Astrophysics, Space Applications and Remote Sensing National Observatory of Athens GR-15236Pen-teliGreece I Kontogiannis Leibniz-Institut für Astrophysik Potsdam (AIP) An der Sternwarte 1614482PotsdamGermany A persistent quiet-Sun small-scale tornado III. Waves October 14, 2020Received / AcceptedarXiv:2010.06327v1 [astro-ph.SR] 13 Oct 2020 Astronomy & Astrophysics manuscript no. tziotziouSun: chromosphere Sun: magnetic fields Sun: photosphere Context. Vortex flows can foster a variety of wave modes. A recent oscillatory analysis of a persistent 1.7 h vortex flow with a significant substructure has suggested the existence of various types of waves within it. Aims. We investigate the nature and characteristics of waves within this quiet-Sun vortex flow, over the course of an uninterrupted 48-min observing time interval, in order to better understand its physics and dynamics. Methods. We used a cross-wavelet spectral analysis between pairs of Hα and Ca ii 8542 Å intensity time series at different wavelengths and, hence, atmospheric heights, acquired with the CRisp Imaging SpectroPolarimeter (CRISP) at the Swedish Solar Telescope (SST), as well as the derived Hα Doppler velocity and full width at half maximum (FWHM) time series. We constructed halftone frequencyphase difference plots and investigated the existence and propagation characteristics of different wave modes. Results. Our analysis suggests the existence of Alfvénic type waves within the vortex flow that propagate upwards with phase speeds of ∼20-30 km s −1 . The dominant wave mode seems to be the fast kink wave mode, however, our analysis also suggests the existence of localised Alfvénic torsional waves, which are related to the dynamics of individual chromospheric swirls that characterise the substructure of the vortex flow. The Hα V-I phase difference analysis seems to imply the existence of a standing wave pattern that is possibly arising from the interference of upwards propagating kink waves with downwards propagating ones that are reflected at the transition region or the corona. Moreover, the results provide further evidence that the central chromospheric swirl drives the dynamics of the vortex flow.Conclusions. This is the first exhaustive phase difference analysis within a vortex flow that explores the nature and dynamics of different wave modes within it. The question, however, of whether, and how, the dissipation of the derived wave modes occurs remains open, and given that such structures are ubiquitous on the solar surface, it's also important to investigate whether they might ultimately play a significant role in the energy budget of the upper layers of the solar atmosphere. Introduction Vortex flows on various, relatively small spatial and temporal scales are ubiquitous in the solar atmosphere of the quiet Sun, having been generated by the turbulent dynamics of solar convection. They are concentrated at the intergranular lanes and formed as a result of angular momentum conservation, which has as a direct consequence on the rapid rotation of the plasma sinking towards the solar interior (Wedemeyer & Steiner 2014). Such flows had already been predicted in theory (Stenflo 1975), revealed through simulations (Nordlund 1985;Stein & Nordlund 2000a,b), and also reported in observations (Brandt et al. 1988;Bonet et al. 2008). Over the past decade, interest in quiet-Sun vortical flows has been revived thanks to their being widely observed in several photospheric and chromospheric lines, as well as the transition region (TR) and the low corona ultraviolet (UV) and extreme ultraviolet (EUV) channels (e.g. Bonet et al. 2010;Attie et al. 2009;Vargas Domínguez et al. 2011;Wedemeyer-Böhm et al. 2012;Park et al. 2016;Tziotziou et al. 2018), while sophisticated numerical simulations have given important results (e.g. Moll et al. 2011a,b;Shelyag et al. 2011;Kitiashvili et al. 2012b;Wedemeyer-Böhm & Rouppe van der Voort 2009;Wedemeyer-Böhm et al. 2012). We note that vortical coherent structures have also been detected in photospheric turbulent flows in plage regions (Chian et al. 2014). Quiet-Sun vortical flows have been found to be associated with motions of magnetic field concentrations referred to as bright points (Bonet et al. 2008) and, thus, they are considered to be immediately associated with magnetic fields. Vortex flows force the footpoints of the magnetic fields to co-rotate (Wedemeyer & Steiner 2014), while the rotation is mediated from the photosphere to the low corona and creates magnetic swirling structures observed at different solar layers (Wedemeyer-Böhm et al. 2012;Tziotziou et al. 2018). We note, however, that non-magnetic vortices with significantly different nature and dynamics than the magnetic ones, have also been reported in the literature (e.g. Stein & Nordlund 1998;Kitiashvili et al. 2012a). Nonetheless, vortex motions are important candidates for the transfer of mass, momentum, and energy, often by waves, from the subsurface to the upper atmospheric layers of the Sun. Magnetohydrodynamic (MHD) waves are considered to be a potential mechanism for the energy transport and for the heating of the solar atmosphere. A&A proofs: manuscript no. tziotziou Magneto-acoustic wave propagation can be triggered by the photospheric horizontal and vertical footpoint motions of localised flux tubes (Fedun et al. 2011a). Analytical works, as well as simulations, have indicated that vortex structures, similarly, can foster a variety of wave modes; both periodic motions at their footpoints and the dynamics of the associated magnetic field do generate shocks and naturally drive different types of MHD waves. The generation of sausage waves has been analytically investigated both in incompressible and compressible magnetically twisted flux tubes by Erdélyi & Fedun (2006, 2007b. Fedun et al. (2011b) used a high-frequency vortex motion as a driver at the footpoint of an open flux tube, with an analytically prescribed magnetic field structure, to simulate the excitation of different types of MHD wave modes, such as sausage, kink and torsional Alfvén waves, within it. In a subsequent work, Fedun et al. (2011c) demonstrated that the plasma structure within a vortex-driven magnetic flux tube can act as a spatial frequency filter for torsional Alfvén waves. Such torsional Alfvénic perturbations, as previously shown by Jess et al. (2009), are responsible for the non-thermal broadening of the Hα line profile, usually observed as variations of its full-width at half-maximum (FWHM). Vigeesh et al. (2012), using a torsional driver, studied the effect of vortex-like motion in a flux tube and reported the generation of slow and fast magnetoacoustic modes. Shelyag et al. (2013) using MHD simulations, including radiative transport and a non-ideal equation of state, identified horizontal motions in magnetic vortices as torsional Alfvénic perturbations. They also steered some discussion within the community as they suggested, based on the use of test particles, that photospheric magnetic field concentrations do not produce magnetic tornadoes or a bathtub effect; this was later numerically challenged by Wedemeyer & Steiner (2014). Finally, several photospheric drivers, including Archimedean and logarithmic velocity spirals, were used by and to investigate the generation of Alfvén, torsional Alfvén waves, and other MHD waves, such as slow kink and slow or fast sausage modes, in magnetic flux tubes. Very few works exist in the literature concerning the observational signatures of waves in vortex flows. Jess et al. (2009) established the existence of torsional Alfvén waves on the magnetic structure around a magnetic bright point by analysing Hα FWHM oscillations. Morton et al. (2013), using high-resolution multi-wavelength ROSA observations from the Dunn Solar Telescope at Sacramento Peak (USA), provided observational evidence (complemented by relevant numerical simulations) for chromospheric torsional Alfvén and kink waves excited by the vortex motions of a strong photospheric magnetic flux concentration. Recently, a detailed oscillatory analysis by Tziotziou et al. (2019) of a long-lived complex vortex flow suggested the presence of magneto-acoustic waves (see below). A similar oscillatory analysis of chromospheric swirls by Shetye et al. (2019) did not provide conclusive evidence that the observed oscillations were either the response to swirling photospheric motions or the result of propagating Alfvénic waves. Liu et al. (2019b), using observations carried out with the Hinode/SOT and Swedish Solar Telescope (SST; Scharmer et al. 2003), provided observational evidence for the excitation of ubiquitous Alfvén pulses by prevalent intensity swirls in the solar photosphere that propagate upwards and reach chromospheric layers. Liu et al. (2019a) further supported the excitation of Alfvén pulses by the co-spatial and co-temporal rotation of photospheric velocity swirls and magnetic swirls found in nu-merical simulations of the solar photosphere with the radiative MHD code Bifrost (Gudiksen et al. 2011). Several techniques have been developed for the observational study of waves on the Sun since the discovery of the 5-min acoustic oscillations by Leighton et al. (1962). One widely used technique for waves that is not, however, very useful in small fields of view (FOVs) or complex structures, is k − ω diagrams, which show the relationship between the wavenumber and temporal frequency (e.g. Ulrich 1970;Deubner 1975;Deubner et al. 1979). With higher, mainly temporal, resolution observations having become available over the years, the phase difference analysis between signals (e.g. timeseries of intensity, velocity, and, more recently, magnetic field) forming at the same or different atmospheric layers have become an important tool for the study of waves (e.g. Mein & Mein 1976;Mein 1977). Phase differences are usually acquired through a spectral Fourier or Wavelet (Torrence & Compo 1998) analysis. These techniques, combined with theoretical or observational knowledge of cutoff frequencies of different waves (the lowest frequency that a wave mode can propagate) and respective formation height differences (where necessary) provide useful information about the existence of different wave modes and their propagation characteristics. In two recent papers, using a multi-wavelength analysis, we meticulously investigated the characteristics and dynamics of a small-scale, 1.7-h persistent vortex flow, observed for the first time in the Hα line centre (LC) (Paper I; Tziotziou et al. 2018), and its oscillatory behaviour (Paper II; Tziotziou et al. 2019). Our analysis revealed a rigidly or quasi-rigidly rotating, funnellike expanding vortex flow that exhibits signatures from low chromospheric heights up to the low corona and comprises at least three recurring, intermittent smaller chromospheric swirls with typical sizes and durations. Derived oscillations, mainly in the range of 3 to 5 min peaking at ∼4 min and extending even up to 10 min at all heights, correspond to the cumulative action of different components such as swaying motions (with periods in the range of 200-220 s), rotation (with periods of ∼270 s for Hα and ∼215 for Ca ii 8542 Å), and waves. The presence of waves, mainly different from acoustic ones, such as magnetoacoustic (e.g. kink) or Alfvén waves, have been suggested from the power behaviour within the vortex flow, compared to a reference quiet-Sun region, as a function of period and height. In this, the third paper of the series, we use a phase difference analysis to further explore the findings of our previous papers, and mainly of Paper II. Our aim is to provide clues about the wave modes related to the vortex flow and of their propagation characteristics. Observations We used the same high-cadence and high-spatial resolution SST CRisp Imaging SpectroPolarimeter (CRISP; Scharmer et al. 2008) Hα and Ca ii 8542 Å datasets, as in Papers I and II (see Table 1 for a summary of the characteristics of the observations). We note, however, that we restricted our analysis during the second observing interval (08:28 UT -09:16 UT) when the vortex flow is more clearly observed (see Paper I). We also used Doppler velocities and the FWHM of the Hα line profile in our analysis, obtained from the Hα profiles according to the methodology presented in the previous two papers, where further details concerning the observations and the datacube processing and coalignment can be found. Figure 1 shows snapshots of the 17.3 ′′ × 19 ′′ region of interest (ROI). The ROI is located almost at the centre of the ac- Table 1. Summary of the used CRISP observations for the present analysis. We note that as in Paper II, no UV, EUV Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) and Helioseismic and Magnetic Imager (HMI;Scherrer et al. 2012) magnetic field observations are used due to their low spatial and temporal resolution compared to CRISP observations. quired 60 ′′ × 60 ′′ CRISP FOV that is centered at S38W10 on the solar disk. It contains the analysed vortex flow (marked with a yellow circle in Fig. 1) with an approximate center at (x, y) = (124.8 ′′ , −592 ′′ ) and a radius of ∼3 ′′ . The 'quiet' Sun region (QSR) and the region with fibril-like structures (FSR) used as comparison regions are also clearly marked in Fig. 1. We note that the QSR, given that the whole ROI is quite dynamic, has been set to be void of large fibril-like structures or small vortex flows, thus, approaching quiet-Sun conditions as closely as possible. Formation heights of the investigated line profiles The formation heights of the two investigated line profiles are particularly pertinent to our analysis. As discussed in Paper I, the Hα LC has a wide formation height range centered slightly above ∼1. Concerning the Ca ii 8542 Å line, its LC is formed at a narrower height range at ∼1-1.1 Mm above the photosphere with the near-LC wavelengths of Ca ii±0.055 Å and Ca ii±0.11 Å forming within a couple of hundred kilometers below (Leenaarts et al. 2009, see their Fig. 4). This LC formation height is quite similar to the values derived by Mein & Mein (1980). Ca ii at ±0.495 Å is formed far lower, at low to mid photospheric heights as reversed granulation is clearly visible at these wavelengths, somewhere between 200 and 500 km according to Fig. 4 of Leenaarts et al. (2009) and the contribution function computed by Cauzzi et al. (2008) and shown in their Fig.5. We note that Rutten et al. (2004) report heights where reversed granulation occurs, between 250-400 km while simulations by Cheung et al. (2007) indicate an even lower height of 130-140 km. Methodology Phase differences, coherency, and halftone images The presence, characteristics, and propagation of waves are examined through a cross-wavelet analysis (Torrence & Compo 1998) between different pairs of signals (e.g. intensities at different heights or intensity-Doppler velocity) on every pixel of the ROI that provides the cross-power, coherence, and, eventually, phase difference as functions of time and period (frequency). A&A proofs: manuscript no. tziotziou For any investigated pair X -Y of time series with respective wavelet transforms W X n and W X n the cross-wavelet transform is defined as W XY n = W X n W Y * n , where W Y * n is the complex conjugate of W Y n , the cross-wavelet power as |W XY n | and the phase difference as φ n = tan −1 [ℑ{W XY n }/ℜ{W XY n }], where ℜ and ℑ are, respectively, the real and the imaginary components of the transform. Although a Fast Fourier Transform could also provide analogous phase differences to those found by the wavelet analysis (Bloomfield et al. 2004), the latter has been chosen as a more suitable method due to the intermittent nature of solar oscillations. The global phase difference corresponds to the time average of the phase difference over the whole time range with crosspower weighting, as introduced by Lites & Chipman (1979). This guarantees that wraparound errors are avoided when angle values just above π are transformed into values just below −π, leading to erroneous phase differences occurring with normal averaging of individual phase differences (see also Tziotziou et al. 2005, for further details). Hereafter, when we discuss about phase differences, we refer to these global phase differences as a function of period (frequency). Finally, coherence provides a measure of the cross-correlation between the two signals and takes its values from zero (indicating no correlation) to unity. However, because random noise always produces a nonzero coherence, a coherence threshold must be determined above which the respective phase difference is considered reliable. For this reason, we adopt the 'floor exceedance coherency approach' described in Bloomfield et al. (2004) that results in a frequencydependent coherence threshold of around 0.6 in most cases. Phase differences corresponding to lower coherence values have been discarded while halftone images, showing the distribution of the cross-power as a function of frequency and phase difference, have been constructed for the remaining, reliable phase differences, following the methodology of Lites & Chipman (1979). To construct these images, the cross-power at every pixel of the considered area is summed over bins that are 3 • wide for the phase difference for each frequency element and normalised to unity (using the maximum derived cross-power). Such halftone images (see also, Kontogiannis et al. 2016) do show the trend of phase difference as a function of frequency, while the corresponding width of the distribution provides a measure of the scatter of the cross-power (Lites et al. 1993). Phase speeds and cut-off frequencies For any frequency f phase lags φ can be easily converted to time lags τ as τ = φ 2π f .(1) In the case of vertically propagating waves between two heights with a known separation, dh, resulting to a phase difference ∆φ, time lags can in turn be related to their phase speed, υ ph . The function, however, between ∆φ, f and dh strongly depends on the atmospheric model used (see e.g. Jafarzadeh et al. 2017, their Equations (1) to (4)). In the simplest case that concerns adiabatic propagation in an isothermal non-stratified atmosphere, the phase speed, υ ph , is equal to υ ph = 2π dh ∆φ f .(2) Small and constant phase differences are an indication of standing or evanescent waves. We note that a monotonic linear increase of phase differences ∆φ as a function of frequency f for a range of frequencies higher than the cut-off frequency (see below), according to Eq. (2), indicates upwards vertical wave propagation with a constant phase speed for this frequency range. However, when such phase differences are derived from photospheric lines, even when the physical conditions of an isothermal atmosphere exist, it has been suggested that the phase difference distribution is not always monotonically increasing with frequency (Mein & Mein 1976;Schmieder 1979;Fleck & Deubner 1989). The propagation of waves is possible above a lowest frequency, known as the cut-off frequency, f 0 . This frequency depends on the wave mode (some modes do not have a cut-off frequency) and on the ambient atmospheric conditions (e.g. temperature, density, magnetic fields). Below f 0 , the expected phase difference for the associated wave mode is equal to zero as its propagation is inhibited. For acoustic waves the cut-off frequency, f 0 is 5.2 mHz and is modified as ∼ f 0 cos θ (Michalitsanos 1973;Suematsu 1990) in the presence of strong, inclined magnetic fields by an angle θ with respect to the solar normal (direction of gravity). In the absence of strong fields, the lowering of the cut-off frequency could also result from significant radiative damping in an isothermal atmosphere (e.g. Worrall 2002). On the other hand, transverse (e.g. kink and Alfvén waves) and longitudinal magnetoacoustic waves (e.g. sausage waves) can have a much lower cut-off frequency than the acoustic one (e.g. Spruit 1981;Kalkofen 1997;McAteer et al. 2002McAteer et al. , 2003Jess et al. 2009;Stangalini et al. 2015). Finally, torsional waves can propagate for any period (Kalkofen 1997). Gravity waves (Mihalas & Toomre 1981) that also have a lower cutoff frequency than acoustic ones (related to the Brunt-Väisälä frequency) are found in much lower atmospheric heights than those investigated in the present analysis (Krijger et al. 2001;Severino et al. 2003;Rutten & Krijger 2003). Phase difference analysis As the ROI is centered at S38W10, the line-of sight (LOS) does not coincide with the vertical to the solar surface. The estimated horizontal pixel offset from the normal to the solar surface is ∼0.1 ′′ per 100 km height difference which is equal to the spatial resolution of the present observations. Hence, projection effects can influence the acquired results and lead to significant errors for signal pairs involving large height separations. Furthermore, the analysis in Paper I indicated that the dynamics within the vortex flow changes considerably as a function of height. The vortex flow and its constituting swirls are clearly visible only in LC and near-LC chromospheric wavelengths in both investigated lines, while for wing wavelengths only some short temporal hints exist for the swirl within the red circle at Hα-0.77 Å (see Fig.4v,w of Paper I) and Ca ii-0.495 Å (see Fig.5f of Paper I). Therefore, for all these reasons, we have restricted our analysis at phase differences between the Hα±0.26 Å -Hα LC, and the Ca ii±0.11 Å -Ca ii 8542 Å LC intensity (I-I) pairs. We note that as the near-LC wavelengths of Ca ii at ±0.055 Å and Ca ii±0.11 Å are both formed very close to the LC, we opted for the latter ones as they give the largest possible height separation. Moreover, in order to minimise the effect of Doppler velocities on intensities, taken at a wavelength distance ∆λ from the LC on either wing of the Hα or Ca ii 8542 Å profiles, we use (similarly to Paper II) their wavelength average (e.g. Hα±0.26 Å is the average intensity at wavelengths -0.26 Å and +0.26 Å). Both selected I-I pairs involve close wavelengths of the investigated line profiles and, hence, at- Phase differences between the Hα±0.26 Å and Hα LC pair within a) the vortex flow area, the areas of swirls within the red, cyan, and green circles and the QSR and FSR areas depicted in Fig. 1 during the second observing time interval, i.e. 08:28 UT -09:16 UT (panels a to f) and b) for the two swirls within the red and cyan circles for specific time intervals (denoted in the respective panels) when these swirls are clearly visible (panels g and h). Both wavelengths are mainly of chromospheric origin and their formation heights are separated by a few hundred kilometers (see Sect. 3). Filled contours represent the cross-power distribution, while white lines denote the position of maximum cross-power, normalised to unity for each frequency element as shown by the vertical bar. The black dashed contour indicates the 50% level from maximum cross-power, and provides a measure of the scatter as it represents the FWHM of the cross-power distribution at each frequency. Cross-power below 10% has been disregarded while all curves have been smoothed with a running average of seven frequency points to remove spurious spikes. The corresponding phase difference of the normalised-to-unity maximum-cross power as a function of frequency, within all considered areas and for the entire considered observing interval only, is also shown in a separate panel (panel i) for an easier and direct comparison. mospheric heights, thus, minimising the influence of projection effects and taking into account the complex vortex dynamics. In addition to the above intensity pairs, we also examine intensity and Doppler velocity (I-V) and velocity and FWHM (V-FWHM) pairs, but only for the Hα line where this analysis is applicable. We note that we reversed the signs of Doppler velocity to correspond to the nomenclature used in early phase difference literature. The I-I pairs that involve phase differences between different layers of the solar atmosphere are used to investigate waves and their propagation characteristics. On the other hand, I-V and V-FWHM pairs, provide physical properties of different wave modes as they relate to response signatures (derived from the line profiles) to processes, such as heating, compression, presence of torsional waves, etc. Our phase difference analysis is mainly based on halftone images (described in Sect. 4) that are related to the vortex flow area and areas of the different individual swirls that constitute its substructure (see Paper I). The QSR and FSR areas are also used for comparison. Such halftone images are derived for the entire second observing interval within all investigated areas. They have also been derived or selected time intervals for the swirls within the red and cyan circles when these are clearly visible in both lines but not for the swirl within the green circle, which is mainly noticeable through its interaction with nearby fibril-like structures (see Paper I). These selected intervals correspond to those of the time-slice images of Figs. 9 and 10 in Paper I; as they are short (∼10 min), the lowest investigated frequency of 2 mHz contains only slightly more than one wave period. However, for the frequency range of interest here (3-5 mHz, see below) and for higher frequencies, the number of periods is considered sufficient for the significance of the acquired results. Before presenting below our results, we note that the oscillatory analysis of Paper II has clearly indicated enhanced power and dominant frequencies of oscillations within the 3-5 mHz frequency band in the entire vortex area. In the three chromospheric swirls, there is a peak at ∼4 mHz in the Hα and at a slightly higher frequency in the Ca ii 8542 Å. Therefore, this 3-5 mHz frequency range (herefter, dominant frequency range) stands as the frequency range of interest. We note that according to the nomenclature of Sect. 4, for any investigated pair X -Y (see titles and captions of all halftone figures hereafter), a positive phase difference means that X leads Y (and vice versa for negative values). A&A proofs: manuscript no. tziotziou As discussed in Papers I and II, the vortex flow comprises the swirl within the red circle and, only partly, the swirls within the cyan and green circles (see also Fig. 1). Parts of the vortex are sometimes free of swirling-related structures and, occasionally, its dynamics is further complicated by the presence of nearby flows from fibril-like structures. Figures 2 and 3 show halftone images for the Hα±0.26 Å -Hα LC and the Ca ii±0.11 Å -Ca ii LC intensity pairs, respectively, for the entire second observing time interval (panels a to f), within the vortex and swirl areas (panels a to d), as well as within the QSR and FSR areas (panels e and f) to be used for comparison. Phase differences within the vortex and swirl areas may contain also significant quiet-Sun contributions due to their intermittent dynamical behaviour. Especially in Hα, they may be occasionally further complicated by the presence of nearby flows from fibril-like structures. For these reasons, phase differences for the swirls within the red and cyan circles, in the particular time intervals when they are continuously visible in both lines, are shown for comparison in panels g and h of Figs. 2 and 3. In all halftone panels, the overall behaviour of the phase differences versus frequency is depicted by the maximum cross-power curve (white lines). These curves are also collectively presented in panel i of Figs. 2 and 3 for easier comparison. When comparing the corresponding halftone images for the considered Hα I-I pair in the vortex area and the areas of individual swirls (Fig. 2), we can see similarities as well as some differences between them. In all three swirls and in the vortex area, there is an increase of the phase difference in the 3-5 mHz frequency range. For the green and cyan swirls as well as for the vortex area, the increase is from ∼0 • to ∼20 • . In the red swirl, there is a small increase from ∼0 • to ∼10 • in the 3-4 mHz frequency range and then a jump of phase difference values to ∼40 • in the 4-5 mHz frequency range, with low cross-power, however. A comparison of the respective panels of Fig. 2 suggests that within the dominant frequency range, it is this swirl within the red circle that seems to dominate the phase difference behaviour within the whole vortex area. Indeed, the high phase differences, up to ∼40 • , around ∼4.5 mHz, with significant cross-power above 0.8, are clearly attributed to this swirl. This reinforces the argument presented in our previous works that this swirl, located at the vortex center, probably serves as a 'central engine' for the entire vortex flow. Differences in the behaviour of the three swirls could also relate to the location of the swirls within the cyan and green circles that is close to the periphery of the vortex flow. Although there is an intermittent character in the appearance of the individual swirls, the described behaviour above seems to be indicative of their dynamics. This can be seen from Fig. 2 (panels g and h), where the behaviour of the phase differences versus the frequency within the two considered swirls, derived for particular time intervals when they are continuously visible, is echoed in the corresponding ones derived for the entire observing time interval considered (Fig. 2, panels b and d). The behaviour of the phase difference versus frequency for the considered Ca ii I-I pair is different from that of the Hα I-I pair. Indeed, Fig. 3 shows that the vortex area, the three swirl ar-eas, and the QSR behave in a similar way when the entire second observing time interval is considered. In all these areas, there is an increase of the phase differences for frequencies ≥2.5 mHz and up to ∼4.0 mHz, where they attain values of ∼20 • . This is expected, to an extent, from the observational analysis of Paper I as a) only some intermittent swirls and not the vortex flow itself are observed in the Ca ii 8542 Å line at these wavelengths (which are both formed lower than the Hα I-I pair); b) these swirls are smaller than the corresponding ones seen in Hα; and c) there is almost no influence from fibril-like structures, the majority of which are barely seen. Therefore, the behaviour of the phase differences versus frequency of all areas within the vortex flow reflect mainly the quiet-Sun behaviour with strong photospheric contribution as it is revealed from the same low cut-off frequency of ∼2.5 mHz and the same general behaviour with the QSR. Minor differences between the individual swirls can be attributed to their particular dynamics. When specific time intervals are considered (Fig. 3, panels g and h), likewise Hα, individual swirls show a behaviour that is reflected in the corresponding panels (Fig. 3, panels b and d) involving the entire considered observing interval. The minimum frequency for which wave propagation is detected (cut-off frequency) can give clues about the wave mode (see Sect. 4.2). In the mainly chromospheric Hα I-I pair (see Fig. 2), the QSR and FSR show a phase difference behaviour versus frequency that is indicative of acoustic propagation and consistent with previous findings (Fleck & Deubner 1989;Deubner & Fleck 1990). More specifically, phase differences in the QSR are small (around 10 • ) but constant, from ∼ 3 mHz and up to the acoustic cut-off frequency of 5.2 mHz, indicative of standing or evanescent waves. Afterwards, they sharply increase as one would expect for acoustic wave propagation in the chromosphere. In the FSR, they are equal to zero up to ∼4.0 mHz suggesting that there is no wave propagation up to this frequency. Afterwards, there is a small but apparent increase of the phase difference due to the lowering of the cut-off frequency. This lowering is due to the presence of inclined magnetic fields related to the fibrilar structures that form the magnetic canopy and allow the propagation of magnetoacoustic waves for frequencies below the acoustic cut-off frequency (Kontogiannis et al. 2010(Kontogiannis et al. , 2014. On the other hand, the behaviour of the phase differences versus frequency within the vortex area and the areas of individual swirls is different from that of the QSR and FSR. The most important result that comes from these images is that the cut-off frequency in the vortices is ∼3.0 mHz, which is well below the chromospheric acoustic cut-off frequency of ∼5.2 mHz and also well below the cut-off frequency of ∼4.0 mHz obtained in the FSR. Propagation for frequencies lower than the acoustic cut-off frequency of 5.2 mHz, as detailed in Sect. 4.2, is an indication of the existence of either non-acoustic wave modes or magnetoacoustic waves that are allowed to propagate due to the lowering of the acoustic cut-off frequency in the presence of inclined magnetic fields or an atmosphere with significant radiative damping. The significantly lower cut-off frequency within the vortex flow and the individual swirls points directly to the existence of nonacoustic wave modes and, more specifically, to Alfvénic type modes. This confirms the results of the oscillatory analysis of Paper II that have clearly suggested the presence of mainly transverse wave modes that are different than the acoustic ones without, however, completely ruling out the presence of some acoustic power. The natural interpretation of the present results, when combined with the swaying motions observed in the area (see e.g. Fig. 9 in Paper I and Fig. 3 in Paper II), is that the areas ex-perience kink MHD oscillations. Kink waves are the only wave modes that can displace the axis of the three individual swirls and of the vortex flow as a whole and have such a lower cut-off frequency. As stated in Sect. 1, the ubiquitous vortex flows can effectively excite kink waves that propagate upwards. The above results provide clear evidence that vortical motions generate kink waves that propagate from the lower layers to the chromosphere. Wave propagation characteristics Despite some obvious differences (discussed in the previous section) in the behaviour of the I-I phase differences versus frequency between the vortex flow and the individual swirls, as well as in the different lines and wavelengths, phase differences, when they are not zero and notwithstanding the significant scatter, are mainly positive with significant cross-power. These prevalent positive phase differences suggest upwards propagation as the lower formed intensity variations lead the higher formed ones. Moreover, linear increases of phase difference with frequency indicate upwards propagation with a constant phase speed. Indeed, in Hα (Fig. 2) and within the dominant frequency range of 3-5 mHz, we see positive phase differences up to ∼40 • with highly variable behaviour, however, as a function of frequency within the swirl denoted with a red circle. Mainly positive phase differences but with lower values (∼10 • -20 • ) are also observed within the swirls denoted with the green and cyan circles despite some negative phase differences for the latter for frequencies lower than 3.5 mHz. As a consequence, positive phase differences are also seen within the whole vortex area. For the Ca ii I-I pair in Fig. 3, likewise Hα, phase differences within the vortex area and individual swirls, are again positive with significant cross-power present but with lower values than in Hα. We note that even in the QSR and FSR, in both lines, we clearly have upwards wave propagation above the respective cut-off frequencies. Figure 4 shows the derived phase speeds, υ ph100 , for a height difference of 100 km used as a reference (an approach chosen as formation height estimates may change in the future with better line formation codes becoming available). Actual values of υ ph are equal to (dh/100 km)×υ ph100 , with dh representing the formation separation height of the two wavelengths of the considered I-I pair. This dh, according to Sect. 3, is in the range of 200-300 km between Hα±0.26 Å and Hα LC and ∼200 km between Ca ii±0.11 Å and Ca ii LC. Weighted-mean phase speeds do not substantially differ within the vortex and the different swirl areas. Actual phase speeds, when the above dh are considered, derived from the lower forming Ca ii I-I pair are generally similar to those derived from the Hα I-I pair and on the order of 20-30 km s −1 or even higher at certain frequencies. Phase speeds in Hα tend to be slightly lower, especially within the swirl denoted with the red circle, when only frequencies above ∼4 mHz are considered. Then the weighted mean υ ph100 becomes 7.6, 5.7, 11 and 7.7 km s −1 within the vortex and the swirls within the red, cyan and green circles, respectively. These derived phase speeds of 20-30 km s −1 are definitely higher than the local sound speed, which is of the order of 10 km s −1 , but are compatible with typical values of Alfvénic type speeds at these atmospheric heights (e.g. De Pontieu et al. 2001;Cranmer & van Ballegooijen 2005;Arber et al. 2016, their Figs. 7, 3, and 2, respectively). Uncertainties in the derivation of phase speeds could rise from the assumed physical conditions (see Sect. 4.2), and from the simplified assumption of vertical wave propagation. Although the actual three-dimensional magnetic structure of vor-A&A proofs: manuscript no. tziotziou Here, υ ph100 has been calculated only for phase differences higher than 10 • within the vortex area (black curves), and the swirls within the red, cyan, and green circles (respective colour curves) shown in Fig. 1. Coloured legends provide the weighted mean υ ph100 values with the respective normalised cross-power as weight. The derivation of actual speeds, υ ph , further requires the actual formation height separation, dh, as they are equal to (dh/100 km)×υ ph100 , according to Eq. (2). tex flows cannot presently be inferred from observations, simulations indicate that the magnetic field is mostly far from vertical and it is, in fact, rather twisted and possibly expanding with height. Wave propagation in such a configuration has only been treated under certain assumptions and for specific wave modes either analytically (e.g. Erdélyi & Fedun 2006, 2007bCheremnykh et al. 2018) but mainly numerically (e.g. Murawski et al. 2018). Results indicate that the various MHD waves are strongly guided by the local magnetic field (see e.g. Bogdan et al. 2003, their Table 4). Therefore, phase differences at the same image pixel employing signals at different heights, for instance, phase differences between different wavelengths of a line profile, should always be treated and explained with extreme caution in terms of wave propagation. I-V phase difference analysis Phase differences between intensity and velocity (I-V) can also be used to infer the properties of waves. Generally, Hα intensity is considered to be a proxy of temperature and density. However, its dependence on temperature is stronger at the wings of the line and weakens towards the LC (Leenaarts et al. 2012), further complicating the interpretation of the I-V phase differences. Considerable work has been done in the past on I-V phase differences for acoustic waves and, especially, for p-modes. It goes back to Whitney (1958) and Holweger & Testerman (1975, their Fig. 6) who showed that for running upwards waves with periods ∼5 min, temperature (as a proxy of intensity) and velocity should be largely in phase and have positive phase differences. For standing waves, on the other hand, intensity should lead velocity by 90 • in the adiabatic limit. In the isothermal case and when the timescale for heat losses is short, the phase dif- ference may reach up to 180 • while negative phase differences between -180 • and -90 • indicate downwards propagation. This phase difference value can be interpreted as a superposition of ascending and descending waves reflected at a chromospheric or transition region boundary with the dense photosphere acting as the lower boundary. Mein & Mein (1976) and Mein (1977), in discussing the V-I phase behaviour of upper chromospheric lines, suggested the generation of standing waves by the transition region. Mein (1977), however, also pointed out the inherent difficulties of such a single reflecting boundary, which was later also heavily debated by Deubner et al. (1996). Fleck & Deubner (1989) also reported the observations of acoustic standing waves in the lower solar chromosphere by measuring a 90 • phase difference between the brightness and velocity oscillations of the Ca ii lines. The formation of a nearly standing wave pattern in the chromosphere through the interference of upwards propagating Alfvén waves with reflected, downwards-propagating ones has been suggested by Hollweg (1981); reflection occurs at atmospheric levels, where the Alfvén speed changes rapidly as we move from the chromosphere to the transition region. Fujimura & Tsuneta (2009), using Hinode/SOT spectropolarimetric observations, considered fluctuations of the LOS magnetic flux and velocity in magnetic flux tubes and found phase differences of ∼ -90 • or ∼ 90 • . They conjectured that these result from the superposition of an ascending and descending kink wave. The former is reflected at the chromospheric-coronal boundary and the ascending and descending superposed waves form standing waves at the line formation layer. Whether the phase is -90 • or 90 • depends on the distance of this layer from the reflecting boundary. Figure 5 shows the acquired phase difference as a function of frequency between the Hα minimum intensity and the Hα LC Doppler velocity within the vortex area and the three individual swirls. Phase differences are always positive meaning that intensity always leads upwards velocity. Moreover, at least for frequencies within the dominant 3-5 mHz range, they are distributed around 90 • . For frequencies higher than ∼5 mHz the phase difference generally drops as a function of frequency, but remains always positive. In some cases, for frequencies above 7 or 8 mHz, it becomes constant, while for the swirl within the red circle it again attains values close to ∼90 • . As already discussed (Sect. 5.1), the observed fluctuations are consistent with Fig. 6. Average phase difference of FWHM oscillations around the dominant frequency of 4 mHz (±0.5 mHz) as a function of distance across the diameters of the swirls within the red (red curve) and cyan circle (cyan curve). The FWHM phase differences are derived relative to each swirl's left edge that is used as a reference point, with the ends of both curves ideally associated with opposite sides of the waveguide in the case of torsional Alfvén perturbations. Only FWHM oscillations at selected time intervals when these swirls are clearly visible (see Fig. 2, panels g and h) have been considered for the phase differences derivation. Overplotted error bars correspond to the respective standard deviation of the FWHM oscillations within the considered frequency range. kink wave modes. The observed I-V phase differences of 90 • (Fig. 5), is an indication of the existence of standing kink waves within the vortex and the individual swirl areas at the formation height of the Hα minimum intensity. These waves can be produced by the upwardly propagating kink waves which, after being reflected in the transition region or corona propagate downwards and form the standing waves in the Hα minimum intensity formation layer. However, standing waves due to the superposition of upwards and downwards propagating magnetoacoustic waves produced by the p-modes cannot be excluded. Reflections of upwardly propagating magnetoacoustic waves have been examined in the case of fibril-like structures like the ones at the west and north-west border of the vortex flow (see Paper I). Such inclined structures form the magnetic canopy and play a crucial role not only in the reflection, but also in the transmission and refraction of waves (Kontogiannis et al. 2014. Unfortunately, as we lack simultaneous high-spatial resolution observations of the magnetic field it is not possible to infer any possible interaction between MHD waves and the magnetic field within the vortex structure, or even with the nearby fibrilrelated magnetic canopy. However, we note that such an investigation would not be an easy task because of the complicated form of the magnetic field lines and the plasma motion inside vortex magnetic structures. FWHM oscillations and phase differences between the Hα Doppler velocity and FWHM In Paper II, we showed that the FWHM parameter derived from the Hα line profiles shows oscillations and enhanced power within the vortex area in the same dominant frequency range as the intensities. Variations of the non-thermal width of a line, therefore, of the FWHM, can be caused by a number of mechanisms, such as Kelvin-Helmholtz instabilities (Kuridze et al. 2015(Kuridze et al. , 2016, transverse displacements, and rotational motion due to kink waves (Goossens et al. 2014) or torsional Alfvén waves (Zaqarashvili 2003;Zaqarashvili & Murawski 2007;Jess et al. 2009). Investigations of the connection between periodic transverse displacements and FWHM variations were mainly performed in spicules at the solar limb, where transverse displacements of their axis correspond to observed LOS Doppler velocities and are easy to measure. In such cases, the largest FWHM is produced at zero displacement from the equilibrium position when the torsional velocity is at its maximum. On the solar disk, kink waves are manifested by the transverse displacements of the magnetic structure axis, which can be observed and measured by an imaging instrument. When the axis of the investigated structure is not parallel to the LOS, as in our case, such motions constitute a component of the derived Doppler velocity. Torsional Alfvén waves, on the other hand, do not cause any bulk displacement of the structure about its central axis or variations in the intensity (i.e. density) but, rather, they cause purely axisymmetric twisting motions and, hence, they are not easily detected. One way to detect them is if they show both red and blue Doppler shifts simultaneously at the opposite edges of the structure, with its axis not in parallel to the LOS. Due to the spatial integration, this creates a periodic non-thermal broadening of the spectral line. In this case, the 180 • phase delay of FWHM oscillations that are produced at the opposite boundaries of the waveguide that outline constant magnetic surfaces (Copil et al. 2008;Van Doorsselaere et al. 2008) are evidence of torsional Alfvén waves. This has been clearly shown in Fig. 4 of Jess et al. (2009), where the phase differences stemming from FWHM oscillations increase from 0 • to 180 • across the diameter of a magnetic bright point. Figure 6 shows a plot of the average phase difference of FWHM oscillations as a function of diameter for the swirls within the red and cyan circles. Only the swirl within the cyan circle shows a monotonic increase of the FWHM phase difference from one edge of the structure to the opposite edge, which nearly reaches a value of 180 • . This result could be suggestive of the presence of torsional Alfvén waves within this swirl. We note that as indicated in Paper I, the rotation of this swirl sometimes changes from counterclockwise to clockwise, which is contrary to the swirl within the red circle that shows a regular clockwise rotation and it is clearly seen only at larger heights than the swirl within the red circle. Both of these effects could also result from the action of torsional waves. The corresponding behaviour for the swirl within the red circle in Fig. 6 differs completely. No monotonic increase of the phase difference exists across its diameter, suggesting that there is no direct indication of torsional A&A proofs: manuscript no. tziotziou Alfvén waves within this swirl,which probably acts as a central driver of the observed flow. We note that in Paper II, variations of the rotational period as a function of the radius were suggested within the vortex flow. Such variations could trigger Kelvin-Helmholtz instabilities that would also manifest themselves as small-scale vortices within a large vortex flow. Although such a possibility cannot be totally ruled out, the size of the observed swirls and the compelling evidence provided above point to the existence of torsional Alfvén waves. Figure 7 shows the derived phase differences for the Hα FWHM -Doppler velocity pair within the vortex area and the different swirls. The halftone images show similarities with the corresponding images shown in (Fig. 5). To our knowledge, no similar observational results exist in literature for a direct comparison while a theoretical interpretation is still missing. The phase differences in all areas attain values of ∼ 60 • -70 • in the frequency range of 3-5 mHz. The FWHM of a line includes thermal and non-thermal broadening. Thermal broadening depends on the temperature inside the structures, while non-thermal broadening is related to turbulent motions, waves, and various inhomogeneities. As stated in Sect. 5.2, Hα intensity has a stronger dependence on temperature at the wings of the line, weakening towards the LC. Differences between Fig. 5 and Fig. 7 may result from different influences of the thermal and non-thermal components on intensity and FWHM. Discussion and conclusions Observations and simulations (see Sect. 1) demonstrate that due to spiralling motions at the photosphere plasma is propagating upwards within vortical magnetic structures, using twisted magnetic field lines as guides. The complex interaction between these rotating plasma flows and vortex-related magnetic fields are thought to be the source of several MHD wave modes. Different, mainly Alfvénic type modes, such as kink waves, sausage modes, and torsional Alfvén waves have been widely suggested in literature to be excited by vortex flows (Fedun et al. 2011b,c;Shelyag et al. 2013;Verth & Jess 2016). The kink mode corresponds to a bulk motion of the plasma within and outside a magnetic structure and, amongst others, displaces the whole structure in the transverse direction. Sausage modes are typically identified by periodic fluctuations of the area of the magnetic structure. Torsional Alfvén modes, on the other hand, can exist independently on each magnetic surface of the structure and do not displace it as a whole (e.g. Erdélyi & Fedun 2007a;Van Doorsselaere et al. 2008;Ruderman & Erdélyi 2009). Despite these behavioural differences, kink modes are considered to be of Alfvénic type as phase speeds are similar to those of Alfvén waves and magnetic tension is also the restoring force. These are characteristics that often lead to a confusion between the two modes in literature. In Paper II, the power and oscillatory analyses of a vortex area have clearly indicated the presence of significant oscillatory power in the range of 3 to 5 mHz, which peaks around 4 mHz, and reflects the cumulative action of different components, such as swaying (transverse) motions, rotation, upwards motions, and waves. Moreover, the behaviour of power within the vortex flow as a function of period and height, when compared to a reference quiet-Sun region, clearly suggested the existence of waves that are substantially different from the acoustic ones, such as magnetoacoustic (e.g. kink) or Alfvén waves. Observational evidence on the types of MHD wave modes that vortex-related flux tubes can support is very important. Not only because magneto-seismology allows us to obtain a diagnostic insight into the properties of the solar plasma, but also because MHD waves channeled through vortex flux tubes are considered to play a key role in the heating of the solar atmosphere as they may carry significant amounts of energy throughout the solar layers (Wedemeyer-Böhm et al. 2012;Liu et al. 2019b). A practical tool for identifying wave modes from observations and their propagation characteristics is a phase difference analysis. In this work, we perform such an analysis between pairs of different parameters within the vortex area and the three individual swirls within it. More specifically, we investigate phase differences between intensity pairs (I-I) at different wavelengths and, hence, sampled layers of the solar atmosphere, in the two observed lines (i.e. Hα and Ca ii 8542 Å), and this analysis is also conducted in a QSR and an FSR for comparison. We also perform a phase difference analysis between pairs of Hα minimum intensity-Doppler velocity (I-V) and FWHM-Doppler velocity (FWHM-V) in the vortex area and the three individual swirls. The behaviour of I-I phase differences with frequency within the vortex-related areas has different characteristics than those within the QSR and the FSR. In the QSR, there is wave propagation for frequencies above the acoustic cut-off frequency of 5.2 mHz, while in the FSR, there is wave propagation for frequencies above a lowered cut-off frequency of ∼ 4 mHz due to the role of inclined magnetic fields in the propagation of magnetoacoustic waves (Suematsu 1990;Kontogiannis et al. 2016). In the vortex area and the individual swirls, the present analysis clearly demonstrates and further supports the existence of other than acoustic wave modes. All results indicate the presence of upwards propagating waves in the frequency range 2.6 -5 mHz, with a cut-off frequency of ∼ 2.6 mHz, which is much lower than the acoustic cut-off frequency of 5.2 mHz. This lowered cut-off frequency cannot be attributed to acoustic waves travelling within inclined magnetic fields or atmospheres with excessive radiative damping. Such a low cut-off frequency is, however, consistent with the theoretical expectations for kink waves (Spruit 1981). In addition, acquired phase speeds of upwards propagating waves are mainly in the range of 20-30 km s −1 which are compatible with Alfvén speeds at the considered solar atmospheric heights and further support the interpretation of the propagating waves as fast kink waves. It is worth remarking that such an interpretation is also in agreement with the evidence for the existence of transverse motions within the vortex flow, reported in the previous two papers (see e.g. Figs.9 to 12 of Paper I and Fig.3 of Paper II). These results fit the picture of a lateral displacement of the axis of the vortex structure that cannot be attributed to other types of waves, such as torsional Alfvén or sausage waves. The existence of sausage waves can be further excluded as: a) no periodic fluctuations of the area of the vortex have been observed, with the vortex area clearly remaining, at least visually, constant (see Paper I); and b) the time-slice images of Fig. 9 of Paper I do not show symmetric spiral variations around the vortex central position as would be expected in this case. Apart from the propagating fast kink waves within the vortex area and the three swirls, revealed from the I-I phase differences, the systematic phase differences between the Hα intensity minimum and the Doppler velocity, which are equal to ∼ 90 • in the same examined areas, are suggestive of the existence of standing waves. These waves may result from the superposition of upwards and downwards propagating waves and are formed at the formation layer of the Hα minimum intensity. We suggest that the ascending waves are kink waves that after being reflected in a boundary found somewhere in the TR or in the corona move downwards with this superposition of counter propagating waves resulting in the formation of standing waves. Standing waves have been detected before in the lower solar atmosphere, for example, by Fujimura & Tsuneta (2009) who examined phase differences between magnetic field and velocity in pores and intergranular magnetic structures and interpreted them as due to kink or sausage waves. However, standing waves due to counter propagating magnetoacoustic waves produced by the p-modes cannot by excluded by the present study. The presence of kink waves cannot exclude the co-presence of torsional Alfvén waves within the structure since such waves could act in individual magnetic surfaces of the structure. Unfortunately, torsional Alfvén waves cannot be easily detected either as intensity or Doppler velocity variations because usually the spatial scale or the temporal periodic variation of the mode are not resolved. Indirectly, the existence itself of substructure in the form of several smaller chromospheric swirls within the conspicuous vortex flow could be attributed to torsional Alfvén waves, as explained in the work of Fedun et al. (2011c). Moreover, oscillations of the FWHM of the Hα line profile, which have been clearly seen and described in Paper II, can be attributed to torsional Alfvén waves (Jess et al. 2009). Oscillations of the FWHM of the Hα profile are mainly caused by periodic fluctuations of non-thermal broadening, which, in turn, can be caused by different reasons, with rotation among them. The phase difference analysis between the FWHM and Doppler velocity oscillations provides a compelling argument that, at least, the swirl within the cyan circle results from the action of torsional Alfvén waves. A phase delay of the FWHM oscillations equal to 180 • , is found on the opposite edges of this particular swirl. Such a phase difference increase from 0 • to 180 • of the FWHM oscillations, found across the diameter of a magnetic element, was attributed to torsional Alfvén waves by Jess et al. (2009). We note, as the simulations indicate, that the main driving force behind such Alfvén waves could mainly be attributed to the rotation of the vortex structure rather than its transverse motion, which is attributed to kink waves. In conclusion, the phase difference analysis performed in this work, combined with the oscillatory analysis (Paper II) and the analysis of the vortex dynamics (Paper I), suggest the coexistence of MHD waves of Alfvénic type both propagating and standing. The dominant type of waves seems to be fast kink waves that propagate upwards and affect the oscillatory behaviour of the entire vortex structure. Localised torsional Alfvén waves related to the dynamics of individual chromospheric swirls within the vortex structure seem to coexist. Moreover, the existence of a standing wave pattern at the height of formation of the Hα minimum intensity is implied by the present analysis. This pattern could arise from the superposition of upwards propagating kink waves with downwards-propagating ones after being reflected at the TR or corona. Numerical simulations have suggested that different MHD wave modes such as kink waves and torsional Alfvén waves may be excited in the same vortex tube and can coexist simultaneously (Fedun et al. 2011b). From observations of spicules at the limb, the coexistence of both kink and torsional Alfvén waves (De Pontieu et al. 2012), as well as propagating and standing transverse oscillations (Okamoto & De Pontieu 2011) have been reported. Both transverse propagating and standing wave modes have also been detected in on-disk chromospheric mottles (Kuridze et al. 2013). As vortex flows act as guides for the propagation of different types of MHD waves, a possible dissipation of the energy carried by them is of great interest for estimating the contribution of vortical structures to the energy budget of the upper layers of the solar atmosphere. Our findings reveal the upwards propagation of fast kink waves with frequencies above 2.6 mHz at the chromospheric heights considered. These waves may carry a considerable amount of energy above the heights of formation of the cores of the Ca ii 8542 Å or Hα lines. Our observations and analysis, however, cannot elucidate whether and where this energy is dissipated; this is certainly not a straightforward task and it is certainly out of the scope of the present study. We find that fast kink modes most probably are partially reflected in a boundary in the TR or corona. It is not clear which amount of the waves propagating upwards reach the corona or whether they are (partially or entirely) dissipated below it and by releasing their energy play a role in heating their local surroundings. It has been shown that reflection occurring in the TR produce in the chromosphere a pattern of counter-propagating waves that are subject to nonlinear wave-wave interactions. The waves decay into turbulence which may significantly increase the heating rates in the chromosphere, while part of the wave energy is transmitted through the TR and produces turbulence in the corona (Matthaeus et al. 1999;van Ballegooijen et al. 2011). Torsional Alfvén waves may also play an important role in the energy propagation through the solar atmosphere and its heating. Soler et al. (2017) investigated the reflection, transmission, and dissipation of torsional Alfvén waves propagating in expanding flux tubes from the photosphere up to the low corona. They have shown that the energy propagation of the torsional waves goes through three different regimes depending on the frequency of the waves. Assuming that the waves are driven below the photosphere, they found that low frequencies are mainly reflected back to the photosphere, while intermediate frequencies are able to be efficiently transmitted to the corona, and high frequencies are completely damped in the chromosphere. The work presented in this series of papers (Paper I, Paper II, and this paper) represents the first exhaustive analysis of the dynamics, oscillations, and waves of a persistent vortex flow. We note, however, that both a theoretical and analytical verification of the acquired results is still missing and continues to be necessary. Our analysis has greatly profited from the high spatial and temporal resolution observations provided by the SST. New high-resolution, multi-line observations in more wavelengths across the line profiles that include simultaneous spectropolarimetric observations are imperative to gaining a broader understanding of the dynamics of swirling structures, which seem to be ubiquitous in the solar atmosphere. Such observations could provide further clues about the generation and propagation of the various types of MHD waves within them and provide answers to the question of whether they play any significant role in the energy budget of the solar atmosphere. Fig. 1 . 1Snapshot of the ROI in Hα LC (first panel), Hα Doppler velocity (second panel), Hα FWHM (third panel), and the Ca ii 8542 Å LC. In both intensity images, black indicates structures in absorption while the grayscaling in each panel is within ±3σ from the respective mean value of the whole image with σ corresponding to the respective standard deviation. The overplotted yellow circle indicates the location of the analysed conspicuous vortex flow, while smaller red, green, and cyan circles denote the approximate location of smaller swirls (substructure), extensively discussed in Papers I and II. The orange and white rectangles in the Hα LC panel show, respectively, the selected QSR and FSR (see text). Fig. 2. Phase differences between the Hα±0.26 Å and Hα LC pair within a) the vortex flow area, the areas of swirls within the red, cyan, and green circles and the QSR and FSR areas depicted in Fig. 1 during the second observing time interval, i.e. 08:28 UT -09:16 UT (panels a to f) and b) for the two swirls within the red and cyan circles for specific time intervals (denoted in the respective panels) when these swirls are clearly visible (panels g and h). Both wavelengths are mainly of chromospheric origin and their formation heights are separated by a few hundred kilometers (see Sect. 3). Filled contours represent the cross-power distribution, while white lines denote the position of maximum cross-power, normalised to unity for each frequency element as shown by the vertical bar. The black dashed contour indicates the 50% level from maximum cross-power, and provides a measure of the scatter as it represents the FWHM of the cross-power distribution at each frequency. Cross-power below 10% has been disregarded while all curves have been smoothed with a running average of seven frequency points to remove spurious spikes. The corresponding phase difference of the normalised-to-unity maximum-cross power as a function of frequency, within all considered areas and for the entire considered observing interval only, is also shown in a separate panel (panel i) for an easier and direct comparison. Fig. 3 . 3Similarly to Fig. 2 halftone images for the Ca ii±0.11 Å and the Ca ii LC pair of the Ca ii 8542 Å line. In the bottom right panel, only the corresponding curves for the vortex area, the QSR and the FSR are shown as curves for respective individual swirl areas are quite similar to these (see text). 5.1. I-I phase difference analysis 5.1.1. Behaviour of the I-I phase differences vs frequency and inferred wave modes Fig. 4 . 4Phase speeds υ ph100 (in km s −1 ) for a height difference of 100 km derived from the phase difference corresponding to the maximum crosspower of the considered I-I pairs (white lines in Figs. 2 and 3) within the dominant frequency range and for different areas. Fig. 5 . 5Similarly toFig. 2halftone images for the Hα minimum intensity -Hα Doppler velocity pair but only for the vortex area and the individual swirl areas for the entire observing interval considered. Fig. 7 . 7Similarly to Fig. 2 halftone images for the Hα FWHM -Doppler velocity pair but only for the vortex area and the individual swirl areas for the entire observing interval considered. Notes.(a) There is only a small temporal offset of ∼2 s between the Ca ii and Hα time series as the two lines were observed sequentially. Only the second 48-min observing interval has been used in the present analysis.Hα 6562.81 Å Ca ii 8542 Å Date of observations June 7, 2014 Observing intervals 07:32 UT -08:21 UT and 08:28 UT -09:16 UT a Wavelengths Hα LC, ±0.26 Å, ±0.77 Å, ±1.03 Å Ca ii LC, ±0.055 Å, ±0.11 Å, ±0.495 Å Temporal cadence 4 s 4 s Pixel size 0.059 ′′ 0.0576 ′′ Hα 120 125 130 135 x [arcsec] −600 −595 −590 y [arcsec] 120 125 130 135 x [arcsec] −8 −4 0 4 8 Doppler velocity (km/s) 120 125 130 135 x [arcsec] 0.9 0.95 1.0 1.15 1.1 Hα FWHM (Å) CaII 8542Å 120 125 130 135 x [arcsec] Article number, page 6 of 12Tziotziou et al.: Waves in a persistent quiet-Sun tornado Acknowledgements. The authors would like to thank the International Space Science Institute (ISSI) in Bern, Switzerland, for the hospitality provided to the members of the team on "The Nature and Physics of Vortex Flows in Solar Plasmas". KT and GT acknowledge support of this work by the project "PROTEAS II" (MIS 5002515), which is implemented under the Action "Reinforcement of the Research and Innovation Infrastructure", funded by the Operational Programme "Competitiveness, Entrepreneurship and Innovation" (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional Development Fund). This research was also supported by IKYDA2020, an action program between the German Academic ExchangeService (Deutscher Akademischer Austauschdienst -DAAD) and the Greek State Scholarship Foundation (I.K.Y). IK acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) grant DE 787/5-1. The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. Many thanks to the referee for the insightful comments and suggestions on improving the paper.Article number, page 11 of 12 A&A proofs: manuscript no. tziotziou . T D Arber, C S Brady, S Shelyag, ApJ. 81794Arber, T. D., Brady, C. S., & Shelyag, S. 2016, ApJ, 817, 94 . R Attie, D E Innes, H E Potts, A&A. 49313Attie, R., Innes, D. E., & Potts, H. E. 2009, A&A, 493, L13 . D S Bloomfield, R T J Mcateer, B W Lites, ApJ. 617623Bloomfield, D. S., McAteer, R. T. J., Lites, B. W., et al. 2004, ApJ, 617, 623 . T J Bogdan, M Carlsson, V H Hansteen, ApJ. 599626Bogdan, T. J., Carlsson, M., Hansteen, V. H., et al. 2003, ApJ, 599, 626 . J A Bonet, I Márquez, J Sánchez Almeida, I Cabello, V Domingo, ApJ. 687131Bonet, J. A., Márquez, I., Sánchez Almeida, J., Cabello, I., & Domingo, V. 2008, ApJ, 687, L131 . J A Bonet, I Márquez, J Sánchez Almeida, ApJ. 723139Bonet, J. A., Márquez, I., Sánchez Almeida, J., et al. 2010, ApJ, 723, L139 . P N Brandt, G B Scharmer, S Ferguson, R A Shine, T D Tarbell, Nature. 335238Brandt, P. N., Scharmer, G. B., Ferguson, S., Shine, R. A., & Tarbell, T. D. 1988, Nature, 335, 238 . G Cauzzi, K P Reardon, H Uitenbroek, A&A. 480515Cauzzi, G., Reardon, K. P., Uitenbroek, H., et al. 2008, A&A, 480, 515 . O Cheremnykh, V Fedun, Y Ladikov-Roev, G Verth, ApJ. 86686Cheremnykh, O., Fedun, V., Ladikov-Roev, Y., & Verth, G. 2018, ApJ, 866, 86 . M C M Cheung, M Schüssler, F Moreno-Insertis, A&A. 4611163Cheung, M. C. M., Schüssler, M., & Moreno-Insertis, F. 2007, A&A, 461, 1163 . A C L Chian, E L Rempel, G Aulanier, ApJ. 78651Chian, A. C. L., Rempel, E. L., Aulanier, G., et al. 2014, ApJ, 786, 51 . P Copil, Y Voitenko, M Goossens, A&A. 478921Copil, P., Voitenko, Y., & Goossens, M. 2008, A&A, 478, 921 . S R Cranmer, A A Van Ballegooijen, ApJS. 156265Cranmer, S. R. & van Ballegooijen, A. A. 2005, ApJS, 156, 265 . B De Pontieu, M Carlsson, Rouppe Van Der, L H M Voort, ApJ. 75212De Pontieu, B., Carlsson, M., Rouppe van der Voort, L. H. M., et al. 2012, ApJ, 752, L12 . B De Pontieu, P C H Martens, H S Hudson, ApJ. 558859De Pontieu, B., Martens, P. C. H., & Hudson, H. S. 2001, ApJ, 558, 859 . F L Deubner, A&A. 44371Deubner, F. L. 1975, A&A, 44, 371 . F L Deubner, B Fleck, A&A. 228506Deubner, F. L. & Fleck, B. 1990, A&A, 228, 506 . F L Deubner, R K Ulrich, E J Rhodes, J , A&A. 72177Deubner, F. L., Ulrich, R. K., & Rhodes, E. J., J. 1979, A&A, 72, 177 . F.-L Deubner, T Waldschik, S Steffens, A&A. 307936Deubner, F.-L., Waldschik, T., & Steffens, S. 1996, A&A, 307, 936 . R Erdélyi, V Fedun, Sol. Phys. 23841Erdélyi, R. & Fedun, V. 2006, Sol. Phys., 238, 41 . R Erdélyi, V Fedun, Science. 3181572Erdélyi, R. & Fedun, V. 2007a, Science, 318, 1572 . R Erdélyi, V Fedun, Sol. Phys. 246101Erdélyi, R. & Fedun, V. 2007b, Sol. Phys., 246, 101 . V Fedun, S Shelyag, R Erdélyi, ApJ. 72717Fedun, V., Shelyag, S., & Erdélyi, R. 2011a, ApJ, 727, 17 . V Fedun, S Shelyag, G Verth, M Mathioudakis, R Erdélyi, Annales Geophysicae. 291029Fedun, V., Shelyag, S., Verth, G., Mathioudakis, M., & Erdélyi, R. 2011b, An- nales Geophysicae, 29, 1029 . V Fedun, G Verth, D B Jess, R Erdélyi, ApJ. 74046Fedun, V., Verth, G., Jess, D. B., & Erdélyi, R. 2011c, ApJ, 740, L46 . B Fleck, F L Deubner, A&A. 224245Fleck, B. & Deubner, F. L. 1989, A&A, 224, 245 . D Fujimura, S Tsuneta, ApJ. 7021443Fujimura, D. & Tsuneta, S. 2009, ApJ, 702, 1443 . M Goossens, R Soler, J Terradas, T Van Doorsselaere, G Verth, ApJ. 7889Goossens, M., Soler, R., Terradas, J., Van Doorsselaere, T., & Verth, G. 2014, ApJ, 788, 9 . B V Gudiksen, M Carlsson, V H Hansteen, A&A. 531154Gudiksen, B. V., Carlsson, M., Hansteen, V. H., et al. 2011, A&A, 531, A154 . J V Hollweg, Sol. Phys. 7025Hollweg, J. V. 1981, Sol. Phys., 70, 25 . H Holweger, L Testerman, Sol. Phys. 43271Holweger, H. & Testerman, L. 1975, Sol. Phys., 43, 271 . S Jafarzadeh, S K Solanki, M Stangalini, ApJS. 22910Jafarzadeh, S., Solanki, S. K., Stangalini, M., et al. 2017, ApJS, 229, 10 . D B Jess, M Mathioudakis, R Erdélyi, Science. 3231582Jess, D. B., Mathioudakis, M., Erdélyi, R., et al. 2009, Science, 323, 1582 . W Kalkofen, ApJ. 486145Kalkofen, W. 1997, ApJ, 486, L145 . I N Kitiashvili, A G Kosovichev, N N Mansour, S K Lele, A A Wray, Phys. Scr. 8618403Kitiashvili, I. N., Kosovichev, A. G., Mansour, N. N., Lele, S. K., & Wray, A. A. 2012a, Phys. Scr, 86, 018403 . I N Kitiashvili, A G Kosovichev, N N Mansour, A A Wray, ApJ. 75121Kitiashvili, I. N., Kosovichev, A. G., Mansour, N. N., & Wray, A. A. 2012b, ApJ, 751, L21 . I Kontogiannis, G Tsiropoula, K Tziotziou, A&A. 56762Kontogiannis, I., Tsiropoula, G., & Tziotziou, K. 2014, A&A, 567, A62 . I Kontogiannis, G Tsiropoula, K Tziotziou, A&A. 585110Kontogiannis, I., Tsiropoula, G., & Tziotziou, K. 2016, A&A, 585, A110 . I Kontogiannis, G Tsiropoula, K Tziotziou, M K Georgoulis, A&A. 52412Kontogiannis, I., Tsiropoula, G., Tziotziou, K., & Georgoulis, M. K. 2010, A&A, 524, A12 . J M Krijger, R J Rutten, B W Lites, A&A. 3791052Krijger, J. M., Rutten, R. J., Lites, B. W., et al. 2001, A&A, 379, 1052 . D Kuridze, V Henriques, M Mathioudakis, ApJ. 80226Kuridze, D., Henriques, V., Mathioudakis, M., et al. 2015, ApJ, 802, 26 . D Kuridze, G Verth, M Mathioudakis, ApJ. 77982Kuridze, D., Verth, G., Mathioudakis, M., et al. 2013, ApJ, 779, 82 . D Kuridze, T V Zaqarashvili, V Henriques, ApJ. 830133Kuridze, D., Zaqarashvili, T. V., Henriques, V., et al. 2016, ApJ, 830, 133 . J Leenaarts, M Carlsson, V Hansteen, &amp; Rouppe Van Der, L Voort, ApJ. 694128Leenaarts, J., Carlsson, M., Hansteen, V., & Rouppe van der Voort, L. 2009, ApJ, 694, L128 . J Leenaarts, M Carlsson, &amp; Rouppe Van Der, L Voort, ApJ. 749136Leenaarts, J., Carlsson, M., & Rouppe van der Voort, L. 2012, ApJ, 749, 136 . R B Leighton, R W Noyes, G W Simon, ApJ. 135474Leighton, R. B., Noyes, R. W., & Simon, G. W. 1962, ApJ, 135, 474 . J R Lemen, A M Title, D J Akin, Sol. Phys. 27517Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17 . B W Lites, E G Chipman, ApJ. 231570Lites, B. W. & Chipman, E. G. 1979, ApJ, 231, 570 . B W Lites, R J Rutten, W Kalkofen, ApJ. 414345Lites, B. W., Rutten, R. J., & Kalkofen, W. 1993, ApJ, 414, 345 . J Liu, M Carlsson, C J Nelson, R Erdélyi, A&A. 63297Liu, J., Carlsson, M., Nelson, C. J., & Erdélyi, R. 2019a, A&A, 632, A97 . J Liu, C J Nelson, B Snow, Y Wang, R Erdélyi, Nature Communications. 103504Liu, J., Nelson, C. J., Snow, B., Wang, Y., & Erdélyi, R. 2019b, Nature Commu- nications, 10, 3504 . W H Matthaeus, G P Zank, S Oughton, D J Mullan, P Dmitruk, ApJ. 52393Matthaeus, W. H., Zank, G. P., Oughton, S., Mullan, D. J., & Dmitruk, P. 1999, ApJ, 523, L93 . R T J Mcateer, P T Gallagher, D R Williams, ApJ. 587806McAteer, R. T. J., Gallagher, P. T., Williams, D. R., et al. 2003, ApJ, 587, 806 . R T J Mcateer, P T Gallagher, D R Williams, ApJ. 567165McAteer, R. T. J., Gallagher, P. T., Williams, D. R., et al. 2002, ApJ, 567, L165 . N Mein, Sol. Phys. 52283Mein, N. 1977, Sol. Phys., 52, 283 . N Mein, P Mein, Sol. Phys. 49231Mein, N. & Mein, P. 1976, Sol. Phys., 49, 231 . N Mein, P Mein, A&A. 8496Mein, N. & Mein, P. 1980, A&A, 84, 96 . A G Michalitsanos, Sol. Phys. 3047Michalitsanos, A. G. 1973, Sol. Phys., 30, 47 . B W Mihalas, J Toomre, ApJ. 249349Mihalas, B. W. & Toomre, J. 1981, ApJ, 249, 349 . R Moll, R H Cameron, M Schüssler, A&A. 533126Moll, R., Cameron, R. H., & Schüssler, M. 2011a, A&A, 533, A126 . R Moll, J Pietarila Graham, J Pratt, ApJ. 73636Moll, R., Pietarila Graham, J., Pratt, J., et al. 2011b, ApJ, 736, 36 . R J Morton, G Verth, V Fedun, S Shelyag, R Erdélyi, ApJ. 76817Morton, R. J., Verth, G., Fedun, V., Shelyag, S., & Erdélyi, R. 2013, ApJ, 768, 17 . S J Mumford, R Erdélyi, MNRAS. 4491679Mumford, S. J. & Erdélyi, R. 2015, MNRAS, 449, 1679 . S J Mumford, V Fedun, R Erdélyi, ApJ. 7996Mumford, S. J., Fedun, V., & Erdélyi, R. 2015, ApJ, 799, 6 . K Murawski, P Kayshap, A K Srivastava, MNRAS. 47477Murawski, K., Kayshap, P., Srivastava, A. K., et al. 2018, MNRAS, 474, 77 . A Nordlund, Sol. Phys. 100209Nordlund, A. 1985, Sol. Phys., 100, 209 . T J Okamoto, B De Pontieu, ApJ. 73624Okamoto, T. J. & De Pontieu, B. 2011, ApJ, 736, L24 . S.-H Park, G Tsiropoula, I Kontogiannis, A&A. 58625Park, S.-H., Tsiropoula, G., Kontogiannis, I., et al. 2016, A&A, 586, A25 . M S Ruderman, R Erdélyi, Space Sci. Rev. 149199Ruderman, M. S. & Erdélyi, R. 2009, Space Sci. Rev., 149, 199 . R J Rutten, A G De Wijn, P Sütterlin, A&A. 416333Rutten, R. J., de Wijn, A. G., & Sütterlin, P. 2004, A&A, 416, 333 . R J Rutten, J M Krijger, A&A. 407735Rutten, R. J. & Krijger, J. M. 2003, A&A, 407, 735 Innovative Telescopes and Instrumentation for Solar Astrophysics. G B Scharmer, K Bjelksjo, T K Korhonen, B Lindberg, B Petterson, Proc. SPIE. S. L. Keil & S. V. AvakyanSPIE4853Scharmer, G. B., Bjelksjo, K., Korhonen, T. K., Lindberg, B., & Petterson, B. 2003, in Proc. SPIE, Vol. 4853, Innovative Telescopes and Instrumentation for Solar Astrophysics, ed. S. L. Keil & S. V. Avakyan, 341-350 . G B Scharmer, G Narayan, T Hillberg, ApJ. 68969Scharmer, G. B., Narayan, G., Hillberg, T., et al. 2008, ApJ, 689, L69 . P H Scherrer, J Schou, R I Bush, Sol. Phys. 275207Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207 . B Schmieder, A&A. 74273Schmieder, B. 1979, A&A, 74, 273 . G Severino, M Oliviero, T Straus, R K Ulrich, Mem. Soc. Astron. Italiana. 74595Severino, G., Oliviero, M., Straus, T., & Ulrich, R. K. 2003, Mem. Soc. As- tron. Italiana, 74, 595 . S Shelyag, P S Cally, A Reid, M Mathioudakis, ApJ. 7764Shelyag, S., Cally, P. S., Reid, A., & Mathioudakis, M. 2013, ApJ, 776, L4 . S Shelyag, P Keys, M Mathioudakis, F P Keenan, A&A. 5265Shelyag, S., Keys, P., Mathioudakis, M., & Keenan, F. P. 2011, A&A, 526, A5 . J Shetye, E Verwichte, M Stangalini, ApJ. 88183Shetye, J., Verwichte, E., Stangalini, M., et al. 2019, ApJ, 881, 83 . R Soler, J Terradas, R Oliver, J L Ballester, ApJ. 84020Soler, R., Terradas, J., Oliver, R., & Ballester, J. L. 2017, ApJ, 840, 20 . H C Spruit, A&A. 98155Spruit, H. C. 1981, A&A, 98, 155 . M Stangalini, F Giannattasio, S Jafarzadeh, A&A. 57717Stangalini, M., Giannattasio, F., & Jafarzadeh, S. 2015, A&A, 577, A17 . R F Stein, Å Nordlund, ApJ. 499914Stein, R. F. & Nordlund, Å. 1998, ApJ, 499, 914 . R F Stein, Å Nordlund, Sol. Phys. 19291Stein, R. F. & Nordlund, Å. 2000a, Sol. Phys., 192, 91 . R F Stein, Å Nordlund, Annals of the New York Academy of Sciences. Turbulence and Convection, ed. J. R. Buchler & H. Kandrup89821Stein, R. F. & Nordlund, Å. 2000b, in Annals of the New York Academy of Sci- ences, Vol. 898, Astrophysical Turbulence and Convection, ed. J. R. Buchler & H. Kandrup, 21 . J O Stenflo, Sol. Phys. 4279Stenflo, J. O. 1975, Sol. Phys., 42, 79 Y Suematsu, Progress of Seismology of the Sun and Stars. Y. Osaki & H. ShibahashiBerlin Springer Verlag367211Lecture Notes in PhysicsSuematsu, Y. 1990, in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 367, Progress of Seismology of the Sun and Stars, ed. Y. Osaki & H. Shibahashi, 211 . C Torrence, G P Compo, American Meteorological Society7961Torrence, C. & Compo, G. P. 1998, Bulletin of the American Meteorological Society, 79, 61 . K Tziotziou, G Tsiropoula, I Kontogiannis, A&A. 623160Tziotziou, K., Tsiropoula, G., & Kontogiannis, I. 2019, A&A, 623, A160 . K Tziotziou, G Tsiropoula, I Kontogiannis, E Scullion, Doyle, J. G. 61851A&ATziotziou, K., Tsiropoula, G., Kontogiannis, I., Scullion, E., & Doyle, J. G. 2018, A&A, 618, A51 . K Tziotziou, G Tsiropoula, P Sütterlin, A&A. 444265Tziotziou, K., Tsiropoula, G., & Sütterlin, P. 2005, A&A, 444, 265 . R K Ulrich, ApJ. 162993Ulrich, R. K. 1970, ApJ, 162, 993 . A A Van Ballegooijen, M Asgari-Targhi, S R Cranmer, E E Deluca, ApJ. 7363van Ballegooijen, A. A., Asgari-Targhi, M., Cranmer, S. R., & DeLuca, E. E. 2011, ApJ, 736, 3 . T Van Doorsselaere, V M Nakariakov, E Verwichte, ApJ. 67673Van Doorsselaere, T., Nakariakov, V. M., & Verwichte, E. 2008, ApJ, 676, L73 . S Vargas Domínguez, J Palacios, L Balmaceda, I Cabello, V Domingo, MNRAS. 416148Vargas Domínguez, S., Palacios, J., Balmaceda, L., Cabello, I., & Domingo, V. 2011, MNRAS, 416, 148 . G Verth, D B Jess, Washington DC American Geophysical Union Geophysical Monograph Series. 216431Verth, G. & Jess, D. B. 2016, Washington DC American Geophysical Union Geophysical Monograph Series, 216, 431 . G Vigeesh, V Fedun, S S Hasan, R Erdélyi, ApJ. 75518Vigeesh, G., Fedun, V., Hasan, S. S., & Erdélyi, R. 2012, ApJ, 755, 18 . S Wedemeyer, O Steiner, PASJ. 6610Wedemeyer, S. & Steiner, O. 2014, PASJ, 66, S10 . S Wedemeyer-Böhm, Rouppe Van Der, L Voort, A&A. 5079Wedemeyer-Böhm, S. & Rouppe van der Voort, L. 2009, A&A, 507, L9 . S Wedemeyer-Böhm, E Scullion, O Steiner, Nature. 486505Wedemeyer-Böhm, S., Scullion, E., Steiner, O., et al. 2012, Nature, 486, 505 . C A Whitney, Smithsonian Contributions to Astrophysics. 2365Whitney, C. A. 1958, Smithsonian Contributions to Astrophysics, 2, 365 . G Worrall, MNRAS. 335628Worrall, G. 2002, MNRAS, 335, 628 . T V Zaqarashvili, A&A. 39915Zaqarashvili, T. V. 2003, A&A, 399, L15 . T V Zaqarashvili, K Murawski, A&A. 470353Zaqarashvili, T. V. & Murawski, K. 2007, A&A, 470, 353
[]
[ "Incomplete Knowledge Graph Alignment", "Incomplete Knowledge Graph Alignment" ]
[ "Vinh Van Tong \nHanoi University of Science and Technology\nVietnam\n", "Thanh Trung Huynh \nGriffith University\nAustralia\n", "Thanh Tam Nguyen \nGriffith University\nAustralia\n", "Hongzhi Yin \nThe University of Queensland\nAustralia\n", "Quoc Viet ", "Hung Nguyen \nGriffith University\nAustralia\n", "Quyet Thang Huynh \nHanoi University of Science and Technology\nVietnam\n" ]
[ "Hanoi University of Science and Technology\nVietnam", "Griffith University\nAustralia", "Griffith University\nAustralia", "The University of Queensland\nAustralia", "Griffith University\nAustralia", "Hanoi University of Science and Technology\nVietnam" ]
[]
Knowledge graph (KG) alignment -the task of recognizing entities referring to the same thing in different KGsis recognized as one of the most important operations in the field of KG construction and completion. However, existing alignment techniques often assume that the input KGs are complete and isomorphic, which is not true due to the real-world heterogeneity in domain, size, and sparsity. In this work, we address the problem of aligning incomplete KGs with representation learning. Our KG embedding framework exploits two feature channels: transitivity-based and proximity-based. The former captures the consistency constraints between entities via translation paths, while the latter captures the neighborhood structure of KGs via an attention guided relation-aware graph neural network. The two feature channels are jointly learned to exchange important features between the input KGs while enforcing the output representations of the input KGs in the same embedding space. Also, we develop a missing links detector that discovers and recovers the missing links in the input KGs during the training process, which helps mitigate the incompleteness issue and thus improve the compatibility of the learned representations. The embeddings then are fused to generate the alignment result, and the high-confidence matched node pairs are updated to the prealigned supervision data to improve the embeddings gradually. Empirical results show that our model is more accurate than the SOTA and is robust against different levels of incompleteness.
null
[ "https://arxiv.org/pdf/2112.09266v2.pdf" ]
247,451,269
2112.09266
88afa8d609745ec47f8882120bf4589192035ad7
Incomplete Knowledge Graph Alignment Vinh Van Tong Hanoi University of Science and Technology Vietnam Thanh Trung Huynh Griffith University Australia Thanh Tam Nguyen Griffith University Australia Hongzhi Yin The University of Queensland Australia Quoc Viet Hung Nguyen Griffith University Australia Quyet Thang Huynh Hanoi University of Science and Technology Vietnam Incomplete Knowledge Graph Alignment Index Terms-knowledge graph alignmentmulti-channel graph neural networksmulti-domain learning Knowledge graph (KG) alignment -the task of recognizing entities referring to the same thing in different KGsis recognized as one of the most important operations in the field of KG construction and completion. However, existing alignment techniques often assume that the input KGs are complete and isomorphic, which is not true due to the real-world heterogeneity in domain, size, and sparsity. In this work, we address the problem of aligning incomplete KGs with representation learning. Our KG embedding framework exploits two feature channels: transitivity-based and proximity-based. The former captures the consistency constraints between entities via translation paths, while the latter captures the neighborhood structure of KGs via an attention guided relation-aware graph neural network. The two feature channels are jointly learned to exchange important features between the input KGs while enforcing the output representations of the input KGs in the same embedding space. Also, we develop a missing links detector that discovers and recovers the missing links in the input KGs during the training process, which helps mitigate the incompleteness issue and thus improve the compatibility of the learned representations. The embeddings then are fused to generate the alignment result, and the high-confidence matched node pairs are updated to the prealigned supervision data to improve the embeddings gradually. Empirical results show that our model is more accurate than the SOTA and is robust against different levels of incompleteness. I. INTRODUCTION Knowledge graphs (KGs) represent the facts about real-world entities in the form of triples head entity, relation, tail entity [1], [2]. Popular knowledge graphs (e.g., DBpedia, YAGO, and BabelNet) are often multilingual, in which each language domain has a separate version [3]. To encourage the knowledge fusion between different domains, knowledge graph alignment -the task of identifying entities in the cross-lingual KGs that refers to the same real-world object -has received great interest from both industry and acadamia [4], [5]. The alignment result can be used for further data enrichment applications such as repairing inconsistencies, filling knowledge gaps, and building cross-lingual KBs [6], [7], [8]. Knowledge graph alignment faces challenges related to efficiency, scalability and richness of incorporated rich information of real-world KGs. Recent approaches solve these challenges by employing graph neural networks (GNNs) [9], [10] to encode the attributional and the relational triples via some message-passing schemes, e.g., GCN-Align [11], RDGCN [12], MUGNN [13], KG-matching [14]. However, existing techniques often assume that the input KGs are nearly identical (isomorphic), which is not true [15], [16], [17]. There is often a considerable gap between the levels of completeness of different monolingual KGs [2], especially between the English domain and other languages [18]. For example, in the DBP15K dataset, the ratio of relational triples between Chinese, Japanese, or French KGs over the English KG is only around 80% [19]. Figure 1 gives an example of incomplete KGs, in which the neighborhood of the two entities referring to the actor Ronald Colman in English and French KGs are inconsistent (his occupation is missing in the English KG while his place of birth is missing in the French KG). Such inconsistencies easily lead to the different representations of the corresponding entities, especially for GNNs where the noise is accumulated over neural layers [20]. In this paper, we address the above challenges by proposing a representation learning framework with multi-channel feature exchange for aligning incomplete knowledge graphs from different domains. To capture the multi-domain nature of KG entities, we develop a graph attention convolutional network that can combine the translated entity name (as the entity's feature) and the relational structure simultaneously. The attention mechanism allows relational importance integration, which helps mitigate the noise by focusing on mutual relations in the input KGs and ignoring the missing ones. Our proposed attention mechanism goes beyond the existing techniques [12], [17] by leveraging relation-aware attentive scoring, which helps the framework to integrate the KG edge information. To guarantee consistency across KGs, we develop an additional embedding component that encodes both entities and relations with the tradition translation constraint [21]. While many recent works neglect this 'seems-to-bestrict' constraint [14], it turns out to strengthen the local information of relational triples and mitigate the information dilution phenomenon in graph convolutional networks [20]. We also develop a missing links detector that consumes the two feature channels to exchange the knowledge from the two input KGs to discover and recover the missing triples. Finally, we combine these dimensions using late-fusion to instantiate the alignment result. We summarise our contributions as follows: • We address the problem of aligning incomplete KGs from different domains and propose a framework called Incomplete Knowledge graphs Aligner via MultI-channel Feature Exchange (IKAMI). The model exploits multiple representations to capture the multi-channel nature of KGs (e.g., relational type, entity name, structural information). This is the first attempt to address the entity alignment and knowledge completion simultaneously, and we argue that this collaboration benefits both tasks, especially the alignment performance. • We employ a translation-based embedding that encodes both entities and relations inspired from the translation model [21]. The translation assumption helps to align both entities and relations and is a great supplement to mitigate the information dilution weakness of the GCNbased embedding [20]. • We propose a graph convolution attention network that efficiently captures relational triples, including the entity name, the relation type, and direction. The attention mechanism allows us to learn the underlying importance of relational triples based on their type, thus focusing on more popular relations and ignoring the ones that appear in only one network. • We propose a missing triples detector that leverages the learned translation-based features to jointly discover and complete the missing links in the two input KGs. First, the potential missing links between the entities are proposed by selecting the ones with high-correlated embeddings, then the correct relation between them is chosen by selecting the appropriate relation embedding. • We design a joint train schedule of the two embedding models to enable the holistic objectives of the embeddings can support each other well. Then, the similarity matrix for each channel is calculated and fused by weighted-sum to return the final result. • We conduct experiments on real-world and synthetic KG datasets to evaluate our scheme. The results show that our framework outperforms other baselines in not only the entity alignment task but also the knowledge completion task by up to 15.2% and 3.5%, respectively. We publish the source code for use by the community. The remainder of the paper is organized as follows. Section II reviews the related works and motivates our approach. Section III presents a problem statement for the joint alignment and completion of incomplete knowledge graphs as well as our approach overview. Section IV describes the architecture of the two feature channels, including a translation-based embedding and a GCN embedding model that captures the relational correlation between the entities of KGs. Section V explains how the current extracted feature channels are iteratively used to instantiate the alignment result and recover the missing triples from the input KGs, which helps to refine the alignment results gradually. Empirical evaluation is given in Section VI, before Section VII presents a summary and conclusions. II. RELATED WORK Knowledge Graph Alignment. Traditional cross-lingual KG alignment approaches often rely on various domainindependent features to discover cross-lingual links [22]. Modern cross-lingual KGs entity aligners exploit the emergence of graph embedding techniques and show promising results [23], [24]. From the two input monolingual KGs, these techniques first embed the entities into low-dimensional vector spaces. The identical entities are then retrieved based on the similarity of the learned representations. The earlier techniques of this paradigm, including MTransE [21], JAPE [25], ITransE [26], and BootEA [27], often employ shallow embedding techniques to embed the entities and the relations using the translation constraint [28], which assumes that the embedding distance between two any entities is equal to the connecting relation. Due to the lack of information integration of shallow models, recent embedding based alignment techniques employ graph convolutional network (GCN) [9] to exploit the complex nature of KGs better. GCN-Align [11] employs GCNs to capture at the same time the relational and the individual entity footprint. RDGCN [12] introduces a two-layers GCN with highway gates to extract attentive interactions between each KG and its dual relation. MUGNN [13] preprocesses the input KGs with grounding rules before forwarding them to a dual-channel GCNs model. KDcoE [29] leverages co-training of a KG embedding and a literal description embedding to enhance the semi-supervised learning of multilingual KG embeddings. MultiKE [30] unifies multiple views of entities to learn embeddings for entity alignment. KG-matching [14] constructs a relation network namely topic entity graph to encode the entities' name by a word-based LSTM. REA [31] proposes a two-component model to enhance the noise robustness of the aligner. AliNet [17] employs entities' distant neighbours to expand their neighbour overlapping [32]. Our work goes beyond the state-of-the-art by enhancing the GNN properties with attention mechanism and entity-relation composition to capture the relational correlation effectively. We also combine the GNN with a translation-based channel to guarantee the knowledge transfer between the input KGs and thus mitigate the incompleteness issue. Similar to our work are GCN-Align and MultiKE, which also use a multi-view scheme, but our GNN channel leverages relation embeddings that better capture relational information. Also, unlike the existing models such as RDGCN and GCN-Align that stacks GNN models, our framework utilizes a non-GNN channel to cover the GNN-based channel and overcome the GNN weaknesses, such as information dilution and noise amplification issue. Knowledge Graph Completion. The Knowledge Graph Completion (KGC) techniques aim to predict the missing triples in the KGs automatically. Existing KGC techniques often embed entities and relations into low-dimensional vectors, then design a score function to check the candidate triplets' plausibility. The earlier methods often employ shallow embedding models with translation-based scoring function [28], [33]. Recently, KGC techniques using deep embedding model and complex score have been proposed, including CNNs based model [34], [35], RNNs based models [36], and GCNs based model [37]. However, they are surprisingly found to be unstable across different datasets due to the overloading of unrelated objectives [38]. Thus, we choose the simple yet powerful translationbased model to supply the main GNN-based channel in our framework. Our work also goes beyond the existing KGC framework by answering at the same time two questions: (i) given any two entities, whether there is a triple connecting them and (ii) which is the relation type of the triple between the two entities; while the existing frameworks often answer only the latter question. It is worth noting that the former question is challenging, as the number of disjoint entities outweighs that of connecting ones and thus might cause low recall in detecting the missing triples. We tackle this challenge by exchanging the useful patterns of high-confidence entities retrieved from the alignment of the two input KGs, thus making the information of the KGs complete for each other. III. INCOMPLETE KNOWLEDGE GRAPH ALIGNMENT In this section, we formulate the problem and discuss the challenges for incomplete knowledge graph alignment as well as our approach overview. A. Problem Formulation Incomplete knowledge graphs (i-KGs). The KG is often denoted as KG = (V, R, E), where V is the set of entities; R is the set of relations and E is the set of triples. The triple h, r, t ∈ E is atomic unit in KG, which depicts a relation r between a head (an entity) h and a tail t (an attribute or another entity). We present the incomplete knowledge graphs by extending the KG notation as i-KG = (V, R, E,Ē), wherē E is the set of missing triples in the i-KG. For brevity's sake, we use i-KG and KG interchangeably in this paper. Incomplete knowledge graph alignment. By generalising the problem setting in related works, i-KG alignment aims to find all of the corresponding entities of two given i-KGs. Without loss of generality, we select one i-KG as the source graph and the other as the target graph, and denote them as KG s = (V s , R s , E s ,Ē s ) and KG t = (V t , R t , E t ,Ē t ) respectively. Note that E s Ē s = E t Ē t , which represents the complete triple facts. Then, for each entity p in the source graph KG s , we aim to recognise its counterpart p (if any) in the target knowledge graph KG t . The corresponding entities (p, p ) also often denoted as anchor links; and existing alignment frameworks often require supervision data in the form of a set of pre-aligned anchor links, denoted by L. Since the corresponding entities reflect the same real-world entity (e.g. a person or concept), the existing alignment techniques often rely on the consistencies, which states that the corresponding entities should maintain similar characteristics across different KGs [11]. The entity consistency states that the entities referring to the same objects should exist in both the KGs and have equivalent name. The relation consistency (a.k.a. the homophily rule) declares that the entities should maintain their relationship characteristics (existence, type, direction). Knowledge graph completion. Given the incomplete knowledge graph i-KG = (V, R, E,Ē), whereĒ is unrevealed, the knowledge graph completion (KGC) task aims to discover all the missing triples h, r, t ∈Ē| h, r, t / ∈ E. While KG alignment and completion have been studied for decades [17], [38], there is little work on jointly solving these problems together. However, doing so is indeed beneficial: missing triples h, r, t ∈Ē in one KG can be recovered by cross-checking another KG via the alignment, which, in turn, can be boosted by recovered links. To the best of our knowledge, this work is a first attempt to solve the joint optimization of KG alignment and completion, which is formally defined as follows. Problem 1 (Joint KG alignment and completion) Given two incomplete knowledge graphs KG s and KG t , the task of joint KG alignment and completion is to: (i) identify all the hidden anchor links between KG s and KG t , and (ii) recover the missing triples in each input KG s and KG t . Table I summarizes the important notations used in this paper. B. Challenges Solving Problem 1 is non-trivial. We argue that an efficient incomplete KG entity alignment framework should overcome the following challenges: C1. Domain gap: As the input KGs are incomplete, there exist inconsistencies between the cross-lingual KGs (i.e. incompatible individual information, inequivalent neighbor set, different number of nodes in each graph). Existing works attempt to tackle this challenge by applying rulebase KG completion as preprocessing step [13], [31], but this fails to leverage the correlation between the two input KGs and requires the addition of pre-aligned relation types information. The margin of translation loss in transitivity-based channel γg The margin of entity alignment loss in proximity-based channel βt Balancing weight for loss terms in transitivity-based channel βg Balancing weight for loss terms in proximity-based channel βtg Balancing weight for alignment matrices of the two channels C2. Task gap: While the incompleteness might cause the inconsistencies, the consistencies (entity consistency, relational consistency) should be respected overall since these constraints guide to finding precisely matched entities w.r.t the KGs specific characteristics (e.g., name equivalence, directional relations). Therefore, handling the two "seem-to-contradict" tasks, KG completion and KG alignment, simultaneously is challenging. C3. Model gap: The neighbourhood structures of KGs provide rich information in various forms (e.g. relational triples, relational directions). Recent works exploit this characteristic by stacking GCNs in their model [11], [12], but the structure of the GCNs are often similar and thus suffer from the same weakness. C. Outline of the Alignment Process To address the above challenges, we argue that an alignment of incomplete knowledge graphs shall happen iteratively. Figure 2 shows an overview of our alignment process. In each step, the input KGs are enriched (missing triples are recovered), improving their alignment simultaneously. Such incremental process allows the hard cases to be more likely solved as the model experiences easier cases and becomes more mature over time. Starting with two incomplete KGs KG s and KG t , the alignment process continuously updates three objects: • Alignment matrix S ∈ R |Vs|×|Vt| that represents the result of alignment between the source and target graphs, where each |V s | and |V t | denotes the numbers of entities in KG s and KG t , respectively. Each component S(p, p ) in the alignment matrix S identifies the alignment level between an entity p ∈ V s and its counterpart entity p ∈ V t . • Alignment seed L that is a set of known aligned entities. • Missing triplesĒ of the KGs themselves. Each iteration of the alignment process comprises the following steps: (1) Representation learning: We first forward the source and target KG networks through two designed feature channels, namely transitivity-based channel and proximity-based channel, to embed the KG entities in different low-dimensional vector spaces. These two channels are parallel: (1.1) Proximity-based channel: is a GCN-based model that unifies the entity name information and topological structure under the same modal. To this end, we overcome the language barrier (C1) by employing the word embedding of the translated entity name. The channel also produce the relation embedding as well as utilize an attention mechanism to exploit the relational information, which helps to detect the proximity of the corresponding entities in many aspects and thus helps to narrow down the language gap (C1) and mitigate the possible noises (C2). The details of this step can be found in Section IV-B. (1.2) Transitivity-based channel is a shallow-based embedding model that enforces the translation constraint between the embeddings of entities and relations in each triple. This constraint helps to emphasize on the topological firstorder proximity information, which thus helps to cover the information dilution issue of GNN based model (C3). Also, the translation constraint encourages the analogy between the source and target KGs embedding space (as the relation type between corresponding entities reflect the same phenomenon), which reduces the noise (C2) and facilitates the feature exchange using the learnt embedding spaces (C1). We discuss about this channel in Section IV-C. (2) Alignment computation: The learned representations are then fused to compute the final alignment result. We first train the Transitivity-based channel to get the representation of entities and relations. Then we train the Proximitybased channel using the input graphs structure as well as entity name embedding. The relation representations of the Transitivity-based channel are also used as input to this channel to allow the relation-aware mechanism of the GNN to work properly. The output of this channel are relation and entity embeddings. The embedding of entities of the two channels then are concatenated and applied a linear transformation to get the final embedding. We then compute the cosine similarity between each pair of entities across two KGs to get the alignment matrix. The highconfidence matched entities then is sampled and added to the alignment seed L, which is used to enhance the quality of the two representation channels in the next iterations. The detailed implementation of this step can be found in Section V-A. (3) Missing triples recovery: We develop a two-step module that leverages the learnt representations are used to recover all possible missing triples in the input KGs. At the first step, a 2-layer perceptron followed by a sigmoid function is used to compute the probability of whether there exists a missing relation between the two entities. If the answer is yes, indicated by the probability being above a pre-defined threshold, we determine which type of relation connecting the entities in the second step. In more details, given the two entities p, q and their transitivity-based embeddings, we choose the relation r whose embedding follows the translation constraint p + r ≈ q. Note that we do not use directly the embeddings of the proximity-based channel due to the incapability of GNN-based model in capturing positional information [20]; but this important channel plays an important role in finding hidden anchor links and update the transitivity-based channel. Further details of this process is presented in V-B The alignment process is stopped when the validation accuracy can not further increase after several consecutive iterations. The final alignment matrix is then used to retrieve the matched entities. IV. FEATURE CHANNEL MODELS A. Pre-processing To efficiently capture the KG relation direction, we preprocess the input KGs by adding inverse triples and self-loop triples as follows: (1) and E = E ∪ { h, r −1 , t | h, r, t ∈ E} ∪ { p, , p |p ∈ V}R = R ∪ R inv ∪ { }(2) where R inv = {r −1 |r ∈ R} represents the inverse relations and denotes the self loop. The inverse edges helps the information can freely propagate in both direction in the learning step, while the adding of the self loop relation enables our GNN to passing message from one node to it-self. B. Transitivity-based channel The main role of this channel is to embed both entities and relations of the two KGs to a same vector space so that those presentations can preserve the structure of the two graphs. To do that, we make use of a well-known translation constraint [28] to all of the triples of the two graphs. We also apply a mapping loss on entities in the alignment seed to make sure the embeddings of the two graphs are in the same vector space. We also find out that by doing this, not only entities but also relations of the two graphs are aligned. Formally, we employ a shallow translation-based embedding as an additional channel to complete the "deep" GNN-based embedding. To this end, for each entity p and relation r, we assign a trainable representation vector h (.) . The backbone of the model is the translation constraint [28], which enforces that for any relational triples h, r, t ∈ E, the following constraint should hold: h h + h r = h t(3) where h h , h r and h t are the embedding of the head, relation and tail entity of the triple, respectively. To integrate this constraint into the model, we employ the dissimilarity measure d t and guarantee that d t (h+r, t) ≈ 0 if h, r, t ∈ E, otherwise d t (h + r, t) >> 0. In our work, we choose d t as Manhattan distance, similar to [21]. Translation loss: Given the translation dissimilarity function d and the supervised triples set E, we optimize the translation loss as follows: L ts = (h,r,t)∈E + (h ,r,t )∈E [γ t + d t (h + r, t) − d t (h + r, t )] + (4) where the [x] + denotes the positive part of x, γ t is a margin hyper-parameter, E is the negative triple set constructed by corrupting either the head or tail in the original triples. Mapping loss: To jointly embed both graphs into the same embedding space, we will minimize the distance between entity pairs in the alignment seeds L: L tm = (p,p )∈L d t (p, p )(5) The final loss function thus has the form: L t = L ts + β tm L tm(6) where β tm ∈ R is a hyper-parameter that scales the important of L tm . Note that the loss function L t (Eq. 6) allows the model to align relations without knowing any pre-aligned relation seeds. This is because the transitivity constraint can guarantee that if the corresponding entities in the input KGs are embedded into a space with the same representations, the same goes with the relation embeddings. The following theorem support this argument: Definition 1 ( -closed entity pair): Given a real number ≥ 0 and two entities p ∈ V 1 and q ∈ V 2 ; an entity pair (p, q) across KGs is called an -closed entity pair if d t (p, q) ≤ . Definition 2 ( -closed relation pair): Given a real number ≥ 0 and two relations r ∈ E 1 and r ∈ E 2 ; a relation pair (r, r ) across KGs is called an -closed relation pair if d t (r, r ) ≤ . Theorem 1 If (p, q) and (p , q ) are two -closed entity pairs, which means d t (p, q) ≤ and d t (p , q ) ≤ , then for any r pq connecting p to q and r p q connecting p to q such that the translation constraint in Eq. 3 is satisfied, then (r pq , r p q ) is a 2 -closed relation pair. Proof: Suppose h p , h q , h p , h q , h rpq , and h r p q are embeddings of p, q, p , q , r pq , and r p q respectively. We have the distance between r pq and r p q is: d t (r pq , r p q ) = ||h rpq − h r p q || L1 As Eq. 3 is satisfied, we can replace h rpq by (h q − h p ) and h r p q by (h q − h p ) to achieve: Thus, we can conclude that (r pq , r p q ) is a 2 -closed relation pair. This valuable characteristic would also be used in the next proximity-based channel to make sure our relationaware mechanism work. d t (r pq , r p q ) = ||(h q − h p ) − (h q − h p )|| L1 = ||(h q − h p ) + (h p − h q )|| L1 ≤ ||h q − h p || L1 + ||h p − h q || L1 = d t (p, q) + d t (p , q ) ≤ + = 2 C. Proximity-based channel In this channel, we take both entity name information and the local neighborhood structure around each entities into considerations. To better alleviate the relation information, we design a specific GNN architecture that allows our model to be relation aware. This GNN has two main innovations namely relation-aware message and relation-aware attention. We apply the mapping loss function on entities alignment seeds. We also introduce a new loss component to map the relation embeddings to the same vector space using the relation representations in the Transitivity-based channel. Formally, our proximity-based channel is designed to unify KGs heterogeneous information under the same modal using deep neural network model consisting of K layers. For the k-th layer, we update the representation h k+1 p ∈ R d of each entity p ∈ V by: h k+1 p = f e   (q,r)∈N (p) α k pqr m k qr  (7) where N (p) = {(q, r)|( p, r, q ∈ E) ∨ ( q, r, p ∈ E)} is the neighbor set of entity p, m k qr ∈ R d denotes the message passing from neighbor entity p to entity q through relation r, α k pqr represents the attention weight that emphasize the importance of the relational message m k qr to p; and f e is a linear transformation followed by a T anh(.) activation function. The innovations of our GNN-based model are twofold: (i) we guarantee the message-passing is relation-aware by learning the relation embedding h k r for each relation r ∈ R and integrating relation semantic into the entity message propagation m k qr and (ii) we design the attention weight α k pqr that further enhance the relation-aware capability of our GNNbased embedding. Relation-aware message: Unlike existing GNN-based techniques that often infer relation embeddings from learnt entity embeddings [17], our technique allows entity and relation embeddings to be learnt independently and jointly contribute to the message passing process by applying the entity-relation composition operations: c k qr = f c h k q , h k r(8) where f c : R d × R d → R d is a composition operator, and c k qr ∈ R d is the composition output vector. We choose the composition operator as substraction function [28] given its simplicity, non-parameter and efficiency. We then compute the message from q to p through r by projecting the composition output based on its direction: m k qr = W λ(r) c k qr (9) W λ(r) =    W I , if r ∈ R inv W S , if r = (self-loop) W O , otherwise Then, along with the entity embdding update in Eq. 7, the relation embedding is updated by: h k+1 r = W k r h k r(10) where W k r ∈ R d×d is a trainable transformation matrix that projects all the relations to the same embedding space and allows them to be utilized in the next layer. Relation-aware attention: Current works often implement their attention mechanisms following GAT [10]. However, GAT layers do not include edge-feature information and have been shown to only compute static attention coefficients. To address this problem, we design our attention score inspiring from GATv2 [39] and make it to be relation-aware by using the pre-defined composition output as follow: α k pqr = exp θ k h k p , c k qr (q * ,r * )∈N (p) exp θ k h k p , c k q * r * (11) θ k (x, y) = a T LeakyReLU W k att [x||y](12) where W k att ∈ R d×2d is a layer-wise attention weight matrix, a ∈ R d is an attention weight vector, and || denotes concatenation. As the composition output vector c k qr contains the information of not only neighbor entity q but also neighbor relation r, our attentive score α k pqr can capture the importance of the message coming from node q to node p conditioned on relation r connecting them. Final embdding: To achieve final embedding h p for entity p and h r for relation r, we concatenate their embeddings at all layers and then use a linear transformation to project them to their final embedding space: h p = W ge [h 0 p ||h 1 p ||...||h K p ](13)h r = W gr [h 0 r ||h 1 r ||...||h K r ](14) where W ge ∈ R d×(K+1)d and W gr ∈ R d×(K+1)d is two linear transformation matrices. Loss function: We use the cosine distance metric to measure the distance between entities across knowledge graphs: d c (p, p ) = 1 − cos (h p , h p )(15) Our objective is to minimize the distance between aligned entity pairs while maximizing the distance between negative entity pairs, using a margin-based scoring function: L gm = (p,p )∈L (p,p )∈L [γ g + d c (p, p ) − d c (p,p )] + (16) where γ g > 0 is a margin hyper-parameter andL is the set of negative entity pairs, which is constructed by replacing one end of each positively aligned pair by its close neighbours according to Eq. 15 [12]. Note that our relation-aware mechanisms can only work if we can reconcile the relation embeddings as pointed out in [17]. Thanks to the aligned relation embeddings at the transitivity-based channel we now can do this by adding the following loss term to transfer the relation alignment results from the transitivity-based channel to the proximity-based channel: L gr = r∈R r ∈R |d ct (r, r ) − d cg (r, r )|(17) where d ct (r, r ) and d cg (r, r ) are the cosine distances between r and r w.r.t translation-based channel and proximity-based channel relation embeddings respectively. We finally combine Eq. 16 and Eq. 17 to get the final loss function of the proximity-based channel: L g = L gm + β g L gr(18) where β g ∈ R is a hyper-parameter weighting the importance of L gr . V. THE COMPLETE ALIGNMENT PROCESS A. Alignment instantiation We use the cosine similarity matrix to compute the similarity between any two entities across KGs: Sim(p, p ) = cos(h p , h p )(19) We then use this function to compute the alignment matrix S for each channel where S(p, q) = Sim(p, q) is proportional to the probability that p is aligned to q. Suppose S t and S g is the alignment matrix of the translation-based channel and the proximity-based channel respectively. We then combine them to form the final alignment matrix as follow: S = β tg S t + (1 − β tg ) S g(20) here β tg is a balancing hyper-parameter. We then use the greedy match algorithm [40] to infer the matching entities from the alignment matrix S. For a fair comparison, we also apply this algorithm to all the baselines. B. Missing triples recovery In a typical KGC problem, we are often given a set of incomplete triples in which two out of three elements are available ( h, r, ? or h, ?, t , ?, r, t ). The task is to predict the missing elements in these triples, which means we know beforehand the position of missing triples. The model thus only needs to predict the missing relation with the searching space size equal to the number of relation |R|. However, our model aims to tackle a more challenging problem in which we do not know that information beforehand. As a result, the model has to find all the missing elements of all possible missing triples with the much larger searching space size of |V| 2 × |R|. To overcome this challenge, we add a two-step module into our model. Firstly, a neural network is built to find all pairs of entities that might have missing relations. We use a 2-layer MLP followed by a sigmoid function to return the probability of how likely some relations between any two entities were missed. For each entity pair p, q , we compute the mentioned probability p e (p, q) for it as follow: p e (p, q) = Sigmoid (M LP (h p ||h q ))(21) After having p e (p, q), we compare this value with the average ν of all the probabilities of pairs which already have relations connecting them. If p e (p, q) ≥ ν, this entity pair will be passed through the second step to predict the missing relations between them. In the second step, since we already have pairs of entities predicted to have missed relations, we only need to apply the tradition KGC methods as mentioned before. Suppose p, q have already predicted to have missing relations, then for each relation r, we compute the dissimilarity value d t (p + r, q). This value will be used to decide whether p, r, q should be filled to the graph. If the dissimilarity value is not larger than the average of this measure of all the positive triples η, which means d(p + r, q) ≤ η, then p, r, q will be allowed to be filled to the graph. Note that we do not use the embeddings of the proximitybased channel in this module because this module employs GNN with fixed initialized word embeddings for entities, which makes this module fail to capture positional information. As a consequence, this module would not perform well when tackling the knowledge completion task. Whereas on the other hand, the transitivity based channel is a shallow architecture with trainable entities representations with higher degree of freedom which allows the module to better capture positional information. Thus, we only make use of the first channel representation for filling missing triples to the graphs. As a result, We update the loss function of the Transitivitybased channel Eq. 4 as follow: L t = L ts + β tc L tc + β tm L tm(22) where β tm , β tc are two real hyper-parameters weighting the importance of L tm and L tc , respectively; L tc is a loss function allowing the first Missing triples recovery module to be trained to produce reasonable p e (.): L tc = (p,q)∈P   −log (p e (p, q)) + q ∝Pn(p) log (p e (p, q ))   (23) where P is the set of entity pairs that already have relations connecting them, and P n (p) is a set of negative examples (i.e., set of p's non-neighbor entities). In our implementation, we first optimize (L ts + β tc L tc ) and then optimize β tm L tm latter. The proximity-based channel also indirectly contributes to the triples recovering process. Because it utilizes name and neighborhood structure information to enhance the module's matching quality. In turn, this matching information can be used to better recover missing relations between entities. Suppose the model successfully recognized p and p , q and q , and r and r are aligned together, p, r, q already in the source KG. Then even if p , r , q is a missing triple in the second graph, our model can easily recovers it. False links correction. During the missing links recovery process, the model might add wrong relations to the KGs when recovering missing triples. Consequently, this action could exert undesirable impacts on the model's performance. Thus, we propose a mechanism allowing our model to correct itself from wrong decisions, preventing it from accumulating noises to the KGs. The intuition is to keep checking the dissimilarity value d t (.) and missing edge probability p e (.) of the filled triples. If an added triple h, r, t fails to satisfy the recovery condition (i.e., d(h + r, t) > η or p(h, t) < ν) at any iteration, it will be removed from the filled triple set. Although this mechanism does not ensure all the wrong decisions would be detected and fixed, it allows the model to be more adaptive. Scaling to large input. To allow our proposed algorithm to scale well to large KGs, we introduce a relaxed version of Missing triple recovery. Firstly, at each iteration, we uniformly sample a small subset V of entities to perform the triple recovery step. Because the complexity of this step is quadratically proportional to the number of entities, this subsampling strategy will significantly reduce the running time of the model. Another benefit is that this mechanism will prevent the model from adding too many triples, most of which can be false as they are inferred from the poor, learned representations of KGs at the early stage of the training process. Consequently, this action may potentially accumulate too much noise to the graphs. Secondly, added triples could be huge as the running process iterates. As a consequence, the false links correction step could cost a significant amount of time because the model has to check every single triple repeatedly. Thus, we allow the model to ignore triples that already pass more than e consecutive checking steps, letting them stay permanently in the triple set. C. Link-augmented training process The training process for the whole model is depicted in Algorithm 1. We preprocess the input KGs in line 3, then initialize the learnable parameters in the two channels in line 4. For the Transitivity-based channel, we initialize entity embeddings and relation embeddings by Xavier initialization. On the other hand, for Proximity-based channel, we employ the node features as pre-trained English word vectors trained by fastText [41]. This initialization is also applied to all the baselines that use entity name embedding. For each training step, we update the entity embeddings and relation embeddings in the two channels by minimizing the loss functions (see Eq. 6, Eq. 18) using Adam optimizer (line [6][7][8][9][10][11]. Then, we periodically update the current alignment matrix S and nominate top c matched entity pairs based on their similarity values. The candidates nominated at least n times are considered as highconfidence anchor links and updated to alignment seed set L. Finally, the optimized matrix S is used to retrieve the aligned pairs. At each 10 epoch, we compute the similarity matrix using Eq. 20 and update the alignment seeds as mentioned earlier. We also evaluate our model's performance during training. The process will stop if the development MRR does not increase in two consecutive evaluation steps. D. Complexity Analysis To analyze the time complexity of our model, we will focus on different parts of our models. In the Transitivity-based channel, the translation constraint loss costs O(|E|), while the mapping loss costs O(|V|). The triples recovery process takes O(|V| 2 × |R|). On the other hand, in the Proximity-based channel, the relation-aware attention and relation-aware message passing Algorithm 1 Training scheme 1: Input: source and target input KGs: KG s and KG t , entity alignment seeds L. 2: Output: optimized alignment matrix S 3: Add inverse and self loop edges to reach KG 4: Initialize the embeddings 5: for epoch e in {1, 2, ..., N } do 6: Update Translation based embedding to minimize Eq. 22 7: for each GNN layer k do 8: Compute layer-wise entity emb. h k e using Eq. 7 9: Compute layer-wise relation emb. h k r using Eq. 10 10: Update final GNN embeddings for entities and relations Eq. 13, Eq. 14 11: Update model parameter to minimize Eq. 18 12: if e ≡ 1 (mod valid epoch) then 13: Update the alignment matrix S using Eq. 20 14: Update the alignment seeds L using S 15: Recover some missing triples following V-B 16: return S costs O(E). Beside, the entities mapping loss and relation mapping loss cost O(|V|) and O(|R| 2 ) respectively. In the alignment computation step, the greedy match and high confidence sampler cost O(|V| 3 ). In sum, the total time complexity is O(|E|+|V| 2 ×|R|+|V| 3 ) VI. EVALUATION In this section, we report the experimental results of our techniques against a sizeable collection of 8 baselines and 8 real-world datasets, covering different aspects such as end-toend comparison, ablation study, data sparsity, labelling effort, and qualitative evidence. A. Experimental Setup Dataset. We use 4 popular benchmarking datasets from [2], including four datasets of two cross-lingual versions: English-French and English-German which were crawled from DBpedia (2016-2020) [42]. The datasets have a total of 240K entities, 2438 relations, and 964543 relational triples. Each dataset consists of two versions; (v1) is the sparse version while (v2) is the dense one. Version V1 was directly obtained using the IDS algorithm [43], while version V2 was created by filtering out low-degree entities and thus being denser than V1. The details of the datasets using in our experiments are shown in Table II. Baselines. We compare the performance of our techniques with 8 SOTA alignment techniques as follows: 1) MTransE [28] is a transitivity-based entity alignment model which employs the translation constraint to learn the representation for each entity and relation in the input KGs using a shallow embedding model. 2) JAPE [44] is a shallow embedding-based technique that generates structure embedding (using a TransE model) and attribute embedding (using a skip-gram model). The learnt embeddings are then used simultaneously to align the entities. 3) KDcoE [29] leverages co-training of a KG embedding and a literal description embedding to enhance the semisupervised learning of multilingual KG embeddings. The KG embedding model jointly trains a translation model with a linear-transformation-based alignment, while the description embedding model employs an attentive GRU encoder (AGRU) to characterize multilingual entity descriptions. 4) GCNAlign [11] is a deep embedding-based technique that employs two GCNs to produce structural and attribute embeddings, which capture the relational and attribute information of the entities. Both embeddings are then used to discover corresponding entities via an alignment seed set [11]. 5) BootEA [27] is a shallow embedding-based technique that generates entity and relation embeddings using translation constraint like MTransE. The model then reconciles the learnt embeddings using alignment seeds, followed by an alignment editing step to reduce the accumulated errors [27]. 6) MultiKE [30] is an embedding-based framework that unifies multiple views of entities to learn embeddings for entity alignment. The model implements three representative views corresponding to name, relation and attribute features, respectively; then entity alignment are retrieved by combining the learnt embeddings in three different strategies. 7) RDGCN [12] is a deep embedding-based technique that employs a two-layers GCN with highway gates to capture the attentive interaction between each KG and its dual relation. The embeddings are then compared directly to obtain the alignment result [12]. 8) Alinet [17] is a deep embedding-based technique that leverages entities' distant neighbours to preprocess the input KGs and expand their neighbour overlapping. The technique then forwards the processed inputs through an attention-based GNN which controls the aggregation of both direct and distant neighborhood information using a gating mechanism. Evaluation metrics. In our experiments, we consider alignment direction from left to right following the setting defined in [2]. We use Hit@m (m = 1, 10) to measure the prediction ability of the techniques based on how accurately true positive alignments are observed in the top-k candidates [45]. Suppose L(p) ∈ V t is the counterpart of entity p ∈ V s : Hit@m = p∈Vs 1 S(p, L(p)) ∈ top-m S(p) #{True anchor links} (24) where 1 hypo = 1 if hypo is true, and 1 hypo = 0 if hypo is false. We also use Mean Reciprocal Rank (MRR) and Mean Rank (MR) under pair-wise setting to measure how high the true anchored links are ranked in the list of alignment candidates: M RR = mean 1 tr (25) M R = mean(tr)(26) where tr is the rank of the matched score of the true target. MR and MRR metrics measure how the models perform at soft alignment scenarios where most of the true target entities have high similarity with the source entities. Hyper-parameter setting. Our multi-channel model requires in total of 16 hyperparameters, which is reported in Table III for reproducibility. Computational environment. The experiments were averaged over 50 runs for each dataset to mitigate the effect of randomness. We used an AMD Ryzen ThreadRipper 3.8 GHz system with 64 GB RAM and four GTX Titan X graphic cards. We implemented our model in Python with Pytorch library. B. End-to-end comparison We report an end-to-end comparison of our alignment model against baseline methods on the real-world datasets in Table IV. It can be seen that our model outperformed the others in all scenarios. Though using a multi-channel mechanism as GCN-Align and RDGCN, the gain of up to 10-20% of Hit@1 and MRR demonstrated the efficiency of relation-aware integration and knowledge transfer mechanism proposed in our work IKAMI, especially for denser version (v2) of the datasets. Also, we achieved much higher results than the transitivitybased model MTransE, which justified the superiority of our proximity GNN-based model. Among the baselines, RDGCN and BootEA, the two deep embedding-based techniques, were the runner-ups. Overall, they achieved up to 93.6% of Hit@1 and 0.969 of MRR over all settings, except the noisy D-W dataset. AliNet and GCN-Align also gave promising results, which demonstrates C. Ablation Study We evaluate the design choices in our model by comparing the performance of the final model with several variants. The variants we consider are: Table V, the full IKAMI model outperformed the other variants in both datasets. In more detail, the full model achieved around 5% and 35% higher of Hit@1 and MRR comparing to the single-channel variants Var1 and Var2, which shows advantages of fusing the two channels and confirms that the add of transitivity channel helps to complete the proximity GNN-based channel. Also, the gain of nearly 30% of IKAMI over the variant Var3 demonstrates the robustness of the innovations introduced in our GNNbased channel comparing to the original GCN. The performance witnessed a dramatic drop when we replaced our GNN architecture with R-GCN (var4). Also, the lack of each attention mechanism (var5), alignment seed update (var6) and swapping mechanism (var7) caused a minor decrease in all the five metrics comparing to the full model. This proves the importance of these technical innovations to our model. D. Robustness to KGs incompleteness We evaluate the robustness of our method against the incompleteness by first investigating the capability of our embeddings to discover missing links (a.k.a KG completion [33]). To this end, from the original KGs pair, we randomly removed 20% triples from the source graph. Then, we recovered the missing triples by selecting the tail entity t that had the closet embeddings to the querying head entity and the relation pair h, r and vice-versa. We compared IKAMI against four baseline KG completion techniques, namely DistMult [34], RotatE [35], TransE [28] and CompGCN [47]. The result is shown in Table VI. It can be seen that IKAMI were either the winner or the first runner-up, despite that our technique was not specialized for this task. Feature exchange between the proximity and transitivity channel can help to reconcile the KGs, which helps to reveal unseen relations from one graph based on similar patterns on the other. As we do not focus heavily on KG completion, interested readers can refer to other baselines [48], [49], [50]. To fully investigate the robustness of the techniques against the KGs incompleteness, we conduct the second experiment where we choose the D-W-V2 as source KG and generate the target KG by removing the triples randomly to generate different levels of noise. The result of the experiment is shown in Figure 5, where we only show the performance of IKAMI and the two best baselines RDGCN and AliNet. In general, all methods suffer performance drop when the noise level increases. Our model outperforms the baseline methods, with the Hit@1 goes from nearly 96% to around 92% when the edges removal ratio goes from 10% to 60%, thanks to the efficient feature exchange mechanism. Our model keeps a margin of about 5% in Hit@1 with the runner-up (RDGCN). The performance of Alinet drops more dramatically than the others, with less than 0.3 of Hit@1 and 0.5 of Hit@10 when the noise level goes up to 60%. E. Saving of labelling effort In this experiment, we evaluated the ability of saving prealigned entities of the techniques by examining their performance under different level of supervision data for the D-W dataset. It can be seen from Figure 4 that our model IKAMI outperformed other baselines for every level of supervision, especially for the lower ones. We achieved a gain of around 20% for the level of 1% comparing to the second best baseline RDGCN. This result demonstrates the capability of the knowledge transform between the KGs using in IKAMI in terms of saving labelling effort. F. Qualitative evidences In this section, we qualitatively interpret our technique by two case studies. First, we visualized the attention coefficient of the relational triples of the entity Ronald Colman in Figure 1 processed by IKAMI. It is clear from Figure 6 that the coefficient for the triples appearing in both KGs outweighed that of the triples appearing in only one KG (e.g. BirthPlace triple, Profession triple). This depicts the capability of our attention mechanism in emphasizing the shared relational triples while mitigating the impact of the noisy ones. Second, list some representative relation alignment generated by the relation embedding from IKAMI between EN and FR KGs. Our technique efficiently captured the underlying semantic of the relation type and aligned them quite accurately, without the need of machine translation. This also highlights the advantage of our relation representation learning and relationaware propagation. Second, we compare the KGC performance of IKAMI with the single-channel transitivity-based technique TransE during the training process. It can be seen from Figure 7 that the fusion with proximity-based channel helps IKAMI not only converged faster but also achieved superior final result against TransE. VII. CONCLUSION Discussions: We provide further key insights as follows: (1) IKAMI can achieve at least 70% performance across all metrics with only 10% of label information, where other baselines fail. (2) IKAMI can still maintain a 90% performance across all metrics with 60% of missing edges, where other baselines fail. (3) The "seem-to-be-strict" translation-based constraint [28] surprisingly helps to strengthen the local signal and thus well-complete GNN-based model. (4) Unlike GNNbased existing works that often infer the relation embedding from entities, the relations should be assigned their own representation. Our relation-aware model in fact can learn underlying semantic and correctly align the relation between cross-lingual KGs without the need of machine translation. Summary: In this paper, we have presented a representation learning framework, IKAMI, for aligning incomplete knowledge graph from different domains. By exchanging multiple feature channels, including transitivity-based features and proximity-based features, between input knowledge graphs via representation learning, the alignment process is performed efficiently and overcomes the heterogeneity and incompleteness of KGs. Experiments show that our method improves various down-stream performances over SOTAs, including alignment, completion, sparsity, and labeling cost. In future work, we plan to incorporate external sources of information (e.g. transfer learning [51], [52]) to further improve the alignment. Fig. 1 : 1Aligning incomplete KGs across domains matrix at k-th layer Wge GNN's final projection matrix for entities Wgr GNN's final projection matrix for relations S Alignment matrix Fig. 2: Framework Overview Fig. 3 : 3Running time (in log scale) on different datasets the power of graph neural networks for entity alignment. On the other hand, the shallow embedding-based method MTransE achieved the lowest values for accuracy.When it comes to the scalability,Figure 3depicts the running time of the techniques. GCN-Align is the fastest because of full-batch setting of GCN. Our model requires the most running time, due to the accuracy-running time tradeoff. Note that our framework allows the users reducing the number of iteration of completion and alignment improvement to reduce the time in sacrificing the alignment accuracy. • Var1: removes the Transitivity-based channel and only keeps the Proximity-based channel • Var2: removes the Proximity-based channel and only keeps the Translation-based channel • Var3: replaces our Proximity-based channel architecture by an original GCN. [9] • Var4: replaces our Proximity-based channel architecture by RGCN [46]. • Var5: does not contain the attention mechanism described in Sec. 3.3. • Var6: does not contain the updating alignment seed described in Sec. 3.4. • Var7: does not contain the swapping mechanism described in Sec. 3.1. We conduct this experiment in the D-W-V1 and D-W-V2 datasets. As from Fig. 5 :Fig. 6 : 56Robustness of graph alignment models against noise on EN-DE-Attention visualisation (EN-FR-V1 dataset). The model pays less attention to noisy relations. Fig. 7 : 7KGC performance comparison between TransE and IKAMI during training TABLE I : ISummary of notation usedSymbols Definition KG = (V, R, E,Ē) Incomplete Knowledge Graph p, q, h, t Entity r Relation hp, hr Entity embedding & Relation embedding V, R, E,Ē Entity set, Relation set, Triple set, Missing triple set N (p) Neighbor set of node p K Number of GNN layers d Embedding dimension γt TABLE II : IIDataset statistics for KG alignmentDatasets KGs V1 V2 #Ent. #Rel. #Rel tr. #Ent. #Rel. #Rel tr. EN-FR-15K EN 15,000 267 47,334 15,000 193 96,318 FR 15,000 210 40,864 15,000 166 80,112 EN-DE-15K EN 15,000 215 47,676 15,000 169 84,867 DE 15,000 131 50,419 15,000 96 92,632 D-W-15K DB 15,000 248 38,265 15,000 167 73,983 WD 15,000 169 42,746 15,000 121 83,365 D-Y-15K DB 15,000 165 30,291 15,000 72 68,063 YG 15,000 28 26,638 15,000 21 60,970 TABLE III : IIIHyper-parameter setting Hyper-parameter name Hyper-parameter value Embedding dim 100 Learning rate 0.1 Mapping weight βt 50 Batch size 1000 Optimizer Adam #negative samples 5 Margin γt 1 Embedding dim 300 #GCN layers K 2 Hidden dim 300 #negative samples 50 Learning rate 0.0005 Attention Leaky ReLU w 0.05 Margin γg 1 Similarity balancing weight βtg 0.4 Eval step 10 TABLE IV : IVEnd-to-end KG alignment performance (bold: winner, underline: first runner-up)Dataset Ver. Metric MTransE GCN-A BootEA RDGCN Alinet JAPE KDcoE MultiKE IKAMI EN-DE V1 Hit@1 .307 .481 .675 .830 .609 .288 .529 .756 .949 Hit@10 .610 .753 .865 .915 .829 .607 .679 .828 .991 MRR .407 .571 .740 .859 .681 .394 .580 .782 .952 MR 223.9 352.3 125.7 67.1 216.7 140.6 124.8 91.5 8.4 V2 Hit@1 .193 .534 .833 .833 .816 .167 .649 .755 .964 Hit@10 .431 .780 .936 .936 .931 .415 .835 .835 .992 MRR .274 .618 .869 .860 .857 .250 .715 .784 .975 MR 193.5 108.0 16.2 74.8 71.1 139.9 16.0 45.2 3.0 EN-FR V1 Hit@1 .247 .338 .507 .755 .387 .263 .581 .749 .907 Hit@10 .563 .680 .794 .880 .829 .595 .721 .843 .992 MRR .351 .451 .603 .800 .487 .372 .628 .782 .935 MR 251.9 562.2 227.7 156.1 483.2 175.6 197.0 97.8 7.2 V2 Hit@1 .240 .414 .660 .847 .580 .292 .730 .864 .978 Hit@10 .240 .796 .906 .934 .877 .624 .869 .924 .998 MRR .336 .542 .745 .880 .689 .402 .778 .885 .986 MR 206.0 131.3 25.7 61.7 94.0 89.1 27.3 12.1 1.2 D-W V1 Hit@1 .259 .364 .572 .515 .470 .250 .247 .411 .724 Hit@10 .541 .648 .793 .717 .703 .541 .473 .583 .911 MRR .354 .461 .649 .584 .552 .348 .325 .468 .793 MR 331.1 765.3 286.3 508.5 575.7 243.7 730.2 275.4 25.3 V2 Hit@1 .271 .506 .821 .623 .741 .262 .405 .495 .857 Hit@10 .584 .818 .950 .805 .925 .581 .720 .724 .984 MRR .376 .612 .867 .684 .807 .368 .515 .569 .900 MR 146.0 146.0 18.4 229.3 72.1 99.0 91.7 38.6 3.0 D-Y V1 Hit@1 .463 .465 .739 .931 .569 .469 .661 .903 .967 Hit@10 .733 .661 .871 .974 .726 .747 .797 .950 .990 MRR .559 .536 .788 .949 .630 .567 .710 .920 .976 MR 245.6 1113.7 365.1 17.8 532.6 211.2 133.3 19.5 3.1 V2 Hit@1 .443 .875 .958 .936 .951 .945 .895 .856 .987 Hit@10 .707 .963 .990 .973 .989 .626 .984 .927 .998 MRR .533 .907 .969 .950 .965 .440 .932 .881 .992 MR 85.2 47.1 4.8 13.8 5.6 82.5 2.1 10.0 1.1 TABLE V : VAblation studyVar D-W-V1 D-W-V2 Hit@1 Hit@10 MRR Hit@1 Hit@10 MRR Var1 .685 .863 .750 .784 .942 .841 Var2 .691 .883 .762 .818 .962 .870 Var3 .716 .903 .783 .828 .960 .873 Var4 .722 .908 .791 .822 .970 .876 Var5 .421 .741 .498 .512 .782 .641 Var6 .468 .752 .515 .556 .799 .663 Var7 .379 .639 .468 .628 .875 .712 Full model .724 .911 .793 .832 .974 .883 TABLE VI : VIKnowledge Graph Completion performanceDataset Ver. Metric DistMult TransE RotatE CompGCN IKAMI EN-FR V1 Hit@1 .177 .239 .251 .324 .485 MRR .311 .365 .381 .421 .621 V2 Hit@1 .193 .195 .205 .314 .329 MRR .337 .340 .357 .413 .469 EN-DE V1 Hit@1 .042 .041 .048 .234 .148 MRR .089 .102 .120 .318 .248 V2 Hit@1 .122 .125 .124 .187 .261 MRR .199 .202 .207 .258 .369 TABLE VII : VIICorrect aligned relations in EN↔FR KGs country ↔ pays (country), birthPlace ↔ lieuNaissance (birth place), deathPlace ↔ lieuMort (dead place) starring ↔ apparaître (starring) , field ↔ domaine (domain) , developer ↔ développeurs (developer) hometown ↔ nationalité (nationality) Knowledge graph embedding: A survey of approaches and applications. Q Wang, Z Mao, B Wang, L Guo, TKDE. 2912Q. Wang, Z. Mao, B. Wang, and L. Guo, "Knowledge graph embedding: A survey of approaches and applications," TKDE, vol. 29, no. 12, pp. 2724-2743, 2017. A benchmarking study of embedding-based entity alignment for knowledge graphs. Z Sun, Q Zhang, W Hu, C Wang, M Chen, F Akrami, C Li, Proc. VLDB Endow. 1312Z. Sun, Q. Zhang, W. Hu, C. Wang, M. Chen, F. Akrami, and C. Li, "A benchmarking study of embedding-based entity alignment for knowledge graphs," Proc. VLDB Endow., vol. 13, no. 12, p. 2326-2340, 2020. Neural latent space model for dynamic networks and temporal knowledge graphs. T Gracious, S Gupta, A Kanthali, R M Castro, A Dukkipati, AAAI. 35T. Gracious, S. Gupta, A. Kanthali, R. M. Castro, and A. Dukkipati, "Neural latent space model for dynamic networks and temporal knowl- edge graphs," in AAAI, vol. 35, no. 5, 2021, pp. 4054-4062. Adaptive network alignment with unsupervised and multi-order convolutional networks. H T Trung, T Van Vinh, N T Tam, H Yin, M Weidlich, N Q V Hung, IEEE 36th International Conference on Data Engineering. H. T. Trung, T. Van Vinh, N. T. Tam, H. Yin, M. Weidlich, and N. Q. V. Hung, "Adaptive network alignment with unsupervised and multi-order convolutional networks," in IEEE 36th International Conference on Data Engineering (ICDE), 2020, pp. 85-96. Pair-linking for collective entity disambiguation: Two could be better than all. M C Phan, A Sun, Y Tay, J Han, C Li, TKDE. 317M. C. Phan, A. Sun, Y. Tay, J. Han, and C. Li, "Pair-linking for collective entity disambiguation: Two could be better than all," TKDE, vol. 31, no. 7, pp. 1383-1396, 2018. Gaussianpath: A bayesian multi-hop reasoning framework for knowledge graph reasoning. G Wan, B Du, AAAI. 35G. Wan and B. Du, "Gaussianpath: A bayesian multi-hop reasoning framework for knowledge graph reasoning," in AAAI, vol. 35, no. 5, 2021, pp. 4393-4401. Dynamic knowledge graph alignment. Y Yan, L Liu, Y Ban, B Jing, H Tong, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35Y. Yan, L. Liu, Y. Ban, B. Jing, and H. Tong, "Dynamic knowledge graph alignment," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 5, 2021, pp. 4564-4572. Variational reasoning for question answering with knowledge graph. Y Zhang, H Dai, Z Kozareva, A J Smola, L Song, AAAI. Y. Zhang, H. Dai, Z. Kozareva, A. J. Smola, and L. Song, "Variational reasoning for question answering with knowledge graph," in AAAI, 2018, pp. 6069-6076. Semi-supervised classification with graph convolutional networks. T N Kipf, M Welling, T. N. Kipf and M. Welling, "Semi-supervised classification with graph convolutional networks," in ICLR, 2017, pp. 1-14. Graph attention networks. P Velickovic, G Cucurull, A Casanova, A Romero, P Liò, Y Bengio, in ICLR. P. Velickovic, G. Cucurull, A. Casanova, A. Romero, P. Liò, and Y. Bengio, "Graph attention networks," in ICLR, 2018, pp. 1-12. Cross-lingual knowledge graph alignment via graph convolutional networks. Z Wang, Q Lv, X Lan, Y Zhang, EMNLP. Z. Wang, Q. Lv, X. Lan, and Y. Zhang, "Cross-lingual knowledge graph alignment via graph convolutional networks," in EMNLP, 2018, pp. 349- 357. Relation-aware entity alignment for heterogeneous knowledge graphs. Y Wu, X Liu, Y Feng, Z Wang, R Yan, D Zhao, IJCAI. Y. Wu, X. Liu, Y. Feng, Z. Wang, R. Yan, and D. Zhao, "Relation-aware entity alignment for heterogeneous knowledge graphs," in IJCAI, 2019, pp. 5278-5284. Multi-channel graph neural network for entity alignment. Y Cao, Z Liu, C Li, Z Liu, J Li, T.-S Chua, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. the 57th Annual Meeting of the Association for Computational LinguisticsY. Cao, Z. Liu, C. Li, Z. Liu, J. Li, and T.-S. Chua, "Multi-channel graph neural network for entity alignment," in Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019, pp. 1452-1461. Cross-lingual knowledge graph alignment via graph matching neural network. K Xu, L Wang, M Yu, Y Feng, Y Song, Z Wang, D Yu, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. the 57th Annual Meeting of the Association for Computational LinguisticsK. Xu, L. Wang, M. Yu, Y. Feng, Y. Song, Z. Wang, and D. Yu, "Cross-lingual knowledge graph alignment via graph matching neural network," in Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019, pp. 3156-3161. A comparative study on network alignment techniques. H T Trung, N T Toan, T Van Vinh, H T Dat, D C Thang, N Q V Hung, A Sattar, Expert Systems with Applications. 140112883H. T. Trung, N. T. Toan, T. Van Vinh, H. T. Dat, D. C. Thang, N. Q. V. Hung, and A. Sattar, "A comparative study on network alignment techniques," Expert Systems with Applications, vol. 140, p. 112883, 2020. Network alignment by representation learning on structure and attribute. T T Huynh, C T Duong, T H Quyet, Q V H Nguyen, A Sattar, Pacific Rim International Conference on Artificial Intelligence. T. T. Huynh, C. T. Duong, T. H. Quyet, Q. V. H. Nguyen, A. Sattar et al., "Network alignment by representation learning on structure and attribute," in Pacific Rim International Conference on Artificial Intelligence, 2019, pp. 698-711. Knowledge graph alignment network with gated multi-hop neighborhood aggregation. Z Sun, C Wang, W Hu, M Chen, J Dai, W Zhang, Y Qu, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence34Z. Sun, C. Wang, W. Hu, M. Chen, J. Dai, W. Zhang, and Y. Qu, "Knowledge graph alignment network with gated multi-hop neighbor- hood aggregation," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, no. 01, 2020, pp. 222-229. An experimental study of state-of-the-art entity alignment approaches. X Zhao, W Zeng, J Tang, W Wang, F Suchanek, IEEE Transactions on Knowledge & Data Engineering. 01X. Zhao, W. Zeng, J. Tang, W. Wang, and F. Suchanek, "An experimental study of state-of-the-art entity alignment approaches," IEEE Transac- tions on Knowledge & Data Engineering, no. 01, pp. 1-1, 2020. Entity alignment for knowledge graphs with multi-order convolutional networks. T T Nguyen, T T Huynh, H Yin, V Van Tong, D Sakong, B Zheng, Q V H Nguyen, IEEE Transactions on Knowledge and Data Engineering. 3213T. T. Nguyen, T. T. Huynh, H. Yin, V. Van Tong, D. Sakong, B. Zheng, and Q. V. H. Nguyen, "Entity alignment for knowledge graphs with multi-order convolutional networks," IEEE Transactions on Knowledge and Data Engineering, vol. 32, no. 13, pp. 1-14, 2021. Representation learning on graphs with jumping knowledge networks. K Xu, C Li, Y Tian, T Sonobe, K Kawarabayashi, S Jegelka, International Conference on Machine Learning. K. Xu, C. Li, Y. Tian, T. Sonobe, K. Kawarabayashi, and S. Jegelka, "Representation learning on graphs with jumping knowledge networks," in International Conference on Machine Learning, 2018, pp. 5449-5458. Multilingual knowledge graph embeddings for cross-lingual knowledge alignment. M Chen, Y Tian, M Yang, C Zaniolo, IJCAI. M. Chen, Y. Tian, M. Yang, and C. Zaniolo, "Multilingual knowledge graph embeddings for cross-lingual knowledge alignment," in IJCAI, 2017, pp. 1511-1517. Alex: Automatic link exploration in linked data. A El-Roby, A Aboulnaga, SIGMOD. A. El-Roby and A. Aboulnaga, "Alex: Automatic link exploration in linked data," in SIGMOD, 2015, pp. 1839-1853. A unified feature selection framework for graph embedding on high dimensional data. M Chen, I W Tsang, M Tan, T J Cham, IEEE Transactions on Knowledge and Data Engineering. 276M. Chen, I. W. Tsang, M. Tan, and T. J. Cham, "A unified feature selection framework for graph embedding on high dimensional data," IEEE Transactions on Knowledge and Data Engineering, vol. 27, no. 6, pp. 1465-1477, 2014. Exploiting centrality information with graph convolutions for network representation learning. H Chen, H Yin, T Chen, Q V H Nguyen, W.-C Peng, X Li, IEEE 35th International Conference on Data Engineering. H. Chen, H. Yin, T. Chen, Q. V. H. Nguyen, W.-C. Peng, and X. Li, "Exploiting centrality information with graph convolutions for network representation learning," in IEEE 35th International Conference on Data Engineering (ICDE), 2019, pp. 590-601. Cross-lingual entity alignment via joint attribute-preserving embedding. Z Sun, W Hu, C Li, International Semantic Web Conference. Z. Sun, W. Hu, and C. Li, "Cross-lingual entity alignment via joint attribute-preserving embedding," in International Semantic Web Confer- ence, 2017, pp. 628-644. Iterative entity alignment via joint knowledge embeddings. H Zhu, R Xie, Z Liu, M Sun, Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence. the Twenty-Sixth International Joint Conference on Artificial IntelligenceH. Zhu, R. Xie, Z. Liu, and M. Sun, "Iterative entity alignment via joint knowledge embeddings." in Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, 2017, pp. 4258- 4264. Bootstrapping entity alignment with knowledge graph embedding. Z Sun, W Hu, Q Zhang, Y Qu, Proceedings of the 27th International Joint Conference on Artificial Intelligence. the 27th International Joint Conference on Artificial IntelligenceZ. Sun, W. Hu, Q. Zhang, and Y. Qu, "Bootstrapping entity alignment with knowledge graph embedding," in Proceedings of the 27th Interna- tional Joint Conference on Artificial Intelligence, 2018, pp. 4396-4402. Translating embeddings for modeling multi-relational data. A Bordes, N Usunier, A Garcia-Durán, J Weston, O Yakhnenko, Proceedings of the 26th International Conference on Neural Information Processing Systems. the 26th International Conference on Neural Information Processing SystemsA. Bordes, N. Usunier, A. Garcia-Durán, J. Weston, and O. Yakhnenko, "Translating embeddings for modeling multi-relational data," in Pro- ceedings of the 26th International Conference on Neural Information Processing Systems, 2013, pp. 2787-2795. Cotraining embeddings of knowledge graphs and entity descriptions for cross-lingual entity alignment. M Chen, Y Tian, K.-W Chang, S Skiena, C Zaniolo, arXiv:1806.06478arXiv preprintM. Chen, Y. Tian, K.-W. Chang, S. Skiena, and C. Zaniolo, "Co- training embeddings of knowledge graphs and entity descriptions for cross-lingual entity alignment," arXiv preprint arXiv:1806.06478, 2018. Multi-view knowledge graph embedding for entity alignment. Q Zhang, Z Sun, W Hu, M Chen, L Guo, Y Qu, IJCAI. Q. Zhang, Z. Sun, W. Hu, M. Chen, L. Guo, and Y. Qu, "Multi-view knowledge graph embedding for entity alignment," IJCAI, 2019. Rea: Robust cross-lingual entity alignment between knowledge graphs. S Pei, L Yu, G Yu, X Zhang, KDD. S. Pei, L. Yu, G. Yu, and X. Zhang, "Rea: Robust cross-lingual entity alignment between knowledge graphs," in KDD, 2020, pp. 2175-2184. Passleaf: A pool-based semisupervised learning framework for uncertain knowledge graph embedding. Z.-M Chen, M.-Y Yeh, T.-W Kuo, in AAAI. 355Z.-M. Chen, M.-Y. Yeh, and T.-W. Kuo, "Passleaf: A pool-based semi- supervised learning framework for uncertain knowledge graph embed- ding," in AAAI, vol. 35, no. 5, 2021, pp. 4019-4026. Knowledge graph embedding by translating on hyperplanes. Z Wang, J Zhang, J Feng, Z Chen, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence28Z. Wang, J. Zhang, J. Feng, and Z. Chen, "Knowledge graph embedding by translating on hyperplanes," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 28, no. 1, 2014. Convolutional 2d knowledge graph embeddings. T Dettmers, P Minervini, P Stenetorp, S Riedel, Thirty-second AAAI conference on artificial intelligence. T. Dettmers, P. Minervini, P. Stenetorp, and S. Riedel, "Convolutional 2d knowledge graph embeddings," in Thirty-second AAAI conference on artificial intelligence, 2018, pp. 1811-1818. A novel embedding model for knowledge base completion based on convolutional neural network. D Q Nguyen, T D Nguyen, D Q Nguyen, D Q Phung, D. Q. Nguyen, T. D. Nguyen, D. Q. Nguyen, and D. Q. Phung, "A novel embedding model for knowledge base completion based on convolutional neural network," pp. 327-333, 2018. Analogical inference for multi-relational embeddings. H Liu, Y Wu, Y Yang, International conference on machine learning. H. Liu, Y. Wu, and Y. Yang, "Analogical inference for multi-relational embeddings," in International conference on machine learning. PMLR, 2017, pp. 2168-2178. End-to-end structure-aware convolutional networks for knowledge base completion. C Shang, Y Tang, J Huang, J Bi, X He, B Zhou, AAAI. 33C. Shang, Y. Tang, J. Huang, J. Bi, X. He, and B. Zhou, "End-to-end structure-aware convolutional networks for knowledge base completion," in AAAI, vol. 33, no. 01, 2019, pp. 3060-3067. A reevaluation of knowledge graph completion methods. Z Sun, S Vashishth, S Sanyal, P Talukdar, Y Yang, ACL. Z. Sun, S. Vashishth, S. Sanyal, P. Talukdar, and Y. Yang, "A re- evaluation of knowledge graph completion methods," ACL, 2020. How attentive are graph attention networks. S Brody, U Alon, E Yahav, arXiv:2105.14491arXiv preprintS. Brody, U. Alon, and E. Yahav, "How attentive are graph attention networks?" arXiv preprint arXiv:2105.14491, 2021. Network similarity decomposition (nsd): A fast and scalable approach to network alignment. G Kollias, S Mohammadi, A Grama, IEEE Transactions on Knowledge and Data Engineering. 2412G. Kollias, S. Mohammadi, and A. Grama, "Network similarity decom- position (nsd): A fast and scalable approach to network alignment," IEEE Transactions on Knowledge and Data Engineering, vol. 24, no. 12, pp. 2232-2243, 2011. Advances in pre-training distributed word representations. T Mikolov, E Grave, P Bojanowski, C Puhrsch, A Joulin, Proceedings of the International Conference on Language Resources and Evaluation (LREC 2018). the International Conference on Language Resources and Evaluation (LREC 2018)T. Mikolov, E. Grave, P. Bojanowski, C. Puhrsch, and A. Joulin, "Ad- vances in pre-training distributed word representations," in Proceedings of the International Conference on Language Resources and Evaluation (LREC 2018), 2018. Dbpedia-a large-scale, multilingual knowledge base extracted from wikipedia. J Lehmann, R Isele, M Jakob, A Jentzsch, D Kontokostas, P N Mendes, S Hellmann, M Morsey, P Van Kleef, S Auer, Semantic web. 6J. Lehmann, R. Isele, M. Jakob, A. Jentzsch, D. Kontokostas, P. N. Mendes, S. Hellmann, M. Morsey, P. Van Kleef, S. Auer et al., "Dbpedia-a large-scale, multilingual knowledge base extracted from wikipedia," Semantic web, vol. 6, no. 2, pp. 167-195, 2015. Yago: A large ontology from wikipedia and wordnet. F M Suchanek, G Kasneci, G Weikum, Journal of Web Semantics. 63F. M. Suchanek, G. Kasneci, and G. Weikum, "Yago: A large ontology from wikipedia and wordnet," Journal of Web Semantics, vol. 6, no. 3, pp. 203-217, 2008. Cross-lingual entity alignment via joint attribute-preserving embedding. Z Sun, W Hu, C Li, International Semantic Web Conference. SpringerZ. Sun, W. Hu, and C. Li, "Cross-lingual entity alignment via joint attribute-preserving embedding," in International Semantic Web Confer- ence. Springer, 2017, pp. 628-644. User identity linkage across online social networks: A review. K Shu, S Wang, J Tang, R Zafarani, H Liu, Acm Sigkdd Explorations Newsletter. 182K. Shu, S. Wang, J. Tang, R. Zafarani, and H. Liu, "User identity linkage across online social networks: A review," Acm Sigkdd Explorations Newsletter, vol. 18, no. 2, pp. 5-17, 2017. Rgcn: Recurrent graph convolutional networks for target-dependent sentiment analysis. J Chen, H Hou, J Gao, Y Ji, T Bai, International Conference on Knowledge Science, Engineering and Management. SpringerJ. Chen, H. Hou, J. Gao, Y. Ji, and T. Bai, "Rgcn: Recurrent graph convolutional networks for target-dependent sentiment analysis," in International Conference on Knowledge Science, Engineering and Man- agement. Springer, 2019, pp. 667-675. Compositionbased multi-relational graph convolutional networks. S Vashishth, S Sanyal, V Nitin, P Talukdar, International Conference on Learning Representations. S. Vashishth, S. Sanyal, V. Nitin, and P. Talukdar, "Composition- based multi-relational graph convolutional networks," in International Conference on Learning Representations, 2019, pp. 1-14. Context-enhanced entity and relation embedding for knowledge graph completion (student abstract). Z Qiao, Z Ning, Y Du, Y Zhou, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35Z. Qiao, Z. Ning, Y. Du, and Y. Zhou, "Context-enhanced entity and relation embedding for knowledge graph completion (student abstract)," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 18, 2021, pp. 15 871-15 872. Parame: Regarding neural network parameters as relation embeddings for knowledge graph completion. F Che, D Zhang, J Tao, M Niu, B Zhao, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence34F. Che, D. Zhang, J. Tao, M. Niu, and B. Zhao, "Parame: Regarding neural network parameters as relation embeddings for knowledge graph completion," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, no. 03, 2020, pp. 2774-2781. Few-shot knowledge graph completion. C Zhang, H Yao, C Huang, M Jiang, Z Li, N V Chawla, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence34C. Zhang, H. Yao, C. Huang, M. Jiang, Z. Li, and N. V. Chawla, "Few-shot knowledge graph completion," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, no. 03, 2020, pp. 3041- 3048. Robust knowledge transfer via hybrid forward on the teacher-student model. L Song, J Wu, M Yang, Q Zhang, Y Li, J Yuan, AAAI. 35L. Song, J. Wu, M. Yang, Q. Zhang, Y. Li, and J. Yuan, "Robust knowledge transfer via hybrid forward on the teacher-student model," in AAAI, vol. 35, no. 3, 2021, pp. 2558-2566. Toward robust long range policy transfer. W.-C Tseng, J.-S Lin, Y.-M Feng, M Sun, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35W.-C. Tseng, J.-S. Lin, Y.-M. Feng, and M. Sun, "Toward robust long range policy transfer," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 11, 2021, pp. 9958-9966.
[]
[ "Co-Processors for Quantum Devices", "Co-Processors for Quantum Devices" ]
[ "Alastair Kay \nDepartment of Mathematics\nRoyal Holloway University of London\nTW20 0EXEghamSurreyUK\n" ]
[ "Department of Mathematics\nRoyal Holloway University of London\nTW20 0EXEghamSurreyUK" ]
[]
Quantum devices, from simple fixed-function tools to the ultimate goal of a universal quantum computer, will require high quality, frequent repetition of a small set of core operations, such as the preparation of entangled states. These tasks are perfectly suited to realisation by a co-processor or supplementary instruction set, as is common practice in modern CPUs. In this paper, we present two quintessentially quantum co-processor functions: production of a GHZ state, and implementation of optimal universal (asymmetric) quantum cloning. Both are based on the evolution of a fixed Hamiltonian. We introduce a new technique for deriving the parameters of these Hamiltonians based on the numerical integration of Toda-like flows. arXiv:1710.04932v4 [quant-ph]
10.1103/physreva.97.032316
[ "https://arxiv.org/pdf/1710.04932v4.pdf" ]
3,633,198
1710.04932
10fda99a32a08522255615d01d37f45c69ba0ba3
Co-Processors for Quantum Devices Alastair Kay Department of Mathematics Royal Holloway University of London TW20 0EXEghamSurreyUK Co-Processors for Quantum Devices (Dated: September 13, 2018) Quantum devices, from simple fixed-function tools to the ultimate goal of a universal quantum computer, will require high quality, frequent repetition of a small set of core operations, such as the preparation of entangled states. These tasks are perfectly suited to realisation by a co-processor or supplementary instruction set, as is common practice in modern CPUs. In this paper, we present two quintessentially quantum co-processor functions: production of a GHZ state, and implementation of optimal universal (asymmetric) quantum cloning. Both are based on the evolution of a fixed Hamiltonian. We introduce a new technique for deriving the parameters of these Hamiltonians based on the numerical integration of Toda-like flows. arXiv:1710.04932v4 [quant-ph] I. INTRODUCTION The task of quantum state synthesis [1,2] lies at the heart of any potential quantum technology -before a quantum protocol can be run, be it a Bell test [3], quantum key distribution [4], quantum cloning [5][6][7], random number generation [8] or quantum computation [9], a non-trivial quantum resource must be prepared. The required resource might be a fixed quantum state such as a Bell state, W -state, cluster [9] or GHZ state, or it might depend on a small input, such as the unknown state of a qubit. The availability of these resource states is the source of the power of quantum technologies. Repeated demands for the same resource state make it vital to concentrate on their accurate functioning. This suggests developing a device that accomplishes that single task, replacing a complex sequence of quantum gates. These might provide the first step in a quantum protocol (i.e. the core functionality of a particular quantum technological device), or operate as a fixed-function subroutine within a quantum computer, much as today's classical processors provide enhanced instruction sets (e.g. SSE or AVX) or co-processors. Our aim is to produce the desired states and transformations by the free evolution of a Hamiltonian whose parameters have been specifically tuned for the task. By doing this directly with the system's Hamiltonian for any relevant experimental scenario, whether this is in the solid state [10,11], trapped ions [12], or even photonic systems [13][14][15], we ensure that the state is produced as accurately and as quickly as possible, reducing the opportunities for external influences to degrade the resource. One special case of this has been extensively studied -perfect quantum state transfer [16][17][18][19][20], wherein an unknown quantum state can be transported between the two extremes of a one-dimensional chain of spins. This example demonstrates the power of the approach -it is twice as fast as the equivalent quantum gate sequence [21] and many of the error modes are relegated to the manufacturing process; they can be identified prior to * [email protected] use and corrected, or simply rejected until a higher quality version is produced [15]. In addition to perfect state transfer, the same device can create graph states [22], of which the cluster states and GHZ states are special cases (albeit in an unusual basis). Minor modifications [19,23,24] have also permitted the creation of Bell states between the extremal sites of the chain. More recently [1,2], new systems have been created to facilitate the synthesis of arbitrary one-excitation states on the chain, such as W -states between subsets of sites, while more exotic interactions have been shown to cause signal amplification, ideal for enhancing measurement signals [25]. In this paper, we develop a new co-processor that creates GHZ states in a particularly straightforward manner, see Section II. Moreover, the one-dimensional transverse Ising model that we introduce is highly appropriate to many experimental scenarios from superconducting qubits [10,11] to trapped ions [12], and demonstrates a reasonable robustness to experimental imperfections (see Section II A). In Section III, we also show how the state synthesis solutions of [1,2] can be combined with the GHZ co-processor to implement optimal universal cloning of one unknown qubit to N clones [7,26,27]. This is the first time that a reliable implementation of optimal universal asymmetric cloning has been proposed (the circuits in [27] were probabilistic in nature), demonstrating that fixed function devices can perform transformations based on a small input space, and realise highly nontrivial quantum properties. In Sec. III A 1, we also introduce the state synthesis problem for uniformly coupled networks (as compared to chains with engineered couplings), and demonstrate that some hypercubes are useful for generating the uniform superpositions (W -states) that are desirable for symmetric cloning. Crucial to the specification of both co-processors is the numerical discovery of appropriate Hamiltonian parameters. We introduce a new technique based on the numerical integration of a differential equation, the Toda flow. This is the main focus of III A, also including discussions of convergence issues in Appendix A. Variants of this [28], and good techniques for its numerical integration [29][30][31], have been extensively studied in the numerical analysis and numerical methods literature. A. Perfect Excitation Transfer Throughout this work, we will rely on many of the insights previously developed in the study perfect state transfer. In essence, the core of this is that there is an N × N tridiagonal matrix h (N ) P ST =           0 J (N ) 1 J (N ) 1 0 J (N ) 2 J (N ) 2 0 J (N ) 3 . . . . . . . . . J (N ) N −2 0 J (N ) N −1 J (N ) N −1 0           . This matrix has a basis {|n } N n=1 , and the coupling strengths are chosen such that e −ih (N ) P ST t0 |n = (−i) N −1 |N + 1 − n(1) The standard solution [17] for these couplings is J (N ) n = n(N − n) and t 0 = π/2. Although there are infinitely many others, some analytic [32], some numeric [19,33], this first solution optimises many desirable features such as speed of transport [21,34,35]. On the other hand, [33] provides an insightful method for producing coupling schemes that are close to some desirable configuration, perhaps imposed by experimental restrictions. The key properties of h (N ) P ST are related to its symmetry and its spectrum [19]. The evolution time has to be long enough such that a relative phase of π (modulo 2π) is generated between neighbouring eigenvectors. As such, the minimum transfer time is related to the minimum eigenvalue gap ∆ via t 0 ≥ π/∆. II. GHZ STATE CREATION Systems such as the transverse Ising are now routinely accessed or simulated in quantum devices [36]. The Nqubit Hamiltonian is H I = N n=1 J n X n + N −1 n=1 B n Z n Z n+1 , where X n and Z n denote the Pauli X and Z matrices respectively acting on qubit n. We will now show how the parameters J n and B n can be tuned so that an initial separable state of |0 ⊗N , which is easily prepared, evolves to a maximally entangled GHZ state in fixed time. The evolution under H I is solved via a Jordan-Wigner transformation [37]. We invoke the Majorana fermions c 2n−1 = X 1 X 2 . . . X n−1 Z n c 2n = X 1 X 2 . . . X n−1 Y n . These evolve under H I independently according to c n (t) = e −iH I t c n e iH I t = 2N m=1 c m m| e −2ih1t |n , where h 1 is the 2N × 2N matrix h 1 = i           0 B 1 −B 1 0 J 1 −J 1 0 B 2 . . . . . . . . . −B N −1 0 J N −1 0 −J N −1 0 B N −B N 0           . Since the Majorana fermions form a basis, specifying the evolution invoked by this matrix h 1 fixes the evolution of the entire system. Moreover, the tridiagonal structure of h 1 is easily transformed into the form of a real symmetric tridiagonal which is well-studied for perfect state transfer [16,17,19]. Recall that J (N ) n are the coupling strengths for a perfect state transfer scheme that has a transfer time t 0 . By making the same identification as [20], This also updates the evolution, J n = J (2N ) 2n B n = J (2N ) 2n−1 , then h (2N ) P ST satisfies Eq. (1), and h (2N ) P ST =           0 B 1 B 1 0 J 1 J 1 0 B 2 . . . . . . . . . B N −1 0 J N −1 0 J N −1 0 B N B N 0          e −ih1t0 |n = D † e −ih (2N ) P ST t0 D |n = (−i) 2N −1 i n−1 D † |2N + 1 − n = (−1) n |2N + 1 − n . It follows that c n (t 0 /2) = (−1) n c 2N +1−n . We can now decompose the initial separable state in terms of the Majorana fermions, |0 0| ⊗N = 1 2 N N −1 n=1 (1 + Z n Z n+1 ) (1 + Z 1 ) = 1 2 N N −1 n=1 (1 + ic 2n c 2n+1 ) (1 + c 1 ). (2) After evolution under H I for time t 0 /2, terms such as c 2n c 2n+1 transform into −c 2N +1−2n c 2N −2n = c 2N −2n c 2N +1−2n , via the anti-commutation of the fermions. Hence, the product involving pairs of fermions is unchanged, and the final state must become 1 2 N N −1 n=1 (1 + ic 2n c 2n+1 ) (1 − c 2N ). This is the same (up to a global phase) as the pure state |GHZ = 1 √ 2 |0 ⊗N − i |1 ⊗N .(3) We have successfully engineered an Ising chain that creates GHZ states by its natural dynamics. The transfer time scales linearly with N if a maximum coupling strength is imposed, which is the best possible scaling for a one-dimensional system [38]. The evolution after subsequent periods of t 0 /2 is readily determined, since (1 + c 1 ) → (1 − c 2N ) → (1 − c 1 ) → (1 + c 2N ) → (1 + c 1 ), so the state evolves as |0 ⊗N → |GHZ → −i |1 ⊗N → −iZ 1 |GHZ → −i |0 ⊗N(4) (indeed, the phase Z 1 might equally well be applied on any qubit). The only part that is not justified is the global phase. However, we know that |0 ⊗N → |GHZ |1 ⊗N → Z 1 |GHZ , so by linearity, we can evaluate |GHZ → −i |1 ⊗N . It is also now straightforward to determine the evolution of other basis states, which we will make use of in Sec. III. Consider an input state |x for x ∈ {0, 1} N . We can alternatively write this as X x |0 ⊗N where X x has Pauli X operators applied on the sites where the bit value of the string x is 1, and identity on the other sites. Each X n = ic 2n−1 c 2n , and therefore evolves to X N +1−n . So, if x R is the reversal of bit string x, we have |x = X x |0 ⊗N → X x R |GHZ .(5) These models are well-suited to near-term experimental realisation. For example, [36] uses Rydberg atoms in a chain to produce a Hamiltonian which can, in principle, be tuned to give an Ising model of up to 51 qubits. The main challenge is to make the fields B n and J n a similar strength. Currently, |B n /J n | ≤ δ 0.1. While incompatible with the standard perfect state transfer couplings [17], other techniques such as [33] can return the couplings for suitable perfect transfer schemes. However, this means that there are two eigenvalues of h 1 which are separated by O(δ N ) (corresponding to the unpaired fermions of the Majorana chain described by Kitaev [39]). Consequently, the time for generating the GHZ state scales as Ω(δ −N ), which is currently prohibitive. A. Robustness of GHZ Synthesis While we have identified the main experimental challenge, the necessary accuracy of engineering for the system parameters could be a concern. This is particularly acute in the case of GHZ state synthesis, because the state in Eq. (2) is described in terms of a large number (up to 2N − 1) of Majorana fermions. If each is transported with some sub-unital fidelity, the overall success of the synthesis could be quite minimal. In fact, the situation is not nearly so bleak, and the system has a good tolerance of these imperfections. A good estimate on the overlap of the output state GHZ| e −iH I t0/2 |0 ⊗N is ≈ 1 + |F | 2 N det (e −ih1t0 h 0 e ih1t0 h 0 − 1), where F = 2N | e −ih1t0 |1 is the single excitation transfer fidelity, and h 0 = N −1 n=1 |2n 2n + 1| − |2n + 1 2n| . This exactly evaluates the evolution of N −1 n=1 (1 + Z n Z n+1 ) , comparing it to itself, by writing it as a fermionic Gaussian state [40,41]. The additional effect of the single excitation c 1 is then approximated, ignoring possible interactions with the Gaussian component. This approximation facilitates numeric simulations, and Fig. 1 demonstrates the effect on a chain of size 21. It must be emphasised that these results are very basic, merely measuring the overlap with the target state. When this device is made, we will characterise the state that is produced, and adapt for its imperfections, such as applying an optimised choice of local unitaries. This can only serve to increase the figure of merit. Or, can we witness the presence of different types of entanglement? In particular, k-body entanglement for k ∼ N . Existing entanglement witnesses are not yet sophisticated enough to be able to discriminate this. On the other hand, one thing that we cannot easily do is replace the perfect transfer couplings with a coupling scheme that achieves nearly perfect transfer (but with a shorter transfer time, making the system less susceptible to noise, and some perturbations), such as those suggested in [42]. Those schemes are tuned specifically for end-to-end transfer. They generate high transfer fidelity between |1 → |2N at a higher speed, at the cost of the transfer fidelity between intermediate sites. However, GHZ synthesis requires high quality transfer for all pairs |n → |2N + 1 − n . For example, the optimal N = 21 solution from [42] has a 1 → 42 excitation transfer fidelity of 0.993 (which is roughly reproduced by the 3% perturbed chains in Fig. 1), but only generates the GHZ state with overlap 0.762 due to the vastly lower transfer fidelities on the middle of the chain, such as for 5 → 38, which is less than 0.4. B. GHZ Creation in the XY Model A generalisation of H I can be written as: H ZY = N n=1 B n X n + N −1 n=1 J n (1+γ n )Z n Z n+1 +J n (1−γ n )Y n Y n+1 . This model is also a free-fermion model, and has the same Majorana fermions as H I . Starting from the same initial state, as described by Eq. (2), the only possible difference is what those Majorana fermions can evolve into, which is again governed by a 2N × 2N matrix h γ , similar to h 1 . We are interested in whether this broader class of Hamiltonians can also produce the GHZ state, again in the hope of improving experimental viability. For pedagogical simplicity, we will fix γ n = γ for all n, although there is no such restriction arising in the mathematics. We could equally well consider the Hadamard-transformed version of this Hamiltonian, which is the more familiar XY model. H XY = N n=1 B n Z n + N −1 n=1 J n (1+γ n )X n X n+1 +J n (1−γ n )Y n Y n+1 . In this case, the initial state would be (H |0 ) ⊗N , and the final state would be H ⊗N |GHZ . We already know two solutions for this matrix. At γ = 1 (vanishing Y Y terms), we have already fixed B n = J (2N ) 2n−1 and J n = J (2N ) 2n /2, while for γ = 0, B n = N and J n = J (N ) n /2 comes from perfect state transfer. Indeed, this last solution is the usual perfect state transfer chain (using H XY ), and it was already observed in [22] that this system is capable of creating the GHZ state (in a non-obvious basis). Solutions for both values of γ have the same eigenvalues, ±1, ±3, ±5, . . . ± (2N − 1). We are hence interested at intermediate values of γ, with the same spectrum. We shall do this by providing a numerical routine to interpolate between the two known solutions. We believe the form of the isospectral transformation is new to the spin chain community, although is well-studied in the numerical methods and analysis literature [28][29][30][31]. We permute the elements of h γ , grouping oddnumbered and even-numbered basis elements together. The matrix then decomposes as h γ = i |0 1| ⊗ X(γ) − i |1 0| ⊗ X T (γ) where X(γ) is a non-symmetric matrix X = N n=1 B n |n n| + N −1 n=1 J n (1 + γ) |n n + 1| + N −1 n=1 J n (1 − γ) |n + 1 n| . The spectrum of h γ is ±λ i where λ i are the singular values of X. Hence, it is sufficient to perform an isospectral transformation on X. To achieve this, we observe that if A and B are anti-Hermitian, then X(t) = e −B Xe A describes an isospectral flow. Taking the derivative, dX dt = XA − BX.(6) So, any small, anti-Hermitian A, B will achieve an isospectral transformation, we just need to select them so that X retains the properties that we want it to: 1. X is tridiagonal, i.e. m| XA |n = m| BX |n for any n, m such that |n − m| > 1. 2. C is centrosymmetric, in the sense n| X |m = N + 1 − m| X |N + 1 − n . We anticipate this being a necessary condition in the same way that it is for state transfer [19], although this is unproven. 3. We require n + 1| X |n / n| X |n + 1 = 1−γ 1+γ to be the same for all n. As conditions on the matrix X, if it satisfies them at any given value, we can ensure they are upheld on subsequent values by imposing them on the derivatives. For a given X, each condition is linear in the coefficients of A and B, and the number of constraints coincides perfectly with the number of coefficients, permitting solution. By performing a numerical integration starting from a known solution for X(0) (which, as already observed, we know for γ = 0, 1) any desired value of γ can be arrived at. Equation (6) can be integrated following two different philosophies. Firstly, one can integrate it directly, i.e. setting the next X to be X → X + δ(XA − BX). The structural aspects of X are preserved exactly, but the isospectral transformation is only accurate to O(δ 2 ) for each step of size δ, giving an overall accuracy of O(δ). Alternatively, one can perform the update X → e −B Xe A . This unitary transformation is isospectral, but the structural properties such as tridiagonality are only accurate to O(δ 2 ). Nevertheless, there is the facility to compensate for any error in the next step, preventing it from accumulating during the integration. Moreover, if δ is shrunk as a solution converges, then the accuracy is arbitrarily good. Throughout this paper, we use first-order (Euler) integrations. While they appear to serve very well, isospectral flows of this form are often challenging to integrate [43], and novel techniques such as Runge-Kutta-Munthe-Kaas have been developed [29,30]. These may be used to improve performance in the future. We conclude that any model H ZY can be tuned to achieve GHZ state generation. An explicit demonstration is given in [44] for 21 qubits and γ = 0.7 (integrating from γ = 0, using the direct method). The modest choice of size derives only from memory limits of simulating a full Hamiltonian for verification. III. OPTIMAL CLONING Production of the GHZ state has shown that although studies of state transfer are often constrained to the single excitation subspace, the same ideas can be applied to generate interesting evolution in multiple excitation sectors. For the GHZ state, this was a fixed input providing a fixed output. Are there other protocols that we might consider? An arbitrary unitary seems to be out of the question, even within the single excitation subspace -if we use a state synthesis routine [1,2] then one can choose the evolution of a particular excitation, say |1 → |ψ . But then, the possible evolution of other input states is tightly constrained. For example, since |2 = H |1 /J 1 , e −iHt |2 = 1 J 1 He −iHt |1 = 1 J 1 H |ψ . So, if |ψ is confined to a small set of sites, |2 evolves only onto those and neighbouring sites. Perfect state transfer demonstrates this -any system that transfers |1 → |N must also transfer |n → |N + 1 − n ; there is no freedom to choose these transformations. Still, there may be interesting protocols that depend upon a small input subspace. Perfect state transfer is one such example, wherein the possible inputs are spanned by a basis of 2 states. Another example is 1 → N cloning. The optimal 1 → N universal asymmetric cloning machine [7,26,27] implements the (not necessarily unique) transformation |0 → A |0 ⊗N + N n=1 β n |1 ⊗(n−1) |0 |1 ⊗(N −n) |1 → A |1 ⊗N + N n=1 β n |0 ⊗(n−1) |1 |0 ⊗(N −n)(7) where A = N n=1 β n B 2 = N n=1 β 2 n , and the β n determine the asymmetry of the cloning quality via the single-copy average fidelities F n = 2 + (β n + A) 2 6 , and satisfy the normalisation A 2 + B 2 = 1. We will now show how this can be implemented using spin chains. There is not a single spin chain that achieves this entire transformation, but we can use them as tools that massively simplify the sequence of quantum gates that need to be applied. To that end, consider a set of M = 2N − 1 qubits. One qubit, k + 2, is the unknown state to be cloned, |ψ , but rotated by a phase gate, and we still aim to produce N clones, on the odd-numbered qubits. The rest are prepared in the separable state: |0 ⊗k (A |0 + iB |1 ) ⊗ √ Z |ψ ⊗ |0 ⊗(M −2−k) . This can be decomposed into the four basis states |0 ⊗k |x |0 ⊗(M −2−k) for x ∈ {0,|0 → A |0 ⊗M + BX k+1 |1 ⊗M |1 → AX k+2 |1 ⊗M + BX k+1 X k+2 |0 ⊗M has been implemented (the inputs being the basis of the qubit to be cloned). Finally, we apply a controlled-NOT controlled from qubit k + 1 and targeting qubit k + 2. This gives the overall transformation |0 → A |0 ⊗M + B |1 ⊗k |0 |1 ⊗(M −1−k) |1 → A |1 ⊗M + B |0 ⊗k |1 |0 ⊗(M −k−1) In fact, the entire transformation up to this point can be implemented by a single Hamiltonian evolution, using a less physically motivated Hamiltonian, based on a tuned 3-body cluster state Hamiltonian [25]. Alternatively, as already observed, one can replace the Ising-generating Hamiltonian with a perfect state transfer Hamiltonian by applying a Hadamard transform before and after. This has the advantage of making the form of the Hamiltonian for the GHZ generation and state synthesis parts the same, up to modification of the coupling strengths, at the cost of adding some local Hadamard gates. From here, we can get our overall desired cloning transformation, creating the N clones on the odd-numbered qubits of the chain, if we can implement |0 ⊗M → |0 ⊗M |1 ⊗M → |1 ⊗M |k + 1 → N n=1 β n |2n − 1 B k + 1 → N n=1 β n 2n − 1 B , where |0 ⊗(k−1) |1 |0 ⊗(M −k) = |k and k = X ⊗M |k . The first two transformations are automatic for an exchange-coupled spin chain H XX = M −1 n=1 J n 2 (X n X n+1 + Y n Y n+1 )(8) because |0 ⊗M and |1 ⊗M are null vectors of H XX . Assume that couplings can be found in order to implement the third transformation in a time t 0 [1,2]. This will be discussed in Sec. III A. Indeed, a suitable solution was found in [2] for an initial state in the middle, k = N − 1, and is reproduced in Fig. 2. For the last condition, observe that [H XX , X ⊗M ] = 0. Hence e −iH XX t0 k = X ⊗M e −iH XX t0 |k , which simply yields that the transformations in the 1 and M − 1 excitation subspaces are essentially identical, and so the final condition will also be satisfied. All of the complexity of producing these clones is conveniently wrapped up in just two helper functions. The corresponding circuit diagrams are contrasted in Figs. 3 and 4. While a quantum circuit for cloning has previously been explicitly stated for small sizes [45], we are not aware of a version, other than probabilistic versions [27], that works deterministically for general 1 → N universal symmetric cloning, let alone the asymmetric case. This is probably because the cloning map in Eq. (7), specialised to symmetric cloning, is not the map usually stated [46,47]: [7] reveals that the cloning map is associated with the ground state of a particular matrix, and the symmetric case is highly degenerate. The version that we have chosen, Eq. (7), extends consistently from the asymmetric case, and lends itself well to implementation with a quantum circuit, as depicted in Fig. 4 provides such a definition for symmetric cloning. This circuit can be modified for asymmetric cloning. Assuming that the architecture exhibits only nearest-neighbour couplings (the only instance where it makes sense to consider implementation via a nearest-neighbour Hamiltonian), the depth of the circuit is N , and comprises O(N 2 ) gates. Why do we only create clones on every second site of the chain? Imagine we have a Hamiltonian like Eq. (8), but including magnetic fields as well. H = M −1 n=1 J n 2 (X n X n+1 + Y n Y n+1 ) + N n=1 B n 2 (1 − Z n ) (9) Let U = (N +1)/2 n=1 X 2n−1 (N −1)/2 n=1 Y 2n . We have that U HU = −H + N n=1 B n 1. Moreover, at t 0 , e −iHt0 = e iHt0 because e −iλnt0 = ±1 for every eigenvalue (neglecting, for simplicity, a possible global phase). Thus, time evolution in the higher excitation subspace is given by a symmetric target spectrum, it suffices to set B n = 0. One can readily verify that although the even-numbered qubits effectively act as ancillas, and although the transformation we implement does not leave them separable, it does not adversely affect the cloning fidelity. Another way to circumvent this restriction is to introduce more parameters to the Hamiltonian. For instance, e −iHt0 |n = −(−i) (N −1)/2+2n U e −i H+|ψ • • √ swap √ swap √ swap clone √ N |0 + |1 • × × √ Z † clone |0 × √ swap √ swap √ Z † clone |0 × Z clone |0 × √ swap √ Z † clone |0 Z clone . . . × √ swap Z . . .H XXZ = M −1 n=1 J n 2 (X n X n+1 + Y n Y n+1 ) + B n Z n Z n+1 commutes with X ⊗N , and gives us the ability to manipulate the diagonal elements in the single excitation subspace via the parameters {B n }. This gives us sufficient control to produce a system that gives clones on every site of the chain. A. Designing State Synthesis Systems While we can use the algorithms of [1,2] to generate figures such as Fig. 2, these are limited by very small radii of convergence. Instead, we would now like to examine if the isospectral flow ideas outlined above can be applied. The two papers [1,2] provide different philosophies for how to produce a useful chain, but for our purposes, much of the calculation in the same. Let us start with a first guess at a Hamiltonian, H 0 (in the single excitation subspace). It has a desired spectrum but its evolution produces e −iH0t0 |φ = |ψ 0 , starting from the separable state |φ (i.e. a single excitation on a particular site), and evolving for a time t 0 , where |ψ 0 is not our target state |ψ t . In practice, we will set |φ = N +1 2 to minimse the evolution time. However, if |ψ t is symmetric, we can reduce the task to finding a chain of half the length, starting with an excitation at site 1 [2]. For that reason, we will typically assume |φ = |1 . How can we make a better guess, H 1 , which should have the same spectrum as H 0 ? Again, we use the isospectral transformation H 1 = e −B H 0 e A for some anti-Hermitian matrices A, B. We have several properties that we want to impose. As in the Ising case, we can rearrange the matrix H 0 into a block structure of {all odd elements, all even elements}, so that H 0 takes the form H 0 = 0 X 0 X T 0 0 . A block-diagonal A = diag(A o A e ) preserves the structure of H 0 , with X 0 evolving as X 0 → e −Ae X 0 e Ao . Next, we want to impose that the tridiagonal structure of H 0 is preserved. This just requires i|Ḣ 0 |j = 0 for all |i−j| > 1. We've already partially achieved this with our block-diagonal of A ensuring that it's true for all |i − j| even. The remainder are simply a set of simultaneous linear equations in the parameters A o and A e : i| X 0 A o − A e X 0 |j = 0∀j = i, i − 1. We are hence building up a set of linear conditions which, so far, just ensure that subsequent matrices maintain the important properties of the initial matrices. Now we must impose that each subsequent iteration moves us towards a better evolution. Since we are updating H 0 → e −A H 0 e A , and assuming A is small, the evolution updates as e −A e −iH0t0 e A |φ ≈ |ψ 0 − A |ψ 0 + e −iH0t0 A |φ . To fix the the new evolution to be |ψ t , we might solve −A |ψ 0 + e −iH0t0 A |1 = |ψ t − |ψ 0 subject to the structure constraints that we have already described. However, a solution of this form is unlikely to keep A is small, it is perhaps better to describe it as a constrained optimisation problem (linear programming): max A: A ≤δ ( ψ t | − ψ 0 |)[e −iH0t0 , A] |φ . Having found A, we update H 0 according to the unitary transformation update (rather than direct integration). For the particular design philosophy of [1], we can go further. There, due the the chosen spectrum, e −iH0t0 = 1 − 2 |λ 0 λ 0 | , the aim is to fix the null vector to |λ t , where and knowing that the null vector of the next iteration is e −A |λ 0 . In successive iterations, we aim to maximise the overlap with |λ t . Hence, we have to solve the linear programming problem |ψ t = (1 − 2 |λ t λ t |) |1 ,min A ∞≤δ λ t | A |λ 0 . Convergence is well motivated -unless there is a reason that either |λ 0 or |λ t must be an null vector of A, we can always find a non-zero value of the overlap, and choosing the sign of A assures that the outcome is always negative, and hence iterates towards an improved solution. The solution must converge, and further justification that it converges globally on the correct solution is given in Appendix A. A typical output is shown in Fig. 5, and is used in [44] to demonstrate the proper functioning of the entire evolution sequence, giving the optimal symmetric cloning fidelity of F = 23 33 . Hypercubes So far, we have discussed engineering a one dimensional chain, choosing the coupling strengths to achieve the evolution that we desire. This perfectly parallels studies of perfect state transfer. However, another avenue for study in perfect state transfer is the set of uniformly coupled graphs that can accomplish the same task. For example, hypercubes of arbitrary dimension k, and side length 2 or 3, achieve perfect state transfer at distances k or 2k respectively [18]. Various other graphs have since been shown to provide perfect transfer, including a variety of modifications of the hypercubes [18,[48][49][50]. Can graphs also be used for the state synthesis tasks that could be useful for quantum cloning? We specifically focus on generating uniform superpositions across some subset of sites, with a preference for those where the phase on each of the superposed sites is the same. Let G be a graph with edges E and N vertices V . The graph Hamiltonian is defined as H G = 1 2 (i,j)∈E (X i X j + Y i Y j ). As before, there is a subspace structure based on the number of excitations, and |0 ⊗N and |1 ⊗N are null vectors, and [H, X ⊗N ] = 0 -we only have to get the evolution in the single excitation subspace correct. If the graph is bipartite, the phase choice can only be consistent if the superposed sites are all part of the same bipartition, by a generalisation of the argument around Eq. (9). In the single excitation subspace, the Hamiltonian is represented by the adjacency matrix, A, of the graph. The conditions on state synthesis [2] are remarkably similar to those of perfect transfer [19,51]: to start from a site n, producing a state |ψ in time t 0 , if |ψ ∈ R N , then λ m |n = ± λ m |ψ for every eigenvector |λ m , and for those eigenvectors for which λ m |n = 0, the eigenvalues λ m must satisfy e −iλmt0 = ±e iφ for some phase φ. This has some further consequences for the spectrum of the adjacency matrix A [52]. For example, with one extra assumption about the nature of the state synthesis task (that all vertices have a perfect revival at the same time), we know that the spectrum for a non-bipartite graph must be integral, while for a bipartite graph, the spectrum is either integral, or rational multiples of √ ∆ for a square-free integer ∆. We will not develop this theory more generally here, but will focus on some special cases that we have found. Several instances of the path P n (i.e. uniformly coupled chain of n vertices) generate uniform superpositions: e −iA(P2)π/4 |1 = |1 − i |2 √ 2 e −iA(P3)π/ √ 8 |2 = |1 + |3 √ 2 e −iA(P3) cos −1 1 √ 3 /( √ 2π) |2 = |1 + i |2 + |3 √ 3 e −i2A(P5)/ √ 27 |3 = |1 + i |2 + i |4 + |5 2 The second of these is ideally suited to 1 → 2 symmetric cloning (and is closely related to a previous construction [53,54]). We can extend these cases by using the hypercube construction [18]. For a graph G, the adjacency matrix of the corresponding k-dimensional hypercube is A(G k ) = k n=1 1 ⊗(n−1) N ⊗ A ⊗ 1 ⊗(k−n) N . This describes independent evolution along each of the k dimensions. Taking a basis x ∈ [N ] k (a k-dimensional vector where each element takes an integer value from 1 to N ), each |x corresponds to a single excitation being on a particular vertex of the graph. Starting from |y , the final amplitude on a vertex |x is k i=1 x i | e −iAt |y i . If A gives a uniform superposition, so does the hypercube. The hypercube of side length 2 (i.e. derived from P 3 ) is particularly compelling. For example, a 3 × 3 square lattice of uniformly coupled qubits generates a uniform superposition of all 9 sites. Or, more applicable to symmetric cloning, P k 3 produces (at a different time) the uniform superposition across all 2 k corners. IV. CONCLUSIONS We have shown how a fixed transverse Ising system can produce a GHZ state, which is a key quantum resource for use in future technologies. This could sit as a stand-alone device, or as a special unit, a co-processor, within a larger quantum device. The fixed-function coprocessor replaces what would otherwise be a complex sequence of unitary gates, with the inaccuracies inherent in the multiple separate steps that have to be taken in its implementation. By extending the results from H I to H ZY , we have potentially opened GHZ synthesis to a much broader range of experiments. Realising that the transformation required to achieve GHZ state synthesis in a transverse Ising model essentially reduces to a statetransfer-like condition on h 1 would also significantly simplify optimal control studies such as [55], moving away from the assumption of perfect engineering. We have also specified a second transformation. This two step procedure, where the first step uses the GHZ synthesiser, implements optimal asymmetric universal cloning of qubits. This is the first time that a nonprobabilistic strategy has been given for these cloning machines. Our transformation is implemented by operations that are local in a one-dimensional chain of qubits. Our use of the GHZ synthesiser to implement a singlecontrol, multiple target controlled-not gate may be of further interest in the relation to the Fourier transform. Both transformations, when restricted to a nearestneighbour architecture, exhibit an optimal O(N ) scaling in run-time, and have essentially identical running times to their quantum circuit equivalents. Central to these results was a new isospectral transformation algorithm, with fine-grained control over directing consecutive iterations. Global convergence of the algorithm is well-motivated. Mathematica scripts that implement the reported results for chains of 21 qubits are available from [44]. The algorithm demonstrates considerable potential for further development, and should be broadly applicable. P ST = Dh 1 D † . FIG. 1 . 1Average overlap with |GHZ for N = 21 when the parameters Bn and Jn are all chosen uniformly at random within a range of ±x% of the value they should be (the 'standard' perfect transfer couplings,[17]), using 1000 samples. FIG. 2 . 2A single excitation, input to the central spin of a 21-qubit spin chain (top) evolves into a uniform superposition over the odd-numbered sites (bottom), as required for optimal symmetric universal 1 → 11 cloning. (FIG. 3 . 3−1) n+m α m |m .This can only be consistent with the desired transform if α m = 0 on every second site, to eliminate the effect of the (−1) n+m term. In doing so, it transpires that for Quantum circuit diagram for quantum cloning when supplemented by two helper Hamiltonians, acting on M = 2N − 1 input qubits. Coefficients are specifically chosen for optimal universal symmetric cloning. The input qubit can be arbitrary. FIG. 4 . 4Quantum circuit diagram for quantum cloning without the two helper Hamiltonians. Moving beyond the symmetric cloner requires replacement of the √ swap with partial swap operations of different amounts. FIG. 5 . 5A single excitation, input to the first spin of a 21qubit spin chain (top) can be caused to evolve into a uniform superposition over the odd-numbered sites (bottom), as required for symmetric 1 → 11 cloning. 1} 2 . We evolve with any M -qubit GHZgenerating H ZY for the generation time. According to Eq. (5), this produces the statesX x1 M −k X x2 M −k−1 |GHZ . Now, we apply a controlled-phase gate between the qubits M − k and M − k − 1. The effect is that if the two bits of x are the same value, this works like a phase gate on the |GHZ state (skipping us on 2 steps in Eq. (4)), while it does nothing if the two bit values are different. Then, we repeat the GHZ evolution. Referring to Eq. (4), this returns X x1 k+1 X x2 k+2 |0 ⊗M if the two bits of x are equal, and X x1 k+1 X x2 k+2 |1 ⊗M otherwise. Overall, Acknowledgements:We would like to thank L. Banchi and G. Coutinho for introductory conversations. This work was supported by EPSRC grant EP/N035097/1.Appendix A: Convergence of Isospectral AlgorithmWe aim to motivate that the algorithm described in Sec. III A has a single point of convergence provided the target null vector, |λ t , can be a null vector of a system with the fixed spectrum.At each step, we iterate with a matrix A, and impose that the elements of A are bounded by some step size δ, A ∞ ≤ δ. It is important that we pick δ such that the second order term in the expansion of λ t | e −A |λ 0 is negligible. Since ψ| A |ψ = 0 for all real-valued states |ψ , the maximum value of λ t | A |λ 0 is achieved withThis is generically possible to fix: A |λ 0 has (N + 1)/2 components, and the (N − 1)/2 free parameters can control the output in the space orthogonal to |λ 0 . By aligning these vectors with the left-and right-maximum singular vectors of A (singular valueThrough the following bounds, we relate δ to σ:where A 2 is the Frobenius norm. Thus, by ensuring that δ = 1 − λ 0 |λ t 2 for small > 0, the second order term is always negligible. Indeed, we can directly bound the value χ = λ t |λ 0 :This solution tends exponentially towards χ = 1. Assuming that A |λ 0 can be picked as specified, we have convergence, and accuracy ε is achieved with an average complexity of O(N 6 log( √ N /ε)), the leading term arising from solving O(N 2 ) linear constraints.Generically, A |λ 0 can be any state in the odd space that is orthogonal to |λ 0 , since there are (N − 1)/2 free parameters with which to achieve this. So, when does this fail? Is this compatible with the observation that some states cannot be the null vector for a tridiagonal system of a particular spectrum[1]?We start by noting that although[1]indicated that none of the vector elements on the odd space can be 0 (as this would give consecutive 0 elements on the complete vector), this was an artificial imposition resulting from requiring non-zero coupling strengths. However, this consideration is not built into the algorithm, so we are not prevented from reaching these forms of |λ t .Since the state A |λ 0 is linear in the free parameters, the space described when the corresponding vectors do not span the space (and are hence linearly dependent on each other) must be a convex space. There is a single inaccessible region, which must therefore include any inaccessible null vectors. The only question is whether this region is tight with that of the inaccessible null vectors.Case StudyIn the absence of universal answers, we investigate the special case of N = 5 and spectrum 0, ±3, ±5, since this is a case where there are forbidden null vectors[1]. The odd space is dimension 3, and there are two free parameters a and b. For a particular h with coupling strengths J 1 , . . . , J 4 ,The ratios γ 1 = J1 J2 and γ 2 = J4 J3 parametrise the possible null vectorsfrom which we can derive that the only time that we do not have access to the whole space is when J 2 1 + J 2 2 = J 2 3 + J 2 4 , and thushaving used the eigenvalue relations (such as J 2 1 + J 2 2 + J 2 3 + J 2 4 = 34) to eliminate the remaining terms. This defines a barrier in the possible space of |λ t that the algorithm cannot cross. Now let us consider the region of |λ t for which there is no h with the correct spectrum and that null vector. Using the explicit eigenvalue relations for the coupling strengths, we can write thatThis has a non-negative solution for J 2 2 when (1 + γ 2 1 )(1 + γ 2 2 ) ≥ 17 2 8 2 .We conclude that our algorithm is capable of converging on any valid null vector for N = 5. If we demand an invalid null vector, the algorithm will converge somewhere on the surface of closest approach, defined by Eq. (A1). It is reasonable to expect similar behaviour in larger spaces, but this remains unproven. . A Kay, 10.1088/1367-2630/aa68f9New J Phys. 1943019A. Kay, New J Phys 19, 043019 (2017). . A Kay, 10.22331/q-2017-08-10-24124A. Kay, Quantum 1, 24 (2017). . J F Clauser, M A Horne, A Shimony, R A Holt, 10.1103/PhysRevLett.23.880Phys. Rev. Lett. 23880J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969). . A K Ekert, 10.1103/PhysRevLett.67.661Phys. Rev. Lett. 67661A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991). . V Bužek, M Hillery, 10.1103/PhysRevA.54.1844Phys. Rev. A. 541844V. Bužek and M. Hillery, Phys. Rev. A 54, 1844 (1996). . R F Werner, 10.1103/PhysRevA.58.1827Phys. Rev. A. 581827R. F. Werner, Phys. Rev. A 58, 1827 (1998). . A Kay, D Kaszlikowski, R Ramanathan, 10.1103/PhysRevLett.103.050501Phys. Rev. Lett. 10350501A. Kay, D. Kaszlikowski, and R. Ramanathan, Phys. Rev. Lett. 103, 050501 (2009). . S Pironio, A Acín, S Massar, A B De La Giroday, D N Matsukevich, P Maunz, S Olmschenk, D Hayes, L Luo, T A Manning, C Monroe, 10.1038/nature09008Nature. 4641021S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Nature 464, 1021 (2010). . R Raussendorf, H J Briegel, 10.1103/PhysRevLett.86.5188Phys. Rev. Lett. 865188R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). . J Majer, J M Chow, J M Gambetta, J Koch, B R Johnson, J A Schreier, L Frunzio, D I Schuster, A A Houck, A Wallraff, A Blais, M H Devoret, S M Girvin, R J Schoelkopf, 10.1038/nature06184Nature. 449443J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Nature 449, 443 (2007). . J H Plantenberg, P C De Groot, C J P M Harmans, J E Mooij, 10.1038/nature05896Nature. 447836J. H. Plantenberg, P. C. de Groot, C. J. P. M. Harmans, and J. E. Mooij, Nature 447, 836 (2007). . R Islam, E E Edwards, K Kim, S Korenblit, C Noh, H Carmichael, G.-D Lin, L.-M Duan, C.-C , R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. . Joseph Wang, J K Freericks, C Monroe, 10.1038/ncomms1374Nat Commun. 2377Joseph Wang, J. K. Freericks, and C. Monroe, Nat Com- mun 2, 377 (2011). . A Perez-Leija, R Keil, A Kay, H Moya-Cessa, S Nolte, L.-C Kwek, B M Rodríguez-Lara, A Szameit, D N Christodoulides, 10.1103/PhysRevA.87.012309Phys. Rev. A. 8712309A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, Phys. Rev. A 87, 012309 (2013). . M Gräfe, R Heilmann, A Perez-Leija, R Keil, F Dreisow, M Heinrich, H Moya-Cessa, S Nolte, D N Christodoulides, A Szameit, 10.1038/nphoton.2014.204Nat. Photon. 8791M. Gräfe, R. Heilmann, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, Nat. Photon. 8, 791 (2014). . R J Chapman, M Santandrea, Z Huang, G Corrielli, A Crespi, M.-H Yung, R Osellame, A Peruzzo, 10.1038/ncomms11339Nat Commun. 711339R. J. Chapman, M. Santandrea, Z. Huang, G. Corrielli, A. Crespi, M.-H. Yung, R. Osellame, and A. Peruzzo, Nat Commun 7, 11339 (2016). . S Bose, 10.1103/PhysRevLett.91.207901Phys. Rev. Lett. 91207901S. Bose, Phys. Rev. Lett. 91, 207901 (2003). . M Christandl, N Datta, A Ekert, A J Landahl, 10.1103/PhysRevLett.92.187902Phys. Rev. Lett. 92187902M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Phys. Rev. Lett. 92, 187902 (2004). . M Christandl, N Datta, T C Dorlas, A Ekert, A Kay, A J Landahl, 10.1103/PhysRevA.71.032312Phys. Rev. A. 7132312M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A 71, 032312 (2005). . A Kay, 10.1142/S0219749910006514Int J Quantum Inf. 8641A. Kay, Int J Quantum Inf. 8, 641 (2010). . C Di Franco, M Paternostro, M S Kim, 10.1103/PhysRevLett.101.230502Phys. Rev. Lett. 101230502C. Di Franco, M. Paternostro, and M. S. Kim, Phys. Rev. Lett. 101, 230502 (2008). . M.-H Yung, 10.1103/PhysRevA.74.030303Phys. Rev. A. 7430303M.-H. Yung, Phys. Rev. A 74, 030303 (2006). . S R Clark, C M Alves, D Jaksch, 10.1088/1367-2630/7/1/124New J. Phys. 7124S. R. Clark, C. M. Alves, and D. Jaksch, New J. Phys. 7, 124 (2005). . L Dai, Y P Feng, L C Kwek, 10.1088/1751-8113/43/3/035302J. Phys. A: Math. Theor. 4335302L. Dai, Y. P. Feng, and L. C. Kwek, J. Phys. A: Math. Theor. 43, 035302 (2010). . V X Genest, L Vinet, A Zhedanov, 10.1016/j.aop.2016.05.009Annals of Physics. 371348V. X. Genest, L. Vinet, and A. Zhedanov, Annals of Physics 371, 348 (2016). . A Kay, 10.1103/PhysRevLett.98.010501Phys. Rev. Lett. 9810501A. Kay, Phys. Rev. Lett. 98, 010501 (2007). . A Kay, R Ramanathan, D Kaszlikowski, Quantum Inf, Comput. 13880A. Kay, R. Ramanathan, and D. Kaszlikowski, Quantum Inf. Comput. 13, 880 (2013). . A Kay, 10.26421/QIC16.11-12Quant Inf. Comput. 16991A. Kay, Quant Inf. Comput 16, 991 (2016). . M T Chu, 10.1016/0024-3795(86)90278-8Linear Algebra and its Applications. 8071M. T. Chu, Linear Algebra and its Applications 80, 71 (1986). H Munthe-Kaas, 10.1016/S0168-9274(98)00030-0Applied Numerical Mathematics Proceedings of the NSF/CBMS Regional Conference on Numerical Analysis of Hamiltonian Differential Equations. 29115H. Munthe-Kaas, Applied Numerical Mathematics Pro- ceedings of the NSF/CBMS Regional Conference on Nu- merical Analysis of Hamiltonian Differential Equations, 29, 115 (1999). . E Celledoni, H Marthinsen, B Owren, 10.1016/j.jcp.2012.12.031Journal of Computational Physics Physics-compatible numerical methods. 2571040E. Celledoni, H. Marthinsen, and B. Owren, Journal of Computational Physics Physics-compatible numerical methods, 257, 1040 (2014). . A Iserles, H Z Munthe-Kaas, S P Nørsett, A Zanna, Acta Numer, 9215A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna, Acta Numer. 9, 215 (2000). . C Albanese, M Christandl, N Datta, A Ekert, 10.1103/PhysRevLett.93.230502Phys. Rev. Lett. 93230502C. Albanese, M. Christandl, N. Datta, and A. Ekert, Phys. Rev. Lett. 93, 230502 (2004). . P Karbach, J Stolze, 10.1103/PhysRevA.72.030301Phys. Rev. A. 7230301P. Karbach and J. Stolze, Phys. Rev. A 72, 030301 (2005). . A Kay, 10.1103/PhysRevA.73.032306Phys. Rev. A. 7332306A. Kay, Phys. Rev. A 73, 032306 (2006). . A Kay, arXiv:1609.01854arXiv:1609.01854quant-ph. quant-phA. Kay, arXiv:1609.01854 [quant-ph] (2016), arXiv:1609.01854 [quant-ph]. . H Bernien, S Schwartz, A Keesling, H Levine, A Omran, H Pichler, S Choi, A S Zibrov, M Endres, M Greiner, V Vuletić, M D Lukin, 10.1038/nature24622Nature. 551579H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 551, 579 (2017). M A Nielsen, Complete notes on fermions and the Jordan-Wigner transform. M. A. Nielsen, "Complete notes on fermions and the Jordan-Wigner transform," (2005). . S Bravyi, M B Hastings, F Verstraete, 10.1103/PhysRevLett.97.050401Phys. Rev. Lett. 9750401S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006). . A Y Kitaev, Phys Usp, 10.1070/1063-7869/44/10S/S2944131A. Y. Kitaev, Phys.-Usp. 44, 131 (2001). S Bravyi, Quantum Inf Comp. 5216S. Bravyi, Quantum Inf Comp 5, 216 (2005). . A Kay, 10.1103/PhysRevLett.107.270502Phys. Rev. Lett. 107270502A. Kay, Phys. Rev. Lett. 107, 270502 (2011). . T J G Apollaro, L Banchi, A Cuccoli, R Vaia, P Verrucchi, 10.1103/PhysRevA.85.052319Phys. Rev. A. 8552319T. J. G. Apollaro, L. Banchi, A. Cuccoli, R. Vaia, and P. Verrucchi, Phys. Rev. A 85, 052319 (2012). . M Calvo, A Iserles, A Zanna, 10.1090/S0025-5718-97-00902-2Math. Comp. 661461M. Calvo, A. Iserles, and A. Zanna, Math. Comp. 66, 1461 (1997). . A Kay, Figshare , 10.6084/m9.figshare.5498020A. Kay, Figshare (2017), 10.6084/m9.figshare.5498020. . V Bužek, S L Braunstein, M Hillery, D Bruß, 10.1103/PhysRevA.56.3446Phys. Rev. A. 563446V. Bužek, S. L. Braunstein, M. Hillery, and D. Bruß, Phys. Rev. A 56, 3446 (1997). . N Gisin, S Massar, 10.1103/PhysRevLett.79.2153Phys. Rev. Lett. 792153N. Gisin and S. Massar, Phys. Rev. Lett. 79, 2153 (1997). . M Murao, D Jonathan, M B Plenio, V Vedral, 10.1103/PhysRevA.59.156Phys. Rev. A. 59156M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, Phys. Rev. A 59, 156 (1999). . R J Angeles-Canul, R Norton, M Opperman, C Paribello, M Russell, C Tamon, Quant Inf. Comput. 10325R. J. Angeles-Canul, R. Norton, M. Opperman, C. Pari- bello, M. Russell, and C. Tamon, Quant Inf. Comput 10, 325 (2010). . A Bernasconi, C Godsil, S Severini, 10.1103/PhysRevA.78.052320Phys. Rev. A. 7852320A. Bernasconi, C. Godsil, and S. Severini, Phys. Rev. A 78, 052320 (2008). . C Facer, J Twamley, J Cresser, 10.1103/PhysRevA.77.012334Phys. Rev. A. 7712334C. Facer, J. Twamley, and J. Cresser, Phys. Rev. A 77, 012334 (2008). . A Kay, 10.1103/PhysRevA.84.022337Phys. Rev. A. 8422337A. Kay, Phys. Rev. A 84, 022337 (2011). Discrete Mathematics Algebraic Graph Theory -A Volume Dedicated to Gert Sabidussi on the Occasion of His 80th Birthday. C , 10.1016/j.disc.2011.06.032312129C. Godsil, Discrete Mathematics Algebraic Graph The- ory -A Volume Dedicated to Gert Sabidussi on the Occasion of His 80th Birthday, 312, 129 (2012). . G De Chiara, R Fazio, C Macchiavello, S Montangero, G M Palma, 10.1103/PhysRevA.70.062308Phys. Rev. A. 7062308G. De Chiara, R. Fazio, C. Macchiavello, S. Montangero, and G. M. Palma, Phys. Rev. A 70, 062308 (2004). . Q Chen, J Cheng, K.-L Wang, J Du, 10.1103/PhysRevA.74.034303Phys. Rev. A. 7434303Q. Chen, J. Cheng, K.-L. Wang, and J. Du, Phys. Rev. A 74, 034303 (2006). . X Wang, A Bayat, S Bose, S G Schirmer, 10.1103/PhysRevA.82.012330Phys. Rev. A. 8212330X. Wang, A. Bayat, S. Bose, and S. G. Schirmer, Phys. Rev. A 82, 012330 (2010).
[]
[ "Pseudo-Hermiticity and weak pseudo-Hermiticity: Equivalence of complementarity and the coordinate transformations in position-dependent mass", "Pseudo-Hermiticity and weak pseudo-Hermiticity: Equivalence of complementarity and the coordinate transformations in position-dependent mass" ]
[ "S.-A Yahiaoui \nDépartement de physique\nFaculté des Sciences\nLPTHIRM\nUniversité Saad DAHLAB de Blida\nBlidaAlgeria\n", "M Bentaiba \nDépartement de physique\nFaculté des Sciences\nLPTHIRM\nUniversité Saad DAHLAB de Blida\nBlidaAlgeria\n" ]
[ "Département de physique\nFaculté des Sciences\nLPTHIRM\nUniversité Saad DAHLAB de Blida\nBlidaAlgeria", "Département de physique\nFaculté des Sciences\nLPTHIRM\nUniversité Saad DAHLAB de Blida\nBlidaAlgeria" ]
[]
The complementarity between the twin concepts of pseudo-Hermiticity and weak pseudo-Hermiticity, established by Bagchi and Quesne [Phys. Lett. A 301 (2002) 173-176], can be understood in terms of coordinate transformations.
null
[ "https://arxiv.org/pdf/0902.0348v1.pdf" ]
18,203,977
0902.0348
fbd7c63df0389828b3bec6e51e151d591773db97
Pseudo-Hermiticity and weak pseudo-Hermiticity: Equivalence of complementarity and the coordinate transformations in position-dependent mass 2 Feb 2009 February 3, 2009 S.-A Yahiaoui Département de physique Faculté des Sciences LPTHIRM Université Saad DAHLAB de Blida BlidaAlgeria M Bentaiba Département de physique Faculté des Sciences LPTHIRM Université Saad DAHLAB de Blida BlidaAlgeria Pseudo-Hermiticity and weak pseudo-Hermiticity: Equivalence of complementarity and the coordinate transformations in position-dependent mass 2 Feb 2009 February 3, 20090365Ca0365Fd Keywords: (Weak) Pseudo-Hermiticitycoordinate transformations The complementarity between the twin concepts of pseudo-Hermiticity and weak pseudo-Hermiticity, established by Bagchi and Quesne [Phys. Lett. A 301 (2002) 173-176], can be understood in terms of coordinate transformations. Introduction In recent years, the concept of pseudo-Hermiticity has attracted much attention on behalf of physicists [1][2][3][4][5][6][7][8][9]. The basic mathematical structure underlying the properties of pseudo-Hermiticity is revealed [3][4][5] and it has been found to be a more general concept then those of Hermiticity and PT -symmetry [10][11][12][13][14][15]. By definition, a linear operator H (here a Hamiltonian) acting in a Hilbert space H is called η-pseudo-Hermitian if it obeys to [3][4][5] ηH = H † η, where η is a Hermitian linear invertible operator and a dagger stands for the adjoint of the corresponding operator. Then (non-Hermitian) Hamiltonian H has a real spectrum [3] if there is an invertible linear operator d : H → H such that η = d † d. As a consequence of this, the reality of the bound-state eigenvalues of H can be associated with η-pseudo-Hermiticity. Note that choosing η = 1 reduces the assumption (1) to the Hermiticity. In a very interesting work [7], Bagchi and Quesne point out that the twin concepts of pseudo-Hermiticity and weak pseudo-Hermiticity are complementary to one another by admitting that it is possible to break up η into two operators, i.e. η + and η − , following combinations η + H = H † η + and η − H = H † η − ,(2) where η ± = η ± η † . The first assumption corresponds to the pseudo-Hermiticity where η + is a second-order differential realization while the second is associated with weak pseudo-Hermiticity and η − is a first-order realization. In the present paper, we take up the study of a complementarity between pseudo-Hermiticity and weak pseudo-Hermiticity under the concept of coordinate transformation and examine how the pseudo-Hermiticity should map to the weak pseudo-Hermiticity. In fact, our primary concern is to point out that the coordinate transformations can be looked upon as a toy model for understanding the complementarity. In this light, the complementarity acquires a mathematical meaning which, unfortunately, was not established in [7]. We end this section by defining a quite formalism used throughout the present work. In the case of a spatially varying mass [16][17][18][19] which will be denoted by M (x) = m 0 m (x), the Hamiltonian proposed by von Roos [16] reads H = 1 4 m α (x) pm β (x) pm γ (x) + m γ (x) pm β (x) pm α (x) + V (x) ,(3) where α, β and γ are three parameters which obey to the relation α + β + γ = −1 in order to grant the classical limit and V (x) = V Re (x) + iV Im (x) ∈ C. Here, p = −i d dx is a momentum with = m 0 = 1, and m (x) is dimensionless-real valued mass. Using the restricted Hamiltonian from the α = γ = 0 and β = −1 constraints [17], the Hamiltonian (3) becomes H = pU 2 (x) p + V (x) ,(4) with U 2 (x) = 1 2m(x) and U (x) ∈ R. The shift on the momentum p in the manner p → p ′ = p − A (x) U (x) ,(5) where A (x) = a (x) + ib (x) ∈ C and a (x), b (x) are real functions, allows to bring the Hamiltonian (4) in the form H → H = p − A (x) U (x) U 2 (x) p − A (x) U (x) + V (x) .(6) 2 Pseudo-Hermiticity generating function As η + = d † d is pseudo-Hermitian and following the ordinary supersymmetric quantum mechanics, the operators d and d † are connecting to the first-order differential realization through [8,9] d = U (x) d dx + Φ (x) , (7.a) d † = −U ′ (x) − U (x) d dx + Φ * (x) , (7.b) where Φ (x) = F (x) + iG (x) ∈ C and F (x), G (x) are real functions. Here, the prime denotes derivative with respect to x. It is obvious that Eqs.(7.a-b) become, under the transformation (5), d → D = U (x) d dx − iA (x) + Φ (x) , (8.a) d † → D † = −U ′ (x) − U (x) d dx + iA * (x) + Φ * (x) , (8.b) and in terms of these, η + is transformed into η + = D † D such as η + = −U 2 (x) d 2 dx 2 − 2K (x) d dx + L (x) ,(9) where K (x) and L (x) are defined as K (x) = U (x) U ′ (x) + iU (x) (G (x) − a (x)) , (10.a) L (x) = Φ * (x) Φ (x) + A * (x) A (x) − [U (x) (iA (x) − Φ (x))] ′ −iΦ * (x) A (x) + iΦ (x) A * (x) . (10.b) Taking the adjoint of Eq. (9), one can easily check that η + is Hermitian; since it is written in the form η + = D † D. On the other hand, the Hamiltonian (6) may be expressed as H = −U 2 (x) d 2 dx 2 − 2M 1 (x) d dx + N 1 (x) + V (x) ,(11) where M 1 (x) = U (x) U ′ (x) − iU (x) A (x) , (12.a) N 1 (x) = i [U (x) A (x)] ′ + A 2 (x) . (12.b) It should be noted that D and D † are two intertwining operators, and then the defining assumption (1) can be generalized into η + H = H † η + . Using Eqs.(9), (11) and the adjoint of Eq.(11) on both sides of the last equation and comparing between their varying differential coefficients, we can recognized from the third-derivative that b (x) = 0, while the second-derivative connects the potential to its conjugate through V (x) = V * (x) − 4iU (x) G ′ (x) .(13) However, the coefficients corresponding to the first-derivative give the shape of the potential, where after integration, we get V (x) = F 2 (x) − G 2 (x) − [U (x) F (x)] ′ − 2iU (x) G ′ (x) + δ,(14) where δ is some constant of integration. The last remaining coefficient corresponds to the null-derivative and gives the pure-imaginary differential equation F 2 (x) − [U (x) F (x)] ′ = G (x) G ′ (x) −F (x) F ′ (x) + 1 2 [U (x) F (x)] ′′ + 1 G ′ (x) 1 4 U 2 (x) G ′′ (x) ′ − G (x) 4 [U (x) U ′′ (x)] ′ + U ′ (x) U (x) 4 G (x) U (x) ′′ + U ′2 (x) U (x) 2 G (x) U (x) ′ − U ′′ (x) U (x) 4 ,(15) which is not easy to solve. However, the η + -orthogonality suggests that the eigenvector, here Ψ (x), is related to H through η + Ψ (x) = 0, or DΨ (x) = 0,(16) leading, after integration, to the ground-state wave function Ψ (x) = Λ (x) ψ (x) = exp i x dy A (x) U (x) ψ (x) = N 0 exp −i x dy F (x) U (x) − i x dy G (x) − a (x) U (x) ,(17) where N 0 is a constant of normalization. The wave function Ψ (x) is then subjected to a gauge transformation in a manner of ψ (x) → Ψ (x) = Λ (x) ψ (x), where Λ (x) = η + (x) [1,7]. Now, using the Schrödinger equation HΨ (x) = EΨ (x) where E = E Re + iE Im , one obtain the differential equation 2F (x) G (x) + U (x) G ′ (x) − U ′ (x) G (x) = −E Im + i (E Re − δ) ,(18) where δ is a constant introduced in Eq. (14). In order to solve suitably Eq. (18), we assume that both sides of Eq.(18) are equal to zero; which requires that E Re = δ and E Im = 0. Therefore, the energy eigenvalues E are real. In these settings, we end up by relating F (x) to G (x) and U (x) through the differential equation F (x) = G (x) 2 U (x) G (x) ′ ,(19) and which proves to be the solution of Eq. (15). Hence, it becomes clear that F (x) (i.e. G (x)) is a generating function leading to identify the potential V (x). Weak pseudo-Hermiticity generating function For the first-order differential realization, η − may be anti-Hermitian and H can be relaxed to be weak pseudo-Hermitian. Then η − can be expressed as η − = U (x) d dx + ϕ (x) ,(20) where ϕ (x) = f (x) + ig (x) ∈ C and f (x), g (x) are real functions. Using Eq.(5), η − and η † − become η − → η − = U (x) d dx − iA (x) + ϕ (x) , (21.a) η † − → η † − = −U ′ (x) − U (x) d dx + iA * (x) + ϕ * (x) . (21.b) As now η − points to weak pseudo-Hermiticity, this amounts to writing η † − = − η − ,(22) which brings to the relation U ′ (x) = 2f (x) + 2b (x) .(23) Letting both sides of η − H = H † η − act on every function and comparing their varying differential coefficients, one deduced from the second-derivative that b (x) = 0, therefore the generating function f (x) in Eq.(23) becomes f (x) = U ′ (x) 2 ,(24) while the first-derivative gives the imaginary part of the potential V Im (x) = iU (x) f ′ (x) − U (x) g ′ (x) − i 2 U (x) U ′′ (x) .(25) The last coefficient corresponds to the null-derivative which gives, after a double integration by parts, the real part of the potential V Re (x) = −g 2 (x) − 1 2 U (x) U ′′ (x) − 1 4 U ′2 (x) + ε,(26) where ε is some constant of integration. In consequence, using Eqs. (22-24), we obtain the potential V (x) = −g 2 (x) − iU (x) g ′ (x) − 1 2 U (x) U ′′ (x) − 1 4 U ′2 (x) + ε.(27) Equivalence of Complementarity-Coordinate transformation In this section, we bring to the notion of the complementarity a mathematical meaning by examining the way in which pseudo-Hermiticity should map into weak pseudo-Hermiticity through the generating functions F (x) and f (x). In fact, it is well known from Eqs. (19) and (24) that both generating functions belong to the same ordinary space representation {X}, then there must be a transformation connecting them. For this reason, we assume that the required transformations are concerned with coordinate transformations (or point canonical transformations.) In mathematical terms, a coordinate transformation x ≡ x (ξ) changes F (x) into f (ξ) in the following way F (x) = G (x) 2 U (x) G (x) ′ x≡x(ξ) −→ f (ξ) = U ′ (ξ) 2 ,(28) where U (x) ≡ U [x (ξ)] = U (ξ). An interesting way to solve this problem, that can be described within coordinate transformation, is to build a differential equation from Eq. (19) and assume that it is maintained invariant if one applies a coordinate transformation. In fact, Eq. (19) can be expressed as U (x) dZ (x) dx = 2F (x) Z (x) ,(29) where Z (x) = U (x) G(x) . It is then obvious that whenever Eq.(29) holds for the set of functions (i.e. U (x), F (x) and Z (x)), similar differential equation will holds for the transformed functions too (i.e. U (ξ), F (ξ) and Z (ξ)) such as U (ξ) dZ (ξ) dξ = 2F (ξ) Z (ξ) ,(30) where F (x) ≡ F [x (ξ)] = F (ξ); idem. for Z (x). Therefore, from Eq.(30), the mass function U (x) is changed in the following way U (x) → U (ξ) = U [x (ξ)] dξ (x) dx .(31) Let us introduce two new functions R (ξ) and S (ξ) related, respectively, to Z (ξ) and F (ξ) by Z (x) → Z (ξ) = Z [x (ξ)] R (ξ) , (32.a) F (x) → F (ξ) = F [x (ξ)] S (ξ) . (32.b) Substituting Eqs.(32.a-b) into Eq.(30) taking into account (31), we get U (x) dZ (x) dx = 2 S (x) F (x) − U (x) d dx ln R (x) Z (x) ,(33) and by identifying it to Eq.(29), one obtain S (x) F (x) = F (x) + U (x) d dx ln R (x),(34) which can be interpreted as a similarity transformation relating F (x) to f (x); i.e. F (x) → f (x) ≡ S (x) F (x) = F (x) + U (x) d dx ln R (x).(35) In this light, let us redefine the coordinate transformation on F (x) following F (x) → F (ξ) = F [x (ξ)] S (ξ) = F [x (ξ)] dξ (x) dx ,(36) and from Eqs. (35) and (19), we get the identity f (ξ) ≡ F [x (ξ)] S (ξ) = G [x (ξ)] S (ξ) 2 U [x (ξ)] G [x (ξ)] ′ .(37) Now in order to recover our result, we assume that the condition G [x (ξ)] S (ξ) = 1 holds, and by defining the generating function G (x) as G [x (ξ)] ≡ S −1 (ξ) = dx (ξ) dξ ,(38) therefore Eq.(37) can be amply simplified, taking into consideration Eq.(30), to f (ξ) ≡ 1 2 U [x (ξ)] dξ (x) dx ′ = U ′ (ξ) 2 .(39) This completes the proof and leads to the identity (28). Conclusion In this paper, we have proposed to give a mathematical meaning for the notion of complementarity between the twin concepts of pseudo-Hermiticity and weak pseudo-Hermiticity within the framework of coordinate transformations, and as a consequence this has opened the way towards understanding the complementarity. Our primary concern in our work implies that all generating functions, whose the associated potentials are related to the pseudo-Hermiticity and weak pseudo-Hermiticity, can be connected into some generalized coordinate transformations. As a concluding remark, we would like to point out the equivalence between the complementarity and coordinate transformations is concerned by a particular choice which the generating function G (x) (i.e. F (x)) can take. . Z Ahmed, Phys. Lett. A. 294287Z. Ahmed, Phys. Lett. A 294 (2002) 287. . B Bagchi, C Quesne, R Roychoudhury, J. Phys. A: Math. Gen. 39127B. Bagchi, C. Quesne, R. Roychoudhury, J. Phys. A: Math. Gen. 39 (2006) L127. . A Mostafazadeh, J. Math. Phys. 43205A. Mostafazadeh, J. Math. Phys. 43 (2002) 205. . A Mostafazadeh, J. Math. Phys. 432814A. Mostafazadeh, J. Math. Phys. 43 (2002) 2814. . A Mostafazadeh, J. Math. Phys. 433944A. Mostafazadeh, J. Math. Phys. 43 (2002) 3944. . A Mostafazadeh, Mod. Phys. Lett. A. 17A. Mostafazadeh, Mod. Phys. Lett. A 17 (2002) 1973. . B Bagchi, C Quesne, Phys. Lett. A. 301173B. Bagchi, C. Quesne, Phys. Lett. A 301 (2002) 173. . O Mustafa, S Habib, Mazharimousavi, Phys. Lett. A. 357295O. Mustafa, S. Habib Mazharimousavi, Phys. Lett. A 357 (2006) 295. . O Mustafa, S Habib, Mazharimousavi, Czech. J. Phys. 56967O. Mustafa, S. Habib Mazharimousavi, Czech. J. Phys. 56 (2006) 967. . C M Bender, S Boettcher, Phys. Rev. Lett. 805243C. M. Bender, S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. . C M Bender, S Boettcher, P N Meisenger, J. Math. Phys. 402201C. M. Bender, S. Boettcher, P. N. Meisenger, J. Math. Phys. 40 (1999) 2201. . C M Bender, G V Dunne, P N Meisenger, Phys. Lett. A. 252272C. M. Bender, G. V. Dunne, P. N. Meisenger, Phys. Lett. A 252 (1999) 272. . S.-A Yahiaoui, O Cherroud, M Bentaiba, J. Math. Phys. 48113503S.-A. Yahiaoui, O. Cherroud, M. Bentaiba, J. Math. Phys. 48 (2007) 113503. . M Bentaiba, S.-A Yahiaoui, L Chetouani, Phys. Lett. A. 331175M. Bentaiba, S.-A. Yahiaoui, L. Chetouani, Phys. Lett. A 331 (2004) 175. . M Bentaiba, L Chetouani, A Mazouz, Phys. Lett. A. 29513M. Bentaiba, L. Chetouani, A. Mazouz, Phys. Lett. A 295 (2002) 13. . O Von Roos, Phys. Rev. B. 277547O. von Roos, Phys. Rev. B 27 (1983) 7547. . D J Bendaniel, C B Duke, Phys. Rev. 152683D. J. BenDaniel, C. B. Duke, Phys. Rev. 152 (1966) 683. Quantum Semiconductor Heterostructures. C Weisbach, B Vinter, Academic PressNew YorkC. Weisbach, B. Vinter, "Quantum Semiconductor Heterostructures", Academic Press, New York, 1993. . S.-A Yahiaoui, M Bentaiba, Int. J. Theo. Phys. on lineS.-A. Yahiaoui, M. Bentaiba, Int. J. Theo. Phys. (2008), on line.
[]
[ "Baryogenesis from Primordial Blackholes after Electroweak Phase Transition", "Baryogenesis from Primordial Blackholes after Electroweak Phase Transition" ]
[ "Niraj Upadhyay \nDepartment of Physics and Astrophysics\nUniversity of Delhi\nDelhi -110 007INDIA\n", "Patrick Das Gupta \nDepartment of Physics and Astrophysics\nUniversity of Delhi\nDelhi -110 007INDIA\n", "R P Saxena \nDepartment of Physics and Astrophysics\nUniversity of Delhi\nDelhi -110 007INDIA\n" ]
[ "Department of Physics and Astrophysics\nUniversity of Delhi\nDelhi -110 007INDIA", "Department of Physics and Astrophysics\nUniversity of Delhi\nDelhi -110 007INDIA", "Department of Physics and Astrophysics\nUniversity of Delhi\nDelhi -110 007INDIA" ]
[]
Incorporating a realistic model for accretion of ultra-relativistic particles by primordial blackholes (PBHs), we study the evolution of an Einstein-de Sitter universe consisting of PBHs embedded in a thermal bath from the epoch ∼ 10 −33 sec to ∼ 5 × 10 −9 sec. In this paper we use Barrow et al's ansatz to model blackhole evaporation in which the modified Hawking temperature goes to zero in the limit of the blackhole attaining a relic state with mass ∼ m pl . Both single mass PBH case as well as the case in which blackhole masses are distributed in the range 8 × 10 2 -3 × 10 5 gm have been considered in our analysis. Blackholes with mass larger than ∼ 10 5 gm appear to survive beyond the electroweak phase transition and, therefore, successfully manage to create baryon excess via X −X emissions, averting the baryon number wash-out due to sphalerons. In this scenario, we find that the contribution to the baryon-to-entropy ratio by PBHs of initial mass m is given by ∼ ǫζ(m/1 gm) −1 , where ǫ and ζ are the CP-violating parameter and the initial mass fraction of the PBHs, respectively. For ǫ larger than ∼ 10 −4 , the observed matter-antimatter asymmetry in the universe can be attributed to the evaporation of PBHs. *
10.1103/physrevd.60.063513
[ "https://export.arxiv.org/pdf/astro-ph/9903253v1.pdf" ]
8,918,558
astro-ph/9903253
27f4da60dce49cc98a35f2ac8b72635c6f762f35
Baryogenesis from Primordial Blackholes after Electroweak Phase Transition arXiv:astro-ph/9903253v1 17 Mar 1999 Niraj Upadhyay Department of Physics and Astrophysics University of Delhi Delhi -110 007INDIA Patrick Das Gupta Department of Physics and Astrophysics University of Delhi Delhi -110 007INDIA R P Saxena Department of Physics and Astrophysics University of Delhi Delhi -110 007INDIA Baryogenesis from Primordial Blackholes after Electroweak Phase Transition arXiv:astro-ph/9903253v1 17 Mar 19991 Incorporating a realistic model for accretion of ultra-relativistic particles by primordial blackholes (PBHs), we study the evolution of an Einstein-de Sitter universe consisting of PBHs embedded in a thermal bath from the epoch ∼ 10 −33 sec to ∼ 5 × 10 −9 sec. In this paper we use Barrow et al's ansatz to model blackhole evaporation in which the modified Hawking temperature goes to zero in the limit of the blackhole attaining a relic state with mass ∼ m pl . Both single mass PBH case as well as the case in which blackhole masses are distributed in the range 8 × 10 2 -3 × 10 5 gm have been considered in our analysis. Blackholes with mass larger than ∼ 10 5 gm appear to survive beyond the electroweak phase transition and, therefore, successfully manage to create baryon excess via X −X emissions, averting the baryon number wash-out due to sphalerons. In this scenario, we find that the contribution to the baryon-to-entropy ratio by PBHs of initial mass m is given by ∼ ǫζ(m/1 gm) −1 , where ǫ and ζ are the CP-violating parameter and the initial mass fraction of the PBHs, respectively. For ǫ larger than ∼ 10 −4 , the observed matter-antimatter asymmetry in the universe can be attributed to the evaporation of PBHs. * Introduction That the Milky Way is essentially made of matter is evident not only from the landings of space probes on Moon and other planets without any disastrous consequences but also from the absence of anti-nuclei in the observed cosmic rays, and from the observations of Faraday rotation [1]. Observational support for absence of significant quantity of anti-matter beyond our Galaxy exists, but it is of indirect nature [1] [2]. Since visible mass in the universe is chiefly in the form of baryonic matter, the inferred matter-antimatter asymmetry essentially boils down to the problem of the origin of baryon asymmetry 1 . The baryon asymmetry is characterized by the baryon-to-photon ratio η = n B /n γ , with n B and n γ being the number densities of net baryons and photons, respectively. According to standard big-bang nucleosynthesis calculations, the predicted abundances of light elements depend only on the free parameter η and are in apparent agreement with the observed abundances provided η lies in the range (2.8 − 4.5) × 10 −10 [3]. Recently, Tytler et al [4] have estimated the baryon-to-photon ratio from the observations of deuterium abundance in a high red-shift quasar absorption system and according to their measurements, log η = −9.18 ± 0.4 ± 0.4 ± 0.2. The esthetically appealing scenario of the universe consisting of equal amount of baryons and anti-baryons at the instant of creation is still compatible with a non-zero η if one invokes Sakharov conditions, namely, of having B, C and CP violating interactions in out-of-thermodynamic equilibrium condition sometime in the early history of the universe [5]. The Grand Unified Theories (GUTs) of fundamental forces incorporate baryon number violating interactions naturally while CP violation can be introduced in such theories in many different ways (it is to be noted that CP -violation added theoretically in GUTs, in general, is not related to the observed CP-violation in the K • −K • system [6]) and therefore it is not surprising that GUTs provide a natural framework for the generation of baryon asymmetry through decay of X −X bosons [7]. However, through the work of Kuzmin, Rubakov and Shaposhnikov [8] it came to be appreciated that baryon number violation can take place during electro-weak phase transition (EWPT) and that such processes could erase baryon-asymmetry generated prior to EWPT era. Use of B-violation in electro-weak theories to produce excess baryons has also been made in the literature [9] but one of the major obstacles in this scenario is the requirement of low Higgs mass which is in direct conflict with the experimental lower limit of m H > 88 GeV [10]. It appears that in the minimal version of electro-weak theory, generating baryon asymmetry may not be possible at all [11] and especially with the discovery of the topquark with a mass around 175 GeV [12] there is hardly any region left in the parameter space of the standard model to produce observed baryon-tophoton ratio [13]. The other major scenario of generating baryon asymmetry is to invoke Hawking evaporation of black-holes. The early sketchy ideas of Hawking and Zeldovich took proper shape with the advent of GUTs, giving rise to a picture of black-holes of small mass emitting X andX bosons thermally which subsequently decay and in the process violate B, C and CP , leading to a production of baryon excess [14]. At the fundamental level, this scenario has an attractive feature in that it combines ideas of black-hole thermodynamics [15][16] on one hand and GUT on the other, to explain the observed matter-antimatter asymmetry in the universe. One of the important ingredients of this picture is the occurrence of mini-black-holes having mass less than 10 14 gm. It is obvious that such black-holes cannot emerge as end products of stellar evolution. However, Zeldovich and Novikov [17] and Hawking [18] argued that primordial black-holes (PBHs) of small mass can be generated from the space-time curvature, and subsequently, Carr [19] showed the possibility of creating PBHs from density fluctuations in the early universe. In the context of inflation, several authors have discussed mechanisms to produce PBHs using the general idea that bubble wall collisions may trap pockets of false vacuum region that subsequently collapse to form black-holes [20]. In a recent work, Nagatani [21] has proposed an interesting blackhole-electroweak mechanism of baryogenesis that requires the presence of a blackhole to create a domain wall around it, leading to genesis of baryon excess without the need of a first order electroweak phase transition. Previous paragraphs of this section indicate that although GUTs can naturally generate baryon asymmetry, any baryon excess generated prior to electro-weak era is erased due to sphaleron transitions, while at the same time, creation of baryon asymmetry solely due to electro-weak processes is fraught with uncertainties as well as the requirement of low Higgs mass, contrary to the experimental situation. Under the present circumstances, it is therefore natural to explore alternate means to explain matter-antimatter asymmetry. Since the existence of PBHs in the early universe is rather generic, one ought to carefully re-examine the mechanism of generating baryon asymmetry through black-hole evaporation. In such a scenario, the crucial point to investigate is whether PBHs survive after the EWPT has taken place, so that the baryon asymmetry created due to their subsequent Hawking evaporation survives, leaving an imprint till the present epoch. The present paper is an attempt to critically examine the evolution of the masses of a collection of PBHs created after the end of inflation, taking into account both the accretion of background matter by the black-holes as well as the mass loss due to Hawking emission. The paper has been organized in the following manner. In Section 2 we discuss processes responsible for the change in a black-hole's mass, and thereafter, we develop a formalism to describe accretion of relativistic matter by mini-black-holes. The subject of black-hole mass spectrum and its evolution is tackled next, in section 3, along with a discussion on the cosmological evolution of a mixture of PBHs and relativistic matter. Section 4 deals with the study of evolution equations numerically as well as a detailed analysis of the numerical solutions pertaining to the survival of PBHs past the EWPT. In section 5, we calculate baryon excess resulting from the decay of X −X bosons emitted by the PBHs during their final stages of Hawking evaporation, and then discuss the implications of these results to the question of matter-antimatter asymmetry. Finally, we end with a brief discussion of the above scenario in section 6. 2 Evolution of the mass of a black-hole Mass loss due to evaporation Bekenstein's conjecture [15] that the area of the event horizon of a black-hole being a measure of its entropy was vindicated by the classic work of Hawking in the early seventies who showed that when quantum effects around a black-hole are included, the black-hole emits particles with a thermal distribution corresponding to a temperature T BH that is proportional to the surface gravity at the event horizon [15] [16], and is given by the relation, T BH = m 2 pl 8πm c 2 k(1) where m and m pl are the mass of the black-hole and the Planck mass, respectively. According to eq. (1), PBHs created in the early universe with a mass ≈ 10 14 gm would be decaying today in a burst of high energy radiation, and there exists in the literature, upper bounds on the abundance of such PBHs from the observed level of cosmic γ-ray flux. As pointed out by Zeldovich and others, the expression in eq. (1) for the Hawking temperature can only be an approximation and is amenable to modifications at Planck scale because of the effects of quantum gravity. In fact, particle physicists have shown from various angles that Hawking evaporation may cease when the black-hole reaches the Planck mass scale leading to a massive relic. In this context, an interesting toy model inspired by superstring theories has been considered by Barrow et al [22] in which the expression for the black-hole temperature has been modified by including correction terms that contain powers of black-hole mass in units of Planck mass. Following Barrow et al's ansatz, one can therefore express the black-hole temperature as T BH = m pl 8π m pl m − κ m pl m n c 2 k (2) where κ is a non-negative constant. For n > 2 and κ ≈ O(1), it is clear that for holes of mass m ≫ m pl , eq. (1) is a limiting case of eq. (2). According to eq. (2), as the hole mass decreases due to evaporation, initially there is a rise in the hole's temperature but as m approaches m pl the temperature starts falling and becomes zero when the mass of the hole reaches the value m rel = κ 1/(n−1) m pl . Therefore, Barrow et al's ansatz implies stable black-hole relics of mass m rel ≈ m pl . To estimate the rate of mass-loss from eq. (2), we may work in the frame-work of radiative transfer, assuming that the hole's event horizon acts like a perfect black-body surface. In such a case, it is easy to show that the energy flux F is related to the energy density ε at the surface of interest in the following manner [23] F = c 4 ε(3) The effective energy-density of ultra-relativistic particles due to Hawking evaporation in the vicinity of the event horizon is related to the temperature of the black-hole by ε = π 2 g BH ⋆ 30 k 4 (hc) 3 T 4 BH (4) where g BH ⋆ = g BH b + (7/8)g BH f is the effective number of degrees of freedom at the temperature T BH , and g BH b and g BH f are the corresponding degrees of freedom for bosons and fermions, respectively. Therefore, the rate of massloss from the event-horizon is given by dm dt = − 1 c 2 F · 4πR 2 S (5) = −α 2 m 2 m pl m − κ m pl m n 4(6) where α 2 = g BH ⋆ c 2 /(30720πh) and κ ≈ 1. In arriving at eq. (6), we have made use of eqs. (2)-(4) as well as the standard result for the Schwarzschild radius R S = 2Gm/c 2 . The calculations that led to eq.(6) were based on modeling the black-hole event-horizon to be the surface of a black-body of radius R S at a thermodynamic temperature T BH . It is, therefore, interesting to compare our result with that of Don Page [24] which is based on rigorous numerical computations for black-holes of mass m > 10 17 gm. According to his calculations, the mass-loss rate for such holes is dm dt = −2.011 × 10 −4h c 4 G 2 m 2(7) If we assume that eq. (7) is valid also for 10 −2 gm < m < 10 17 gm, then comparing (6) and (7) one obtains g BH ⋆ ≈ 20, which is not too unreasonable since for holes of mass 10 −2 gm one expects g BH ⋆ to be as high as ≈ 100 (in most GUTs). Accretion of relativistic matter by a mini blackhole The temperature of the universe is expected to be extremely high just after the end of inflation, and therefore matter during that period will be in the form of ultra-relativistic particles. For particles with de Broglie wavelength λ ≪ R S , the capture cross-section corresponding to a Schwarzschild black- [25]. When the de Broglie wavelength of a particle is larger than R s , the capture cross-section is likely to be negligible as the blackhole sees an incident wave rather than a point particle. For high energy particles with λ ≪ R S , we will make use of the geometric optics approximation in which any such ultra-relativistic particle hitting a fictitious sphere of radius r c around the hole will be absorbed. hole is ∼ πr 2 c where r c = (3 √ 3/2)R S If I ν represents the specific intensity of such particles corresponding to energy hν and if dA is an area element on this fictitious sphere then the rate at which energy is accreted by the hole per unit range of ν per unit area is given by dE ν dtdνdA = dΩ cos θI ν = πI ν(8) Since the effective area of capture is 4πr 2 c , the rate at which energy is accreted in the frequency range [ν, ν + dν] is given by dE ν dt = (2πr c ) 2 I ν dν(9) To obtain the total rate of accretion of energy we integrate eq.(9) over frequency keeping in mind that geometric optics approximation requires the lower limit of integration ν min to be a few times c/r c . For ultra-relativistic particles, momentum is p ≈ hν/c so that the number density of particles of species A in the frequency-range (ν, ν + dν) takes the form n A (ν)dν = 4πg A c 3 ν 2 dν e hν/kT ± 1 (10) where g A is the spin-degeneracy factor for the A th species and the +(−) sign refers to fermions (bosons). In eq. (10) T is the temperature of the universe. Therefore, the specific intensity I ν A corresponding to the species A is given by [23] I ν A = chνn A (ν) 4π = g A c 2 hν 3 e hν/kT ± 1(11) Making use of eqs. (9) and (11), we can express the net rate of energy accretion by a hole in the following manner dE dt = 2πr c c 2 g uni b ∞ ν min hν 3 e hν/kT − 1 dν + g uni f ∞ ν min hν 3 e hν/kT + 1 dν(12) where ν min = α 1 c/r c is the lower frequency cut-off, α 1 being a number of the order of 10 (this takes care of the fact that only particles with λ ≪ R S are considered to have been captured by the blackhole). In eq.(12), g uni b and g uni f are the total bosonic and fermionic degrees of freedom, respectively, for the cosmic soup. These are to be distinguished from g BH b and g BH f introduced in section (2.1). The rate at which the hole's mass grows as a result of accretion is dm dt = 405 π 3 c 5 ε R G 2 m 2 g uni b g uni ⋆ ∞ x min x 3 e x − 1 dx + g uni f g uni ⋆ ∞ x min x 3 e x + 1 dx(13) where x min = hν min /kT . In obtaining the above equation, we have made use of a change of variable in eq.(12) along with r c = (3 √ 3/2)R S . We note that ε R appearing in eq.(13) is the energy-density of the background relativistic particles, ε R = π 2 g uni ⋆ (kT ) 4 /(30h 3 c 3 ), g uni ⋆ being the temperature-dependent effective spin-degeneracy factor and is equal to g uni b + 7/8g uni f . From eq.(13) it is evident that accretion plays an important role for massive PBHs at early epochs when the temperature of the universe is very high so that energy density ε R of the relativistic particles is large while x min is small. This is easy to understand from a physical point of view in the sense that only when the temperature is large that there are sufficient number of particles with de Broglie wavelength much less than the Schwarzschild radius of the PBHs, ready to be accreted. By the same token, when the hole-mass reaches a size of the order of m pl , neither accretion nor quantum evaporation is significant. Blackhole mass spectrum and evolution of the universe It is evident that the mass distribution of PBHs is intimately linked to the mechanism of their production. Several authors [19,20,26,27] in the literature have discussed blackhole mass spectrum from diverse angles. Since the mass spectrum is sensitive to production mechanisms and, since so far no particular model of PBH creation has been singled out, we adopt a very general procedure in this paper to analyse the evolution of blackhole mass spectrum. We consider a distribution function N(m, t) such that N(m, t)dm represents number-density of PBHs with mass in the range (m, m + dm) at the cosmic epoch t. We assume that the creation of PBHs stopped after a cosmic epoch t pbh so that at later times in a given comoving volume the number of holes remain the same while their masses change due to a combination of Hawking radiation and accretion of background matter. Note that we are working under the assumption that the ultimate state of a PBH along the course of its evolution is a stable relic of mass ≈ m pl , i.e. a hole does not disappear completely as the original Hawking radiation mechanism would demand. Also, since the mass m of a hole changes with time, the mass distribution function at time t and at time t + dt are related as a 3 (t)N(m, t)dm = a 3 (t + dt)N(m ′ , t + dt)dm ′(14) where m ′ is related to m through m ′ = m +ṁdt and a(t) is the FRW scalefactor at cosmic epoch t. Making a Taylor expansion of quantities at the RHS of eq. (14), and using the relation dm ′ = dm 1 + ∂ṁ ∂m dt (15) we obtain ∂N ∂t + 3ȧ a N + ∂ ∂m (Nṁ) = 0(16) With the help of the mass distribution function N(m, t), we can also obtain an expression for the mass-density associated with PBHs as ρ BH (t) = ∞ m rel mN(m, t)dm(17) It is useful to express the black-hole mass-distribution as N(m, t) = N 0 (t)f (m, t)(18)where N 0 ∝ a −3 (t) so that ∞ m rel f (m, t)dm is independent of time. With the help of eq.(18) it can be easily shown that eq.(16) reduces to ∂f ∂t + ∂ ∂m (ṁf ) = 0(19) Essentially, f (m, t)dm represents the number of black-holes with mass in the interval (m, m + dm) in a unit coordinate volume at the cosmic epoch t, while the dilution of black-hole number density due to the expansion of the universe is taken care of by the factor N 0 (t) = A/a 3 (t). Differentiating eq.(17) with respect to t and then making use of eqs. (18) and (19) it can be shown that dρ BH dt + 3ȧ a ρ BH = N 0 ∞ m relṁ f (m, t)dm(20) Since the total energy-momentum tensor is divergence free, we also have the equation [28]: c 2 d dt (ρ R + ρ BH )a 3 + 3p R a 2ȧ = 0(21) where ρ R = ε R /c 2 is the mass density of radiation. Here we have assumed that the black-holes possess negligible peculiar speeds so that their contribution to pressure is insignificant. Using p R = c 2 ρ R /3 and eq.(21) in eq. (20) we obtain: dρ R dt + 4ȧ a ρ R = −N 0 ∞ m relṁ f (m, t)dm(22) Eq (22) just reflects, as is to be expected, the fact that an effective black-hole mass loss (or gain) would imply ρ R ∝ a −4−α where α(t) is negative(positive) because of black-holes acting as source (sink) of radiation. In any mechanism of PBH production, the actual masses of the blackholes will be distributed in a discrete fashion, and therefore without loss of generality the distribution function can be expressed as f (m, t) = K i=1 β i δ(m − m i (t))(23) where β i are constant weights corresponding to m i , and K is the number of distinct black-hole masses. It can easily be ascertained that the distribution function in eq(23) indeed is a solution of eq (19), since ∂ ∂t δ(m − m i (t)) =ṁ i m − m i (t) δ(m − m i (t))(24) and ∂ ∂m [ṁf (m, t)] = − K i=1 β iṁ i m − m i (t) δ(m − m i (t))(25) Consequently, we have: dρ R dt + 4ρ Rȧ a = −N 0 (t) K i=1 β iṁi(26) The manner in which the individual mass m i of a PBH changes with time depends on the combination of Hawking evaporation rate and the accretion of background relativistic matter as discussed in section 2. Therefore, making use of eq (6) and eq (13) in the context of a PBH with mass m i we get the following result: dm i dt = 405 π 3 c 3 ρ R G 2 m 2 i g uni b g uni ⋆ ∞ x min x 3 dx e x −1 + g uni f g uni ⋆ ∞ x min x 3 dx e x +1(27)−α 2 m 2 i m pl m i − κ m pl m i n 4 From eqs. (17), (18) and (23), the mass density ρ BH associated with the PBHs can be written as: ρ BH (t) = N 0 (t) K i=1 β i m i (t)(28) The evolution of the scale-factor a(t) then follows from the flat FRW Einstein equation: ȧ a 2 = 8πG 3 ρ R + N 0 (t) K i=1 β i m i (t)(29) In writing down the above equation, we have adopted the inflationary paradigm according to which universe in the post-inflationary phase is described essentially by a flat FRW model. In this paper, the evolution of the universe is determined by three coupled differential equations (26), (27) and (29) along with the fact that N 0 (t) ∝ a −3 (t). Numerical Evolution In this section, we solve 2 + K coupled non-linear, first order differential equations (26), (27) and (29), set up in the preceding section, numerically using Hemming's fourth-order, double precision predictor-corrector method. To begin with, we fix N 0 (t) by demanding that β i m i (t 0 )N 0 (t 0 ) represents the initial fraction ζ i of total mass density ρ(t 0 ) that lies in blackholes having initial mass m i (t 0 ) so that, β i m i (t 0 )N 0 (t 0 ) = ζ i ρ(t 0 )(30) As N 0 (t) ∝ a −3 (t), we have, from eq(30), N 0 (t) = a 3 (t 0 ) a 3 (t) ζ i ρ(t 0 ) β i m i (t 0 )(31) Since N 0 (t) is independent of i, we obtain the following relation between ζ i and β i , ζ i ∝ β i m i (t 0 )(32) where the constant of proportionality in eq(32) can be determined from the following identity, ρ R (t 0 ) = ρ(t 0 ) − K i=1 ζ i ρ(t 0 )(33) leading to the following expression, proportionality constant = K i=1 β i m i (t 0 ) −1 1 − ρ R (t 0 ) ρ(t 0 )(34) Substituting eq(31) in eqs (26) and (29), we obtain, dρ R dt + 4ρ Rȧ a = − a 3 (t 0 ) a 3 (t) ρ(t 0 ) K i=1 ζ iṁ i m i (t 0 ) (35) ȧ a 2 = 8πG 3 ρ R + a 3 (t 0 ) a 3 (t) ρ(t 0 ) K i=1 ζ i m i (t) m i (t 0 ) ,(36) respectively. In our original formulation (see section 3), the blackhole initial massspectrum was completely specified by the set of numbers {β i , m i (t 0 ); i = 1, . . . K}. Equivalently, since β i and ζ i are related by equation (32), we may as well specify the spectrum by the set {ζ i , m i (t 0 ); i = 1, . . . K}. For the purpose of numerical evolution, it is convenient to cast equations (27) , (35) and (36) in terms of dimensionless quantities defined below τ = t Gρ 0 (37) α(τ ) = a(t) a 0 (38) R(τ ) = ρ R (t) ρ 0 α 4 (39) M i (τ ) = m i (t) m i (t 0 )(40) where ρ 0 ≡ ρ(t 0 ) and a 0 ≡ a(t 0 ). In terms of the above quantities, the system of differential equations assumes the following form: α ′ = 1 α 8π 3 (R + α K i=1 ζ i M i ) (41) R ′ = −α K i=1 ζ i M ′ i (42) M ′ i = m i (t 0 )M 2 i (Gρ 0 ) −1/2 (43)   405 π 3 c 3 G 2 ρ 0 Rα −4 J i − α 2 m pl m i (t 0 ) 1 M i − κ m pl m i (t 0 ) 1 M i n 4   where prime denotes differentiation with respect to τ and for convenience we have introduced J i ≡ J(x 0 i , T ) = g uni b g uni ⋆ ∞ x 0 i x 3 dx e x − 1 + g uni f g uni ⋆ ∞ x 0 i x 3 dx e x + 1 with x 0 i = h kT α 1 c r ci and r ci = 3 √ 3Gm i (t) c 2 We choose t 0 to be the cosmic-epoch when inflation ends ≈ 10 −33 sec., and set ρ 0 = 10 56 GeV 4 , which is the density expected at GUT scale. In our numerical evolution program, the actual values used for the following parameters are listed below: α 1 = 10 κ = 0.1 n = 3 g BH b = g BH f = g uni b = g uni f = 50 First we consider the case when K = 1, i.e. at the end of inflation a fraction ζ of matter lies in blackholes, all with initial mass m 0 = m 0 (t 0 ). We study different models by varying ζ in the range 10 −3 to 10 −1 while m 0 runs through the range 10 3 to 5×10 5 gm. In fig (1) we plot a(t) for a typical choice of ζ and m 0 . The plots of a(t) for a radiation-dominated (RD) FRW universe (a ∼ t 1/2 ) and a matter-dominated (MD) FRW universe (a ∼ t 2/3 ) are also given in the same figure. The initial behaviour of the system is that of a RD universe but soon the evolution of the scale-factor a becomes similar to that in an MD universe, and subsequently, as the PBHs evaporate, the dynamics becomes RD again. This is because initially energy density of relativistic matter gets depleted owing to accretion by PBHs, resulting in its decrease faster than the kinematic rate a −4 (see fig (1)) so that the dominant contribution from 'dust' like blackholes drives a faster expansion rate. We have also compared our results with approximate estimates obtained by assuming that a(t) ∼ t 2/3 from t = t 0 (end of inflation) to t = t EW P T (epoch of EWPT) and that a(t) ∼ t 1/2 afterwards. For the range of parameters 10 3 < m 0 < 10 5 (gm) and 0.001 < ζ < 0.1, the estimates agree with our numerical results to within an order of magnitude. In fig (2) we plot a typical mass m as a function of time. It is evident from the figure that growth of blackhole mass due to accretion takes place only in the initial period when the temperature and density of the universe is very high. This is anyway expected since the de Broglie wavelength λ of a typical particle just after the end of inflation is ∼ 10 −28 cm, while the R S for a blackhole of mass as low as ∼ 100 gm is ∼ 10 −26 cm leading to a substantial accretion because of λ < R S criteria. At intermediate times the curve flattens out reflecting a balance between accretion and Hawking evaporation. During this phase, the dynamics is essentially MD since radiation loses out in the competition because of the expansion of the universe as well as its attenuation due to accretion by PBHs. Towards the end, Hawking evaporation begins to dominate the evolution of PBH mass as the accretion automatically gets switched off due to the decrease in temperature and density of background radiation. In fig (3) we plot, for a typical choice of parameters ζ = 0.01 and m 0 = 2.5 × 10 5 , the ambient temperature of the universe T as well as the Hawking temperature T BH of the black-hole. The straight line portion of the curve has slope equal to −2/3. Thus T falls, at intermediate times, as if the dynamics of the universe was akin to that of a MD universe. At later times, when evaporation becomes the dominant process in the evolution of the holes, the universe at first starts cooling at a slower rate, but eventually reheats due to rapid evaporation of the blackholes (the reheat portion is not included in the figure). The point at which EWPT occurs is marked by an arrow in the figure, (i.e. T is ∼ 100 GeV at this instant of time) corresponding to a value of ∼ 2 × 10 −13 sec. We note that the epoch of EWPT is considerably lower than the standard value of ∼ 10 −10 sec obtained from the time-temperature relation in big-bang models. The reason for this is not hard to understand, as depletion of radiation by the accreting PBHs leads to a MD phase causing the T to decline faster than the usual t −1/2 fall. Now, the amount of reheating should be such that the temperature of the universe does not rise above the EWPT temperature (∼ 100 GeV), because, otherwise the sphaleron processes will be re-ignited, leading once again to a washing out of BAU generated. This, in effect, constrains our parameters ζ and m 0 . In fig (4) we plot the combination of ζ and m 0 for which the re-heat temperature is 100 GeV, and these points are empirically fitted with a curve. From the numerical evolution, we find that the region lying below to the right of the curve consists of those values of (ζ,m 0 ) for which reheat temperature remains below 100 GeV. While the region lying left of the curve consists of those combinations for which PBHs evaporate away, reaching the relic state before EWPT, and therefore are not of any use as far as baryogenesis is concerned. From fig (4) it is evident that for ζ lying in the interval (10 −5 ,0.1) a PBH with initial mass less than ∼ 2×10 5 gm converges to the relic state before EWPT, and hence does not contribute to generation of baryons. Therefore, we find that if the initial PBH mass spectrum is a deltafunction peaking at the mass m 0 , baryogenesis through blackhole evaporation is viable only when the initial mass of the PBHs exceeds ∼ 2 × 10 5 gm for reasonably low values of ζ. Next, we consider the case in which blackhole masses at time t 0 are distributed in a pseudo-Maxwellian manner as shown in fig (5). The blackhole masses fall in a range from 8 × 10 2 gm to 3 × 10 5 gm, with 22 distinct mass values contributing to a total fraction 22 i ζ i ∼ 0.09 of the mass density of the universe just after the end of inflation. From the plot fig (6) it is apparent that blackholes with larger initial mass accrete background hot matter at higher rates than those with smaller initial mass, as expected from the fact that higher mass PBHs have larger cross-section for absorbing matter. We find that those PBHs with initial mass greater than 1.5×10 5 gm reach X −X emitting phase after the epoch 1.9 × 10 −11 sec, the instant at which EWPT takes place for this spectrum of masses. Once again we find that EWPT oc-curs sooner than that in the standard model. There are 7 of such blackhole masses which finally contribute to the production of baryon excess. Because of the wide distribution of blackhole masses, the instants at which the PBHs reach the relic mass are staggered, hence no sharp re-heating takes place in our analysis, rather the temperature of the universe falls at a slower rate till the largest size blackhole (with initial mass = 3×10 5 gm) evaporates, leaving behind a relic mass around the epoch ∼ 3 × 10 −9 sec, when the temperature of the universe is ∼ 9 GeV. The decline of temperature with time is shown in fig (7). Even in the case of blackhole mass distribution, K being larger than 1, in principle, one can constrain the parameter space (ζ i , m i (t 0 )) from the requirement of re-heating less than 100 GeV (as undertaken when K = 1, see fig (4)), however the exercise is enormously time consuming, and is beyond the scope of the present paper. Baryogenesis We saw in the previous section that for ζ ≈ 0.01, blackholes created with mass less than ≈ 2 × 10 5 gm evaporate and reach the relic state before the EWPT and hence their contribution to baryon asymmetry is doubtful due to the expected B-violation induced by sphalerons. However, blackholes with initial mass larger than ≈ 2.5 × 10 5 gm certainly ought to be considered as sources of baryogenesis since they reach the X −X emission phase well past the EWPT. In this section, we proceed to estimate the quantity of excess baryons resulting from blackholes whose Hawking temperature reaches GUT scale after EWPT. Representing the specific intensity of X-bosons radiated with energy hν from a blackhole by I X ν , we have the relation (e.g., see [23]) I X ν = u X ν (Ω) v(44) where u X ν (Ω) is the specific energy density and v is the speed of the emanating X-bosons. With v = c   1 − mc 2 hν 2   1/2(45) and u X ν (Ω) = hν 3 c 4 vg X e hν/kT BH (m) − 1(46) we may express I X ν as I X ν = hν 3 c 2   1 − mc 2 hν 2   g X e hν/kT BH (m) − 1(47) It is to be noted that g X and T BH (m) are the spin degeneracy factor of X-bosons and Hawking temperature of a blackhole of mass m, respectively. The flux-density of X-bosons at a distance r from the blackhole is given by F X ν = πI X ν R 2 S r 2(48) Therefore, from equations (47) and (48) the rate of emission of X-bosons from a blackhole of mass m is derived to be dN X (m) dt = ∞ m X c 2 /h F X ν · 4πr 2 dν hν (49) = 4πR 2 S c 4 g X m 3 X h 3 I(y i )(50) where m X is the mass of the X-boson and I(y i ) = ∞ 1 y 2 − 1 e y/y i − 1 dy (51) while y i (t) ≡ kT BH (m i (t)) m X c 2(52) Since, at any given cosmic epoch t, the number density of blackholes with mass lying in the interval (m, m + dm) is N 0 (t)f (m, t) (see eq (18)), the rate at which X andX bosons are generated in a unit proper volume is given by dn XX dt = 2N 0 (t) dN X (m) dt f (m, t)dm(53) In eq (53) the factor 2 arises because we have included production ofXbosons as well. Making use of the form given in eq (23) we can express eq (53) as dn XX dt = 2N 0 (t) K i=1 β i dN X (m i ) dt(54) The lifetime of a X-boson τ X = Γ −1 X turns out to be ≈ 10 −36 sec when m X ≈ 10 14 GeV [29] which is negligible in comparison with the time scales over which blackhole mass changes or the universe expands appreciably. Hence, the rate of increase of net baryon number in a unit proper volume is ≈ ǫ dn XX dt with ǫ ≡ Γ(X → ql) − Γ(X →ql) Γ tot(55) being the net baryon number generated by the decay of a pair of X andX [30]. If n B (t) represents net baryon number density at the cosmic epoch t then d dt (a 3 (t)n B (t)) = ǫa 3 (t) dn XX dt(56) Employing eq (31) and (54) in (56) and then integrating the latter, we obtain a 3 (t)n B (t) − a 3 (t EW P T )n B (t EW P T ) = 2ǫ(ρ 0 a 3 0 ) K i=1 ζ i m i (t 0 ) t t EW P T dN X (m i ) dt ′ dt ′ (57) Assuming that prior to blackhole baryogenesis, the net baryon number in the universe is zero (i.e. n B (t EW P T ) = 0) and making use of eq (50) in eq (57), we get the following expression for the net baryon number density at any time, n B (t) = ρ 0 a 3 0 a 3 (t) 4G 2 πh 3 · ǫg X m 3 X · K i=1 ζ i m i (t 0 ) t t EW P T dt ′ m 2 i (t ′ )I(y i (t ′ )) (58) After EWPT has taken place, the evolution of PBH mass is totally dominated by eq (6) since the de Broglie wavelength λ of a typical particle is larger than ∼ 10 −15 cm, while the R S corresponding to a blackhole of mass as high as ∼ 10 7 gm is only ∼ 10 −21 cm. Hence, using eq (6) we can change the variable of integration in eq(58) from t ′ to m i (t ′ ) so that, t t EW P T m 2 i (t ′ )I(y i (t ′ ))dt ′ = 2m 4 pl α 2 (8πm X ) 4 H(m i (lower), m i (t EW P T ))(59) where H is defined to be, H(m i (lower), m i (t EW P T )) ≡ m i (t EW P T ) m i (lower) 1 y 2 i y i ∞ k=1 e −k/y i k 3 + ∞ k=1 e −k/y i k 2 dm i (60) In obtaining eqs (59) and (60) we have used the series equivalent of the integral given in eq(51).The value of m i (lower) is set by requiring y i to be 10 −3 since the series given in eq(60) is negligibly small for smaller values of y i . This automatically takes into account the fact that only those PBHs matter for BAU that are capable of emitting X −X after EWPT. For PBH masses larger than 10 5 m pl , the value of H is 2.7 × 10 −2 and becomes insensitive to the exact value of m i (t EW P T ) thereafter. Therefore, for the 7 PBHs that survive the EWPT, we have, 22 i=16 ζ i m i (t 0 ) H(m i (lower), m i (t EW P T )) = 1.97 × 10 −9(61) The entropy density of the universe at any epoch t is given by, s = 2π 2 45 g uni ⋆ k 4 (hc) 3 T 3 (t)(62) We estimate the baryon-to-entropy ratio at t ∼ 3 × 10 −9 sec, when all the PBHs settle on to the relic state, by making use of eqs(58),(59),(60),(61) and (62), n B (t) s(t) = 7.5 × 10 −8 ǫg X g BH ⋆ 100 −1 g uni ⋆ 100 −1(63) The contribution to baryon-to-entropy ratio by PBHs with initial mass m 0 and initial mass fraction ζ goes roughly as, n B s ≈ ǫζg X m 0 1 gm −1 g BH ⋆ 100 −1 g uni ⋆ 100 −1(64) Hence, in the case of a delta-function mass spectrum with ζ ≈ 0.01 and m 0 ≈ 2.5 × 10 5 gm, one obtains a baryon-to-entropy ratio of ≈ 4 × 10 −8 ǫ, with g X = 1. Thus one may use eq(64) along with the value of n B /s ≈ 10 −11 , that follows from observations, to put a constraint on ǫζ/m 0 . This implies that one requires the CP-violating parameter ǫ to be around ∼ 10 −4 to generate excess baryons from evaporating PBHs. Discussions To study the evolution of PBHs, in the early universe, that undergo accretion along with steady mass loss due to Hawking evaporation, we have laid down a formalism which can handle any blackhole mass spectrum that can be decomposed as a sum of weighted δ-functions. Accretion of ambient hot matter by a blackhole has been modeled in the limit of geometric approximation, so that only those particles with de Broglie wavelength less than about a tenth of Schwarzschild radius are considered for absorption by the blackhole. The evolution of a flat FRW universe and the PBHs has been studied numerically to find conditions under which blackholes survive past the electroweak phase transition in order that their subsequent evaporation leads to baryogenesis. The basic picture which emerges is the following. In the case of a blackhole mass spectrum that peaks sharply at a single mass value m 0 , when ζ (the initial mass fraction of PBHs) is of the order of 1%, PBHs with initial mass m 0 less than about 2.3 × 10 5 gm evaporate before EWPT. Therefore, only PBHs with m 0 greater than this critical value need be considered for generation of BAU. Here, we wish to point out that the model of accretion which one considers can make an immense difference in the final result of the analysis. If one uses a simple spherical model of accretion in which the capture-cross section is just πR 2 S and with no de Broglie wavelength based cutoff then blackholes of initial mass m 0 ≈ 10 3 gm can successfully live past the EWPT, and eventually contribute to the BAU (see Majumdar et al. in [14]). While on using the same set of parameters with a wavelength based cutoff model of accretion, we find that PBHs of such small initial mass do not survive beyond the EWPT. For reasonable choice of parameters, we find that in the case of PBHs with a distribution of mass ranging from 8 × 10 2 -3 × 10 5 gm, blackholes with initial mass larger than about ∼ 10 5 gm reach the relic state much after EWPT. Because of the presence of blackholes with mass less than 10 5 gm that evaporate at a faster rate, pumping in energetic particles into the surrounding medium, the ambient temperature in this case declines at a slower rate, and hence EWPT takes place later than in the case when all PBHs had the same mass of 2.5 × 10 5 gm. As described in sections (3) and (4), the evolution of mass spectrum is totally determined by the manner in which individual blackhole masses change with time, β i or equivalently ζ i remaining fixed for all times. As an illustration, we have shown the evolution of mass spectrum in fig (6) for a particular set of ζ i . We wish to point out that accretion is important only during the initial stages just after the end of inflation when the temperature of the universe is ∼ 10 13 GeV, causing an increase in the mass of a blackhole by a factor of ∼ 4. There are two factors responsible for a blackhole of initial mass of ∼ 10 5 gm to live after the EWPT. One being the increase in the mass due to accretion, while the other is the occurance of EWPT sooner than that in a model in which there is no depletion of radiation due to PBHs acting as sinks. For blackholes with mass less than ∼ 10 5 gm, accretion is less due to the reduction in capture cross-section because of which the rate of depletion of radiation is not large leading to a delayed occurance of EWPT, after the blackholes have reached the final relic state. Barrow et al's ansatz [22] which has been used in this paper to take into account expected modification of Hawking emission becomes important only when the mass of evaporating blackholes fall below ∼ 10m pl . Our numerical results are not sensitive to the exact form of modified blackhole temperature. For baryogenesis, significant quantity of X −X are emitted only during the phase when blackhole temperature is ∼ T GU T , because of which the integral H is not sensitive to the upper limit m(t EW P T ) so long as the latter is larger than 10 5 m pl . Therefore, the final expression for baryon-to-entropy ratio turns out to be rather simple (see eq (64)), implying that if at the end of inflation 1% of total matter goes into creating PBHs with initial mass 2.5×10 5 gm then this scenario can successfully lead to BAU provided the CP-violating parameter ǫ is over 10 −4 . Thus production of baryon excess through blackhole evaporation is a viable alternative to GUTs or electroweak baryogenesis, although there is no denying that because of the presence of parameters like ζ i and m i (t 0 ) whose values a priori are uncertain, this scenario cannot provide meaningful constraint on the value of ǫ. Figure Captions Figure 1 : 1The evolution of the scale-factor a(t) for ζ = 0.01, m 0 = 2.5 × 10 5 gm. The plots a ∼ t 1/2 and a ∼ t 2/3 are provided for comparision. Figure 2 : 2The evolution of the mass m(t) of the PBHs for a typical choice ζ = 0.01, m 0 = 2.5 × 10 5 gm. Figure 3 :Figure 4 :Figure 5 : 345The temperature T of the background thermal bath and the Hawking temperature T BH for a typical choice ζ = 0.01 and m 0 = 2.5 × 10 5 gm. The instant of EWPT is marked by an arrow. The combinations ζ and m 0 for which the reheat temperature = T EW P T = 100 GeV. The region with acceptable reheat temperatures < 100 GeV is indicated in the figure. The analytical fit with dotted line is purely empirical. Black-hole mass spectrum: plot of ζ i against m i (t 0 ). Figure 6 : 6The evolution of the masses m i (t) of a collection of PBH masses distributed according to the spectrum shown in fig (5). Figure 7 : 7The cooling of the universe for the case where PBH masses are distributed according to the spectrum displayed in fig (5). The epoch of EWPT is marked by an arrow, and it takes place at 1.9 × 10 −11 sec. If neutrinos are massive then the gravitating mass may as well be dominated by leptons. However, there is hardly any direct measure of the lepton number of the universe. AcknowledgementsWe wish to thank Dr. Amitabha Mukherjee and Dr. Archan Majumdar for useful suggestions. It is a pleasure to thank Harvinder Kaur Jassal, Hatem Widyan and Abha Dev for going through the LATEX file of this paper carefully. One of us (NU) would like to thank the University Grants Commission, New Delhi, for financial support. . G Steigman, Ann. Rev. Astr. Ap. 14339G. Steigman, Ann. Rev. Astr. Ap 14, 339 (1976) . A De Rujula, Nucl. Phys. B(Proc.Suppl.). 48514A. de Rujula, Nucl. Phys. B(Proc.Suppl.)48, 514 (1996) . K A T Olive &amp; S, Scully Int, Jour. of Mod. Phy. A. 11409K. A. Olive & S. T. Scully Int. Jour. of Mod. Phy. A 11, 409 (1996) . David Tytler, Xiao - , Ming Fan, Scott Burles, Nature. 381207David Tytler, Xiao-Ming Fan and Scott Burles, Nature 381, 207 (1996) . A D Sakharov, JETP Letters. 524A. D. Sakharov, JETP Letters 5, 24 (1967) CP -violation and baryogenesis. R D Peccei, CP -violation in Particle Physics & Astrophysics. J. Tran Thanh VanR. D. Peccei, "CP -violation and baryogenesis", CP -violation in Particle Physics & Astrophysics, ed. J. Tran Thanh Van, Editions Frontieres, (1989); . A Masiero, R N D Mohapatra &amp; R, Peccei, Phys. Lett. B. 108111A. Masiero, R. N. Mohapatra & R. D. Peccei, Phys. Lett. B 108, 111 (1982) . E W For, A D Kolb, &amp; A Linde, Riotto, Phys. Rev. Lett. 774290For example, see E. W. Kolb, A. D. Linde & A. Riotto, Phys. Rev. Lett 77, 4290 (1996) . V A Kuzmin, V A Rubakov, M E , Shaposhnikov Phys. Lett. 15536V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov Phys. Lett. B155, 36 (1985) Electro-weak Baryogenesis, hep-ph/9803479. To appear in Rev. of Mod. Phys. Mark Trodden, Mark Trodden, Electro-weak Baryogenesis, hep-ph/9803479. To appear in Rev. of Mod. Phys. October 1999. . A S Majumdar, S K Sethi, S Mahajan, A Mukherjee, N Panchapkesan, R P Saxena, Mod. Phys. Letters A. 9459A. S. Majumdar, S. K. Sethi, S. Mahajan, A. Mukherjee, N. Panchapkesan and R. P. Saxena, Mod. Phys. Letters A 9, 459 (1994); . K Kajantie, M Laine, K Rummukainen, &amp; M Shaposhkinov, Phys. Rev. Lett. 772887K. Kajantie, M. Laine, K. Rummukainen & M. Sha- poshkinov Phys. Rev. Lett. 77 2887 (1996); . M Gurtler, E M Ilgenfritz, &amp; A Schiller, Phys. Rev. D. 563888M. Gurtler, E. M. Il- genfritz & A. Schiller Phys. Rev. D 56 3888 (1997) Particle Physics Booklet, American Institute of Physics. Particle Data GroupParticle Data Group, Particle Physics Booklet, American Insti- tute of Physics, (July 1996) . K Kajantic, M Laine, K Rammukainen, M Shaposhnikov, Nucl. Phys. B. 466189K. Kajantic, M. Laine, K. Rammukainen and M. Shaposhnikov, Nucl. Phys. B 466, 189 (1996) . . B Ya, Zeldovich, JETP Lett. 2425Ya. B. Zeldovich, JETP Lett. 24, 25 (1976); Barrow. J , Mon. Not. Roy. Astr. Soc. 192427J. D. Bar- row, Mon. Not. Roy. Astr. Soc. 192, 427 (1980); . D Lindley, Mon. Not. Roy. Astr. Soc. 198317D. Lind- ley, Mon. Not. Roy. Astr. Soc. 198, 317 (1981); . D Lindley, Mon. Not. Roy. Astr. Soc. 199775D. Lindley, Mon. Not. Roy. Astr. Soc. 199, 775 (1982); . J D Barrow, E J Copeland, E W R Kolb &amp; A, Liddle, Phys. Rev. D. 43984J. D. Barrow, E. J. Copeland, E. W. Kolb & A. R. Liddle, Phys. Rev. D 43, 984 (1991); . A S Majumdar, P Gupta, &amp; R P Saxena, Int. Jour. of Mod. Phys. D. 4517A. S. Majumdar, P. Das Gupta & R. P. Saxena, Int. Jour. of Mod. Phys. D 4, 517 (1995) . L D Bekenstein, Phys. Rev. D. 72333L. D. Bekenstein, Phys. Rev. D 7, 2333 (1973) . S W Hawking, Nature. 24840S. W. Hawking, Nature 248, 40 (1974); . S W Hawking, Comm. Math. Phys. 43199S. W. Hawking, Comm. Math. Phys. 43, 199 (1975) . B Ya, D Zeldovich &amp; I, Novikov, Soviet Astronomy -AJ. 10602Ya. B. Zeldovich & I. D. Novikov, Soviet Astronomy -AJ 10, 602 (1967); . S W Hawking, Mon. Not. R. Astr. Soc. 15275S. W. Hawking, Mon. Not. R. Astr. Soc. 152, 75 (1971) . B J Carr, Astr. Jour. 2011B. J. Carr, Astr. Jour. 201, 1 (1975) . K Sato, M Sasaki, H Kodama, &amp; K Maeda, Prog. Theor. Phys. Progress Letters. 651443K. Sato, M. Sasaki, H. Kodama & K. Maeda, Prog. Theor. Phys. Progress Letters 65, 1443 (1981); . H Kodama, M Sasaki, K Sato, &amp; K Maeda, Prog. Theor. Phys. 662052H. Kodama, M. Sasaki, K. Sato & K. Maeda, Prog. Theor. Phys. 66, 2052 (1981); . H Kodama, M Sasaki, &amp; K Sato, Prog. Theor. Phys. 681979H. Ko- dama, M. Sasaki & K. Sato, Prog. Theor. Phys. 68, 1979 (1982); . S W Hawking, I G Moss, &amp; J M Stewart, Phy. Rev. D. 262681S. W. Hawking, I. G. Moss & J. M. Stewart, Phy. Rev. D 26, 2681 (1982); . L J D H Hall &amp; S, Hsu, Phys. Rev. Lett. 642848L. J. Hall & S. D. H. Hsu, Phys. Rev. Lett. 64, 2848 (1990) . Y Nagatani, Phy. Rev. D. 5941301Y. Nagatani, Phy. Rev. D 59, 041301 (1999) . J D Barrow, E J R Copeland &amp; A, Liddle, Phys. Rev. D. 46645J. D. Barrow, E. J. Copeland & A. R. Liddle, Phys. Rev. D 46, 645 (1992) G B P Rybicki &amp; A, Lightman, Radiative processes in Astrophysics. John Wiley & SonsG. B. Rybicki & A. P. Lightman, Radiative processes in Astro- physics, John Wiley & Sons (1979) . Don Page, Phys. Rev. D. 13198Don Page, Phys. Rev. D 13, 198 (1976) L D M Landau &amp; E, Lifshitz, The Classical Theory of Fields. Pergamon PressL. D. Landau & E. M. Lifshitz, The Classical Theory of Fields, Pergamon Press, (1987) . I D Novikov, A G Polnarev, A A Starobinsky, Ya, Astron. Astrophys. 80B. ZeldovichI. D. Novikov, A. G. Polnarev, A. A. Starobinsky & Ya. B. Zel- dovich, Astron. Astrophys. 80, 104 (1979) . C R Evans, &amp; J S Coleman, Phys. Rev. Lett. 721782C. R. Evans & J. S. Coleman, Phys. Rev. Lett. 72, 1782 (1994) S Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & SonsS. Weinberg, Gravitation and Cosmology: Principles and Appli- cations of the General Theory of Relativity, John Wiley & Sons (1972) . E W Kolb, A &amp; I I Riotto, Tkachev, hep-ph/9801306E. W. Kolb, A. Riotto & I. I. Tkachev hep-ph/9801306 Figure 1: The evolution of the scale-factor a(t) for ζ = 0.01, m 0 = 2.5× 10 5 gm. t 2/3 are provided for comparisionFigure 1: The evolution of the scale-factor a(t) for ζ = 0.01, m 0 = 2.5× 10 5 gm. The plots a ∼ t 1/2 and a ∼ t 2/3 are provided for comparision. Figure 2: The evolution of the mass m(t) of the PBHs for a typical choice ζ = 0.01, m 0 = 2.5 × 10 5 gm. Figure 2: The evolution of the mass m(t) of the PBHs for a typical choice ζ = 0.01, m 0 = 2.5 × 10 5 gm. The temperature T of the background thermal bath and the Hawking temperature T BH for a typical choice ζ = 0.01 and m 0 = 2.5 × 10 5 gm. The instant of EWPT is marked by an arrow. Figure. 3Figure 3: The temperature T of the background thermal bath and the Hawking temperature T BH for a typical choice ζ = 0.01 and m 0 = 2.5 × 10 5 gm. The instant of EWPT is marked by an arrow. The region with acceptable reheat temperatures < 100 GeV is indicated in the figure. The analytical fit with dotted line is purely empirical. The combinations ζ and m 0 for which the reheat temperature = T EW P T = 100 GeV. Figure 4: The combinations ζ and m 0 for which the reheat temperature = T EW P T = 100 GeV. The region with acceptable reheat temperatures < 100 GeV is indicated in the figure. The analytical fit with dotted line is purely empirical. Black-hole mass spectrum: plot of ζ i against m i (t 0 ). Figure. 5Figure 5: Black-hole mass spectrum: plot of ζ i against m i (t 0 ). Figure 6: The evolution of the masses m i (t) of a collection of PBH masses distributed according to the spectrum shown in fig. Figure 6: The evolution of the masses m i (t) of a collection of PBH masses distributed according to the spectrum shown in fig (5). The cooling of the universe for the case where PBH masses are distributed according to the spectrum displayed in fig (5). The epoch of EWPT is marked by an arrow, and it takes place at 1.9 × 10 −11 sec. 7Figure 7: The cooling of the universe for the case where PBH masses are distributed according to the spectrum displayed in fig (5). The epoch of EWPT is marked by an arrow, and it takes place at 1.9 × 10 −11 sec.
[]
[ "Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools", "Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools" ]
[ "Takeshi Ogasawara *[email protected] \nIBM Research-Tokyo\nTokyoJapan\n", "Yinhe Cheng \nIBM Systems\nAustinTXUnited States of America\n", "Tzy-Hwa Kathy Tzeng \nIBM Systems\nPoughkeepsieNYUnited States of America\n" ]
[ "IBM Research-Tokyo\nTokyoJapan", "IBM Systems\nAustinTXUnited States of America", "IBM Systems\nPoughkeepsieNYUnited States of America" ]
[]
This paper introduces a high-throughput software tool framework called sam2bam that enables users to significantly speed up pre-processing for next-generation sequencing data. The sam2bam is especially efficient on single-node multi-core large-memory systems. It can reduce the runtime of data pre-processing in marking duplicate reads on a single node system by 156-186x compared with de facto standard tools. The sam2bam consists of parallel software components that can fully utilize multiple processors, available memory, highbandwidth storage, and hardware compression accelerators, if available. The sam2bam provides file format conversion between well-known genome file formats, from SAM to BAM, as a basic feature. Additional features such as analyzing, filtering, and converting input data are provided by using plug-in tools, e.g., duplicate marking, which can be attached to sam2bam at runtime. We demonstrated that sam2bam could significantly reduce the runtime of next generation sequencing (NGS) data pre-processing from about two hours to about one minute for a whole-exome data set on a 16-core single-node system using up to 130 GB of memory. The sam2bam could reduce the runtime of NGS data preprocessing from about 20 hours to about nine minutes for a whole-genome sequencing data set on the same system using up to 711 GB of memory.
10.1371/journal.pone.0167100
null
135,526
1608.01753
74647dd35bad63039aa77da0956a80d7d2087d70
Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools Takeshi Ogasawara *[email protected] IBM Research-Tokyo TokyoJapan Yinhe Cheng IBM Systems AustinTXUnited States of America Tzy-Hwa Kathy Tzeng IBM Systems PoughkeepsieNYUnited States of America Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools RESEARCH ARTICLE This paper introduces a high-throughput software tool framework called sam2bam that enables users to significantly speed up pre-processing for next-generation sequencing data. The sam2bam is especially efficient on single-node multi-core large-memory systems. It can reduce the runtime of data pre-processing in marking duplicate reads on a single node system by 156-186x compared with de facto standard tools. The sam2bam consists of parallel software components that can fully utilize multiple processors, available memory, highbandwidth storage, and hardware compression accelerators, if available. The sam2bam provides file format conversion between well-known genome file formats, from SAM to BAM, as a basic feature. Additional features such as analyzing, filtering, and converting input data are provided by using plug-in tools, e.g., duplicate marking, which can be attached to sam2bam at runtime. We demonstrated that sam2bam could significantly reduce the runtime of next generation sequencing (NGS) data pre-processing from about two hours to about one minute for a whole-exome data set on a 16-core single-node system using up to 130 GB of memory. The sam2bam could reduce the runtime of NGS data preprocessing from about 20 hours to about nine minutes for a whole-genome sequencing data set on the same system using up to 711 GB of memory. Introduction The rapid advance of sequencing technology and its falling cost are driving the use of nextgeneration sequencing (NGS) in a great variety of domains. The volume of data generated by NGS was projected to double every five months [1]. It is highly desirable to be able to quickly process the raw data from a sequencer to the format (e.g. VCF, CNV) that is ready to be integrated with knowledge base and other data sources for further analysis. Typical 50x coverage of whole genome sequencing (WGS) can easily generate up to 500-GB FASTQ files. The data processes on modern computers are still very time-consuming for such huge data sets. The processing time for the Broad's Genome Analysis Toolkit (GATK) Best Practices pipeline [2] from reference alignment to variant calling can take up to a day or days to finish [3] for typical 50x coverage of WGS data. Genome data analysis pipelines involve data pre-processing steps before variant calling, which are necessary to achieve accurate variant calling. They scan input data, analyze reads, and filter out reads that can affect accuracy. The pre-processing steps take FASTQ-format [4] files as input and produce compressed binary files in BAM formats [5], which are widely accepted as common file formats to represent aligned sequenced data. The calibrated BAM files are then used in variant discovery to identify sites where the data display variations that are relative to the reference genome. The pre-processing steps for SAM parsing, sorting, duplicate marking, and BAM file compression can take tens of hours for a WGS SAM file. Their total runtime dominates the whole pre-processing workflow (explained in the next sub-section) and is a clear performance bottleneck. The purpose of sam2bam is to improve the efficiency of this pre-processing through fully utilizing available CPUs and memory. The overall architecture of the tools should be redesigned so that computer resources are fully utilized to significantly reduce the runtime of such data pre-processing steps (e.g., by 100x). Current major software tools are single-or partially multi-threaded. Partially multithreaded tools usually have to wait for data generated from single-threaded components because every component (single-or multi-threaded) is executed one by one. Therefore, they can not fully utilize multiple CPUs that are available at all times. Single-threaded components become bottlenecks in performance on multi-CPU systems. For example, let us assume that 80% of the runtime is executed by multi-threads and the remaining time is executed by a single thread. Speed-up for such tools by using multiple CPUs is limited to 5x even if hundreds of CPUs are available on the system. The sam2bam simultaneously executes functional components (e.g., file I/O, SAM parsing, and data compression) to achieve further speed-ups on such many-CPU systems, instead of executing the components one-by-one in a large loop. The components are combined as a pipeline. In addition, most steps are multi-threaded. An appropriate number of CPUs are allocated to each component so that no components become a bottleneck in the pipeline. The sam2bam can achieve more than 100x speed-up on a single node system with these redesigned frameworks for NGS data pre-processing. Pre-processing Steps for Variant Discovery Pre-processing steps are necessary to prepare data for analysis to maximize the accuracy of variant discovery. Pre-procesing is also recommended in GATK Best Practices [6]. Pre-processing starts with FASTQ-format files and ends in a calibrated BAM file. Pre-processing for the DNA data usually involves five steps. 1. Mapping sequence reads to reference genome This step is usually done by BWA mem [7] or other reference alignment tools and SAM files are generated. 2. Sorting sequence reads based on coordinates This step is usually done by Picard SortSam or samtools sort. Some tools that are used in the following steps, such as when the Picard MarkDuplicates tool requires the sorted input files. 3. Marking duplicate alignments This step is commonly done by the Picard MarkDuplicates tool to remove the alignments of duplicate reads. 4. Performing local realignment around indels This is usually done by GATK RealignerTar-getCreator and IndelRealigner tools to reduce artifacts produced in regions around the indels. 5. Recalibrating the base quality score This step is usually done by GATK BaseRecalibrator and PrintReads tools to improve the accuracy of base quality scores that the variant calling step relies on. Preprocessing is very time-cosuming that usually requires tens of hours for a WGS dataset and hours for a whole exome (WEX) dataset. The sorting and duplicate marking steps took most of the runtime for the five steps of preprocessing, which usually range from 60-70% of the total pre-processing time, depending on the tools that were used and the test case size. Therefore, the sorting and duplicate marking steps were identified as a bottleneck in the overall pre-processing steps, and sam2bam was focused on improving the performance of these two steps by redesigning the framework to take advantage of multiple CPU cores, large-capacity memory, and hardware accelerators that are available on modern computers. Design and Implementation The main design goal of sam2bam was to provide a high-throughput framework to process genome files at a rate of gigabytes per second (GB/s). The framework consists of a data pipeline that converts the data format from SAM to BAM as outlined in Figs 1 and 2. Data format conversion is divided into multiple steps. Many of these steps are multi-threaded, while a few steps that order the data stream are single-threaded. More CPUs are allocated for more complex steps so that such steps are not bottlenecks in performance. Each step continuously processes data by using CPUs as long as the data are driven from the previous step. While sam2bam provides a data processing pipeline, it uses the samtools/high-throughput sequencing library (HTSLIB) [5] for the data structures and utility functions to handle the SAM and BAM data formats. Plug-in Codes Plug-in codes that analyze, filter, and modify data can be attached to sam2bam at runtime. We can develop the plug-in codes and run them on the high-throughput framework. There are three types of plug-in codes. • Filter The filter plug-in code can be inserted as an additional step of the pipeline (Fig 3). It analyzes input data and determines if the data meet the criteria that the plug-in has. For example, with a filter plug-in that only includes read alignments that overlap a given region of the reference genome, the BAM file that is produced only includes alignments that overlap the specified region. • Accelerator The accelerator plug-in code improves the target pipeline step by using hardware accelerators, such as field-programmable gate arrays (FPGAs). Compression acceleration is currently supported in sam2bam. It produces a compressed BAM file using the standard compression library (or zlib) [8]. If sam2bam detects an accelerator, it automatically offloads compression to hardware while carrying out compression with software (Fig 4). An accelerator that provides the same application programming interface (API) as zlib can be enabled by using the accelerator plug-in. • Analyzer The analyzer plug-in code analyzes a set of read alignments and modifies them. If the analyzer plug-in code is attached to sam2bam, the latter runs the first half of the pipeline that parses data in the SAM format and pools the alignment information in the system. How the alignment information is pooled will be explained in the next subsection (Pipeline Configuration: items 4 and 5). When all alignment information is pooled in the system, the analyzer plug-in scans the pooled information, analyzes it, and generates output on the basis of the analysis (Fig 5). When the analyzer plug-in has completed analysis, the pooled data are Architecture for sam2bam with analyzer plug-ins. Alignment database is created when analyzer plug-ins are enabled. Binary alignments that are produced by SAM parsing are placed in either main memory or external storage so that they can later be used for generating compressed BAM files by using second half of pipeline. Alignment database has summarized information on binary alignments that is used by analyzer plug-ins. transferred to the second half of the pipeline to produce a compressed BAM file. For example, with an analyzer plug-in for marking duplicate alignments, the alignments that were mapped to the same reference region are identified and are marked as duplicates except for the one that has the highest level of quality. Pipeline Configuration The entire process of sam2bam conversion is split into functional stages such as file reading, SAM line detection, and SAM parsing. The stages run in parallel. Each stage can also process multiple data blocks in parallel. Each filter plug-in transfers binary alignments that meet criteria of filter to go to next step. sam2bam can use multiple filter plug-ins at a time. Each plug-in is executed by multiple threads. In this example, SAM parsing generates three binary alignments. Filter 1 first filters out Alignment 1, and then Filter 2 filters out Alignment 2. doi:10.1371/journal.pone.0167100.g003 Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools 1. Reading a SAM file Data blocks (e.g., 64 KB each) are read from a SAM file, which is a sequence of text lines (called SAM lines). Data blocks do not always end at the boundaries between SAM lines. This step adjusts the boundaries between the blocks so that the new blocks end at the SAM line boundaries by scanning each block from the end to the beginning until the first new-line character is found to enable multi-threads to independently process the blocks in the next step. 2. Splitting a block into SAM lines SAM lines are extracted from the data blocks transferred from the previous stage. The data blocks are scanned from the beginning to the end to find new-line characters, which separate SAM lines. A major effort in this stage is to find these characters by using a function of the standard C-language library (memchr). The performance of the function is optimized by using vector instructions, if they are available [9]. A scanned data block is transferred to the next pipeline stage, which parses SAM lines with the positions where SAM lines start. 3. Parsing SAM lines Binary alignment records are created from the SAM lines. The sam2bam locates a SAM line in the data block at each line position calculated by the previous stage, parses the SAM line by using a library function of Samtools (sam_parse1), and creates a binary alignment record in the data format used in Samtools (bam1_t). The binary alignment record contains the same contents as the corresponding SAM/BAM alignment. Each binary record has a global sequence number, which is created on the basis of the block number and a local number of the record within the block. This global sequence number is used when analysis tools need to know which record has appeared first in the input file for a given set of records. Virtualizing binary alignment records The sam2bam supports two modes: all the binary alignment records are placed in the main memory (memory mode) or in external storage (storage mode). If there is sufficient memory, it is recommended that the memory mode be run to obtain the best performance; otherwise, the storage mode can be used. The binary alignment records are moved to either the main memory or the external storage in this step. The binary alignment records are managed in the later stages in the virtual space of sam2bam. A virtual address is assigned to each binary alignment record. A virtual address can be translated into either a main memory address or an offset of the file in external storage. If sam2bam is only invoked with plug-ins that access streaming input data but not pooled information, the input data are transferred from this step to the last step 7. In that case, only the memory mode is enabled. Moreover, steps 5 and 6 are skipped. 5. Building alignment database Some plug-ins can start analysis after all the input data are read. An alignment database is built for such plug-ins to provide pooled information on the alignments in this step. Each database entry is represented by a data record called baminfo, which is unique to sam2bam. Baminfo is created from each binary alignment record but it only contains information that can be used by the analyzer plug-ins in step 6. Long string data in the binary alignment records are not copied into baminfo. The long string data include reference sequence names, concise idiosyncratic gapped alignment report (CIGAR), sequences, base qualities, and optional fields. If any analyzer plug-in needs any of the omitted data in step 6, the plug-in summarizes the data and saves them into baminfo when baminfo is created. The baminfo records are arranged in the reference genome position space to construct the alignment database. The baminfo records can be looked up from the alignment database by using the mapping positions of clipped sequences. The baminfo records that have the same mapping position are gathered and are found in the database by using a single lookup. If there is an analysis that needs the functionality of the baminfo records being looked up by using an unclipped mapping position, the baminfo records are arranged by using the unclipped position as well as the clipped position. 6. Analyzing baminfo records The plug-in code undertakes any analysis by using information in the alignment database. Duplicate marking is an example of such code. Multithreading can accelerate this step by decomposing the input space into sub spaces. For example, we can split the data set of the alignment database into N Ã B blocks and allocate B blocks to each of the N threads. 7. Writing a compressed BAM file This step groups the binary alignment records into 64-KB blocks (BAM blocks), compresses them, and writes a sequence of the compressed BAM blocks to a BAM file. This step produces a sorted BAM file if the alignment database is available. The sam2bam does not require a separate step for sorting. If some plug-ins set up the alignment database, this step obtains input by scanning the alignment database in the order of sort keys and writes the alignments to the BAM file in the sort-key order. Therefore, all the output from sam2bam is automatically sorted. In contrast, if no analyzer plug-ins are used, the output is not sorted. There is a dispatcher that splits a sequence of the binary alignment records into 64-KB subsequences by only using their virtual addresses and their sizes. The dispatcher obtains input data by traversing the alignment database or receiving them from step 4. The dispatcher is single-threaded to ensure the order of the binary alignment records in each BAM block is maintained and the order of BAM blocks in the output. It does not construct actual BAM blocks by copying the data to avoid performance bottlenecks in the dispatcher. It instead transfers each set of virtual addresses that construct a single BAM block to a thread pool for compression. If the dispatcher receives the input data from step 4, the binary alignment records in the physical memory provide their record sizes. Otherwise, the dispatcher traverses baminfo records in the alignment database to obtain the virtual addresses and record sizes for the binary alignment records. Multi-threads are used for compressing BAM blocks to accelerate this step. BAM blocks are constructed and compressed in parallel. The compression method is gzip, which is widely used in the real world [5]. If hardware compression accelerators are available in the system, sam2bam dynamically loads their library codes that can be called via the standard zlib application programming interface (API) and it can offload compression to the accelerators. Results and Discussion To demonstrate that sam2bam can significantly reduce the runtime in marking duplicate alignments, we compared the runtime of sam2bam versus Picard [10], which is a widely-used tool set that is also recommended in GATK best practices [11]. Experimental Environment Benchmark Data Sets. Two data sets in the SAM format were used to evaluate performance. The first was 150x coverage of WEX data, and second was 50x coverage of WGS data. The WEX data were part of the 1000 Genome Project data [12]. The input SAM file size was 52 GB. The WGS data were part of the Cancer Genome Atlas (TCGA) Benchmark 4 dataset, i.e., G15512.HCC1954.1 [13]. The input SAM file size was 546 GB. The SAM files that we used were created by running the Burrows-Wheeler Aligner (BWA) [7] for the FASTQ-format data converted from the original BAM files. Sam2bam Configuration. The sam2bam handled the SAM and BAM data formats by calling the modified code of samtools. The original code for samtools was obtained from its development repository as of August 2015. Two plug-ins were created for the performance evaluation: an analyzer plug-in for marking duplicate alignments and a compression accelerator plug-in. The source code for sam2bam and the instructions on how to build sam2bam are available from a GitHub repository (https://github.com/t-ogasawara/sam-to-bam). The analyzer plug-in was enabled to mark duplicate alignments. The sam2bam created the alignment database, as was explained earlier in Section Pipeline Configuration: item 5. The analyzer plug-in traversed the alignment database by using the unclipped position to find the candidates of duplicate alignments and find alignments that had the same beginning and end positions. It also found their pairs by using mate information that is available in the alignment database. The plug-in further used the same criteria as Picard MarkDuplicates [14] for the candidates to select one alignment from duplicates. The duplicates were analyzed in parallel by assigning the segmented unclipped position regions to threads. This analyzer plug-in was provided as a pre-built library for POWER8 systems and was installed when sam2bam was built. The accelerator plug-in enabled the use of a hardware compression card. Part of the multithreads for compression offloaded compression tasks to the hardware card instead of performing compression with software. This accelerator plug-in was also constructed when sam2bam was built. Picard Tools. Two Picard tools, SortSam and MarkDuplicates, have been suggested to mark duplicate alignments on GATK Best Practices [11]. SortSam first takes a SAM file as input, sorts the alignments, and writes the result to a BAM file. MarkDuplicates then takes the produced BAM file as the input, marks duplicate alignments, and writes the result to another BAM file. The BAM files are compressed by default. We used Picard tools (version 2.1.1) [10] in the BioBuilds package (version 2016-04) [15]. OpenJDK 1.8.0_72-internal was used to run the Picard tools with 21 GB of Java heap memory. Hardware. The runtime of the target programs and the maximum size of the memory that was used by the programs during program execution were measured by using a command, /usr/bin/time. The programs were run on a single node of IBM Power Systems S822LC [16] that had 16 POWER8-based CPU cores [17], where 128 logical processors were available (eight logical processors per core) with 1 TB of memory. The machine was attached to high performance storage, i.e., an IBM Elastic Storage Server (ESS) GL4 via a Mellanox FDR switch [18]. A hardware card that provided FPGA-based zlib acceleration [19] was attached to the machine and this could speed up compression of the BAM data. The operating system was Ubuntu 14.04.1. The theoretical maximum performance for the storage mode of sam2bam that was explained in Pipeline Configuration was measured by using the file system in the main memory (/dev/shm), which simulated an ideal high-performance device (e.g., a solid state drive (SSD)). Such a device is mandatory to achieve high levels of performance in the storage mode since sam2bam in the storage mode performs a huge number of I/O operations that are not always sequential accesses. Performance Evaluation with Whole Exome Data The sam2bam demonstrated more than 100 times better performance than Picard and finished in about one minute in both memory and storage modes while Picard needed more than two hours ( Table 1). The sam2bam benefited from multi-threading and pipelining, which was explained in Pipeline Configuration. The sam2bam read a SAM file for SAM parsing and parsed it at rates of 1.7-2.0 GB/s. This high level of performance was due to multi-threading and pipelining (we will discuss performance without them in S1 Text). The runtime of duplicate marking was 5% of the total runtime. BAM blocks were compressed at rates of 1.5-1.7 GB/s, including a rate of additional 0.9 GB/s with hardware compression. Such high throughput was achieved by pipelining as well as multi-threading (we will discuss performance without pipelining in S2 Text). The performance of the framework on which alignments were analyzed and processed is critical for high performance tools. The runtime of a Picard tool that converts the file format from SAM to BAM is 92% that of SortSam and 70% that of MarkDuplicates for Picard (we will discuss the details in S3 Text). Performance Evaluation with Whole Genome Sequencing Data The sam2bam demonstrated 156 times better performance than Picard. The sam2bam finished in about 9 minutes in the memory mode while Picard needed more than 20 hours ( Table 2). The storage mode was 81% slower than the memory mode for WGS data, while the memory and storage modes demonstrated similar performance for WEX data. This slowdown was mainly due to slowdown in BAM block compression in the storage mode (37% of throughput in the memory mode). We collected system-level profiles to analyze the slowdown in BAM block compression. The profiles indicated that the computation time in the operating system was significantly increased by 33 times in the storage mode using the WGS data, but it was only increased by 151% when using the WEX data. We need to further investigate additional activities undertaken by the operating system to address the slowdown with the WGS data in the storage mode. min Although sam2bam was more than 186 times faster than the standard tools, it required more memory than Picard in the memory mode. The sam2bam reduced the maximum memory size by placing binary alignments in external storage instead of in the main memory in the storage mode. The performance of sam2bam with data compression by using both software and hardware was 43% better than that of sam2bam with software-only compression. The Java heap size for Picard was sufficient since the time spent in garbage collection of the Java heap was negligible (about 0.63% of the total runtime). doi:10.1371/journal.pone.0167100.t001 Accuracy of Duplicate Marking The accuracy of duplicate marking for sam2bam could be evaluated by measuring the number of alignments that Picard MarkDuplicates marked but sam2bam did not and also by measuring the number of alignments that sam2bam marked but PicardMarkDuplicates did not. Outputs were compared between Picard MarkDuplicates and sam2bam (we will discuss how we compared the outputs in S4 Text) to evaluate the accuracy of duplicate marking. If sam2bam and Picard MarkDuplicates marked the same sets of alignments, sam2bam could be considered to be accurate and could be used as a fast alternative to Picard MarkDuplicates. We tested and verified that duplicate marking by sam2bam was accurate, based on the experimental results obtained from WEX and WGS data sets. There were 16 million duplicate alignments for the WEX data set and 188 million for the WGS data set. These alignments were the same between sam2bam and Picard MarkDuplicates when making the comparison explained in S4 Text. Supporting Information Fig 1 . 1Architecture for sam2bam without analyzer plug-ins. Pipeline is configured with plug-in code that filters out data. Gray boxes indicate steps in pipeline. Steps that have multiple boxes are multi-threaded. Blue boxes denote files in storage. Light-blue boxes denote data in memory. Light green boxes denote plug-in code.doi:10.1371/journal.pone.0167100.g001Fig 2. Fig 3 . 3Filter plug-ins. Fig 4 . 4Accelerator plug-in. Compression threads create uncompressed blocks by copying data at virtual addresses. Compression threads that use accelerator plug-in transfer blocks to hardware accelerator, while remaining compression threads compress blocks by using software library code. Compressed blocks are written to BAM file.doi:10.1371/journal.pone.0167100.g004Fig 5. Analyzer plug-ins. Analyzer plug-in codes obtain field values of alignments from database and save analysis results in database. Each analyzer plug-in code can use multiple threads. In this example, three threads execute analyzer plug-in. doi:10.1371/journal.pone.0167100.g005 S1 Text. Advantage of multi-threaded and pipelined SAM parsing. (PDF) S2 Text. Evaluation of performance of multi-threaded and pipelined generation of compressed BAM file. (PDF) S3 Text. Discussion on performance of file format conversion framework. (PDF) S4 Text. Methodology of comparing marked duplicates between tools. (PDF) Author Contributions Table 1 . 1Runtimes and maximum memory sizes for marking duplicates on 52 GB WEX data.SAM parsing, sorting Duplicate marking Total runtime Table 2 . 2Runtimes and maximum memory sizes for marking duplicates on 546GB WGS data. doi:10.1371/journal.pone.0167100.t002 Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools PLOS ONE | DOI:10.1371/journal.pone.0167100 November 18, 2016SAM parsing, sorting Duplicate marking Total runtime PLOS ONE | DOI:10.1371/journal.pone.0167100 November 18, 2016 Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools PLOS ONE | DOI:10.1371/journal.pone.0167100 November 18, 2016 7 / 11 Next-generation sequencing: adjusting to data overload. M Baker, 10.1038/nmeth0710-495Nat Methods. 77Baker M. Next-generation sequencing: adjusting to data overload. Nat Methods. 2010; 7(7):495-9. doi: 10.1038/nmeth0710-495 From FastQ data to high-confidence variant calls: the Genome Analysis Toolkit best practices pipeline. G A Der Auwera, M O Carneiro, C Hartl, R Poplin, G Del Angel, A Levy-Moonshine, 10.1002/0471250953.bi1110s4325431634Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools PLOS ONE. 11Sam2bam: High-Performance Framework for NGS Data Preprocessing Tools PLOS ONE | DOI:10.1371/journal.pone.0167100 November 18, 2016 2. Van der Auwera GA, Carneiro MO, Hartl C, Poplin R, del Angel G, Levy-Moonshine A, et al. From FastQ data to high-confidence variant calls: the Genome Analysis Toolkit best practices pipeline. Curr Protoc Bioinformatics. 2013; 11(1110):11.10.1-11.10.33. doi: 10.1002/0471250953.bi1110s43 PMID: 25431634 Infrastructure for GATK Best Practices Pipeline Deployment. A Prabhakaran, B Shifaw, M Naik, P Narvaez, G Van Der Auwera, G Powley, Prabhakaran A, Shifaw B, Naik M, Narvaez P, Van der Auwera G, Powley G, et al. Infrastructure for GATK Best Practices Pipeline Deployment;. Available: http://www.intel.com/content/dam/www/public/ us/en/documents/white-papers/deploying-gatk-best-practices-paper.pdf. The Sanger FASTQ file format for sequences with quality scores, and the Solexa/Illumina FASTQ variants. Pja Cock, C J Fields, N Goto, M L Heuer, P M Rice, 10.1093/nar/gkp113720015970Nucleic Acids Research. 386Cock PJA, Fields CJ, Goto N, Heuer ML, Rice PM. The Sanger FASTQ file format for sequences with quality scores, and the Solexa/Illumina FASTQ variants. Nucleic Acids Research. 2010; 38(6):1767- 1771. doi: 10.1093/nar/gkp1137 PMID: 20015970 The Sequence Alignment/Map format and SAMtools. H Li, B Handsaker, A Wysoker, T Fennell, J Ruan, N Homer, 10.1093/bioinformatics/btp35219505943Bioinformatics. 2516Li H, Handsaker B, Wysoker A, Fennell T, Ruan J, Homer N, et al. The Sequence Alignment/Map format and SAMtools. Bioinformatics. 2009; 25(16):2078-2079. doi: 10.1093/bioinformatics/btp352 PMID: 19505943 GATK Best Practices-Pre-processing. GATK Best Practices-Pre-processing;. Available: https://www.broadinstitute.org/gatk/guide/bp_step. php?p=1. Fast and accurate short read alignment with Burrows-Wheeler transform. H Li, R Durbin, 10.1093/bioinformatics/btp32419451168Bioinformatics. 2514Li H, Durbin R. Fast and accurate short read alignment with Burrows-Wheeler transform. Bioinformat- ics. 2009; 25(14):1754-1760. doi: 10.1093/bioinformatics/btp324 PMID: 19451168 The SAM/BAM format specification working group. Sequence Alignment/Map Format Specification. The SAM/BAM format specification working group. Sequence Alignment/Map Format Specification; 2015. Available: http://github.com/samtools/sam-spec. Optimization of GLIBC's memcpy, memmove and memchr functions. IBM developerWorks. Optimization of GLIBC's memcpy, memmove and memchr functions. IBM developerWorks;. Available: https://www.ibm.com/developerworks/community/wikis/home/wiki/W51a7ffcf4dfd_4b40_9d82_ 446ebc23c550/page/Porting%20Story%20%231. . ; Picard, Available, Picard;. Available: http://broadinstitute.github.io/picard/. Map and mark duplicates. (howto) Map and mark duplicates;. Available: http://gatkforums.broadinstitute.org/gatk/discussion/ 2799/howto-map-and-mark-duplicates. NA12878 exome alignment. NA12878 exome alignment;. Available: ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/phase3/data/ NA12878/exome_alignment/NA12878.mapped.ILLUMINA.bwa.CEU.exome.20121211.bam. The Cancer Genome Atlas (TCGA) Benchmark 4 dataset. The Cancer Genome Atlas (TCGA) Benchmark 4 dataset;. Available: https://cghub.ucsc.edu/datasets/ benchmark_download.html. Overview of the dedup function of bamUtil. Overview of the dedup function of bamUtil;. Available: http://genome.sph.umich.edu/wiki/BamUtil: _dedup. . Biobuilds, Org, BioBuilds.org;. Available: https://biobuilds.org/downloads/. IBM Power Systems S822LC technical overview and introduction. IBM Power Systems S822LC technical overview and introduction;. Available: http://www.redbooks.ibm. com/redpapers/pdfs/redp5283.pdf. IBM POWER8 processor core microarchitecture. B Sinharoy, J A Van Norstrand, R J Eickemeyer, H Q Le, J Leenstra, D Q Nguyen, 10.1147/JRD.2014.2376112IBM Journal of Research and Development. 591Sinharoy B, Van Norstrand JA, Eickemeyer RJ, Le HQ, Leenstra J, Nguyen DQ, et al. IBM POWER8 processor core microarchitecture. IBM Journal of Research and Development. 2015; 59(1):2:1-2:21. doi: 10.1147/JRD.2014.2376112 Introducing the Elastic Storage Server for Power. 18. Introducing the Elastic Storage Server for Power;. Available: http://www.ibm.com/support/ knowledgecenter/SSYSP8_4.0.0/com.ibm.ess.v4r0.deploy.doc/bl8dep_intro.htm. A hardware accelerated version of zLib based compression/decompression RFC1950/RFC1951/ RFC1952 with help of an FPGA based PCIe card. A hardware accelerated version of zLib based compression/decompression RFC1950/RFC1951/ RFC1952 with help of an FPGA based PCIe card;. Available: https://github.com/ibm-genwqe/genwqe- user.
[ "https://github.com/t-ogasawara/sam-to-bam).", "http://github.com/samtools/sam-spec.", "https://github.com/ibm-genwqe/genwqe-" ]
[ "An embedded-atom method model for liquid Co, Nb, Zr and supercooled binary alloys", "An embedded-atom method model for liquid Co, Nb, Zr and supercooled binary alloys" ]
[ "Pascal Thibaudeau \nNanochemistry Research Institute Department of Applied Chemistry\nCommissariatà l'Energie Atomique Le Ripault\nBP 16F-37260MontsFrance\n", "Julian D Gale \nCurtin University of Technology\nGPO Box U19876845PerthWesternAustralia\n" ]
[ "Nanochemistry Research Institute Department of Applied Chemistry\nCommissariatà l'Energie Atomique Le Ripault\nBP 16F-37260MontsFrance", "Curtin University of Technology\nGPO Box U19876845PerthWesternAustralia" ]
[]
The parameters of many-body potentials for Co, Nb and Zr metals, based on the embeddedatom method, have been systematically derived. The analytical potential scheme allows us to reproduce correctly the cohesive energies and structural properties of the pure metals and selected alloys making use of a small set of parameters. With a pair potential going smoothly to zero for a sufficient cutoff radius, radial partial and bond angular distribution functions for Co, Nb, Zr and alloys are computed using molecular dynamics simulations that ensure good quantitative agreement with the available experimental data up to the melting point. Atomic short range order is analysed in the light of consecutive Gaussian function decomposition and Honeycutt-Andersen indices. PACS numbers: 61.25.Mv, 71.15.Mb Today we are not surprised that a non-crystalline solid orders magnetically. It is known that, with few important exceptions, the amorphous and crystalline phases of the same material do not differ very much magnetically. In the last three decades, the discovery and large scale investigation of rapidly solidified alloys have made possible a new scenario in basic and applied magnetism. The unusual behaviour of the bulk metallic magnetic glasses (BMMGs) -formed by supercooling the liquid state of certain metallic magnetic alloys -usually occurs for systems containing atoms that exhibit a well-known sensitivity to the immediate neighbourhood [1, 2]. However, despite the large number of technological applications of BMMGs, the detailed origin of the links between structural and magneticproperties has yet to be established, but is strongly dependent on the atomic short-range order[3].Nowadays, atomic-scale simulations of solids based on interatomic potentials are routinely performed to explore the short-range order of BMMGs[4,5]as a precursor to the study of the magnetic ordering, which is evidently beyond such methods. However, the construction of realistic n-body potentials is mandatory to any simulations. When applied to metals and alloys, several major methods and extensions of construction of such potentials have been established from density functional theory, i.e., the embedded-atom method (EAM) [6] or tight-binding second moment approximation, i.e., the Finnis-Sinclair (FS) model [7] and related models[8,9]. The two models have very similar computational requirements, and the names are often used interchangably, however there are some distinctions which come to the fore when considering multicomponent alloys[10]. Because of the parameters involved in these model potentials, the EAM method was first used to study simple metals in their relevant crystalline structures[11,12]. Extensions to binary alloys were formulated [13] with special attention to hcp-fcc [14] and hcp-bcc [15] systems. Because they are often based on spin-less approximations of electronic density, such effective model potentials traditionally neglect the magnetic ordering [7, 10, 16, 17]. Notable exceptions have been published recently by Dudarev et al. [18] and Ackland et al.[19,20], but such methods are still in their infancy. Even in spin-less schemes, these effective potentials may be able to represent the compositional ordering which is, at least in the localised magnetism picture, a prerequisite for understanding the magnetic behaviour. Existing magnetic potentials concentrate on the of 1551, whereas the 1541 and 1431 indices are more characteristic of a distorted icosahedral local order. Up to the distance cutoff corresponding to first minimum of each g(r),Table IVreports the HA analysis in the liquid state of each simple elements. The microscopic analysis index Co
null
[ "https://arxiv.org/pdf/0809.0198v1.pdf" ]
115,701,990
0809.0198
745fa978560f248b60723465c25378881cbf2cd4
An embedded-atom method model for liquid Co, Nb, Zr and supercooled binary alloys 1 Sep 2008 (Dated: September 1, 2008) Pascal Thibaudeau Nanochemistry Research Institute Department of Applied Chemistry Commissariatà l'Energie Atomique Le Ripault BP 16F-37260MontsFrance Julian D Gale Curtin University of Technology GPO Box U19876845PerthWesternAustralia An embedded-atom method model for liquid Co, Nb, Zr and supercooled binary alloys 1 Sep 2008 (Dated: September 1, 2008)PACS numbers: 6125Mv, 7115Mb The parameters of many-body potentials for Co, Nb and Zr metals, based on the embeddedatom method, have been systematically derived. The analytical potential scheme allows us to reproduce correctly the cohesive energies and structural properties of the pure metals and selected alloys making use of a small set of parameters. With a pair potential going smoothly to zero for a sufficient cutoff radius, radial partial and bond angular distribution functions for Co, Nb, Zr and alloys are computed using molecular dynamics simulations that ensure good quantitative agreement with the available experimental data up to the melting point. Atomic short range order is analysed in the light of consecutive Gaussian function decomposition and Honeycutt-Andersen indices. PACS numbers: 61.25.Mv, 71.15.Mb Today we are not surprised that a non-crystalline solid orders magnetically. It is known that, with few important exceptions, the amorphous and crystalline phases of the same material do not differ very much magnetically. In the last three decades, the discovery and large scale investigation of rapidly solidified alloys have made possible a new scenario in basic and applied magnetism. The unusual behaviour of the bulk metallic magnetic glasses (BMMGs) -formed by supercooling the liquid state of certain metallic magnetic alloys -usually occurs for systems containing atoms that exhibit a well-known sensitivity to the immediate neighbourhood [1, 2]. However, despite the large number of technological applications of BMMGs, the detailed origin of the links between structural and magneticproperties has yet to be established, but is strongly dependent on the atomic short-range order[3].Nowadays, atomic-scale simulations of solids based on interatomic potentials are routinely performed to explore the short-range order of BMMGs[4,5]as a precursor to the study of the magnetic ordering, which is evidently beyond such methods. However, the construction of realistic n-body potentials is mandatory to any simulations. When applied to metals and alloys, several major methods and extensions of construction of such potentials have been established from density functional theory, i.e., the embedded-atom method (EAM) [6] or tight-binding second moment approximation, i.e., the Finnis-Sinclair (FS) model [7] and related models[8,9]. The two models have very similar computational requirements, and the names are often used interchangably, however there are some distinctions which come to the fore when considering multicomponent alloys[10]. Because of the parameters involved in these model potentials, the EAM method was first used to study simple metals in their relevant crystalline structures[11,12]. Extensions to binary alloys were formulated [13] with special attention to hcp-fcc [14] and hcp-bcc [15] systems. Because they are often based on spin-less approximations of electronic density, such effective model potentials traditionally neglect the magnetic ordering [7, 10, 16, 17]. Notable exceptions have been published recently by Dudarev et al. [18] and Ackland et al.[19,20], but such methods are still in their infancy. Even in spin-less schemes, these effective potentials may be able to represent the compositional ordering which is, at least in the localised magnetism picture, a prerequisite for understanding the magnetic behaviour. Existing magnetic potentials concentrate on the of 1551, whereas the 1541 and 1431 indices are more characteristic of a distorted icosahedral local order. Up to the distance cutoff corresponding to first minimum of each g(r),Table IVreports the HA analysis in the liquid state of each simple elements. The microscopic analysis index Co I. INTRODUCTION one-site magnetism, for which the energetics reduces in form to a simple embedded-atomtype potential, explaining the success of standard EAM schemes on magnetic materials. Supercooled Co 1−x Nb x and Co 1−x Zr x magnetic metallic glasses in the cobalt-rich region represent test cases for both hcp-bcc and hcp-hcp metal sub-systems. These compounds have been experimentally studied because they are strong ferromagnets, as revealed by a high value of the exchange constant [21]. Consequently, large short-range compositional inhomogeneities should induce significant variations of the long-range magnetic order. The present work considers both the construction of embedded-atom potentials for such materials, as well as the application to molecular dynamics (MD) simulations in order to assess one of the issues in the large field of these BMMGs, namely short-range order quasicrystallinity. II. METHODOLOGY FOR POTENTIALS In the FS method, the total internal energy E of a N-atom system and the electron density, ρ(R i ), for an atom located at R i due to all other atoms are given as; E = 1 2 N i,j,(i =j) φ(r ij ) − N i=1 F (ρ(R i )),(1)ρ(R i ) = N j =i f (r ij ),(2) where f (r ij ) is the electron density at atom i due to atom j as a function of the distance between them, r ij = R i −R j is the separation distance between atoms i and j, F (ρ(R i )) is the energy to embed atom i in an electron density ρ(R i ), and φ(r ij ) is a two-body potential between atoms i and j. As long as an angular independent formulation is considered, the electron density is a radial function only. For an alloy model, an embedding function, F , has to be specified for each atomic species supplemented by an atomic electron-density function, f , and a two-body potential, φ, specified for each possible combination of atomic species. For uncompressed metals, Gupta [22] and Tomànek et al. [23] have shown that the host electron density can be represented as an exponentially decreasing function of the distance to better account for atomic relaxation near impurities and surfaces. In this approximation, f is given as; f (r) = f e exp(−χ(r/r e − 1)), where f e is a scaling factor determined by the cohesive energy, E c , and the atomic volume, r e is the nearest-neighbour distance in the relevant pair of atoms and χ is an adjustable parameter. Analysing the interatomic interactions in effective-medium theory, Jacobsen et al. [24] have shown that if an exponential form is chosen for the density function, then the interatomic potential, φ, should also be an exponential function of the distance. In this study, the interatomic potentials of all the pairs considered are defined by a potential of the form; φ(r) =              A exp(−r/r 0 ) 0 ≤ r ≤ r 1 , 5 i=0 a i r i r 1 ≤ r ≤ r m , 0 r m ≤ r,(4) where the interaction is designed so as to go smoothly to zero at the distance r m according to a polynomial spline function. The potentials are constructed subject to the constraints that the radial functions and their first and second derivatives must be continuous at the boundary points, and also that the function must have a stationary point at r m . Once A, r 0 , r 1 and r m are fixed, this procedure ensures that the coefficients, {a i }, are uniquely determined by solving a simple 6 × 6 linear system of equations. These coefficients are reported in Table I for completeness. For all the pairs of atoms, r 1 = 2.5Å is kept fixed and corresponds to a typical radius where the stiff repulsive part of the potential ends in metals [25,26,27]. Because of the screening in metals, the stationary point is located at least between the second and third nearest neighbours for the lowest energy crystal phases as previously noted [28]. Since the embedding energy is assumed to be independent of the source of the electron density and the hopping integrals are a function only of a radial distance between atoms, the embedding functional F is taken as F [ρ(r)] = ρ(r).(5) This functional form gives a band energy proportional to the square root of the second moment of the electron density of states [22]. However, moments of higher order cannot be expressed in such a simple analytical form and a more complex method must be applied [29]. Johnson [13] has considered that since the electron density at any location is taken as a linear superposition of atomic electron densities, this function should be taken directly from monoatomic models with a relative scaling factor between elements for an alloy model. On the other hand, Finnis and Sinclair [7] and Cleri and Rosato [30] have considered mixed pair electron-density functions not necessarily connected to the atomic ones, removing the alloy scaling factors. For hcp-Co and hcp-Zr, the parameters for the atomic electron-density and the interatomic potential are fitted in order to reproduce the experimental cohesive energies, the unit cell parameters and the five independent elastic constants of these systems as given by Cleri and Rosato [30]. Moreover, cell parameters and elastic constants of fcc-Co and bcc-Zr are also included during the fitting procedure as taken from references [31,32,33] and references therein. Generally, for a small cutoff distance the largest number of interacting neighbours per atom in the crystalline structure leads to the more stable phase. Ducastelle [34] has shown that in the second-moment approximation, with interactions restricted to the nearest neighbours, the cohesive energy for the hcp and fcc phases is the same and the c/a ratio is equal to the ideal value. Hence, it is necessary to go up to at least the fourth-moment approximation to discriminate between the fcc and hcp phases and to give a value of c/a different from 2 2/3. In our case, since the c/a ratios are not taken to be the ideal one and because the potentials have a very short range, the cutoff distance of atomic electron density should be larger than that of the potential. So for the electron density, the cutoff distance is taken to be 4.87Å, thus including up to seven shells of neighbours within hcp-Co, and three for both hcp-Zr and bcc-Nb, which are all the stable phases for each pure element. A check is also performed to ensure that at least three shells of neighbours have been included for the high pressure/high temperature structures since this is necessary to keep the relative energies of each phases in the correct order [33]. Moreover, it has been observed that in incorporating the elastic constants of the bcc-Zr phase in the database, the fitting of both the cell parameters and cohesive energy of the hcp-Zr is rather poor with this cutoff radius, so only the hcp-Co, fcc-Co, hcp-Zr and bcc-Nb elastic constants are included during the fit. For Nb, the cohesive energy, the lattice parameter and the three independent elastic constants of the bcc-Nb phase are taken from reference [35] and the theoretical fcc-Nb cell parameter and cohesive energy are also included [36]. No elastic constants of the fcc-Nb were found to incorporate into the training set of observables. The selection of the functional form taken in Eq.(1)-(2) is extended to AB alloys based on the second-moment form that has been applied to Zr [33,37]. The embedding function, atomic electron-density function and two-body potential are assumed to be of the same form as in Eq.(3)-(4), with φ AB and φ BA assumed to be equal. The alloy potentials and atomic electron-density functions are determined independently of the monoatomic counterparts if sufficient data are available. However, it is known that for equilibrium immiscible systems it is a challenging task to fit cross potentials, since there is often insufficient experimental data related to the respective alloy compounds. In order to circumnavigate this problem, density functional calculations have been performed on selected intermetallic structures using the Quantum-ESPRESSO package [38]. For these calculations, nonlocal ultrasoft pseudo-potentials are employed in combination with a plane-wave basis set. The generalized-gradient approximation, as parametrized by Perdew, Burke and Ernzerhof [39], is selected for the exchange and correlation term. For the Brillouin zone sampling, a 12×12×12 Monkhorst-Pack mesh is used for the k-point summation in the self-consistent calculations [40] for all the primitive cells, which leads to converged structural parameters to within 0.1% of the cell parameter. Thus the lattice constants and cohesive energies of several Co-(Zr,Nb) and Zr-Nb crystalline structures reported in Table (III) are obtained and then included in the fitting procedure for the Co-(Zr,Nb) and Zr-Nb cross potentials and atomic electron-density functions. The parameters of the fitted terms are listed in Table I. The overall fitting procedure is performed in two separate steps. First, the densities and potentials are derived for the simple metals. Once the corresponding parameters are obtained, the fit is applied to selected binary metals for cross-densities and potentials without altering the terms for the simple metals. The same cutoff radius is kept constant during all the steps and the calculations are performed within the GULP computer code [41]. In Table II, a list of some basic physical properties as computed by the present set of potentials and the corresponding experimental values are shown for Co, Nb and Zr. The fit correctly reproduces the structures and properties of hcp-Co, fcc-Co, hcp-Zr and bcc-Nb. The absolute average percentage difference between calculation and experiment is found to be 0.6%, 0.7% and 8%, for the cell parameters, cohesive energies elastic constants, respectively. However, the elastic constants of fcc-Nb produced a negative Young's modulus, as expected, in this excited locally unstable structural phase [44]. The properties of bcc-Zr are reproduced with a sufficient accuracy, except for the C 44 elastic constant where the largest percentage error of 45% occurs. The potentials fitted by Willaime et al. [33] also exhibit such a discrepancy though with a much larger error. This may be corrected by relaxing the constraint of the square-root form of the embedding functional and considering more neighbours [12] or by including explicit angularly dependent terms in the potentials at a cost of additional parameters [45]. In Table III, a list of some basic physical properties fitted from these potentials and the corresponding ab initio calculated values for selected binary alloys is shown. The B2, C15 and L1 2 denominations are relative to the strukturbericht structural types classification [48], such as B2 is the CsCl structure type, C15 is the MgCu 2 structure type and L1 2 is the Cu 3 Au type. The fitting procedure appears to correctly reproduce the ab initio derived phase stability order and lattice parameters. However, the absolute cohesive energies are reproduced to a lesser extent. This may be also improved consistently by increasing the cutoff radius on Zr and Nb electron density terms, which includes more neighbours in the total energy sums of Eq.(1). III. APPLICATION TO SIMPLE METALS The validity of this potential in describing the atomic interactions can be illustrated outside the original systems used for parametrisation by considering liquid cobalt. An MD [46], (2) experimental value of 6.774Å [47]. The B2, C15 and L1 2 classification is relative to the strukturbericht structural types classification [48]. polynomial that fits a sub-range of several points. For all the calculated radial functions, a third-degree, five-point smoothing procedure is applied several times on the data until convergence [49]. MD simulations are repeated on three different atomic configurations at the same temperature in order to sample more accurately the configurational space and an overall averaged RPDF is computed. This averaged RPDF compares very well to the experimental data [50] as shown in Fig. 1. The agreement is similar to that presented by Bhuiyan et al. [51] and more recently by Han and coworkers [52] using much more elaborate EAM potentials, but in these works no quantitative analysis of the RPDFs was performed. However, the first peak of the RPDF g(r) appears to be asymmetric and might be composed of more than one atomic shell. Following Kita et al. [53], a decomposition of rg(r) in consecutive Gaussian functions is applied. The insert in Fig. 1 shows the decomposition of rg(r) at 1670 K in six Gaussian functions where all 18 parameters are allowed to vary freely during the fit. The first coordination shell is defined by a cutoff distance r c , which is taken to be the first minimum of g(r). For this temperature, r c = 3.46Å. The first peak is composed of three Gaussian subpeaks located at r 1 = 2.407Å, r 2 = 2.645Å and r 3 = 3.095Å, respectively. The coordination number N c is calculated by integrating the Gaussian function according to N c = 4π 3/2 n 0 Aσr i , where n 0 is the atomic density, A is the amplitude of the Gaussian function, σ is the square root of the variance, and r i is the maximum radius. For each subpeak, N c is equal to 3.46, 4.71 and 3.81 with a sum of 11.98. For a temperature of 1800K, which is slightly greater than the experimental melting point, the sum decreases to 11.57. These coordination numbers are close to the experimental ones found in liquid Co [50] (12.5±0.5 at 1670 K and 12.1±0.5 at 1800 K) and consistent with those calculated for other metallic systems [54]. Such a high value of the coordination number and the possibility of decomposing the first peak of g(r) is an indication that the short-range order of the liquid Co is more complex than the one given by a simple icosahedral ordering as suggested by Holland et al. [50]. Moreover, performing MD simulations for these two temperatures allows us to For this metal at that temperature, the first peak is composed of two Gaussian functions located at r 1 = 3.091Å and r 2 = 3.574Å, respectively. The ratio r 2 /r 1 = 1.156 is close to that of the two first nearest-neighbour distances for a bcc lattice 2/ √ 3 = 1.1547. The coordination number N c for each subpeak is 6.10 and 5.59 with a sum of 11.69. This value is close to the experimental one of 11.9 ± 0.5 found in liquid Zr [55]. Even if the ratio of the radii tends to favour a bcc lattice, the corresponding coordination is very different. MD simulations are repeated under the same conditions for pure Nb to a higher temperature of 2750 K, which is the experimental melting point. The simulated RPDF of this liquid state is shown in Fig. 3. For this metal, the first peak of rg(r) is strongly asymmetric and is composed of three Gaussian functions up to r c = 4.0Å, located at r 1 = 2.691Å, r 2 = 3.141Å and r 3 = 3.840Å. The ratio r 2 /r 1 = 1.167 is greater than that of the two first nearest-neighbour distances for a bcc lattice. The coordination number N c for each subpeak is 2.95, 7.06 and 3.10 with a sum of 13.11. To our knowledge, no experimental data on the radial distribution function of pure Nb is available in order to compare with. The short-range order can also be examined by calculating the bond angle distribution functions g(θ) that represent the angle between the bonds connecting a central atom to two neighbouring atoms, as illustrated in Fig. 4 for liquid cobalt, zirconium and niobium. The angle is calculated for pairs of interatomic distances given by a cutoff corresponding to the first minimum of the RPDF (i.e. 3.5Å for Co, 4.3Å for Zr and 4.1Å for Nb). In the case of Zr, the calculated distribution exhibits a prominent peak near θ = 57 [58]. Using first-principles molecular dynamics simulations, Jakse and Pasturel [59] have concluded there exists competition between a polyhedral and bcc-type short-range order in liquid and supercooled Zr, whereas Kim and Kelton [58] have reported no regular dominant cluster type that can describe the experimental liquid structure of transition metals, including Zr. This is supported by the values of the HA indices reported in Table IV which are not very different from each element in the liquid state. The abundance of the 1661 pairs indicates that bcc order is very low, but slightly increases when going from Co to Nb. Interestingly, the lowest energy geometrical structures of magnetic cobalt clusters mainly follow an icosahedral growth pattern with some cubic-type structures at some particular sizes [60]. For liquid Zr, these low values have been reported both experimentally [58] and using first-principles molecular dynamics simulations [59]. Furthermore for liquid Zr, Jakse and Pasturel have performed HA analysis on the inherent structures and found an abundance of the 1551 pairs in the liquid state that is twice the value found here. Such structures indicate the presence of perfect icosahedra as a local minima of the potential energy surface. For instance on liquid Nb, the 1551 index goes to 0.30±0.01 on inherent structure. IV. APPLICATION TO BINARY ALLOYS To validate the quality and transferability of our potentials, lattice constants and atomic internal positions of several crystal structures not entering in the fit have been calculated minimising the free energy at T=300K, and compared with the experimental values [61,62]. The results are summarised in Table V for the varying degrees of freedom according to the corresponding space group and the agreement is generally good with an average absolute error of 1.76%. To complete the validation, the elastic constants of CoNb and Co 3 Nb are calculated ab initio applying finite differences to the stress tensor and compared with those calculated analytically by our potentials. The results are shown in Table VI and the agreement is satisfying. As for liquids, MD simulations are performed on Co 0.9 Zr 0.1 for a 300 atom system with periodic boundary conditions. First, the atoms are placed randomly into the simulation cell using a hard sphere criterion based on their atomic radii and the cell volume is then adjusted according to the phenomenological Miedema theory [63], which gives good estimates for the experimental volume of glasses of these alloys. Then at constant pressure and a temperature of 1800 K (higher than the liquidus phase boundary for this alloy composition), MD has been run for 200 ps to simulate the liquid phase. The sample is then quenched at a rate of 7.5 × 10 14 Ks −1 and maintained at 300 K for at least a further 200 ps. In Fig. 5, the simulated RPDFs are computed and compared against experiment [64]. The simulated partial distribution functions for cobalt compare well to experiment including the position of the first and second peaks. However, the experimental Zr-Zr RPDF does not exhibit a structural trend, whereas our simulation does. Rößler and Teichler have reported similar results in their study of atomic mobilities and structural properties of supercooled amorphous Co 1−x Zr x using very different interatomic potentials [4]. This suggests a possible lack of resolution both in X-ray diffraction and EXAFS because of a low contribution in the spectra of these minority atoms, as previously anticipated [4]. However, our potentials seem to reproduced correctly the position of the first peak for Zr-Zr and the double peak character of the second peak in the Co-Co and Co-Zr distributions in comparison with ref. [4]. The maximum of the Co-Co peak is simulated to occur at 2.44Å, compared with an experimental result of 2.42Å [64]. Using the Gaussian subpeak analysis, this first peak is found to be MD simulations have also been performed for Co 0.9 Nb 0.1 under the same conditions. The reported RPDFs are shown in Fig. 6. In Co 0.9 Nb 0.1 and Co 0.9 Zr 0.1 , the Co-Co first peak distance is calculated to be at 2.437Å and is not affected by the non-magnetic added atoms at such a low concentration. The situation changes for the second and third peaks with a much more distinct third shell in Co 0.9 Nb 0.1 than in Co 0.9 Zr 0.1 . Interestingly, the Co-Nb (resp.Co-Zr) first neighbour equilibrium distance is 2.49Å (resp.2.79Å). This is lower than the simple prediction related to their corresponding atomic radii (2.70Å (resp.2.85Å)) [65]. However, Jamet et al. [66] have reported a Co-Nb distance of 2.58Å in studying cobalt nanoparticles embedded in a niobium matrix. The simulated correlation in the minority pairs is structured in the Co-Nb system, as for Co-Zr. However, in both cases the first shell of neighbours seems to be depleted of their atoms to fill the second or third shells. It is doubtful that these structures are an artefact of the low number of atoms considered in our simulations because Rößler and Teichler have simulated systems more than twice the size of ours and found the same behaviour. Up to r c = 3.1Å, the bond angular distributions of these two systems are calculated and shown in Fig. 7. These distributions exhibit wellstructured peaks suggesting more crystalline environments including a 150 • distinct angle in the Co 0.9 Nb 0.1 . As the concentration of minority atoms is low, these distributions are dominated by hcp-like Co clusters upon cooling. This trend is more pronounced for added Nb atoms than Zr atoms probably because of their corresponding atomic radii. To assess such hypothesis, the HA indices are calculated and reported in Experimental data from [55] are shown by points. The insert shows the simulated rg(r) at 2290 K and its analysis in Gaussian subpeaks (dashed curves). (open circles) [64]. For each atomic pair, the insert shows the simulated rg(r) at 300 K and its analysis in Gaussian peaks (dashed curves). predict a variation of the density with the temperature of dρ/dT = −9.68 10 −4 gcm −3 K −1 in good agreement with the experimentally reported value of dρ/dT = −9.88 10 −4 gcm −3 K −1 value [48]. MD simulations were repeated under the same conditions for pure Zr to a higher temperature of 2290 K, above the experimental melting point. The experimental RPDF is compared against the simulation of this liquid state, shown in Fig. 2. • (close to an equilateral triangle), a broader maximum near θ = 109 • and a rather flat maximum near θ = 150 • . In the case of Co, the first peak is broader and close to 55 • whereas the second peak enlarges but remains at 109 • . For Nb, the situation is different with two broader peaks near 50 • and 99 • . Viewing the structure in terms of dominant clusters, the bond angle with the highest density at the nearest neighbour distance are for a regular icosahedron 63.4 • and 116.4 • , while for fcc the prominent angles are 60 • , 90 • and 120 • . For hcp, angles of 109.471 • and 146.443 • are added when the ratio c/a = 2 2/3 but they are less frequent. For a bcc lattice, the prominent angles are 70.53 • and 109.471 • . In the Zr structure, the first peak tends to favour the fcc and hcp structures while the second peak tends to favour the hcp and bcc. This means that the dominant structure should be hcp. However, the angles of 90 • and 120 • are not so strong whereas some defective icosahedron angles should be there too. This suggests a predominantly distorted icosahedral character. For the same reasons, the case of Co also favours defective icosahedron as well. For Nb as the first peak is much more located near 50 • , this suggests much more intriguing short-range structures with less neighbours.To assess more quantitatively the local structures in amorphous alloys, Honeycutt and Andersen (HA) analysis has been proven to successfully differentiate face-centered cubic, hexagonal close packed, icosahedron and binary bcc structures[56,57]. To perform such analysis, a set of four indices is constructed for each pair: (i) the first index denotes to what peak of the RPDF, g(r), the pair under consideration belongs; (ii) the second index represents the number of near neighbours shared by this pair; (iii) the third index counts the number of nearest-neigbour bonds among the shared neighbours; (iv) and a fourth index is used to differentiate configurations with the same three indices, but with a different topology.For instance, fcc crystals are fully described by four pairs, such as 1421, 2101, 2211 and 2441, whereas hcp crystals also contains the 1422 and 2331 pairs in addition. Moreover, the 1441, 1661, 2101, 2211, and 2441 are the only pairs in a perfect bcc crystal. Icosahedral order is described by Mackay icosahedra, composed of twinned, distorted fcc tetrahedra with an index composed of 3 shells at 2.43Å, 2.55Å and 2.88Å with coordination numbers of 4.42, 4.83 and 2.30. The total coordination number up to r c = 3.12Å is equal to 11.55 in comparison with 10.90 found experimentally. In our simulations a narrower scattering of the positions of the first 3 subpeaks is observed, in contrast to the liquid state, which is anticipated during the cooling. In the closest crystalline form, Co 23 Zr 6 , the highest coordination number is obtained for a pair of cobalt atoms located at 2.4334Å in a cubic cluster and other local structures with 4 and 3 neighbours are also found at 2.36Å, 2.51Å, 2.51Å and 2.81Å [62]. This suggests tetrahedral clustering below r c or defective icosahedra up to r c in this supercooled alloy. On the other hand, the maximum of the first peak in Co-Zr is simulated to be at 2.79Å, in excellent agreement with the reported experimental distance of 2.79Å [64]. The analysis through Gaussian functions reveals 2 subpeaks located at 2.71Å and 3.00Å with a coordination number of 8.68 and 8.40, respectively. This also suggests a clustering of the bcc-type at very short range. clear short-range order is shown in agreement with available experiments for the majority pairs. The situation is different for the minority pairs within the Co-rich region. In this region, both simulated Zr-Zr and Nb-Nb RPDFs are correlated and exhibit transfers of atoms from the first shells of neighbours to the second and third. The Honeycutt-Andersen analysis exhibits mainly both distorted and pure icosahedral orders of various degrees in competition with other crystalline orders in the liquid phases and supercooled alloys. JDG would like to thank the Government of Western Australia for a Premier's Research Fellowship. FIG. 1 : 1Simulated and experimental RPDF g(r) of cobalt at a temperature of 1670 K, below the melting point. Experimental data from[50] are shown by points. The insert shows the simulated rg(r) at 1670 K and its analysis in Gaussian peaks (dashed curves). FIG. 2 : 2Simulated and experimental RPDF g(r) of liquid zirconium at a temperature of 2290 K. FIG. 3 : 3Simulated RPDF g(r) of liquid niobium at a temperature of 2750 K. The insert shows the simulated rg(r) and its analysis in seven Gaussian subpeaks (dashed curves). FIG. 4 :FIG. 5 : 45Bond angle distribution at T = 2290 K for liquid Zr (solid line), at T = 1800 K for liquid Co (dashed line) and at T = 2750 K for liquid Nb (dotted line). The peaks in the bond angle distribution for perfect icosahedral order are indicated by the vertical lines. Radial partial distribution functions of Co 0.9 Zr 0.1 at 300 K compared with experiment FIG. 6 :FIG. 7 : 67Simulated radial partial distribution functions of Co 0.9 Nb 0.1 quenched at 300K. Bond angular distribution functions of Co 0.9 Zr 0.1 and Co 0.9 Nb 0.1 quenched at 300K. TABLE I : IPotential and atomic electron-density parameters for (Co,Nb,Zr) systems. hcp-Co fcc-Co hcp-Zr bcc-Zr bcc-Nb fcc-Nba(Å) Fitted 2.5065 3.5414 3.1865 3.5358 3.3138 4.2198 Experiment (1) 2.507 3.544 3.2317 3.574 3.3 4.23 c(Å) Fitted 4.0606 - 5.2035 - - - Experiment (1) 4.0689 - 5.1476 - - - E c Fitted -4.402 -4.360 -6.192 -5.970 -7.577 -7.383 (eV/atom) Experiment (1) -4.386 - -6.167 -6.13 -7.57 -7.39 C 11 (GPa) Fitted 315 255 150 103 245 101 Experiment (2) 319 242 154 104 245 - C 12 (GPa) Fitted 155 159 80 89 132 122 Experiment (2) 166 160 67 93 132 - C 44 (GPa) Fitted 78 128 33 70 28 25 Experiment (2) 82 128 36 38 28 - C 13 (GPa) Fitted 111 159 52 89 132 122 Experiment (2) 102 160 65 93 132 - C 33 (GPa) Fitted 373 255 177 103 245 101 Experiment (2) 373 242 172 104 245 - C 66 (GPa) Fitted 80 128 35 70 28 25 Experiment (2) 77 128 44 38 28 - TABLE II : IIPhysical properties for Co, Zr and Nb simple metals as fitted with a cutoff radius of 4.87Å.(1) Cohesive energies and lattice parameters are taken from Kittel[42],(2) elastic constants are taken from Simmons and Wang[43]. simulation with 300 atoms, which allows an individual description up to the 13 th nearestneighbors in hcp-Co, and periodic boundary conditions was first performed within the isobaric, isothermal ensemble (NPT) at a temperature of 1670 K, which is slightly below the experimental melting point. The simulation was run for 200 ps with a time step of 0.1 fs to simulate the radial pair distribution function (RPDF). Once thermal equilibrium is reached, the RPDFs are sampled every 0.2 ps to produce an average. As these RPDFs are subject to statistical noise, a smoothing formula is used to replace each RPDF by a least-squaresa (Å) E c (eV/atom) Structure Fitted ab initio Fitted ab initio CoZr (B2) 3.1612 (1) 3.1753 -6.9368 -6.9850 Co 2 Zr (C15) 6.8944 6.9040 -6.8659 -6.7946 Co 3 Zr (L1 2 ) 3.7058 3.7189 -6.3484 -6.5120 CoNb (B2) 2.9502 3.0523 -8.3599 -8.1750 Co 2 Nb (C15) 6.6038 6.7357 (2) -7.5813 -7.6012 Co 3 Nb (L1 2 ) 3.5233 3.6289 -7.2423 -7.2425 ZrNb (B2) 3.4380 3.4380 -8.6865 -8.6200 Zr 2 Nb (C15) 7.9961 8.0208 -8.2544 -7.7621 Nb 2 Zr (C15) 7.8805 7.8565 -9.1236 -9.1725 TABLE III : IIILattice parameters and cohesive energies for selected binary metals in their corresponding symmetry as computed with a cutoff radius of 4.87Å. (1) experimental value of 3.181Å TABLE IV : IVHoneycutt and Andersen analysis of the simulations in the liquid state for Co emerging from the data of Table IV indicates that the short-range order of the liquid state is dominated by distorted icosaheral and icosahedral structures since the 1541, 1431 and 1551 indices respectively are large as anticipated. The high value of the 2331 pairs is also an indication of the icosahedral order. The small distortion from perfect icosahedral order observed in the angular distributions of these liquids suggests that the local icosahedral order should dominate. However, small distortion form a perfect tetrahedron does not form different HA indices from those for a perfect icosahedron. Our HA analysis shows that the icosahedral distortion is larger than reported by the bond angle distribution curves. This result is in agreement with the experimental investigation on liquid Ti, Zr and Ni conducted by Kim and Kelton(T=1670K), Zr (T=2290K) and Nb (T=2750K). TABLE V : VLattice constants and atomic internal positions of Co 7 Nb 6 and Co 23 Zr 6 calculated with our potentials at 300K and compared to the experimental values [61, 62]. CoNb (D 2 ) Co 3 Nb (L 12 ) ab initio EAM ab initio EAM C 11 (GPa) 251 242 368 357 C 12 (GPa) 173 143 164 194 C 44 (GPa) 60 71 160 131 TABLE VI : VIab-initio elastic constants of selected binaries as compared to the present set of potentials. Table VII . VIIIn Co 0.9 Zr 0.1 theindex Co 0.9 Zr 0.1 Co 0.9 Nb 0.1 1311 0.02 0.06 1321 0.03 0.03 1421 0.02 0.06 1422 0.04 0.11 1431 0.14 0.21 1441 0.05 0.01 1541 0.15 0.17 1551 0.24 0.16 1661 0.08 0.03 2101 1.49 1.57 2211 0.81 0.87 2321 0.20 0.20 2331 0.70 0.62 2441 0.10 0.13 TABLE VII : VIIHoneycutt and Andersen analysis of the simulations for supercooled Co 0.9 Zr 0.1 and Co 0.9 Nb 0.1 at 300K. The absolute error bars of the abundances are 0.01. dominant pairs are 1551 and 1541 exhibiting more icosahedral than distorted icosahedralorder. This tendency is inverted in Co 0.9 Nb 0.1 system with a larger amount of 1431 pairs favouring distorted icosahedral order. This is explained by a smaller difference in atomic radii between Co and Nb than Co and Zr because an atomic size difference of approximately 10% can relieve spatial frustration and stabilise the icosahedral structure[67]. In Co 0.9 Nb 0.1 the 1422 pairs are abundant indicating an hcp order which is less present in Co 0.9 Zr 0.1 system.Even if the value is rather low, the 1661 pairs are of importance indicating a slight bcc order, as anticipated. It is observed that while useful for gaining understanding of the evolution of the dominant short-range order, the single cluster model cannot capture the richness of the supercooled binary alloys structures.V. CONCLUSIONA fitting procedure has been performed to consistently derive a self-consistent set of manybody parameters for Co, Zr and Nb simple metals and selected alloys, including validation against first principles results where there are gaps in the experimental data. Combined with MD simulations, these parameters allow us to calculate RPDFs and bond angular distributions in the liquid phase for Co, Zr and Nb. Applied to supercooled binary alloys, . A Inoue, B Shen, H Koshiba, H Kato, A R Yavari, Nature Materials. 2661A. Inoue, B. Shen, H. Koshiba, H. Kato, and A.R. Yavari. Nature Materials, 2:661, 2003. . A R Yavari, Nature Materials. 6181A.R. Yavari. Nature Materials, 6:181, 2007. . D B Miracle, Nature Materials. 3697D. B. Miracle. Nature Materials, 3:697, 2004. . U K Rößler, H Teichler, Phys. Rev. E. 61394U.K. Rößler and H. Teichler. Phys. Rev. E, 61:394, 2000. . D Wolf, V Yamanakov, S R Phillpot, A Mukherjee, H Gleiter, Acta Materialia. 531D. Wolf, V. Yamanakov, S.R. Phillpot, A. Mukherjee, and H. Gleiter. Acta Materialia, 53:1, 2004. . M S Daw, M I Baskes, Phys. Rev. B. 296443M.S. Daw and M.I. Baskes. Phys. Rev. B, 29:6443, 1984. . M W Finnis, J E Sinclair, Phil. Mag. A. 5045M.W. Finnis and J.E. Sinclair. Phil. Mag. A, 50:45, 1984. . Ch, J Hausleitner, Hafner, Phys. Rev. B. 45115Ch. Hausleitner and J. Hafner. Phys. Rev. B, 45:115, 1992. . J Philips, A E Zou, M Carlsson, Widom, Phys. Rev. B. 499322R Philips, J. Zou, A.E. Carlsson, and M. Widom. Phys. Rev. B, 49:9322, 1994. Interatomic Forces in Condensed Matter. M W Finnis, Oxford University PressOxfordM.W. Finnis. Interatomic Forces in Condensed Matter. Oxford University Press, Oxford, 2003. . D J Oh, R A Johnson, J. Mater. Res. 3471D.J. Oh and R.A. Johnson. J. Mater. Res., 3:471, 1988. . R A Johnson, D J Oh, J. Mater. Res. 41195R.A. Johnson and D.J. Oh. J. Mater. Res., 4:1195, 1989. . R A Johnson, Phys. Rev. B. 3912554R.A. Johnson. Phys. Rev. B, 39:12554, 1989. . J Cai, Y Y Ye, Phys. Rev. B. 548398J. Cai and Y.Y. Ye. Phys. Rev. B, 54:8398, 1996. . R F Zhang, Y Kong, B X Liu, Phys. Rev. B. 71214102R.F. Zhang, Y. Kong, and B.X. Liu. Phys. Rev. B, 71:214102, 2005. . G D Ackland, D J Bacon, A F Calder, T Harry, Phil. Mag. A. 75713G.D. Ackland, D.J. Bacon, A.F. Calder, and T. Harry. Phil. Mag. A, 75:713, 1997. . M I Mendelev, S Han, D J Srolovitz, G D Ackland, D Y Sun, M Asta, Phil. Mag. 833977M.I. Mendelev, S. Han, D.J. Srolovitz, G.D. Ackland, D.Y. Sun, and M. Asta. Phil. Mag., 83:3977, 2003. . S L Dudarev, P M Derlet, J. Phys.: Condens. Matter. 177097S.L. Dudarev and P.M. Derlet. J. Phys.: Condens. Matter, 17:7097, 2005. . G D Ackland, S K Reed, Phys. Rev. B. 67174108G.D. Ackland and S.K. Reed. Phys. Rev. B, 67:174108, 2003. . G D Ackland, Phys. Rev. Lett. 9715502G.D. Ackland. Phys. Rev. Lett., 97:015502, 2006. . G Suran, M Naili, J C S Rivoire, Levy, J. Appl. Phys. 675649G. Suran, M. Naili, M Rivoire, and J.C.S. Levy. J. Appl. Phys., 67:5649, 1990. . R P Gupta, Phys. Rev. B. 236265R.P. Gupta. Phys. Rev. B, 23:6265, 1981. . D Tománek, A A Aligia, C A Balseiro, Phys. Rev. B. 325051D. Tománek, A.A. Aligia, and C.A. Balseiro. Phys. Rev. B, 32:5051, 1985. . K W Jacobsen, J K Nøskkov, M J Puska, Phys. Rev. B. 357423K.W. Jacobsen, J.K. Nøskkov, and M.J. Puska. Phys. Rev. B, 35:7423, 1987. . X D Dai, Y Kong, J H Li, B X Liu, J. Phys.: Condens. Matter. 184527X.D. Dai, Y. Kong, J.H. Li, and B.X. Liu. J. Phys.: Condens. Matter, 18:4527, 2006. . O Yifang, Z Bangwei, L Shuzhi, J Zhanpeng, Z. Phys. B -Condensed Matter. 101161O. Yifang, Z. Bangwei, L. Shuzhi, and J. Zhanpeng. Z. Phys. B -Condensed Matter, 101:161, 1996. . Ch, J Hausleitner, Hafner, J. Phys.: Condens. Matter. 26651Ch. Hausleitner and J. Hafner. J. Phys.: Condens. Matter, 2:6651, 1990. . Z Bangwei, O Yifang, Phys. Rev. B. 483022Z. Bangwei and O. Yifang. Phys. Rev. B, 48:3022, 1993. The Recursion Method and Its Applications. P Turchi, F Ducastelle, Springer-VerlagBerlinP. Turchi and F. Ducastelle. The Recursion Method and Its Applications. Springer-Verlag, Berlin, 1985. . F Cleri, V Rosato, Phys. Rev. B. 4822F. Cleri and V. Rosato. Phys. Rev. B, 48:22, 1993. . C Yoo, P Söderlind, H Cynn, J. Phys.: Condens. Matter. 10311C.S Yoo, P. Söderlind, and H. Cynn. J. Phys.: Condens. Matter, 10:L311, 1998. . P Modak, A K Verma, R S Rao, B K Godwal, R Jeanloz, Phys. Rev. B. 7412103P. Modak, A.K. Verma, R.S. Rao, B.K. Godwal, and R. Jeanloz. Phys. Rev. B, 74:012103, 2006. . F Willaime, C Massobrio, Phys. Rev. B. 4311653F. Willaime and C. Massobrio. Phys. Rev. B, 43:11653, 1991. . F Ducastelle, Orsay, FranceUniversité de Paris-SudPhD thesisF. Ducastelle. PhD thesis, Université de Paris-Sud, Orsay, France, 1972. . R Pasianot, D Farkas, E J Savino, Phys. Rev. B. 436952R. Pasianot, D. Farkas, and E.J. Savino. Phys. Rev. B, 43:6952, 1991. . J Häglund, A Fernandez-Guillermet, G Grimvall, M Körling, Phys. Rev. B. 4811685J. Häglund, A. Fernandez-Guillermet, G Grimvall, and M. Körling. Phys. Rev. B, 48:11685, 1993. . F Willaime, C Massobrio, Phys. Rev. Lett. 632244F. Willaime and C. Massobrio. Phys. Rev. Lett., 63:2244, 1989. . S Baroni, A Corso, S De Gironcoli, P Giannozzi, C Cavazzoni, G Ballabio, S Scandolo, G Chiarotti, P Focher, A Pasquarello, K Laasonen, A Trave, R Car, N Marzari, A Kokalj, S. Baroni, A. Dal Corso, S. de Gironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pasquarello, K. Laasonen, A. Trave, R. Car, N. Marzari, and A. Kokalj. http://www.pwscf.org. . J P Perdew, K Burke, M Ernzerhof, Phys. Rev. Lett. 781396J.P. Perdew, K. Burke, and M. Ernzerhof. Phys. Rev. Lett., 78:1396, 1996. . H J Monkhorst, J D Pack, Phys. Rev. B. 135188H.J. Monkhorst and J.D. Pack. Phys. Rev. B, 13:5188, 1976. . J D Gale, A L , Molecular Simulation. 29291J.D. Gale and A.L. Rohl. Molecular Simulation, 29:291, 2003. Introduction to the Solid State Physics. C Kittel, WileyNew YorkC. Kittel. Introduction to the Solid State Physics. Wiley, New York, 1966. Single Crystal Elastic Constants and Calculated Aggregated Properties. G Simmons, H Wang, MIT PressCambridgeG. Simmons and H. Wang. Single Crystal Elastic Constants and Calculated Aggregated Prop- erties. MIT Press, Cambridge, 1971. . P J Craievich, J M Sanchez, R E Watson, M Weinert, Phys. Rev. B. 552787P.J. Craievich, J.M. Sanchez, R.E. Watson, and M. Weinert. Phys. Rev. B, 55(2):787, 1997. . M I Baskes, R A Johnson, Modelling Simul. Mater. Sci. Eng. 2147M.I. Baskes and R.A. Johnson. Modelling Simul. Mater. Sci. Eng., 2:147, 1994. . K H J Buschow, Journal of the Less-Common Metals. 85221K. H. J. Buschow. Journal of the Less-Common Metals, 85:221, 1982. . J K Pargeter, W Hume-Rothery, Journal of the Less-Common Metals. 12366J.K. Pargeter and W. Hume-Rothery. Journal of the Less-Common Metals, 12:366, 1967. . C J Smithell, Smithells Metals Reference Book. Butterworths. C.J. Smithell. Smithells Metals Reference Book. Butterworths, London, 1983. Introduction to Numerial Analysis. F B Hildebrand, McGraw-HillF.B. Hildebrand. Introduction to Numerial Analysis. McGraw-Hill, 1965. Buslaps, and S. Reutzel. D Holland-Moritz, T Schenk, R Bellissent, V Simonet, K Funakoshi, J M Merino, T , J. Non-Cryst. Solids. 47D. Holland-Moritz, T. Schenk, R. Bellissent, V. Simonet, K. Funakoshi, J.M. Merino, T. Bus- laps, and S. Reutzel. J. Non-Cryst. Solids, 312-314:47, 2002. . G M Bhuiyan, M Silbert, M J Stott, Phys. Rev. B. 53636G.M. Bhuiyan, M. Silbert, and M.J. Stott. Phys. Rev. B, 53:636, 1995. . X J Han, J Z Wang, M Chen, Z Y Guo, J. Phys.: Condens. Matter. 162565X.J. Han, J.Z. Wang, M. Chen, and Z.Y. Guo. J. Phys.: Condens. Matter, 16:2565, 2004. . J B Kita, Z Van Zydveld, T Morita, Iida, J. Phys.: Condens. Matter. 6811Y Kita, J.B. Van Zydveld, Z. Morita, and T. Iida. J. Phys.: Condens. Matter, 6:811, 1994. . G Kresse, J Hafner, Phys. Rev. B. 4813115G. Kresse and J. Hafner. Phys. Rev. B, 48:13115, 1993. . T Schenk, D Holland-Moritz, V Simonet, R Bellissent, D M Herlach, Phys. Rev. Lett. 8975507T. Schenk, D. Holland-Moritz, V. Simonet, R. Bellissent, and D. M. Herlach. Phys. Rev. Lett., 89:075507, 2002. . J D Honeycutt, H C Andersen, J. Phys. Chem. 914950J.D. Honeycutt and H.C. Andersen. J. Phys. Chem., 91:4950, 1987. . H Jonsson, H C Andersen, Phys. Rev. Lett. 602295H. Jonsson and H.C. Andersen. Phys. Rev. Lett., 60:2295, 1988. . T H Kim, K F Kelton, J. Chem. Phys. 12654513T.H. Kim and K.F. Kelton. J. Chem. Phys., 126:054513, 2007. . N Jakse, Pasturel, Phys. Rev. Lett. 91195501N Jakse and A Pasturel. Phys. Rev. Lett., 91:195501, 2003. . J L Rodriguez-Lopez, F Aguilera-Granja, K Michaelian, A Vega, J. of Alloys and Compounds. 36993J.L. Rodriguez-Lopez, F. Aguilera-Granja, K. Michaelian, and A. Vega. J. of Alloys and Compounds, 369:93, 2004. . A K Shurin, P I Kripyakevich, E I Gladyshevskii, Kristallografiya, 10414A.K. Shurin, P.I. Kripyakevich, and E.I. Gladyshevskii. Kristallografiya, 10(3):414, 1965. Yu B Kuz&apos;ma, V Ya, Yu V Markiv, R V Voroshilov, Skolozdra, Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy. 2259Yu.B. Kuz'ma, V.Ya. Markiv, Yu.V. Voroshilov, and R.V. Skolozdra. Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy, 2:259, 1966. . A R Miedema, P F De Châtel, F R De Boer, Physica B. 1001A.R. Miedema, P.F. de Châtel, and F.R. de Boer. Physica B, 100:1, 1980. . Yu A Babanov, A F Sidorenko, A V Ryazhkin, V R Shvetsov, J Moessinger, H Kronmueller, Nucl. Inst. and Meth. in Phys. Res. A. 405400Yu.A. Babanov, A.F. Sidorenko, A.V. Ryazhkin, V.R. Shvetsov, J. Moessinger, and H. Kron- mueller. Nucl. Inst. and Meth. in Phys. Res. A, 405:400, 1998. . B Vainshtein, V M Fridkin, V L Indenbom, Modern Crystallography. 2SpringerStructure of Crystals. 3 rd editionB.K Vainshtein, V.M. Fridkin, and V.L. Indenbom. Structure of Crystals, volume 2 of Modern Crystallography. Springer, 3 rd edition, 2000. . M Jamet, P Dupuis, G Mélinon, A Guiraud, W Pérez, A Wernsdorfer, B Traverse, Baguenard, Phys. Rev. B. 62493M. Jamet, V Dupuis, P. Mélinon, G. Guiraud, A. Pérez, W. Wernsdorfer, A. Traverse, and B. Baguenard. Phys. Rev. B, 62:493, 2000. . D R Nelson, F Spaepen, page 1. Academic. 42D.R. Nelson and F. Spaepen. volume 42, page 1. Academic, Boston, 1989.
[]
[ "Photometric Observations of 107P/Wilson-Harrington", "Photometric Observations of 107P/Wilson-Harrington" ]
[ "Seitaro Urakawa \nBisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan\n", "Shin-Ichiro Okumura \nBisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan\n", "Kota Nishiyama \nBisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan\n", "Tsuyoshi Sakamoto \nBisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan\n", "Noritsugu Takahashi \nBisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan\n", "Shinsuke Abe \nInstitute of Astronomy\nNational Central University\n300 Jhongda Road32001JhongliTaoyuanTaiwan\n", "Masateru Ishiguro \nDepartment of Physics and Astronomy\nSeoul National University\n599 Gwanak-ro, Gwanak-gu151-742SeoulRepublic of Korea\n", "Kohei Kitazato \nResearch Center for Advanced Information Science and Technology\nUniversity of Aizu\nAizu-Wakamatsu965-8580FukushimaJapan\n", "Daisuke Kuroda \nOkayama Astrophysical Observatory\nNAOJ\n3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan\n", "Sunao Hasegawa \nInstitute of Space and Astronautical Science\nJapan Aerospace Exploration Agency\n3-1-1 Yoshinodai, Chuo-ku252-5210SagamiharaKanagawaJapan\n", "Kouji Ohta \nDepartment of Astronomy\nKyoto University\nSakyo-ku606-8502KyotoJapan\n", "Nobuyuki Kawai \nDepartment of Physics\nTokyo Institute of Technology\n2-12-1 Ookayama, Meguro-ku152-8551TokyoJapan\n", "Yasuhiro Shimizu \nOkayama Astrophysical Observatory\nNAOJ\n3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan\n", "Shogo Nagayama \nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n", "Kenshi Yanagisawa \nOkayama Astrophysical Observatory\nNAOJ\n3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan\n", "Michitoshi Yoshida \nDepartment of Physical Science\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan\n", "Makoto Yoshikawa \nBisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan\n\nInstitute of Space and Astronautical Science\nJapan Aerospace Exploration Agency\n3-1-1 Yoshinodai, Chuo-ku252-5210SagamiharaKanagawaJapan\n" ]
[ "Bisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan", "Bisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan", "Bisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan", "Bisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan", "Bisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan", "Institute of Astronomy\nNational Central University\n300 Jhongda Road32001JhongliTaoyuanTaiwan", "Department of Physics and Astronomy\nSeoul National University\n599 Gwanak-ro, Gwanak-gu151-742SeoulRepublic of Korea", "Research Center for Advanced Information Science and Technology\nUniversity of Aizu\nAizu-Wakamatsu965-8580FukushimaJapan", "Okayama Astrophysical Observatory\nNAOJ\n3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan", "Institute of Space and Astronautical Science\nJapan Aerospace Exploration Agency\n3-1-1 Yoshinodai, Chuo-ku252-5210SagamiharaKanagawaJapan", "Department of Astronomy\nKyoto University\nSakyo-ku606-8502KyotoJapan", "Department of Physics\nTokyo Institute of Technology\n2-12-1 Ookayama, Meguro-ku152-8551TokyoJapan", "Okayama Astrophysical Observatory\nNAOJ\n3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan", "National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan", "Okayama Astrophysical Observatory\nNAOJ\n3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan", "Department of Physical Science\nHiroshima University\n1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan", "Bisei Spaceguard Center\nSpaceguard Association\n1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan", "Institute of Space and Astronautical Science\nJapan Aerospace Exploration Agency\n3-1-1 Yoshinodai, Chuo-ku252-5210SagamiharaKanagawaJapan" ]
[]
We present lightcurve observations and multiband photometry for 107P/Wilson-Harrington using five small-and medium-sized telescopes. The lightcurve has shown a periodicity of 0.2979 day (7.15 hour) and 0.0993 day (2.38 hour), which has a commensurability of 3:1. The physical properties of the lightcurve indicate two models: (1) 107P/Wilson-Harrington is a tumbling object with a sidereal rotation period of 0.2979 day and a precession period of 0.0993 day. The shape has a long axis mode (LAM) of L 1 :L 2 :L 3 = 1.0:1.0:1.6. The direction of the total rotational angular momentum is around λ = 310 • , β = −10 • , or λ = 132 • , β = −17 • . The nutation angle is approximately constant at 65 • . (2) 107P/Wilson-Harrington is not a tumbler. The sidereal rotation period is 0.2979 day. The shape is nearly spherical but slightly hexagonal with a short axis mode (SAM) of L 1 :L 2 :L 3 = 1.5:1.5:1.0. The pole orientation is around λ = 330 • , β = −27 • . In addition, the model includes the possibility of binary hosting. For both models, the sense of rotation is retrograde. Furthermore, multiband photometry indicates that the taxonomy class of 107P/Wilson-Harrington is C-type. No clear rotational color variations are confirmed on the surface.
10.1016/j.icarus.2011.06.044
[ "https://arxiv.org/pdf/1106.5238v1.pdf" ]
51,814,373
1106.5238
01f2d0ab4eb4a4499bd804421fd6e9aca1fb4434
Photometric Observations of 107P/Wilson-Harrington 26 Jun 2011 Seitaro Urakawa Bisei Spaceguard Center Spaceguard Association 1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan Shin-Ichiro Okumura Bisei Spaceguard Center Spaceguard Association 1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan Kota Nishiyama Bisei Spaceguard Center Spaceguard Association 1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan Tsuyoshi Sakamoto Bisei Spaceguard Center Spaceguard Association 1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan Noritsugu Takahashi Bisei Spaceguard Center Spaceguard Association 1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan Shinsuke Abe Institute of Astronomy National Central University 300 Jhongda Road32001JhongliTaoyuanTaiwan Masateru Ishiguro Department of Physics and Astronomy Seoul National University 599 Gwanak-ro, Gwanak-gu151-742SeoulRepublic of Korea Kohei Kitazato Research Center for Advanced Information Science and Technology University of Aizu Aizu-Wakamatsu965-8580FukushimaJapan Daisuke Kuroda Okayama Astrophysical Observatory NAOJ 3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan Sunao Hasegawa Institute of Space and Astronautical Science Japan Aerospace Exploration Agency 3-1-1 Yoshinodai, Chuo-ku252-5210SagamiharaKanagawaJapan Kouji Ohta Department of Astronomy Kyoto University Sakyo-ku606-8502KyotoJapan Nobuyuki Kawai Department of Physics Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku152-8551TokyoJapan Yasuhiro Shimizu Okayama Astrophysical Observatory NAOJ 3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan Shogo Nagayama National Astronomical Observatory of Japan 2-21-1 Osawa181-8588MitakaTokyoJapan Kenshi Yanagisawa Okayama Astrophysical Observatory NAOJ 3037-5, 719-0232Honjo, KamogataAsakuchi, OkayamaJapan Michitoshi Yoshida Department of Physical Science Hiroshima University 1-3-1 Kagamiyama, Higashi-Hiroshima739-8526HiroshimaJapan Makoto Yoshikawa Bisei Spaceguard Center Spaceguard Association 1716-3, 714-1411Okura, Bisei, IbaraOkayamaJapan, Japan Institute of Space and Astronautical Science Japan Aerospace Exploration Agency 3-1-1 Yoshinodai, Chuo-ku252-5210SagamiharaKanagawaJapan Photometric Observations of 107P/Wilson-Harrington 26 Jun 2011Preprint submitted to Icarus June 28, 2011Asteroids rotationCometsPhotometrySatellites of asteroids We present lightcurve observations and multiband photometry for 107P/Wilson-Harrington using five small-and medium-sized telescopes. The lightcurve has shown a periodicity of 0.2979 day (7.15 hour) and 0.0993 day (2.38 hour), which has a commensurability of 3:1. The physical properties of the lightcurve indicate two models: (1) 107P/Wilson-Harrington is a tumbling object with a sidereal rotation period of 0.2979 day and a precession period of 0.0993 day. The shape has a long axis mode (LAM) of L 1 :L 2 :L 3 = 1.0:1.0:1.6. The direction of the total rotational angular momentum is around λ = 310 • , β = −10 • , or λ = 132 • , β = −17 • . The nutation angle is approximately constant at 65 • . (2) 107P/Wilson-Harrington is not a tumbler. The sidereal rotation period is 0.2979 day. The shape is nearly spherical but slightly hexagonal with a short axis mode (SAM) of L 1 :L 2 :L 3 = 1.5:1.5:1.0. The pole orientation is around λ = 330 • , β = −27 • . In addition, the model includes the possibility of binary hosting. For both models, the sense of rotation is retrograde. Furthermore, multiband photometry indicates that the taxonomy class of 107P/Wilson-Harrington is C-type. No clear rotational color variations are confirmed on the surface. Introduction Asteroids and comets are primordial bodies that formed in the earliest stage of the solar system. Their rotational states, shapes, and material reflect the collisions, disruptions, and chemical processes since then to the present. Some small solar system bodies exhibit behavior such as that shown by both comets and asteroids (so-called, comet-asteroid transition objects). As an example, near-earth object (NEO) (3200) Phaethon shows signs of past cometary activity because it is thought to be associated with the Geminid (Gustafson, 1989). Dynamical numerical simulations and spectral observations for (3200) Phaethon support (2) Pallas, which is outer main belt asteroids, is the most likely parent body of (3200) Phaethon (Clark et al. 2010;de León et al., 2010). Meanwhile, objects that display cometary activities in the main-belt asteroid (MBA) region have recently been discovered. They are classified as main-belt comets (MBCs) (Hsieh and Jewitt, 2006); the MBCs are 133P/Elst-Pizzaro , P/2005 U1 (Read et al., 2005), 176P/LINEAR (Hsieh et al., 2011), P/2008 R1 (Garrad) (Jewitt et al., 2009), P/2010 A2 (LINEAR) (Birtwhistle et al., 2010), P/2010 R2 (La Sagra) (Marsden et al., 2010), and (596) Scheila (Bodewits et al., 2011;Jewitt et al., 2011). One possible activation mechanism for MBCs is impacts with small (e.g., meter-sized) objects (Toth, 2000;Díaz and Gil-Hutton, 2008;Jewitt et al., 2010;Snodgrass et al., 2010;Bodewits et al., 2011;Jewitt et al., 2011). The other activation mechanisms are rotational-fissions due to the spin-up by Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effects , and thermal influences (Jewitt et al., 2009). Interesting properties of MBCs are their dynamical origin and possible function as reservoirs for water-ice and organics. A numerical integration by Haghighipour (2009) states that the origin of 133P/Elst-Pizarro, 176P/LINEAR, and P/2005 U1 (Read) is concordant with the Themis family of asteroids. Compared with all asteroids, the Themis family of asteroids includes B-type asteroids at a relatively high population rate. Some B-type asteroids in the Themis family seem to have experienced aqueous alterations (Yang and Jewitt, 2010;Clark et al., 2010). (3200) Phaethon is also a B-type asteroid and shows the existence of aqueous alteration materials (Licandro et al., 2007). In addition, water-ice and organics are detected on the surface of (24) Themis (Rivkin and Emery, 2010) and (65) Cybele, which orbits along the outer edge of the main belt (Licandro et al., 2011). The study of comet-asteroid transition objects provides keys to the dynamical origin and evolution of NEOs, the mutual collisions of small solar system bodies, the material differences between asteroids and comets, and the origin of Earth's water. This study's purpose is to obtain the rotational states, shape model, and rotational color variations for 107P/Wilson-Harrington (also know as (4015) Wilson-Harrington; hereafter 107P), which is a representative comet-asteroid transition object. 107P was discovered accompanied by a faint cometary tail at Palomar Observatory in 1949 (Fernandez et al., 1997). The object, however, could not be tracked because of insufficient observations to determine an accurate orbit. Later, a near-earth asteroid 1979VA (= 4015) was discovered. Subsequent observations identified asteroid (4015) 1979VA and 107P as the same object. Despite a devoted search, no cometary activity has been detected since the initial observation of 107P (Chamberlin et al., 1996;Lowry and Weissman, 2003;Ishiguro et al., 2011). 107P is an Apollo asteroid whose orbital parameters are a = 2.639 AU, e = 0.624, i = 2.785 • , and the Tisserand parameters (T J ) = 3.08. A numerical simulation by Bottke et al. (2002) mentions that there is a 4 % chance that 107P has a JFC origin and a 65% chance it has an origin in the outer main-belt region. Taxonomically, it is categorized as a CF-type (Tholen, 1989). The reflectance spectrum in the region 3800-6200Å is similar to (3200) Phaethon (Chamberlin et al., 1996). The thermal properties of 107P have been investigated by mid-infrared photometry with NASA's Spitzer Telescope (Licandro et al., 2009). These observations show that the beaming parameter, the diameter, and the albedo are η = 1.39 ± 0.26, D = 3.46 ± 0.32 km, and p v = 0.059 ± 0.011, respectively. The rotational period of 107P has been reported to be 3.556 hour and 6.10 ± 0.05 hour by Harris and Young (1983) and Osip et al. (1995), respectively. Osip et al. (1995) ascribes the difference of the two reports to the noisy data of Harris and Young (1983) because of the weather conditions. The few days' observation in both reports, however, is not enough to determine the correct rotational period. Longer observations are required to derive the correct rotational period and other physical properties. We hypothesize that 107P migrates to the NEO region from the outer main-belt region inhabited by six of seven known MBCs, and impacts with small objects could eject dust and/or expose sub-surface ice that then trigger 107P's cometary activity. Post-MBC, 107P is capable of becoming host to water-ice, organics, and aqueous alteration materials. In this hypothesis, the impacts' influence would be apparent in the rotational states and/or the surface color variations. We had an opportunity to observe 107P from August 2009 to March 2010. Our long observation campaigns enable us to derive the rotational states, shape model, and rotational color variations. Furthermore, the orbit of 107P makes it accessible by spacecraft. A more advanced sample return mission from a D-type asteroid or an asteroid-comet transition object is envisioned in Japan. One candidate is 107P (Yoshikawa et al., 2008). Clarification of the physical properties of 107P is important to the design of the future mission. If we are able to obtain 107P's physical properties, the data will be useful to revise the physical model of Licandro et al. (2009), similar to the Hayabusa-2 target 162173 (1999JU3) whose physical model was reconstructed by both thermal observations and the lightcurve (Müller et al., 2011). This paper is organized as follows. In Section 2, we describe the observations made and the data reduction. In Section 3, we mention the rotational states and shape model of 107P. In Section 4, we focus on the possibility of tumbling motion and the existence of a binary. Finally, we summarize the physical model of 107P and discuss the feasibility of a sample return mission. Observations and Data Reduction Observations We conducted the photometric observation campaigns of 107P with five small-and medium-sized telescopes. The observational circumstances and the states of 107P are listed in Table 1 and Table 2, respectively. All telescopes were operated with the non-sidereal tracking mode. The longest-term observation of this campaign was carried out using the 1.0 m f/3 telescope at the Bisei Spaceguard Center (BSGC 1 ) from September 6, 2009 to March 11, 2010. The detector consisted of four CCD chips with 4096 × 2048 pixels. We used one CCD chip to obtain as many images as possible by shortening the processing time. The field of view (FOV) for one CCD chip is 1.14 • × 0.57 • with a pixel resolution of 1.0 ′′ . The exposure time varied from 30 s to 600 s according to the observational situations. Individual images were taken with a commercially available short-pass (long-wavecut) filter, the effective wavelength of which ranged from 490 nm to 910 nm. We denote the filter as W in Table 1. In order to investigate rotational 1 BSGC is administrated by the Japan Space Forum. color variations, multiband photometry was conducted using a Sloan Digital Sky Survey (SDSS) g ′ , r ′ , i ′ , z ′ filter on December 17, 2009. One set of observations was made using three consecutive images for each filter. The filters were changed in the following sequence: three g ′ images → three r ′ images → three i ′ images → three z ′ images. We repeated this sequence four times. The second-longest-term observation used the 0.5 m f/6.5 Multicolor Imaging Telescope for Survey and Monstrous Explosions (MITSuME) (Kotani et al., 2005) with an exposure time of 120-300 s. The fourth observation was made using the Lulin One-meter Telescope (LOT) (Huang et al., 2005) in Taiwan on December 7-10, 2009. The CCD consists of a 1340 × 1300 array, and the FOV covers the area of 11.5 ′ × 11.2 ′ . The pixel resolution and f-number are 0.51 ′′ /pixel and 8, respectively. The images were obtained using a Johnson-Cousins R c filter with an exposure time of 90 s. The photometric precision was insufficient after February 7, 2010. We did not use these data for the estimation of rotational periods and shape models. These data were utilized for the trend confirmation of lightcurves that were obtained from the other day's data and the monitoring of cometary activity. Data reduction All images were bias and flat-field corrected. When using the data of OAO for the derivation of lightcurve, we stacked four images to compensate for the poor flux. All observation times were corrected using the light travel time from 107P to the Earth. By using the IRAF/APPHOT 2 package, we measured the raw magnitude of 107P and from three to seven reference stars that were bright enough compared with 107P. We set aperture radius to 1.5 × FWHM for 107P and reference stars images, respectively. Since the reference star images are slightly elongated by the non-sidereal tracking, the aperture radius is larger than that of 107P. We calibrated the magnitude fluctuations due to the change of atmospheric conditions as follows, F i c (t) = F i o (t) − F i r (t).(1) Here, F i c (t) is the lightcurve by rotation of 107P in i-th observation day, F i o (t) is the raw magnitude of 107P, F i r (t) is the averaged raw magnitude of reference stars and represents the change of atmospheric conditions, and t is the observational time. Next, we define the averaged magnitude of F i c (t) in each night as the normalized (zero) magnitude. The lightcurve by rotation of 107P can be rewritten as F i wh (t) = F i c (t) − F i c ,(2) where F i c is the averaged magnitude of F i c (t). Since the averaged magnitude is normalized to zero magnitude for all nights, we can connect the different night's lightcurve with little regard for the difference of absolute magnitude. In addition, the difference of reference stars each night does not affect the periodicity of lightcurve. The problem of this procedure could include the offset between different nights when the short observation time per day poses the detection of a specific peak (bottom) in the lightcurve. However, the peaks and bottoms in the lightcurve have been detected evenly (See Fig. 3) because the observation time per day is long enough from November 2009 to February 2010 when the data are utilized for the analysis. Thus, the offset is negligible. Furthermore, the apparent magnitude change of 107P is gradual up to 1.0 magnitude per month ( Table 2). The change does not act on the derivation of rotational period that is expected to be from three to seven hours according to past reports. In contrast to the relative photometry of the lightcurve, more photometric precision is required to detect the rotational color variation by multiband photometry. In order to improve the photometric precision, we averaged three consecutive images of 107P for the BSGC's data and 14-16 consecutive images of 107P for the OAO's data. We also measured the flux of ten standard stars from SDSS data Release 7 (Abazajian et al., 2009), whose stars were imaged simultaneously in the same frame as 107P (Table 3). These objects have magnitudes of about 14 mag to 16 mag in the r ′ -band and classification code 1 (= primary), quality flag 3 (= good), and object class 6 (= star). We evaluated atmospheric extinction coefficients and conversion factors to standardize the SDSS system for each filter. The atmospheric extinction coefficient was calculated by the magnitude variations of the standard stars for the change in airmass. Extra-atmospheric instrumental magnitudes of both 107P and the standard stars were derived using the obtained atmospheric extinction coefficient. The conversion factors were estimated by comparing the extra-atmospheric instrumental magnitudes with the magnitude of standard stars. In BSGC's observation, the multi-color images were not obtained simultaneously. The brightness of 107P by the rotation changes inevitably during the filter switch. We defined the time of the third r ′ images in each sequence as a standard time, and then calibrated the amount of brightness change for the standard time. The amount of brightness change was estimated by the fitting curve of the lightcurve. In order to compare the OAO's data with BSGC's, the R c and I c magnitudes obtained at OAO were converted to r ′ and i ′ magnitudes using the conversion equations proposed by Jordi et al. (2006). Results Rotational states Since the "Standard Feature (SF )" that is, the flux peaks and/or bottoms in lightcurves, shifts along the phase of lightcurves due to changes in the geometric relationship between the Earth, 107P, and the Sun during the long-term observations, the estimation of the sidereal rotation period for 107P requires a short-term observation within a few weeks. We use the data from December 7 to 22, when the phase-shift is small. The photometric precision of 107P and the observational implementation time per day are enough to make clear the sidereal rotational period during the term. Assuming double-peak lightcurves, a period analysis is carried out with a Lomb-Scargle periodgram (Lomb, 1976;Scargle, 1982). Lightcurves mainly represent the light scattering cross section of objects. When we assume that an object is a symmetric ellipsoidal body, almost the same cross section appears in every half rotation. Therefore, it is difficult On the other hand, the period of 0.0993 day may be the precession period. If an object has tumbling motions, the lightcurve is dominated by two periods: one, P ψ , for the rotation about the extremal axis of the object as an inertia ellipsoid, and the other, P φ , for the precession about the total rotational angular momentum vector. When frequencies are defined as 2f ψ = P −1 ψ and 2f φ = P −1 φ , the lightcurve periodicity of tumbling objects appears at frequencies that are a linear combination of f ψ and f φ (Kaasalainen, 2001). When we assume that P ψ is 0.2979 day and P φ is 0.0993 day, the frequency 4f ψ = 2(f φ − f ψ ) = 6.713 day −1 approximately corresponds to the inverse of period of Harris and Young (1983) and one half of our rotational period of 0.2979 day. The existence of periodicity of the linear combination of two periods shows circumstantial evidence for tumbling. We make the shape model of 107P in the following subsection and discuss the feasibility of tumbling motion in Section 4.1. Direction of total rotational angular momentum and shape model Above, we suggested the possibility of tumbling motion. If 107P is a tumbler, the pole orientation does not accord with the direction of total rotational angular momentum and is not stable. What we can obtain is not the pole orientation but the direction of total rotational angular momentum. The direction of total rotational angular momentum of 107P can be estimated using the "epoch method" (Magnusson, 1986) or the "lightcurve inversion method" Kaasalainen et al., , 2002. The "amplitude method" is also proposed as an alternative method (Magnusson 1986). However, we cannot adopt the amplitude method because of the small amplitude change during the observational term. The epoch method determines the direction of total rotational angular momentum by minimizing the phase-shift of the SF in lightcurves. We select a lightcurve peak around the phase of 0.01 at the bottom of Fig. 2 as the SF because the peak is better observed than any other feature. Moreover, we use the data obtained from November 5, 2009 to February 5, 2010. The identification of the lightcurve peak is difficult from the data of another term because of the photometric error. The phase-shift can be written as T i − T 0 P ψ − n i = θ i − θ 0 2π ,(3) where T 0 is the time at the first SF , T i is the time at the i-th SF , P ψ is the sidereal rotational period, and n i is the number of rotations between T 0 and T i . θ 0 and θ i are the projected directions of the phase angle bisector (P AB) in the plane that is perpendicular to the direction of total rotational angular momentum at T 0 and T i , respectively. Table 4 shows the epoch of SF s. The left hand of Eq. (3) is estimated from the observations; the right hand is theoretically calculated from the tentative direction of total rotational angular momentum and the orbital information of 107P. We define δ with the following equation δ = N i T i − T 0 P ψ − n i − θ i − θ 0 2π 2 /(N − 1),(4) where N is the number of the epoch; here N = 4. We can estimate the direction of total rotational angular momentum by seeking the minimum of δ. The lightcurve inversion method derives the most adequate shape model and the corresponding direction of total rotational angular momentum. When carrying out the lightcurve inversion method, we set the initial conditions for the direction of total rotational angular momentum and the sidereal rotation period to that of 0.2979 day. We seek the least deviation between the observational lightcurve and the reconstructed lightcurve from the shape model by scanning the direction of total rotational angular momentum in steps of 1 • in ecliptic longitude and latitude. We use the data of November 16, 1979 (Harris andYoung, 1983); December 1-2, 1992 (Osip et al., 1995); and our high photometric precision data (the error is less than 0.05 mag), which was obtained from November 7, 2009 to January 18, 2010. Fig. 5 shows the deviation map of the direction of total rotational angular momentum using the lightcurve inversion method. In addition to the candidates of the epoch method, we have found three other candidates: C (λ = 330 • , β = −27 • ), D (λ = 328 • , β = −61 • ) , and E (λ = 167 • , β = 7 • ). However, we exclude the candidates D and E because they are less compatible with the epoch method. Note that the observational data was obtained in the phase angle from 21 • to 77 • . There is no data in a low phase angle. Furthermore, the direction of rotational angular momentum has an uncertainty of typically more than 5 • . For example, though the ground-based observation of Itokawa showed that the pole orientation was Kaasalainen et al., 2003), the Hayabusa spacecraft revealed that the pole orientation of Itokawa was λ = 128.5 • , β = −89.66 • (Demura et al., 2006). Moreover, we can see from Table 4 that the SF in the lightcurve shifts to the negative direction with time. As a corollary, the direction of total rotational angular momentum of the three candidates is south of the ecliptic plane. This indicates that the sense of sidereal rotation is retrograde. The lightcurve of Fig. 6 has been calibrated for the phase-shift. λ = 355 • , β = −84 • ( The six peaks and the periodicity in the lightcurve become clearer than in the lightcurve of 0.2979 day period in Fig. 2. This result adds to the evidence of the retrograde rotation. Next, we make the three shape models of 107P for the directions of total rotational angular momentum A, B and C. The shape model A is shown in Fig. 7. Here, L 1 , L 2 , and L 3 are defined as the normalized axis length when 107P is a triaxial ellipsoid body. The axes satisfy the relationship L 1 ≤ L 2 ≤ L 3 . L 3 is a sidereal rotation axis of the shape model A. We have found that the normalized axis lengths L 1 , L 2 , and L 3 are around 1.0, 1.0, and 1.6, respectively. The axis ratio of the shape model B is around the same as that of the shape model A. The shape model A and B indicate a so-called long axis mode (LAM). Some previous studies have described the motion of a force-free asymmetric rigid body (Samarasinha and A'Hearn, 1991;Kaasalainen, 2001). Now, L 1 ≃ L 2 indicates that the equations of force-free precession are simplified to the following, ψ = cos θ M I 3 −φ ,(5)φ = M I 1 ,(6)I 1 = µ 20 (L 2 2 + L 2 3 ),(7)I 3 = µ 20 (L 2 1 + L 2 2 ).(8) Here, ψ, φ, θ are the Euler angles of sidereal rotation, precession, and and L 2 should be longer than 3 L 3 ifφ = −3ψ and L 1 ≃ L 2 are correct. These situations are inconsistent with our results. Therefore, 107P of the shape model C is a non-precession object rather than a precessional object. Taxonomic class and rotational color variations The taxonomic class and rotational color variations for 107P were investigated by a color-color diagram. We note that the classification of subclasses, such as B, F, or G-type, is difficult using multiband photometry. We conducted the multiband photometry eight times (P hase-1 to -8). The obtained color-color diagram and the color index are shown in Fig. 9 and Table 5. We utilized the z ′ images of P hase-6 for those of P hase-7 due to the poor weather conditions in P hase-7. The color-color diagram macroscopically shows that 107P is a C-type (including B, F, G-type) asteroid. The colors of P hase-2 and P hase-6 indicate typical C-type features in the three color indices. The others are slightly reddish features like an X-type asteroid in the color index of r ′ − i ′ . Though only the g ′ − r ′ of P hase-3 barely exceed the one-sigma of mean color index in Table 5, it is difficult to assert the detection of the rotational color variation due to the photometric error. In addition to it, the long observation term of ∼0.15 in phase (∼1.0 hour) for each sequence obscures the detection of rotational color variation. In order to confirm the color variations, follow-up observations and/or exploration by spacecraft are needed. of i ′ − z ′ . Mean shows the arithmetic average and standard deviation of each color index. (Ivezić et al., 2001). X-type asteroids include E, M, and P-type asteroids. The case of low-albedo asteroids indicates P-type. Obs term [Phase] Observatory g ′ − r ′ r ′ − i ′ i ′ − z ′ Discussion Tumbling We discuss the possibility of tumbling. There were some reports in which asteroid lightcurves indicated tumbling, e.g., (253) Mathilde (Mottola et al., 1995), (3288) Seleucus (Harris et al., 1999), and (4179) Toutatis (Spencer et al., 1995;Kryszczyńska et al., 1999). Pravec et al. (2005) assessed the validity of tumbling for these asteroids based on whether the lightcurves could be approximated with two-dimensional Fourier series and the physical model of tumbling could be constructed. The two-dimensional Fourier series is described in the following form F m (t) = C 0 + m j=1 C j0 cos 2πj P ψ t + S j0 sin 2πj P ψ t + m k=1 m j=−m C jk cos 2πj P ψ + 2πk P φ t + S jk sin 2πj P ψ + 2πk P φ t ,(9) where m is the order, C 0 is the mean reduced light flux, C jk and S jk are the Fourier coefficients for the linear combination of the two frequencies P −1 ψ and P −1 φ , respectively, and t is the time. Substituting m = 2, P ψ = 0.2979 day, and P φ = 0.0993 day for 107P, we obtain a fitting curve, as shown in However, since the sidereal rotation period and the processing period have a commensurability of 3:1, the lightcurve can also be reconstructed using the one-dimensional Fourier series of sixth order. As we mentioned in Section 3.2, the physical model is possibly constructed using a LAM of L 1 :L 2 :L 3 = 1.0:1.0:1.6, (λ = 310 • , β = −10 • ) or (λ = 132 • , β = −17 • ), θ = 65 • , P ψ = 0.2979 day, and P φ = 0.0993 day. Although Pravec et al. (2005) mentions that a tumbling asteroid generally does not return to the same orientation in any single period, the approximately equal length of L 1 and L 2 for 107P suggests a negligible change for the nutation angle. Therefore, 107P can return to the same orientation every 0.2979 day. These circumstances imply that 107P might be a tumbling object. Assuming 107P is a tumbler, external forces are required to trigger the motion. Impacts of small objects, tidal encounters with planets, and YORP effects are suggested by Pravec et al. (2005). Though 107P is a NEO, the object did not have an encounter with Earth in 1949. In the case of km-size objects, the efficient onset of tumbling by YORP requires a longer timescale than that of collision with small objects (Vokrouhlický et al., 2007). Therefore, we propose the impact of small objects as a probable cause for tumbling of 107P. The orbital origin of 107P has a high possibility of being from the outer MBA region inhabited by MBCs. One possibility is that the cometary activities of MBCs are caused by impacts of small objects. We can consider that 107P is originally an object like an MBC and impacts with small objects in the NEO region could eject dust and/or expose sub-surface ice that then trigger 107P's cometary activity. When we suppose that the collisional excitation happened in 1949, the damping timescale (Harris, 1994) is expressed as τ = P 3 ψ C 3 D 2 ,(10) where D is the mean diameter of tumblers in kilometer units and C is a constant of about 17 (uncertain by about a factor of 2.5). The units of P ψ and τ are hours and billion (10 9 ) of years, respectively. Since the damping timescale of around 6.2×10 6 yr is long enough, 107P would continue tumbling even if the impact occurred before 1949. Binary asteroids We describe the situation in which 107P hosts a binary. In order to confirm the existence of a binary, the detection of mutual eclipse events is required in the lightcurve. The mutual eclipse events were not detected in the observations of Harris and Young (1983) and Osip et al. (1995) 6 as the primary (secondary) eclipse and the secondary (primary) eclipse, respectively, the orbital period of the binary is 0.2979 day. Supposing a circular orbit and negligible mass for the binary, the semi-major axis is described as a = GMP 2 orb 4π 2 1 3 ,(11) where G is the gravitational constant, M is the mass of 107P, and P orb is the orbital period of the binary. For the sake of simplicity, when assuming that 107P is spherical with the diameter of 3.46 km (Licandro et al., 2009) and a typical density of 2 g/cm 3 , the semi-major axis is around 3.65 km. In the case of the same albedo for 107P and the binary, the flux decrease of the total eclipse (A mut ) satisfies the following relationship (Polishook et al., 2011) A mut = 2.5log 1 + R s R p 2 ,(12) where R s is the radius of the binary and R p is the radius of 107P. Since the typical flux decrease is ∼0.05 mag in Fig. 6, the radius of the binary is around 0.4 km. When we assume the orbital plane of 107P accords with the line of sight from an observer, the inclination of the binary as an occulter satisfies sin i < R p + R s a .(13) Here, i is the inclination of the binary for the orbital plane of 107P is less than 36 • in the 107P system. If i is zero, the eclipse duration is estimated to be ∼0.05 day. The term is around one-sixth of the orbital period and consistent with the interval of lightcurve peaks of Fig. 6. Moreover, the binary hypothesis indicates that the double-peak period of the lightcurve without the eclipse becomes 0.1490 day. As we mentioned in Section 3.1, the period of 0.1490 day as the sidereal rotation of 107P is not likely. Alternatively, the lightcurve without the eclipse might be a quadruple-peak lightcurve whose period is 0.2979 day. Though the quadruple-peak lightcurve is rare, the period could compatibly account for all the past reports. In addition, the situation shows that the sidereal rotation of 107P and the orbital periods of the binary are locked with 0.2979 day. The period of 0.0993 day is explained by the period between the egress time of the primary (secondary) eclipse and the ingress time of the secondary (primary) eclipse. The promising mechanisms for formation of asteroid binaries are the rotational-fission due to the spin-up by YORP effects (Scheeres, 2007;Pravec and Harris, 2007;Walsh et al., 2008), tidal encounter with planets (Richardson et al., 1998;Bottke et al., 1999;Walsh and Richardson, 2006), and the escaping ejecta by the collisions (Durda et al., 2004;Polishook et al., 2011). The mechanisms have a lot in common with the cause of tumbling. Fissions and collisions in every mechanism can trigger 107P's cometary activity. The possible existence of a binary is consistent with the past cometary activity. Summary This study revealed the physical properties of 107P by a photometric observation campaign. We detected the lightcurve periodicity to be 0.2979 day by retrograde motion. Impacts of small objects are suggested as a cause for the tumbling and comet activity. Alternatively, the past comet activity itself is thought to be a cause of the tumbling, like a 1P/Halley (Samarasinha and A'Hearn, 1991). 2. 107P is not a tumbler. The sidereal rotation period is 0.2979 day. The shape is roughly spherical but slightly hexagonal with a SAM of L 1 :L 2 :L 3 = 1.5:1.0:1.0. The pole orientation is around λ = 330 • , β = −27 • . The sense of rotation is retrograde. The lightcurve of commensurability would reflect a discriminative appearance like (2867) Steins, which has been explored by the Rosetta spacecraft (Keller et al., 2010). Otherwise, the lightcurve also indicates the possibility of hosting a binary whose orbital period is 0.2979 day. The existence of a binary is also consistent with the past cometary activity. Finally, we describe the mission feasibility for 107P. The orbit accessible by spacecraft makes 107P a promising target for a sample-return mission. If 107P is not a tumbling object, the moderate rotational period of 0.2979 day would enable us to obtain a sample by the touchdown of a spacecraft, whereas touchdown on 107P would require a difficult maneuver if 107P is a tumbler. In that case, a multi-fly-by mission that combines with the sample return mission for another target would become a hopeful plan. 2 IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. Fig. 1 : 1The power spectrum from the period analysis shows four period candidates of 0.0993 day, 0.2294 day, 0.2591 day and 0.2979 day(Fig. 1). A typical error of 0Power spectrum for the sidereal rotation period of 107P by assuming the double-peak lightcurve. The calculation is carried out on data obtained from December 7 to 22.day corresponds to ± 0.005 hour. Though the most significant candidate is 0.2591 day, we conclude that 0.2979 day (≃ 7.15 hour) is the sidereal rotation period of 107P for the following reasons. First, the amount of the amplitude in the folded lightcurve with 0.2591 day is not stable in the same phase. That is to say, the different amplitudes overlap on a specific phase. For example, the lightcurve peaks and bottoms overlap around the phase of 0.2-0.4 in the folded lightcurve with 0.2591 day(Fig. 2: Top). In the case of the folded lightcurve with 0.2979 day, the same amplitudes appear periodically (Fig. 2: Bottom. A few lightcurves each night are also shown inFig. 3). The periods of 0.2591 day and 0.2979 day correspond approximately to 3.86 and 3.36 cycles per day, respectively. The difference is just 0.5 cycles per day. Fig. 2 :Fig. 3 : 23Lightcurve of 107P. (Top) The lightcurve is folded with 0.2591 day. The peak and bottom of the lightcuve overlap around the phase between 0.2 and 0.4. (Bottom) The lightcurve is folded with 0.2979 day. The same amplitudes appear periodically. (Top) Lightcurve in December 7, 2009. (Bottom) Lightcurve in December 8, 2009. The phase corresponding to Fig. 2 is added to the top scale of the figures. to distinguish the difference of the half rotation using the short observation time, which is comparable with the sidereal rotation period. We call the indistinctive period a pseudo-period. The periods of 0.2591 day and 0.2294 day (= 4.36 cycles per day) are the pseudo-period of 0.2979 day. Second, the period of 0.2979 day is able to explain the previous reports about the sidereal rotation period of 107P. Since the period of 0.2979 day is around twice the period of Harris and Young (1983) (0.1482 day ≃ 3.556 hour), their data show enough periodicity in 0.2979 day. Needless to say, assuming the lightcurve of 107P has a triple-peak, the period of 0.1490 day (≃ 3.58hour) is also a candidate for the sidereal rotation period. However, the possibility of 0.1490 day is eliminated by the inconsistency with the data ofOsip et al. (1995).Harris and Young (1983) would recognize their lightcurve as the typical double-peak with the period of 3.556 hour, because the third amplitude of flux in their lightcurve was not detected. Moreover, the period ofOsip et al. (1995) (0.2542 day = 6.1 ± 0.05 hour) is approximately the same as 0.2591 day (≃ 6.22 hour). Our data set also has a sufficiently high significance level around the period of 0.2542 day. As we mentioned above, however, the period of around 0.2591 day is a pseudo-period. Since the observation term ofOsip et al. (1995) was only two days, the demarcation of a pseudo-period would have been difficult. Third, we focus on the lightcurve of 0.2979 day as having an unusual six peaks. The period of 0.0993 day is just one third that of 0.2979 day. If 107P has a typical double-peak lightcurve, the period of 0.0993 day is the sidereal rotation period. However, the amplitudes overlap the different three peaks and bottoms in the folded lightcurve with a period of 0.0993 day. Thus, we exclude the period of 0.0993 day as the sidereal rotation period. Fig. 4 Fig. 4 : 44shows the δ map that is obtained by scanning the celestial sphere with a trial axis in steps of 1 • in ecliptic longitude and latitude. Two candidates δ maps for the direction of total rotational angular momentum by the epoch method.are found near the directions, A (λ = 310 • , β = −10 • ) and B (λ = 132 • , β = −17 • ). Since the epoch method generally derives two solutions with around the same significance level, we cannot determine a unique solution. Fig. 5 : 5Deviation maps for the direction of total rotational angular momentum by the lightcurve inversion method. Fig. 6 : 6Calibrated lightcurve for phase-shift. The fitting curve is described by two-order two-dimensional Fourier series. Fig. 7 : 7Shape model A of 107P. (Left) Pole-on view. (Center) Equatorial view from the right side of the pole-on image. (Right) Equatorial view from the bottom of the pole-on image. nutation, respectively. M is the total rotational angular momentum in an inertial frame. I 1 and I 3 express the inertia moment of a triaxial ellipsoid by using mass µ. Moreover, the equations show that the motion of the external axis about M occurs as a constant rate. From these equations andφ = 3ψ, the nutation is negligible, and the angle θ is constant around 65 • . A tilted, rugby-ball-shaped body rotates with a period of 0.0993 day about the total rotational angular momentum, and with a period of 0.2979 day about the external axes of 107P itself. Alternatively, we can assume a case that has the sidereal rotation of 0.0993 day and the precession period of 0.2979 day. Substitutingψ = 3φ, there is no solution for the nutation angle. Therefore, the assumption is not adequate.Meanwhile, as we show inFig. 8, the normalized axis lengths for the shape model C are around 1.5, 1.5, and 1.0. Here, the axes satisfy the relationship L 1 ≥ L 2 ≥ L 3 . L 3 is a sidereal rotation axis. Thus, the shape model C is a short axis mode (SAM). The rotation and precession of a SAM are in opposite direction from I 1 < I 3 . We calculate the nutation angle by assumingφ = −3ψ. However, there is no solution for the nutation angle. To obtain the solution for the nutation angle,φ should be less than −3.62ψ if the axis lengths of the shape model C are correct, or the axis lengths of L 1 Fig. 8 : 8Shape model C of 107P. (Left) Pole-on view. (Center) Equatorial view from the right side of the pole-on image. (Right) Equatorial view from the bottom of the pole-on image. Fig. 9 : 9Color-color diagram of 107P. Letters in the figure represent the taxonomic classes of asteroids on the color-color diagram Fig. 6 . 6The fitting curve adequately reconstructs the trend of the lightcurve. because of the viewing angle, the lower photometric precision, or the absence of the binary. On the other hand, we detected the around same flux decrease in every 0.50 phase. Therefore, the existence of the binary is conceivable as the other interpretation of the shape model C. If we define the flux decrease around the phase of 0.15 (or 0.30, 0.45) and 0.65 (or 0.80, 0.95) in Fig. day and 0.0993 day with a commensurability of 3:1. The multiband photometry indicates that the taxonomy class of 107P is C-type. No clear rotational color variations are confirmed on the surface. We suggested two models to explain the different interpretations of the lightcurve periodicity. 1. The commensurability reflects tumbling with the sidereal rotation period of 0.2979 day and the precession period of 0.0993 day. The shape is a LAM of L 1 :L 2 :L 3 = 1.0:1.0:1.6. Around the same length of L 1 and L 2 shows the nutation angle is approximately constant at 65 • . The direction of total rotational angular momentum is around λ = 310 • , β = −10 • , or λ = 132 • , β = −17 • . 107P returns to the same orientation every 0.2979 SDSS g ′ , Johnson-Cousins R c and I c ) image simultaneously. The detector is 1024 × 1024 pixels CCD with FOV of 26 ′ × 26 ′ (1.52 ′′ /pixel). The images were taken with an exposure time of 120 s.at Okayama Astrophysical Observatory (OAO) from November 7, 2009 to December 21, 2009. The telescope is capable of obtaining a three-color (To search for the rotational color variation, we used the data of December 17 because they could be compared with the observations of the BSGC and the photometric precision of the other day's data was not sufficient to detect the color variation. The third observation was carried out using a 1.05 m f/3.1 Schmidt telescope with 2048 × 2048 pixels CCD at Kiso Observatory on August 17, 19, and 20 and December 12, 2009. This instrument provides a FOV of 50 ′ × 50 ′ (1.46 ′′ /pixel). The images were obtained using a Kron-Cousins R c filter Table 1 : 1Observation states.Observatory Year/Mon/Day Exp time(s) Filter BSGC 2009/12/17 300 g ′ , r ′ , i ′ , z ′ BSGC † 2009/09/6,7,9,10,15,16,19, 2009/10/8,10,28 30-180 W BSGC 2009/11/3,5-7,11,14, 2009/12/5,7-9,19,22 60-300 W BSGC 2010/01/3,6-8,14-18,22,23 180-300 W BSGC † 2010/02/3-5,7,9,16,18,19, 2010/03/11 240-600 W OAO 2009/11/7,14,15,18-21,23, 120 g ′ , Rc, Ic OAO 2009/12/1,2,6,7,14,16-21 120 g ′ , Rc, Ic KISO † 2009/08/17,19,20, 2009/12/12 120-300 Rc LOT 2009/12/7-10 90 Rc UH88 2009/12/19 60 Rc † A sufficient number of data was not obtained from August to October because the altitude of 107P fell below 25 • about 30 minutes from the observation start and 107P overlapped stars of the galactic plane. Table 2 : 2States of 107P in each month. Phase angle (Sun-107P-observer). Apparent magnitude. This value is estimated using UCAC 2 catalog stars that are taken in the same field with 107P.Year/Mon/Day ∆ [AU] a R [AU] b α [ • ] c Sky motion [ ′′ /min] m d 2009/08/17-20 0.687-0.684 1.309-1.286 49.8-51.4 0.45-0.55 17.7-17.7 2009/09/6-19 0.653-0.612 1.162-1.084 59.9-66.0 1.09-1.49 17.9-17.9 2009/10/8-28 0.529-0.434 1.008-0.995 73.8-77.2 2.18-3.26 17.3-17.5 2009/11/3-23 0.410-0.382 1.006-1.083 76.3-65.4 3.65-4.52 17.1-16.7 2009/12/1-22 0.401-0.543 1.130-1.277 59.2-46.3 4.31-2.96 16.6-17.0 2010/01/3-23 0.670-0.936 1.373-1.539 42.0-37.8 2.32-1.70 17.4-18.4 2010/02/3-19 1.105-1.372 1.632-1.767 36.1-33.8 1.51-1.34 18.8-19.7 2010/03/11 1.732 1.932 30.8 1.22 20.0 a Object to observer distance. b Heliocentric distance. c d Table 3 : 3Standard stars in SDSS-7.Ra[ • ] Dec[ • ] g ′ r ′ i ′ z ′ 12.051081 8.615352 15.891 ± 0.003 15.279 ± 0.003 15.057 ± 0.003 14.913 ± 0.005 12.246608 8.604861 15.209 ± 0.003 14.756 ± 0.003 14.598 ± 0.003 14.525 ± 0.004 12.073952 8.515561 14.536 ± 0.003 14.215 ± 0.003 14.115 ± 0.003 14.074 ± 0.004 12.011267 8.489718 15.766 ± 0.004 14.891 ± 0.004 14.587 ± 0.003 14.444 ± 0.004 12.275938 8.545386 16.158 ± 0.003 15.566 ± 0.003 15.349 ± 0.004 15.230 ± 0.005 11.898784 8.539119 14.947 ± 0.003 14.554 ± 0.003 14.418 ± 0.003 14.366 ± 0.004 12.365509 8.569521 14.587 ± 0.003 14.037 ± 0.003 13.839 ± 0.003 13.713 ± 0.003 12.374946 8.538940 16.005 ± 0.003 15.555 ± 0.003 15.396 ± 0.004 15.307 ± 0.005 11.796300 8.569376 15.308 ± 0.003 14.641 ± 0.003 14.413 ± 0.003 14.287 ± 0.004 12.371825 8.478789 15.763 ± 0.004 15.298 ± 0.004 15.156 ± 0.004 15.099 ± 0.005 Table 4 : 4Epoch of Standard Feature (SF ) and the amount of phase-shift.Year/Mon/Day Ecliptic longitude Ecliptic latitude Amount of phase-shift(PAB)[ • ] (PAB)[ • ] (×10 −2 ) 2009/11/5 341.958 4.611 0 2009/12/10 30.902 2.805 −5.305 2009/12/20 54.706 1.094 −6.113 2010/01/3 57.829 0.879 −7.794 2010/02/5 76.114 −0.202 −8.506 Table 5 : 5Color index of 107P. The observation term (Obs term) of each sequence is expressed as the rotational phase in the lightcurve. Since the data of the OAO are obtained with three bands, there is no color index AcknowledgmentsWe appreciate the observational campaign participants, especially Dr.Ryosuke Nakamura and Dr. Masanao Abe, for their dedicated coordination of the campaign and encouragement. We also acknowledge the Japan Space The Seventh Data Release of the Sloan Digital Sky Survey. K N Abazajian, Astrophys. J. S. 182Abazajian, K.N., et al., 2009. The Seventh Data Release of the Sloan Digital Sky Survey. Astrophys. J. S 182, 543-558. P Birtwhistle, W H Ryan, H Sato, E C Beshore, K Kadota, Comet P/2010 A2 (LINEAR). IAU Circ. 9105. 1Birtwhistle, P., Ryan, W.H., Sato, H., Beshore, E.C., Kadota, K., 2010. Comet P/2010 A2 (LINEAR). IAU Circ. 9105, 1. Collisional Excavation of Asteroid (596) Scheila. D Bodewits, Astrophys. J. L. 7333Bodewits, D., et al., 2011. Collisional Excavation of Asteroid (596) Scheila. Astrophys. J. L 733, L3. . W F Bottke, D C Richardson, P Michel, S G Love, 1620Bottke, W.F., Richardson, D.C., Michel, P., Love, S.G., 1999. 1620 Shaped by Planetary Tides?. 433 Geographos, Eros, Astron. J. 117Geographos and 433 Eros: Shaped by Planetary Tides? Astron. J. 117, 1921-1928. Debiased Orbital and Absolute Magnitude Distribution of the Near-Earth Objects. W F Bottke, Icarus. 156Bottke, W.F., et al., 2002. Debiased Orbital and Absolute Magnitude Distribution of the Near-Earth Objects. Icarus 156, 399-433. 4015 Wilson-Harrington, 2201 Oljato, and 3200 Phaethon: Search for CN Emission. A B Chamberlin, L A Mcfadden, R Schulz, D G Schleicher, S J Bus, Icarus. 119Chamberlin, A.B., McFadden, L.A., Schulz, R., Schleicher, D.G., Bus, S.J., 1996. 4015 Wilson-Harrington, 2201 Oljato, and 3200 Phaethon: Search for CN Emission. Icarus 119, 173-181. Spectroscopy of B-type asteroids: Subgroups and meteorite analogs. B E Clark, Journal of Geophysical Research. 1156005Clark, B.E., et al., 2010. Spectroscopy of B-type asteroids: Subgroups and meteorite analogs. Journal of Geophysical Research 115, E06005. Origin of the near-Earth asteroid Phaethon and the Geminids meteor shower. J De León, H Campins, K Tsiganis, A Morbidelli, J Licandoro, Astron. Astrophys. 51326de León, J., Campins, H., Tsiganis, K., Morbidelli, A., Licandoro, J., 2010. Origin of the near-Earth asteroid Phaethon and the Geminids meteor shower. Astron. Astrophys. 513, A26. Pole and Global Shape of 25143 Itokawa. H Demura, Science. 312Demura, H., et al., 2006. Pole and Global Shape of 25143 Itokawa. Science 312, 1347-1349. Collisional activation of asteroids in cometary orbits. C G Díaz, R Gil-Hutton, Astron. Astrophys. 487Díaz, C.G., Gil-Hutton, R., 2008. Collisional activation of asteroids in cometary orbits. Astron. Astrophys. 487, 363-367. The formation of asteroid satellites in large impacts: results from numerical simulations. D D Durda, Icarus. 170Durda, D.D., et al., 2004. The formation of asteroid satellites in large impacts: results from numerical simulations. Icarus 170. 243-257. Comet P/1996 N2 (Elst-Pizarro). E W Elst, IAU Circ. 64561Elst, E.W., et al., 1996. Comet P/1996 N2 (Elst-Pizarro). IAU Circ. 6456, 1 Analysis of POSS Images of Comet-Asteroid Transition Object 107P/1949 W1 (Wilson-Harrington). Y R Fernandez, L A Mcfadden, C M Lisse, E F Helin, A B Chamberlin, Icarus. 128Fernandez, Y.R., McFadden, L.A., Lisse, C.M., Helin, E.F., Chamberlin, A.B., 1997. Analysis of POSS Images of Comet-Asteroid Transition Object 107P/1949 W1 (Wilson-Harrington). Icarus 128, 114-126. Geminid meteoroids traced to cometary activity on Phaethon. B A S Gustafson, Astron. Astrophys. 225Gustafson, B.A.S., 1989. Geminid meteoroids traced to cometary activity on Phaethon. Astron. Astrophys. 225, 533-540. Dynamical constraints on the origin of Main Belt comets. N Haghighipour, Meteoritics & Planetary Science. 44Haghighipour, N., 2009. Dynamical constraints on the origin of Main Belt comets. Meteoritics & Planetary Science 44, 1863-1869. Asteroid rotation. IV. A W Harris, J W Young, Icarus. 54Harris, A.W., Young, J.W., 1983. Asteroid rotation. IV. Icarus 54, 59-109. Tumbling asteroids. A W Harris, Icarus. 107Harris, A.W., 1994. Tumbling asteroids. Icarus 107, 209-211. Asteroid Lightcurve Observations from 1981 to 1983. A W Harris, J W Young, E Bowell, D J Tholen, Icarus. 142Harris, A.W., Young, J.W., Bowell, E., Tholen, D.J., 1999. Asteroid Lightcurve Observations from 1981 to 1983. Icarus 142, 173-201. A Population of Comets in the Main Asteroid Belt. H H Hsieh, D Jewitt, Science. 312Hsieh, H.H., Jewitt, D. 2006. A Population of Comets in the Main Asteroid Belt. Science 312, 561-563. Physical Properties of Main-Belt Comet 176P/LINEAR. H H Hsieh, M Ishiguro, P Lacerda, D Jewitt, Astron. J. acceptedHsieh, H.H., Ishiguro, M., Lacerda, P., Jewitt, D. 2011. Physical Properties of Main-Belt Comet 176P/LINEAR. Astron. J. (accepted). GRBs Optical follow-up observation at Lulin observatory. K Y Huang, Taiwan. Il Nuovo Cimento C. 28Huang, K.Y., et al., 2005. GRBs Optical follow-up observation at Lulin observatory, Taiwan. Il Nuovo Cimento C 28, 731-734. . M Ishiguro, Search for the Comet Activity of 107P/(4015Ishiguro, M., et al., 2011. Search for the Comet Activity of 107P/(4015) . Wilson-Harrington, Apparition. Astrophys. J. 726101Wilson-Harrington during 2009/2010 Apparition. Astrophys. J. 726, 101. Solar System Objects Observed in the Sloan Digital Sky Survey Commissioning Data. Z Ivezić, Astron. J. 122Ivezić, Z., et al., 2001. Solar System Objects Observed in the Sloan Digital Sky Survey Commissioning Data. Astron. J. 122, 2749-2784. Main-Belt Comet P/2008 R1 (Garradd). D Jewitt, B Yang, N Haghighipour, Astron. J. 137Jewitt, D., Yang, B., Haghighipour, N., 2009. Main-Belt Comet P/2008 R1 (Garradd). Astron. J. 137, 4313-4321. A recent disruption of the main-belt asteroid P/2010 A2. D Jewitt, H Weaver, J Agarwal, M Mutchler, M Drahus, Nature. 467Jewitt, D., Weaver, H., Agarwal, J., Mutchler, M., Drahus, M., 2010. A recent disruption of the main-belt asteroid P/2010 A2. Nature 467, 817-819. . D Jewitt, H Weaver, M Mutchler, S Larson, J Agarwal, Hubble Space Telescope Observations of Main-belt CometsJewitt, D., Weaver, H., Mutchler, M., Larson, S., Agarwal, J., 2011. Hubble Space Telescope Observations of Main-belt Comets (596) Scheila. . Astrophys. J. L. 7334Astrophys. J. L 733, L4. Empirical color transformations between SDSS photometry and other photometric system. K Jordi, E K Grebel, K Ammon, AstronJordi, K., Grebel, E.K., Ammon, K., 2006. Empirical color transformations between SDSS photometry and other photometric system. Astron. . Astrophys. 460Astrophys. 460, 339-347. Interpretation of lightcurves of precessing asteroids. M Kaasalainen, Astron. Astrophys. 376Kaasalainen, M., 2001. Interpretation of lightcurves of precessing asteroids. Astron. Astrophys. 376, 302-309. Optimization Methods for Asteroid Lightcurve Inversion. I. Shape Determination. M Kaasalainen, J Torppa, Icarus. 153Kaasalainen, M., Torppa, J., 2001. Optimization Methods for Asteroid Lightcurve Inversion. I. Shape Determination. Icarus 153, 24-36. Optimization Methods for Asteroid Lightcurve Inversion. II. The Complete Inverse Problem. M Kaasalainen, J Torppa, K Muinonen, Icarus. 153Kaasalainen, M., Torppa, J., Muinonen, K. 2001. Optimization Methods for Asteroid Lightcurve Inversion. II. The Complete Inverse Problem. Icarus 153, 37-51. Asteroid Models from Disk-integrated Data. M Kaasalainen, S Mottola, M Fulchignoni, Bottke., W.F., Cellino, A., Paolicchi, P., Binzel, R.P.Asteroid III. Univ. of Arizona pressTucsonKaasalainen, M., Mottola, S., Fulchignoni, M. 2002. Asteroid Models from Disk-integrated Data. In: Bottke., W.F., Cellino, A., Paolicchi, P., Binzel, R.P. (Eds.), Asteroid III. Univ. of Arizona press, Tucson, pp. 139-150. CCD photometry and model of MUSES-C target (25143) 1998 SF36. M Kaasalainen, Astron. Astrophys. 405Kaasalainen, M., et al., 2003. CCD photometry and model of MUSES-C target (25143) 1998 SF36. Astron. Astrophys. 405, 29-32. E-Type Asteroid (2867) Steins as Imaged by OSIRIS on Board Rosetta. H U Keller, Science. 327Keller, H. U., et al., 2010. E-Type Asteroid (2867) Steins as Imaged by OSIRIS on Board Rosetta. Science 327, 190-193. MITSuME-Multicolor Imaging Telescopes for Survey and Monstrous Explosions. T Kotani, Il Nuovo Cimento C. 28Kotani, T., et al., 2005. MITSuME-Multicolor Imaging Telescopes for Survey and Monstrous Explosions. Il Nuovo Cimento C 28, 755-758. Relation between rotation and lightcurve of 4179 Toutatis. A Kryszczyńska, T Kwiatkowski, S Breiter, T Micha Lowski, Astron. Astrophys. 345Kryszczyńska, A., Kwiatkowski, T., Breiter, S., Micha lowski, T., 1999. Relation between rotation and lightcurve of 4179 Toutatis. Astron. Astrophys. 345, 643-645. . J Licandro, H Campins, T Mothé-Diniz, N Pinilla-Alonso, J De León, The nature of comet-asteroid transition object (3200) PhaethonLicandro, J., Campins, H., Mothé-Diniz, T., Pinilla-Alonso, N., de León, J., 2007. The nature of comet-asteroid transition object (3200) Phaethon. . Astron. Astrophys. 461Astron. Astrophys. 461, 751-757. Spitzer observations of the asteroid-comet transition object and potential spacecraft target. J Licandro, 107P. 4015Licandro, J., et al., 2009. Spitzer observations of the asteroid-comet transition object and potential spacecraft target 107P (4015) Wilson-Harrington. . Astron. Astrophys. 507Astron. Astrophys. 507, 1667-1670. 65) Cybele: detection of small silicate grains, water-ice, and organics. J Licandro, Astron. Astrophys. 52534Licandro, J., et al., 2011. (65) Cybele: detection of small silicate grains, water-ice, and organics. Astron. Astrophys. 525, A34. Least-squares frequency analysis of unequally spaced data. N R Lomb, Astrophysics and Space Science. 39Lomb, N.R., 1976. Least-squares frequency analysis of unequally spaced data. Astrophysics and Space Science 39, 447-462. CCD observations of distant comets from Palomar and Steward Observatories. S C Lowry, P R Weissman, Icarus. 164Lowry, S.C., Weissman, P.R., 2003. CCD observations of distant comets from Palomar and Steward Observatories. Icarus 164, 492-503. Distribution of spin axes and senses of rotation for 20 large asteroids. P Magnusson, Icarus. 68Magnusson, P., 1986. Distribution of spin axes and senses of rotation for 20 large asteroids. Icarus 68, 1-39. Comet P/2010 R2 (la Sagra). B G Marsden, IAU Circ. 91691Marsden, B.G., et al., 2010. Comet P/2010 R2 (la Sagra). IAU Circ. 9169, 1. The slow rotation of 253 Mathilde. S Mottola, Planetary and Space Science. 43Mottola, S., et al., 1995. The slow rotation of 253 Mathilde. Planetary and Space Science 43, 1609-1613. JU3), a potential flyby and rendezvous target for interplanetary missions. T G Müller, Astron. Astrophys. 525145Thermo-physical properties of 162173Müller, T.G., et al., 2011. Thermo-physical properties of 162173 (1999 JU3), a potential flyby and rendezvous target for interplanetary missions. Astron. Astrophys. 525, A145 The rotation state of 4015. D Osip, H Campins, D G Schleicher, Osip, D., Campins, H., Schleicher, D.G., 1995. The rotation state of 4015 Revisiting origins for the near-Earth asteroids. Wilson-Harrington, Icarus. 114Wilson-Harrington: Revisiting origins for the near-Earth asteroids. Icarus 114, 423-426. Rotation periods of binary asteroids with large separations -Confronting the Escaping Ejecta Binaries model with observations. D Polishook, N Brosch, Prialnik, Icarus. 212Polishook, D., Brosch, N., Prialnik., 2011. Rotation periods of binary asteroids with large separations -Confronting the Escaping Ejecta Binaries model with observations. Icarus 212, 167-174. Tumbling asteroids. P Pravec, Icarus. 173Pravec, P., et al., 2005. Tumbling asteroids. Icarus 173, 108-131. Binary asteroid population. 1. Angular momentum content. P Pravec, A W Harris, Icarus. 190Pravec, P., Harris, A.W., 2007. Binary asteroid population. 1. Angular momentum content. Icarus 190, 250-259. M I Read, T H Bressi, T Gehrels, J V Scotti, E J Christensen, Comet P/2005 U1 (Read). IAU Circ. 86241Read, M.I., Bressi, T.H., Gehrels, T., Scotti, J.V., Christensen, E.J., 2005. Comet P/2005 U1 (Read). IAU Circ. 8624, 1. Tidal Distortion and Disruption of Earth-Crossing Asteroids. D C Richardson, W F Bottke, S G Love, Icarus. 134Richardson, D.C., Bottke, W.F., Love, S.G., 1998. Tidal Distortion and Disruption of Earth-Crossing Asteroids. Icarus 134, 47-76. Detection of ice and organics on an asteroidal surface. A S Rivkin, J P Emery, Nature. 464Rivkin, A.S., Emery, J.P., 2010. Detection of ice and organics on an asteroidal surface. Nature 464, 1322-1323. Observational and dynamical constraints on the rotation of comet P/Halley. N H Samarasinha, M F Hearn, Icarus. 93Samarasinha, N.H., A'Hearn, M.F., 1991. Observational and dynamical constraints on the rotation of comet P/Halley. Icarus 93, 194-225. Studies in astronomical time series analysis. II -Statistical aspects of spectral analysis of unevenly spaced data. D J Scargle, Astrophys. J. 263Scargle, D.J., 1982. Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data. Astrophys. J. 263, 835-853. Rotational fission of contact binary asteroids. D J Scheeres, Icarus. 189Scheeres, D.J., 2007. Rotational fission of contact binary asteroids. Icarus 189, 370-385. A collision in 2009 as the origin ogf the debris trail of asteroid P/2010 A2. C Snodgrass, Nature. 467Snodgrass, C., et al., 2010. A collision in 2009 as the origin ogf the debris trail of asteroid P/2010 A2. Nature 467, 814-816. The lightcurve of 4179 Toutatis: Evidence for complex rotation. J R Spencer, Icarus. 117Spencer, J.R., et al., 1995. The lightcurve of 4179 Toutatis: Evidence for complex rotation. Icarus 117, 71-89. Asteroid taxonomic classifications. D J Tholen, Binzel, R.P., Gehrels, T., Matthews, M.S.Asteroid II. Univ. of Arizona pressTucsonTholen, D.J., 1989. Asteroid taxonomic classifications. In: Binzel, R.P., Gehrels, T., Matthews, M.S. (Eds.), Asteroid II. Univ. of Arizona press, Tucson, pp. 1139-1150. Impact-generated activity period of the asteroid 7968. I Toth, Toth, I., 2000. Impact-generated activity period of the asteroid 7968 Identification of the asteroid 427 Galene as the most probable parent body of the impactors. Elst-Pizarro, Astron. Astrophys. 360Elst-Pizarro in 1996: Identification of the asteroid 427 Galene as the most probable parent body of the impactors. Astron. Astrophys. 360, 375-380. Generalized YORP evolution: Onset of tumbling and new asymptotic states. D Vokrouhlický, S Breiter, Nesvorný, W F Bottke, Icarus. 191Vokrouhlický, D., Breiter, S., Nesvorný, Bottke, W.F., 2007. Generalized YORP evolution: Onset of tumbling and new asymptotic states. Icarus 191, 636-650. Binary near-Earth asteroid formation: Rubble pile model of tidal disruptions. K J Walsh, D C Richardson, Icarus. 180Walsh, K.J., Richardson, D.C., 2006. Binary near-Earth asteroid formation: Rubble pile model of tidal disruptions. Icarus 180, 201-216. Rotational breakup as the origin of small binary asteroids. K J Walsh, D C Richardson, P Michel, Nature. 454Walsh, K.J., Richardson, D.C., Michel,P., 2008. Rotational breakup as the origin of small binary asteroids. Nature 454, 188-191. Identification of Magnetite in B-type Asteroids. B Yang, D Jewitt, Astron. J. 140Yang, B., Jewitt, D., 2010. Identification of Magnetite in B-type Asteroids. Astron. J. 140, 692-698. Japan's Future Plans for Missions to Primitive Bodies: Hayabusa-2, Hayabusa-Mk2, and Marco Polo. M Yoshikawa, H Yano, J Kawaguchi, Small Bosy Exploration Wg. 1747Lunar. Planet. Sci. XXXIX. abstractYoshikawa, M., Yano, H., Kawaguchi, J., Hayabusa-2 Pre-Project Team., Small Bosy Exploration Wg., 2008. Japan's Future Plans for Missions to Primitive Bodies: Hayabusa-2, Hayabusa-Mk2, and Marco Polo. Lunar. Planet. Sci. XXXIX, 1747 (abstract).
[]
[ "Microstate Counting via Bethe Ansätze in the 4d N = 1 Superconformal Index", "Microstate Counting via Bethe Ansätze in the 4d N = 1 Superconformal Index" ]
[ "Alfredo González Lezcano \nSISSA International School for Advanced Studies Via Bonomea 265\nTrieste and INFN\nsezione di Trieste a\n34136\n", "Leopoldo A Pando Zayas \nThe Abdus Salam International Centre for Theoretical Physics Strada Costiera 11\n34014TriesteItaly\n\nLeinweber Center for Theoretical Physics\nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMIUSA\n" ]
[ "SISSA International School for Advanced Studies Via Bonomea 265\nTrieste and INFN\nsezione di Trieste a\n34136", "The Abdus Salam International Centre for Theoretical Physics Strada Costiera 11\n34014TriesteItaly", "Leinweber Center for Theoretical Physics\nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMIUSA" ]
[]
We study the superconfomal index of four-dimensional toric quiver gauge theories using a Bethe-Ansatz approach recently pioneered by Benini and Milan. Relying on a particular set of solutions to the corresponding Bethe Ansatz equations we evaluate the superconformal index in the large N limit, thus avoiding to take any Cardy-like limit. We present explicit results for theories arising as a stack of N D3 branes at the tip of toric Calabi-Yau cones: the conifold theory, the suspended pinch point gauge theory, the first del Pezzo theory and Y p,q quiver gauge theories. For generic quiver gauge theories we find a particular correction to the superconformal index in the Cardy limit that happens to vanish in the case of N = 4 supersymmetric Yang-Mills. We estimate how such correction affects the entropy of the would be dual electrically charged rotating AdS 5 black holes.
null
[ "https://arxiv.org/pdf/1907.12841v3.pdf" ]
198,985,629
1907.12841
067d44038537aa2c58923d1ef33851e8e53582ff
Microstate Counting via Bethe Ansätze in the 4d N = 1 Superconformal Index Jul 2019 Alfredo González Lezcano SISSA International School for Advanced Studies Via Bonomea 265 Trieste and INFN sezione di Trieste a 34136 Leopoldo A Pando Zayas The Abdus Salam International Centre for Theoretical Physics Strada Costiera 11 34014TriesteItaly Leinweber Center for Theoretical Physics Department of Physics University of Michigan 48109Ann ArborMIUSA Microstate Counting via Bethe Ansätze in the 4d N = 1 Superconformal Index Jul 2019 We study the superconfomal index of four-dimensional toric quiver gauge theories using a Bethe-Ansatz approach recently pioneered by Benini and Milan. Relying on a particular set of solutions to the corresponding Bethe Ansatz equations we evaluate the superconformal index in the large N limit, thus avoiding to take any Cardy-like limit. We present explicit results for theories arising as a stack of N D3 branes at the tip of toric Calabi-Yau cones: the conifold theory, the suspended pinch point gauge theory, the first del Pezzo theory and Y p,q quiver gauge theories. For generic quiver gauge theories we find a particular correction to the superconformal index in the Cardy limit that happens to vanish in the case of N = 4 supersymmetric Yang-Mills. We estimate how such correction affects the entropy of the would be dual electrically charged rotating AdS 5 black holes. Introduction The understanding of the quantum microstates responsible for the entropy of black holes has long been one of the central questions in the path to a quantum theory of gravity. In the context of the AdS/CFT correspondence it has recently been shown that the entropy of certain asymptotically AdS 4 black holes admits a microscopic explanation in terms of a topologically twisted field theory [1] (see [2,3] for reviews with extensive lists of references). More recently the question of microstates for asymptotically AdS 5 black holes dual to N = 4 supersymmetric Yang-Mills (SYM) which was originally tackled in [4] has been revisited providing a microscopic entropy matching using various approaches. A broader interpretation of localization was successfully put forward in [5] while an analysis of the free-field partition function in a particular limit led to the entropy in [6] (see also [7]). Both these groups relied on a particular Cardy-like limit to evaluate the path integral. Another approach, put forward by Benini and Milan in [8], attacked the superconformal index using a Bethe Ansatz approach developed in [9]. One key advantage of this approach is that it does not require taking the Cardy limit and thus opens the door for a more in-depth understanding of the superconformal index. In this brief note we simply generalize the large N results obtained for N = 4 SYM using the Bethe-Ansatz approach to a large class of N = 1 4d supersymmetric field theories. Other recent studies demonstrating that the Cardy-like limit of the superconformal index of 4d N = 4 SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographically dual AdS 5 rotating black holes were presented in [10,11]. Such analysis has by now been extended to generic N = 1 supersymmetric gauge theories [12,13] including a particular description specialized to arbitrary N = 1 toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data [14]. In this note we verify that a class of holonomies to the Bethe-Ansatz equations used prominently in [8], namely, those of the form u i − u j = τ N (i − j) (where u i are the holonomies of the gauge group) exploited for N = 4 SYM can be generalized to evaluate the superconformal index of generic N = 1 four-dimensional superconformal field theories. The rest of the note is organized as follows. In section 2 we show that a particular class of holonomies solves the Bethe-Ansatz equation for generic 4d N = 1 gauge theories and proceed to evaluate the superconformal index in the large N limit where we find a correction to the Cardy like limit. Section 3 works out explicitly the index for a number of superconformal field theories. In section 4 we estimate how this leading O(N 2 ) correction in the superconformal index modifies the entropy of the would be dual AdS 5 black holes. We conclude in 5. Bethe-Ansatz approach to the superconformal index In this section we generalize the solutions to the Bethe-Ansatz type equations proposed in [8,9] to evaluate the sueperconformal index of N = 4 SYM to generic 4d N = 1 supersymmetric gauge theories. For concreteness we will work in the context of toric quiver gauge theories which are naturally decorated with extra global and baryonic symmetries but the results apply more generally to 4d N = 1 supersymmetric gauge theories. Consider a generic N = 1 theory with semi-simple gauge group G, flavor symmetry G F and non-anomalous U (1) R R-symmetry. The matter content of this theory is taken to be n χ chiral multiplets Φ a in representations R a of G, with flavor weights ω a in some representation R F of G F and superconformal R-charge r a . Let us start by introducing the following quantities which are related to global fugacities and holonomies in the Cartan of the gauge group: p = e 2πiτ , q = e 2πiσ , v α = e 2πiξα , z i = e 2πiu i (2.1) and the R-charge chemical potential which is fixed by supersymmetry to: ν R = 1 2 (τ + σ) . (2.2) With the above data, the integral representation for the superconformal index can be written as I (p, q; v) = (p; p) rk(G) ∞ (q; q) rk(G) ∞ W G ∮ T rk(G) ∏ nχ a=1 ∏ ρa∈Ra Γ e (pq) ra 2 z ρa v ωa ; p, q ∏ α∈∆ Γ e (z α ; p, q) rk(G) i=1 dz i 2πiz i . (2.3) The integration variables z i parametrize the maximal torus of the gauge group G and the integration contour is the product of rk (G) unit circles. Following standard notation, ρ a are the weights of the representation R a , α parametrizes the roots of G and W G is the order of the Weyl group. The notation adopted also denotes z ρa ≡ ∏ rk(G) i=1 z ρ i a i and v ωa = ∏ rk(G F ) α=1 v ω α a α . The other functions involved in the expression for the superconformal index are the Elliptic Gamma function 4) and the q-Pochhamer symbol Γ e (z; p, q) = ∞ m,n=0 1 − p m+1 q n+1 z 1 − p m q n z , p < 1, q < 1,(2.(z; q) ∞ = ∞ n=0 (1 − zq n ) , q < 1. (2.5) An interesting result of [9] and [8] is to rewrite the above superconformal index in terms of solutions to certain Bethe-Ansatz like system of equations taking the generic form of Q i (u; ξ, ν R , ω) = 1 ∀ i = 1, ..., rk (G) (2.6) where ω is such that rτ = sσ with r and s coprime integer numbers (in practice we will evaluate the equations for r = s). Furthermore, the "Bethe-Ansatz operator" is defined as: Q i (u; ξ, ν R , ω) = nχ a=1 ρa∈Ra P (ρ a (u) + ω a (ξ) + r a ν R ; ω) ρ i a ,(2.7) where P (u; ω) = e −πi u 2 ω +πiu θ 0 (u; ω) . (2.8) Thus, P (ρ a (u) + ω a (ξ) + r a ν R ; ω) = e −πi 1 ω (ρa(u)+ωa(ξ)+raν R ) 2 +πi(ρa(u)+ωa(ξ)+raν R ) θ 0 (ρ a (u) + ω a (ξ) + r a ν R ; ω) , (2.9) where: θ 0 (u; ω) = e 2πiu ; e 2πiω ∞ e 2πi(ω−u) ; e 2πiω ∞ . (2.10) Now we would like to evaluate the Bethe-Ansatz equations for the case of a toric quiver gauge theory. Toric quiver gauge theories describe the low energy dynamics of a stack of N D3 branes probing the tip of a toric Calabi-Yau singularity; there is by now a vast literature detailing how to construct a supersymmetric field theory given toric data (see, for example [15,16]). Consider a toric quiver gauge theory whose gauge group G has n v simple factors (in all the N = 1 quiver gauge theories we will deal with, the number of simple factors coincides with the number of vector multiplets). We focus, for concreteness, on the case in which all the gauge group factors are SU(N a ), a goes from 1 to n v , with N a = N ∀ a, the same numerical value for all nodes. In these theories the weight vectors ρ are such that for any bi-fundamental field Φ ab (notice that in the more generic notation used in [9], the index a of Φ a would now split into ab): ρ Φ ab ij (u) ≡ u ab ij ≡ u a i − u b j . (2.11) Let us now evaluate the operator P (u; ω) for a generic Φ ab (when Φ ab transforms in the adjoint representation of G then in this notation a = b): Q ia (u; ξ, τ, σ, ω) = (a,b) j b ρ (a,b) ij P u ia − u j b + d−1 l=1 q l (a,b) ∆ l ρ (a,b) ij , (2.12) where (a, b) run over all the fields Φ ab for a fixed a. The d −1 fugacities correspond to the flavor symmetries appearing in the generic toric gauge theories that we will study, d is the number of external points of the toric diagram that are related to the quivers defining the theory [14]. If we denote ⟨a, b⟩ ≡ (a, b) ρ (a,b) ij >0 , which implies: Q ia (u; ξ, τ, σ, ω) = ⟨a,b⟩ j b ρ ⟨a,b⟩ ij ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ P u ia − u j b + ∑ d−1 l=1 q l ⟨a,b⟩ ∆ l P u j b − u ia + ∑ d−1 l=1 q l ⟨b,a⟩ ∆ l ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ρ ⟨a,b⟩ ij (2.13) = ⟨a,b⟩ j b ρ ⟨a,b⟩ ij ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e −2πi −u ia +u j b θ 0 −u j b + u ia + ∑ d−1 l=1 q l ⟨a,b⟩ ∆ l ; ω θ 0 u ia − u j b + ∑ d−1 l=1 q l ⟨b,a⟩ ∆ l ; ω ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ρ ⟨a,b⟩ ij = e −2πi ∑ j b u ia −u j b ⟨a,b⟩ j b θ 0 −u ia + u j b + ∑ d−1 l=1 q l ⟨a,b⟩ ∆ l ; ω θ 0 −u j b + u ia + ∑ d−1 l=1 q l ⟨b,a⟩ ∆ l ; ω . Let us now introduce a Lagrange multiplier λ a that accounts for the constraint ensuring the condition ∑ i u a i = 0 [8], with its help, equation (2.13) can be written as: Q ia (u; ξ, τ, σ, ω) = e 2πi ∑ b λ b −∑ j b u ab ij ⟨a,b⟩ j b θ 0 −u ab ij + ∑ d−1 l=1 q l ⟨a,b⟩ ∆ l ; ω θ 0 −u ba ji + ∑ d−1 l=1 q l ⟨b,a⟩ ∆ l ; ω , (2.14) where we have denoted u ia −u j b ≡ u ab ij . Restricting ourselves to the case with τ = σ, we would like to propose a set of u ab ij that makes (2.14) equal to 1, thus solving the Bethe-Ansatz equation (2.6). It is natural to make an attempt with a direct generalization of the type of solution encountered in [8], namely: u ab ij = τ N (i a − i b ) and λ a = (N − 1) 2. These solutions appeared first in [17] while evaluating the topologically twisted of 4d N = 1 theories on T 2 × S 2 in the high temperature limit; it was later shown in [18] that such configuration provides an exact solution on the Bethe-Ansatz equations. Consider one generic factor entering in (2.14) for a fixed value of b: j b θ 0 u ab ij + ∆ ab ; ω θ 0 −u ab ij + ∆ ab ; ω u ab ij = τ N (ia−i b ) = ∏ ia−1 k=0 θ 0 τ N k + ∆ ab × ∏ −1 k=ia−N θ 0 τ N k + ∆ ab ∏ N −ia k=0 θ 0 τ N k + ∆ ab × ∏ −1 k=ia−1 θ 0 τ N k + ∆ ab (2.15) with ∆ ab ≡ d−1 l=1 q l ⟨a,b⟩ ∆ l j b θ 0 u ab ij + ∆ ab ; ω θ 0 −u ab ij + ∆ ab ; ω = ∏ N −1 k=0 θ 0 τ N k + ∆ ab × ∏ −1 k=ia−N −e 2πiτ k N e 2πi∆ ab ∏ N −1 k=0 θ 0 τ N k + ∆ ab × ∏ −1 k=1−ia −e 2πiτ k N e 2πi∆ ab = (−1) N −1 e 2πi∆ ab N −2ia+1 e 2πiτ ia− N+1 2 . In (2.15) we have used the following properties of the θ 0 function: θ 0 (u + n + mτ ; τ ) = −e −2πmiu−πimτ (m−1) θ 0 (u; τ ) (2.16) θ 0 (u; τ ) = θ 0 (τ − u; τ ) = −e 2πiu θ 0 (−u; τ ) . Inserting (2.15) back into (2.14) leads to multiplying all the results obtained in (2.15) for all n a values of b connected with a via some field Φ ab : Q ia (u; ξ, τ ) = e 2πi na N−1 2 −∑ j b τ N (ia−j b ) (−1) na(N −1) e 2πi ∑ b ∆ ab N −2ia+1 e 2πinaτ ia− N+1 2 (2.17) na b=1 ∆ ab = 0 ⇓ Q ia (u; ξ, τ ) = e 2πi na N−1 2 −∑ j b τ N (ia−j b ) (−1) na(N −1) e 2πinaτ ia− N+1 2 = e 2πi na N−1 2 −na τ N N ia− N(N−1) 2 (−1) na(N −1) e 2πinaτ ia− N+1 2 = 1 ∎ . Evaluation of the index The formula for the index reads: I (p, q; v) = κ G û∈M BAE Z tot (û; ξ, ν R , rω, sω) H (û; , ξ, ν R , ω) −1 (2.18) κ G = (p; p) rk(G) ∞ (q; q) rk(G) ∞ W G Z tot (u; ξ, ν R , rω, sω) = rs {m ia }=1 Z (u − mω; ξ, ν R , rω, sω) Z (u; ξ, ν R , rω, sω) = ∏ Φ ab ∏ ia≠j b Γ e u ia − u j b + ∑ d−1 l=1 q l ⟨a,b⟩ ∆ l ; τ, σ ∏ α∈∆ Γ e (α (u) ; τ, σ) H (u; ξ, ν R , ω) = det 1 2πi ∂Q ia (u; , ξ, ν R , ω) ∂u j b iaj b . Dominant contributions to the index in the large N limit will come from terms analogous to the dominating the expression obtained in [8] for the N = 4 SYM theory. This implies that in order to investigate the large N limit of (2.18), we only need to consider the following term: Γ e u ab ij + ∆ ab ; τ, τ = e −πiQ u ab ij +∆ ab ;τ,τ θ 0 u ab ij +∆ ab τ ; − 1 τ × ∞ k=0 ψ k+1+u ab ij τ ψ k−u ab ij −∆ ab τ (2.19) Q (u; τ, σ) = u 3 3τ σ − τ + σ − 1 2τ σ u 2 + (τ + σ) 2 + τ σ − 3 (τ + σ) + 1 6τ σ u + (τ + σ − 1) (τ + σ − τ σ) 12τ σ Q (u + ∆; τ, τ ) = u 3 3τ 2 + u 2 ∆ τ 2 − 2τ − 1 2τ 2 + u 1 − 6τ + 5τ 2 6τ 2 + ∆ 2 τ 2 − 2τ − 1 τ 2 ∆ − − ∆ 2 2τ 2 (2τ − 1) + ∆ 6τ 2 5τ 2 − 6τ + 1 + 1 12τ 2 (2τ − 1) 2τ − τ 2 + ∆ 3 τ 2 . Now recalling that solutions of the form u ab ij = τ N (i a − j b ) satisfy the Bethe-Ansatz equations, we proceed to analyze the large N limit. Thus, the leading contribution in N to log I takes the form: [8], the sum ∑ v is carried over the n v vector multiplets and n χ is the number of chiral fields. The conservation of U(1) charges implies ∑ Φ ab [∆ ab ] τ = 0, which allows us to eliminate every linear term in [∆ ab ] τ appearing in (2.21), therefore we can write: log I = − iπN 2 3τ 2 Φ ab ([∆ ab ] τ − τ ) [∆ ab ] τ − τ + 1 2 ([∆ ab ] τ − τ + 1) − (2.20) − iπN 2 3τ 2 v τ τ − 1 2 (τ − 1) = − iπN 2 3τ 2 Φ ab [∆ ab ] τ [∆ ab ] τ + 1 2 ([∆ ab ] τ + 1) + 3τ 2 [∆ ab ] τ − 3τ [∆ ab ] 2 τ − 3τ [∆ ab ] τ + + iπN 2 3τ 2 (n χ − n v ) τ τ − 1 2 (τ − 1) , where [∆ ab ] τ is defined such that [∆] τ = ∆ mod 1log I = − iπN 2 3τ 2 Φ ab [∆ ab ] τ [∆ ab ] τ + 1 2 ([∆ ab ] τ + 1) − 3τ [∆ ab ] 2 τ + (2.21) + iπN 2 3τ 2 (n χ − n v ) τ τ − 1 2 (τ − 1) . Let us now analyse the properties of the function we have obtained. Recalling that: [∆ + 1] τ = [∆] τ , (2.22) [−∆] τ = − [∆] τ − 1 [∆ + τ ] τ = [∆] τ + τ, Then, it is possible to write: K(∆, τ ) ≡ [∆] τ [∆] τ + 1 2 ([∆] τ + 1) − 3τ [∆] 2 τ = 1 2 2∆ 3 − 3 ∆ ∆ + ∆ − 6τ ∆ ∆ ,(2.23) which holds when ∆ < 1. Equation (2.21) is very similar to the one obtained in [14] when analyzed n the Cardy-like limit of the index, however, there is an extra contribution of the form iπN 2 3τ 2 (n χ − n v ) τ τ − 1 2 (τ − 1) which is still of order O(N 2 ) but sub-leading when τ → 0. Notice that at this point there is no dependence on the holonomies of the gauge groups since we have already evaluated in the solutions of the Bethe-Ansatz equations. Finally we have: log I = − iπN 2 3τ 2 Φ ab K (∆ ab , τ ) + iπN 2 3τ 2 (n χ − n v ) τ τ − 1 2 (τ − 1) . (2.24) Defining ∆ d such that: ∑ d I=1 ∆ I − 2τ = −1 [8], it can be shown that log I can be writen as: log I = − iπN 2 6τ 2 C IJK ∆ I ∆ J ∆ K + iπN 2 3τ 2 (n χ − n v ) τ τ − 1 2 (τ − 1) (2.25) The coefficients C IJK in (2.25) correspond, as pointed out in [14], to the Chern-Simons couplings of the holographic dual gravitational description as elucidated in [19]. It is worth noting that the new contribution to the index cancels in the prototypical example of N = 4 SYM since when written in N = 1 language, n χ = n v = 3. In that case, as demonstrated in [8], the large N and Cardy limits coincide. The correction we highlight in equation (2.25) becomes important for theories with n χ ≠ n v which are generic in the space of toric quiver gauge theories. In the following section we proceed to evaluate the superconformal index for various models, some of them recently discussed in a similar context in [14], and compare our results with (2.25). We will, in one very relevant example, explore how the new contribution affects the would-be macroscopic entropy of the dual black holes. The superconformal index of various SCFT's We will test our result (2.21) in various cases in each of which we follow the prescription of charge assignment used in [14]. We will restrict ourselves to the regime of fugacities ∆ i of the d − 1 U(1) global symmetries such that: 0 ≤ ∆ i ≤ 1 2 ∀i, 0 ≤ d−1 i=1 ∆ i , ≤ 1 (3.1) which in our case will be useful to evaluate the function K(∆, τ ) using equation (2.23). This regime also coincides with the one in which the existence of a universal saddle point in which all the holonomies vanish according to the analysis carried in [14] can be ensured. The conifold theory We would like to study the index in the large N limit and thus investigate it beyond the Cardylike limit. We will explicitly evaluate the correction determined in equation (2.25). To do so we start with one of the simplest examples of toric quiver gauge theories -the conifold theory [20] whose quiver diagram is given below. We take the ranks of all the gauge groups equal (N 1 = N 2 = N) and the sub-index in N i helps describe the representations of the matter fields: N 1 N 2 A 1 , A 2 B 1 , B 2 The superpotential is W ∝ ǫ ij ǫ kl Tr A i B k A j B l (3.2) The global charges of the conformal field theory are: a U(1) R factor, two SU(2) factors and finally there is a U(1) B baryonic symmetry. A fascinating fact about this theory is that it admits a gravity dual in terms of strings in AdS 5 × T 1,1 . The isometries of T 1,1 realize the mesonic symmetries of the field theory in terms of the isommetries of CP 1 × CP 1 ; the U(1) B baryonic symmetry is associated to the unique non-trivial three-cycle of the geometry. It is worth pointing out that the rotating electrically charged black holes dual to the superconformal index have not yet been constructed on the supergravity side, and that remains an outstanding problem. We use the basis for the charges suggested by the toric diagram discussed in [14] and we summarize them in the following table: Field U(1) R U(1) 1 U(1) 2 U(1) 3 A 1 1/2 1 0 0 A 2 1/2 0 0 1 B 1 1/2 0 1 0 B 2 1/2 -1 -1 -1 With this information we are ready to evaluate equation (2.21): log I = − iπN 2 3τ 2 [K (∆ 1 , τ ) + K(∆ 2 , τ ) + K(∆ 3 , τ ) + K (−∆ 1 − ∆ 2 − ∆ 3 , τ )] + (3.3) + iπN 2 3τ 2 2τ τ − 1 2 (τ − 1) = − iπN 2 τ 2 −∆ 2 1 (∆ 2 + ∆ 3 ) − ∆ 2 ∆ 3 (1 − 2τ + ∆ 2 + ∆ 3 ) − ∆ 1 (∆ 2 + ∆ 3 ) (1 − 2τ + ∆ 2 + ∆ 3 ) + + iπN 2 3τ 2 2τ τ − 1 2 (τ − 1) , where we have used n χ = 4 and n v = 2. After imposing the condition ∑ d I=1 ∆ I − 2τ = −1 yields : log I = − iπN 2 τ 2 [∆ 2 ∆ 3 ∆ 4 + ∆ 1 ∆ 3 ∆ 4 + ∆ 1 ∆ 2 ∆ 3 + ∆ 1 ∆ 2 ∆ 4 ] + (3.4) + iπN 2 3τ 2 2τ τ − 1 2 (τ − 1) . We see that log I presents the behavior proposed in (2.25) with the explicit O(N 2 ) correction to the index. The Suspended Pinch Point The suspended pinch point (SPP) gauge theory corresponds to the near horizon limit of a stack of N D3 branes probing the tip of the conical singularity , x 2 y = wz. The SPP gauge theory is described by the following quiver N 1 N 2 N 3 φ X 32 , X 23 X 12 , X 21 X 31 , X 13 All the ranks are taken to be the same with N 1 = N 2 = N 3 = N and the sub-indices are meant to help understand the representation properties of the matter fields. The superpotential is W = Tr [X 21 X 12 X 23 X 32 − X 32 X 23 X 31 X 13 + X 13 X 31 φ − X 12 X 21 φ] . (3.5) Each X ij transforms in the N representation of the index i-th node and in the N of the j-th node. The field φ transforms in the adjoint representation of the corresponding gauge group. The charge assignment for the U(1) R and the extra U(1) i global symmetries can be taken as: Field U(1) R U(1) 1 U(1) 2 U(1) 3 U(1) 4 φ 4/5 1 1 0 0 X 12 2/5 0 0 0 1 X 21 4/5 -1 -1 0 -1 X 23 2/5 0 1 0 0 X 32 2/5 1 0 0 0 X 31 4/5 -1 -1 -1 0 X 13 2/5 0 0 1 0 The next step is to use this information and perform the evaluation (2.21). log I = − iπN 2 3τ 2 [K (∆ 1 + ∆ 2 , τ ) + K(∆ 4 , τ ) + K(−∆ 1 − ∆ 2 − ∆ 4 , τ ) + K (∆ 2 , τ ) + K(∆ 1 , τ ) + (3.6) + K(−∆ 1 − ∆ 2 − ∆ 3 , τ ) + K(∆ 4 , τ )] + iπN 2 3τ 2 4τ τ − 1 2 (τ − 1) = − iπN 2 τ 2 [−∆ 2 1 (∆ 2 + ∆ 3 + ∆ 4 ) + ∆ 1 (1 − 2τ + ∆ 2 + ∆ 3 )(−∆ 2 − ∆ 3 ) + (2τ − 1 − 2∆ 2 )∆ 4 − ∆ 2 4 + + ∆ 2 (1 − 2τ + ∆ 2 )∆ 3 + ∆ 2 3 + ∆ 4 (1 − 2τ + ∆ 2 + ∆ 4 ) ] + iπN 2 3τ 2 4τ τ − 1 2 (τ − 1) , where we have used: n χ = 7 and n v = 3. Now we use: ∑ 5 I=1 ∆ I − 2τ = −1 we introduce a fifth fugacity ∆ 5 that permits us to rewrite (3.6) in the following, more symmetric, way: log I = − iπN 2 τ 2 [2∆ 2 ∆ 3 ∆ 4 + ∆ 2 ∆ 3 ∆ 5 + ∆ 2 ∆ 4 ∆ 5 + 2∆ 1 ∆ 3 ∆ 4 + ∆ 5 ∆ 3 ∆ 2 + (3.7) + ∆ 1 ∆ 4 ∆ 5 + ∆ 1 ∆ 2 ∆ 3 + ∆ 1 ∆ 2 ∆ 4 + ∆ 1 ∆ 2 ∆ 5 ] + + iπN 2 3τ 2 4τ τ − 1 2 (τ − 1) . This result is in agreement with equation (2.25) which is what is expected from toric geometry and reinforces the validity of the analysis of [14] which was limited to the Cardy-Like limit but contains a correction. The dP 1 theory We consider now the theory arising from a stack of N D3 branes at the tip of the complex Calabi-Yau cone whose base is the first del Pezzo surface. The quiver associated to this theory is : N 1 N 2 N 3 N 4 X (α) 23 X 12 X 13 X (α) 41 X 42 X (α) 34 , X (3) 34 where N 1 = N 2 = N 3 = N 4 = N and the superpotential is given by: W = ǫ αβ Tr X (α) 34 X (β) 41 X 13 − X (α) 34 X (β) 23 X 42 + X 12 X (3) 34 X (α) 41 X (β) 23 . (3.8) The charge assignment specified by the toric data is given by: Field U(1) R U(1) 1 U(1) 2 U(1) 3 X 12 1/2 0 0 1 X (1) 23 1/2 0 1 0 X (2) 23 1/2 -1 -1 -1 X (1) 34 1 0 1 1 X (2) 34 1 -1 -1 0 X (3) 1/2 1 0 0 X (1) 41 1/2 0 1 0 X (2) 41 1/2 -1 -1 -1 X 13 1/2 1 0 0 X 41 1/2 1 0 0 Let us now evaluate the leading in N part of the superconformal index according to (2.21) : log I = − iπN 2 3τ 2 [2K(∆ 1 , τ ) + K (∆ 3 , τ ) + 2K(∆ 2 , τ ) + 2K (−∆ 1 − ∆ 2 − ∆ 3 , τ ) + (3.9) + K(∆ 2 + ∆ 3 , τ ) + K(−∆ 1 − ∆ 2 , τ )] + + iπN 2 τ 2 2τ τ − 1 2 (τ − 1) = − iπN 2 τ 2 [−(2∆ 1 + ∆ 2 )∆ 2 3 − 3∆ 1 ∆ 2 (∆ 1 + ∆ 2 + 1) − − ∆ 2 2 + 4∆ 1 ∆ 2 + ∆ 2 + 2∆ 1 (∆ 1 + 1) ∆ 3 + τ (6∆ 1 ∆ 2 + 2(2∆ 1 + ∆ 2 )∆ 3 ] + iπN 2 τ 2 2τ τ − 1 2 (τ − 1) . We notice that n χ = 10 and n v = 4. Introducing now ∆ 4 via the constraint ∆ 1 +∆ 2 +∆ 3 +∆ 4 −2τ = −1 we obtain: log I = − iπN 2 τ 2 [2∆ 1 ∆ 2 ∆ 3 + 3∆ 1 ∆ 2 ∆ 4 + 2∆ 1 ∆ 3 ∆ 4 + 2∆ 2 ∆ 3 ∆ 4 ] + (3.10) + iπN 2 τ 2 2τ τ − 1 2 (τ − 1) 3.4 Y p,q quiver gauge theories The Y pq model corresponds to quiver gauge theories with 2p gauge groups and a chiral field content of bifundamental fields. The charge assignment and the corresponding multiplicity of the fields are shown below: Multiplicity U(1) 1 U(1) 2 U(1) 3 U(1) R p + q 1 0 0 1/2 p 0 1 0 1/2 p − q 0 0 1 1/2 p -1 -1 -1 1/2 q 0 1 1 1 q -1 -1 0 1 where we have used the prescription of [14]. Now we evaluate the leading, order O(N 2 ), part of the superconformal index (2.21): log I = − iπN 2 3τ 2 [(p + q)K (∆ 1 , τ ) + pK (∆ 2 , τ ) + (p − q)K (∆ 3 , τ ) + (3.11) + pK (−∆ 1 − ∆ 2 − ∆ 3 , τ ) + qK (∆ 2 + ∆ 3 , τ ) + qK (−∆ 1 − ∆ 2 , τ )] + iπN 2 3τ 2 2(p + q)τ τ − 1 2 (τ − 1) = − iπN 2 3τ 2 [(p + q)K (∆ 1 , τ ) + pK (∆ 2 , τ ) + (p − q)K (∆ 3 , τ ) + + pK (−∆ 1 − ∆ 2 − ∆ 3 , τ ) + qK (∆ 2 + ∆ 3 , τ ) + qK (−∆ 1 − ∆ 2 , τ )] + iπN 2 3τ 2 2(p + q)τ τ − 1 2 (τ − 1) = − iπN 2 3τ 2 [∆ 2 (∆ 1 − ∆ 3 )(−q)(∆ 1 + ∆ 2 + ∆ 3 − 2τ + 1) − − p (∆ 2 + ∆ 3 ∆ 2 1 + (∆ 2 + ∆ 3 ∆ 1 (∆ 2 + ∆ 3 − 2τ + 1) + ∆ 2 ∆ 3 (∆ 2 + ∆ 3 − 2τ + 1) ] + + iπN 2 3τ 2 2(p + q)τ τ − 1 2 (τ − 1) . Here n χ = 4p + 2q and n v = 2p. Finally we eliminate τ from (3.11) using ∑ 4 I=1 ∆ I − 2τ = −1, which successfully reproduce the structure of (2.25): log I = − iπN 2 3τ 2 [p∆ 1 ∆ 2 ∆ 3 + (p + q)∆ 1 ∆ 2 ∆ 4 + p∆ 1 ∆ 3 ∆ 4 + (p − q)∆ 2 ∆ 3 ∆ 4 ] + (3.12) + iπN 2 3τ 2 2(p + q)τ τ − 1 2 (τ − 1) . Corrections to the dual black hole entropy Let us now investigate how the extra term in (2.25) modifies the entropy obtained by taking the Legendre transform of log I. Our starting point it to organize the computation as to maximally take advantage of the scaling properties (with respect to ∆ I and τ ) of the superconformal index in the Cardy limit. Thus we write: log I = S E + S τ (4.1) S E ≡ − iπN 2 6τ 2 C IJK ∆ I ∆ J ∆ K S τ ≡ iπN 2 3τ 2 (n χ − n v ) τ τ − 1 2 (τ − 1) Since S τ is independent of ∆ I we have: ∂ log I ∂∆ I = ∂S E ∂∆ I (4.2) ∂ log I ∂τ = ∂S E ∂τ + ∂S τ ∂τ The Legendre transform leading to the entropy can be written as: S (Q, Λ) = log I + 2πi d I=1 ∆ I Q I − 2τ J + 2πiΛ d I=1 ∆ I − 2τ + 1 , (4.3) where Λ is a Lagrange multiplier imposing the constraint. The extremization condition implies: ∂S ∂∆ I = 0, ⇒ ∂ log I ∂∆ I = −2πi (Q I + Λ) (4.4) ∂S ∂τ = 0 ⇒ ∂S E ∂τ = −4πi J − Λ J ≡ J + 1 4πi ∂S τ ∂τ . The homogeneity of S E leads to the important relation: S E = d I=1 ∆ I ∂S E ∂∆ I + τ ∂S E ∂τ . (4.5) Following [14], we insert (4.5) in (4.1) and evaluating on the extremization solutions we find: S (Q, J) = 2πiΛ(Q, J) + S τ − 4πiτ J − J (4.6) = 2πiΛ(Q, J) + S τ − τ ∂S τ ∂τ = 2πiΛ(Q, J) + iπN 2 3 (n χ − n v ) 1 τ (J) − 3 2 . The properties of S E allow us to reconstruct S 2 E from suitable combinations of products of its derivatives with respect to ∆ I which generically leads to a cubic equation to determine L(Q, J). To make the analysis somehow concrete we now focus on a particular model. Entropy corrections to the Y p,p theory Let us track the implications of our correction in the particular case of gauge theories obtained as a stack of D3 branes placed at the cone over Y p,p , equivalently, a stack of N D3-branes placed at the tip of C 3 Z 2p , which leads to an orbifold of N = 4 SYM. The gravity dual is expected to be simply related to string theory on AdS 5 × Y p,p and one can expect to be able to read off the entropy on the gravity side readily following the extremization prescription put forward in [21]. Our goal in this section is to estimate the correction to the entropy in the Cardy limit -S Cardy -due to the correction we have computed for the superconformal index. If we specialize equation (3.12) to (p = q) case we obtain: log I = − iπpN 2 3τ 2 [∆ 1 ∆ 2 ∆ 3 + 2∆ 1 ∆ 2 ∆ 4 + ∆ 1 ∆ 3 ∆ 4 ] + (4.7) + iπN 2 3τ 2 2pτ τ − 1 2 (τ − 1) . This particular model -the Y p,p quiver gauge theory -has been recently analyzed in the Cardy limit in [14], where the second term in equation (4.7) was not considered . A further simplification (seding the baryonic charge Q 2 → 0 and further renaming Q 4 → Q 2 ) was previously presented in [13]. In the notation of [14] which we follow in this note the above simplifications lead to an entropy of the form S Cardy = 2π Q 1 Q 3 + Q 1 Q 4 + Q 3 Q 4 − p N 2 2 2J. (4.8) This is quite similar to the N = 4 SYM expression for the entropy and will guide our intuition for corrections. Let us now return to the treatment explain at the beginning of this section. It can be shown that S E in our case (the first line in equation( 4.7)) satisfies: 0 = ∂S E ∂∆ 1 2 2 ∂S E ∂∆ 3 + ∂S E ∂∆ 4 ∂S E ∂∆ 2 − ∂S E ∂∆ 2 2 − 2 ∂S E ∂∆ 3 − ∂S E ∂∆ 4 2 + (4.9) + 4pN 2 ∂S E ∂τ 2 . Using equation (4.1), we can obtain a cubic equation for Λ in the same spirit that [14], however we are able to keep track of the modification produced in the entropy by the presence of S τ in (2.25), hence, we have: 0 = (Q 1 + Λ)[2 (2(Q 3 + Λ) + (Λ + Q 4 )) (Λ + Q 2 ) − (Λ + Q 2 ) 2 − (4.10) − (2(Λ + Q 3 ) − (Λ + Q 4 )) 2 ] + 4pN 2 (Λ −J) 2 . The solution of (4.10) when plugged into equation (4.6) leads to an entropy of the form: S (Q, Λ) = 2π (Q 1 + Q 3 )(Q 2 + Q 4 ) + Q 4 Q 2 2 − Q 2 2 4 − Q 2 3 − Q 2 4 4 − 2N 2 pJ (4.11) + iπN 2 3 (n χ − n v ) 1 τ − 3 2 = 2π (Q 1 + Q 3 )(Q 2 + Q 4 ) + Q 4 Q 2 2 − Q 2 2 4 − Q 2 3 − Q 2 4 4 − 2N 2 p J + N 2 12 (n χ − n v ) 1 − 1 2τ 2 (J) + iπN 2 3 (n χ − n v ) 1 τ (J) − 3 2 In the above expression we have left (n χ −n v ) explicitly in the corrections to highlight its effect. The angular velocity τ (J) appears only formally, it should be substituted by the extremization procedure, we have indicated such operation as τ (J). The most dramatic effect is a shift in the angular momentum. Conclusions In this brief note we have explored the superconformal index following the Bethe Ansatz approach introduced by Benini and Milan [9]. We have shown that a class of solutions can be extended to solve the Bethe Ansatz equation for a large class of 4d N = 1 supersymmetric gauge theories. The Bethe Ansatz approach has the advantage that it does not require to take the Cardy limit and therefore provides a more complete large N expression. Indeed, for generic toric quiver gauge theories we determined that there is a contribution of the order O(N 2 ) which is sub-leading in the Cardy limit. We hope that more work along this direction might eventually allow to understand the growth of states in the index in a more systematic fashion. For example, by exploiting the Bethe Ansatz approach to the topologically twisted index a systematic study of 1 N corrections for the ABJM index was performed in [22]; a similar study for a Chern-Simons matter theory dual to massive IIA black holes was reported in [23]. Such understanding of 1 N corrections will naturally translate into interesting aspects in the dual quantum gravity side for AdS 5 black holes. For example, the statistical entropy of certain magnetically charged AdS 4 black holes has recently been given a microscopic explanation in terms of the topologically twisted index [1] (see [2,3] for a reviews with comprehensive lists of references). The investigation of sub-leading (logarithmic in N) corrections such as those performed recently [24,25] have helped clarify the nature of the degrees of freedom on the gravitational side of the duality. One would hope for similar developments in the context of AdS 5 black holes. There are many other interesting open problems. At the technical level, it would be interesting to generalize the Bethe Ansatz approach to arbitrary fugacities such that a general expression depending on both angular momenta can be achieved. There is little doubt that such generalization will yield the expected results but it will clarify the inner workings of the evaluation of the superconformal index. In this manuscript we have completely avoided the subtle discussion concerning the space of solutions of the Bethe Ansatz equations, we limited ourselves to just one class and showed that it yields a contribution sufficient to extract the dual black hole entropy and its potential corrections. It would be very illuminating to have a better understanding of all the solutions and how one should weight their contributions to the index. Finally, it is an important open problem to construct explicitly the black holes dual to the field theories discussed in this manuscript. Our computation, as well as those in a number of recent publications [12,13,14], shows that it is relatively easy to find the superconformal index in a large class of supersymmetric four-dimensional field theories some of which have known supergravity dual. Moreover, using the entropy formula one can evaluate the entropy and realize that it corresponds to that of large black holes in AdS 5 . However, the explicit black hole construction on the gravity side is still in its infancy, not much is known beyond the AdS 5 black holes dual to N = 4 SYM (and some of its orbifolds). It remains an outstanding challenge for the supergravity community to explicitly construct rotating electrically charged black holes which could be understood as dual of available field theory results. One particular example that comes to mind among the class discussed in this note would be the black holes in asymptotically AdS 5 × T 1,1 and, more generally, AdS 5 × Y p,q . AcknowledgmentsWe are thankful to Antonio Amariti, Francesco Benini, Alejandro Cabo-Bizet, Ivan Garozzo, Gabriele Lo Monaco, Jun Nian, Paolo Milan and Alberto Zaffaroni. LPZ is partially supported by the U.S. Department of Energy under grant de-sc0007859. Black hole microstates in AdS 4 from supersymmetric localization. F Benini, K Hristov, A Zaffaroni, 10.1007/JHEP05(2016)0541511.04085JHEP. 0554F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS 4 from supersymmetric localization, JHEP 05 (2016) 054 [1511.04085]. Black hole microstates and supersymmetric localization. S M Hosseini, 2018-02. 1803.01863Milan Bicocca UPh.D. thesisS. M. Hosseini, Black hole microstates and supersymmetric localization, Ph.D. thesis, Milan Bicocca U., 2018-02. 1803.01863. A Zaffaroni, Lectures on AdS Black Holes, Holography and Localization. A. Zaffaroni, Lectures on AdS Black Holes, Holography and Localization, 2019, 1902.07176. An Index for 4 dimensional super conformal theories. J Kinney, J M Maldacena, S Minwalla, S Raju, 10.1007/s00220-007-0258-7hep-th/0510251Commun. Math. Phys. 275209J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251]. Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes. A Cabo-Bizet, D Cassani, D Martelli, S Murthy, 1810.11442A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes, 1810.11442. Large AdS black holes from QFT. S Choi, J Kim, S Kim, J Nahmgoong, 1810.12067S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, 1810.12067. S Choi, J Kim, S Kim, J Nahmgoong, 1811.08646Comments on deconfinement in AdS/CFT. S. Choi, J. Kim, S. Kim and J. Nahmgoong, Comments on deconfinement in AdS/CFT, 1811.08646. F Benini, P Milan, 1812.09613Black holes in 4d N = 4 Super-Yang-Mills. F. Benini and P. Milan, Black holes in 4d N = 4 Super-Yang-Mills, 1812.09613. A Bethe Ansatz type formula for the superconformal index. F Benini, P Milan, 1811.04107F. Benini and P. Milan, A Bethe Ansatz type formula for the superconformal index, 1811.04107. A Ardehali, 10.1007/JHEP06(2019)1341902.06619Cardy-like asymptotics of the 4d N = 4 index and AdS 5 blackholes. 134A. Arabi Ardehali, Cardy-like asymptotics of the 4d N = 4 index and AdS 5 blackholes, JHEP 06 (2019) 134 [1902.06619]. Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula. M Honda, 8091M. Honda, Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, 1901.08091. The asymptotic growth of states of the 4d N=1 superconformal index, Submitted to. A Cabo-Bizet, D Cassani, D Martelli, S Murthy, 1904.05865J. High Energy Phys. A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The asymptotic growth of states of the 4d N=1 superconformal index, Submitted to: J. High Energy Phys. (2019) [1904.05865]. J Kim, S Kim, J Song, A 4d N = 1 Cardy Formula. 3455J. Kim, S. Kim and J. Song, A 4d N = 1 Cardy Formula, 1904.03455. Entropy function from toric geometry. A Amariti, I Garozzo, G. Lo Monaco, A. Amariti, I. Garozzo and G. Lo Monaco, Entropy function from toric geometry, 1904.10009. An Infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals. S Benvenuti, S Franco, A Hanany, D Martelli, J Sparks, 10.1088/1126-6708/2005/06/064hep-th/0411264JHEP. 0664S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, An Infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, JHEP 06 (2005) 064 [hep-th/0411264]. Gauge theories from toric geometry and brane tilings. S Franco, A Hanany, D Martelli, J Sparks, D Vegh, B Wecht, 10.1088/1126-6708/2006/01/128hep-th/0505211JHEP. 01128S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211]. The Cardy limit of the topologically twisted index and black strings in AdS 5. S M Hosseini, A Nedelin, A Zaffaroni, 10.1007/JHEP04(2017)0141611.09374JHEP. 0414S. M. Hosseini, A. Nedelin and A. Zaffaroni, The Cardy limit of the topologically twisted index and black strings in AdS 5 , JHEP 04 (2017) 014 [1611.09374]. The topologically twisted index of N = 4 super-Yang-Mills on T 2 × S 2 and the elliptic genus. J Hong, J T Liu, 10.1007/JHEP07(2018)0181804.04592JHEP. 0718J. Hong and J. T. Liu, The topologically twisted index of N = 4 super-Yang-Mills on T 2 × S 2 and the elliptic genus, JHEP 07 (2018) 018 [1804.04592]. Triangle anomalies from Einstein manifolds. S Benvenuti, L A Pando Zayas, Y Tachikawa, 10.4310/ATMP.2006.v10.n3.a4hep-th/0601054Adv. Theor. Math. Phys. 10395S. Benvenuti, L. A. Pando Zayas and Y. Tachikawa, Triangle anomalies from Einstein manifolds, Adv. Theor. Math. Phys. 10 (2006) 395 [hep-th/0601054]. Superconformal field theory on three-branes at a Calabi-Yau singularity. I R Klebanov, E Witten, 10.1016/S0550-3213(98)00654-3hep-th/9807080Nucl. Phys. 536199I. R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B536 (1998) 199 [hep-th/9807080]. An extremization principle for the entropy of rotating BPS black holes in AdS 5. S M Hosseini, K Hristov, A Zaffaroni, 10.1007/JHEP07(2017)1061705.05383JHEP. 07106S. M. Hosseini, K. Hristov and A. Zaffaroni, An extremization principle for the entropy of rotating BPS black holes in AdS 5 , JHEP 07 (2017) 106 [1705.05383]. Toward Microstate Counting Beyond Large N in Localization and the Dual One-loop Quantum Supergravity. J T Liu, L A Zayas, V Rathee, W Zhao, 10.1007/JHEP01(2018)0261707.04197JHEP. 0126J. T. Liu, L. A. Pando Zayas, V. Rathee and W. Zhao, Toward Microstate Counting Beyond Large N in Localization and the Dual One-loop Quantum Supergravity, JHEP 01 (2018) 026 [1707.04197]. . J T Liu, L A Pando Zayas, S Zhou, Subleading Microstate Counting in the Dual to Massive Type IIA, 1808.10445J. T. Liu, L. A. Pando Zayas and S. Zhou, Subleading Microstate Counting in the Dual to Massive Type IIA, 1808.10445. One-Loop Test of Quantum Black Holes in antide Sitter Space. J T Liu, L A Zayas, V Rathee, W Zhao, 10.1103/PhysRevLett.120.221602Phys. Rev. Lett. 1202216021711.01076J. T. Liu, L. A. Pando Zayas, V. Rathee and W. Zhao, One-Loop Test of Quantum Black Holes in antide Sitter Space, Phys. Rev. Lett. 120 (2018) 221602 [1711.01076]. Precision Microstate Counting for the Entropy of Wrapped M5-branes. D Gang, N Kim, L A Pando Zayas, 1559D. Gang, N. Kim and L. A. Pando Zayas, Precision Microstate Counting for the Entropy of Wrapped M5-branes, 1905.01559.
[]
[ "BIFURCATION OF LIMIT CYCLES IN PIECEWISE QUADRATIC DIFFERENTIAL SYSTEMS WITH AN INVARIANT STRAIGHT LINE", "BIFURCATION OF LIMIT CYCLES IN PIECEWISE QUADRATIC DIFFERENTIAL SYSTEMS WITH AN INVARIANT STRAIGHT LINE" ]
[ "Leonardo P C Da ", "Cruz And ", "Joan Torregrosa " ]
[]
[]
We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weakfoci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude.
10.1016/j.jmaa.2022.126256
[ "https://arxiv.org/pdf/2204.00085v1.pdf" ]
247,922,470
2204.00085
092569cdd950bd7747db7c504fb928fac5b920c2
BIFURCATION OF LIMIT CYCLES IN PIECEWISE QUADRATIC DIFFERENTIAL SYSTEMS WITH AN INVARIANT STRAIGHT LINE 31 Mar 2022 Leonardo P C Da Cruz And Joan Torregrosa BIFURCATION OF LIMIT CYCLES IN PIECEWISE QUADRATIC DIFFERENTIAL SYSTEMS WITH AN INVARIANT STRAIGHT LINE 31 Mar 2022arXiv:2204.00085v1 [math.DS] We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weakfoci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude. Introduction In past years, a big interest in the study of the dynamics of piecewise systems has emerged, due to the fact that many real phenomena can be modeled with this class of systems. For example, the existence and uniqueness of periodic orbits or the existence of a continuum of periodic orbits. These problems appear in many areas of research. In particular in electrical and mechanical engineering, in control theory, and even in the analysis of genetic networks. See for example [1,10]. Usually, the simplest models are defined via planar piecewise polynomial vector fields Z = (Z + , Z − ) in the following way. Taking 0 as a regular value of the function h : R 2 → R, we denote the discontinuity curve by Σ = h −1 (0) and the two regions it delimits by Σ ± = {±h(x, y) > 0}. So, the piecewise vector field can be written as Z ± : (ẋ,ẏ) = (X ± (x, y), Y ± (x, y)), for (x, y) ∈ Σ ± , where X ± and Y ± are polynomials of degree n in Σ ± . The above piecewise vector field is continuous when it satisfies Z + = Z − on the separation curve Σ. Otherwise we will say that it is discontinuous. The local trajectories of Z on Σ was stated by Filippov in [11] (see Figure 1). The points on Σ where both vectors fields simultaneously point outward or inward from Σ define the escaping (Σ e ) and sliding region (Σ s ), respectively. The interior of its complement on Σ defines the crossing region (Σ c ), and the boundary of these regions is constituted by tangential points of Z ± with Σ. As this work is restricted to the study of limit cycles of crossing type, that we will refer to them only as limit cycles, we do not recall here the precise definition of the vector field on Σ e and Σ s . Let Z ± h denote the derivative of the function h in the direction of the vector Z ± that is, Z ± h(p) = ∇h(p), Z ± (p) . Notice that p ∈ Σ c provided that Z + h(p) · Z − h(p) > 0, p ∈ Σ e ∪ Σ s provided that Z + h(p) · Z − h(p) < 0, and p in Σ is a tangential point of Z ± provided that Z + h(p)Z − h(p) = 0. We say that p ∈ Σ is a pseudo-equilibrium of Z, if p is either a tangential point or an equilibrium of Z + or Z − . We call p ∈ Σ an invisible fold of Z + (resp. Z − ) if p is a tangential point of Z + (resp. Z − ) and (Z + ) 2 h(p) < 0 (resp. (Z − ) 2 h(p) > 0). Let us consider that both differential systems in (1) (when we thought them separately) have an equilibrium point at the origin such that the eigenvalues of their Jacobian matrices at 0 have zero real part. For simplicity, we will consider only the cases when the linear part Figure 1. Definition of the vector field on Σ following Filippov's convention in the sewing, escaping, and sliding regions. of each system is written in its normal form. Hence, after a time rescaling if necessary, (1) writes as Z ± = ẋ = −y + n k=2 P ± k (x, y), y = x + n k=2 Q ± k (x, y), if (x, y) ∈ Σ ± ,(2) being Σ ± = {(x, y) : ±h(x, y) > 0} and h a C 1 function for which 0 is a regular value. Obviously, for the problem that we would like to study, the time orientation is taken in order that the origin has a monodromic character. In this paper, we will assume that the discontinuity curve Σ = h −1 (0) is a straight line passing through the origin. In fact, we will take them being one of the coordinates axes. As usual P ± k , Q ± k denote homogeneous polynomials of degree k. As in the analytic scenario, the problem of distinguishing whether the origin of (2) is a center or a focus is also known as Poincaré center problem, centerfocus problem, or just center problem. In the piecewise polynomial class, there are other center problems that are not considered in this work. This is the case, for example, when we consider a pseudo-equilibrium point of fold-fold type. In addition to the difficulty of increasing the number of parameters, when we have fixed the degree of a vector field, we have to consider other types of centers appearing in the nonsmooth scenario. We will deal with this point in Section 3. A very related problem is the analysis of the number of limit cycles bifurcating from the origin. It is well-known that linear vector fields have no limit cycles. But this is not the case in a piecewise scenario. Freire, Ponce, Rodrigo and Torres prove in [12] that only one limit cycle exists in continuous piecewise linear differential systems. For discontinuous piecewise linear differential systems Freire, Ponce, and Torres in [14] prove that only two limit cycles of small amplitude bifurcate from the origin. They also prove that a third (big) limit cycle exists. Also for this class of differential systems, the center problem from a monodromic equilibrium point but near the infinity is solved in [13], where the number of limit cycles bifurcating from the infinity is also considered. The authors prove that at least three limit cycles bifurcate from infinity. Both problems are related to using a transformation that moves the infinity to the origin. All the limit cycles are in fact crossing limit cycles, because they cut the separation straight line Σ. For quadratic vector fields, Bautin showed in [3] that the maximum number of limit cycles of small amplitude near an equilibrium point is three and, moreover, this upper bound is reached. For quadratic discontinuous differential systems, this problem is studied in [17,21]. The work of Bautin is very appreciated because increasing the degree, these problems remain open. Thus, it is quite natural to restrict the study to some special families. For example, the planar quadratic vector fields have an invariant straight line. Cherkas, Zhilevich, and Rychkov, see [6,7,26], proved that this family has only one limit cycle. For more details on this problem see [8] and [27]. They prove that the canonical form of such systems is ẋ = −y + dx + l x 2 + m x y + n y 2 , y = x + bx y, being d ∈ [0, 2), m ≥ 0, and b = 0. We notice that, with these conditions, the origin is an equilibrium point of monodromic non-degenerate type. We will analyze this problem in the piecewise framework with 2 zones separated by a straight line passing through the origin. Hence, the dimension of the space of parameters (d, l, m, n, b) ∈ R 5 will be doubled, (d 1 , d 2 , l 1 , l 2 , m 1 , m 2 , n 1 , n 2 , b 1 , b 2 ) ∈ R 10 . But, as we will see, the number of limit cycles will increase much more than doubled. A first natural extension is to consider that the invariant straight line is the same in both zones. Therefore we will assume b 1 = b 2 = b. As in [8], after the rescaling (x, y) → (−x/b, −y/b) if necessary, we can assume b = −1. This change of variables does not modify the crossing limit cycles and the dimension of the parameter space decreases to 8. An interesting phenomenon is that the number of limit cycles increases and it depends on the separation straight line. We will study two situations but fixing the canonical form (3): firstly when the separation straight line is the horizontal axis and secondly when it is the vertical axis. As we will see in the following results, the highest number of limit cycles is obtained for the second situation in the discontinuous case while in the continuous case is for the first one. Consequently, the first piecewise quadratic differential system is Z j = ẋ = −y + d j x + l j x 2 + m j x y + n j y 2 , y = x(1 − y), if (x, y) ∈ Σ H j ,(4) where the discontinuity straight line is Σ H = {(x, y) : y = 0} and Σ H j = {(−1) j y < 0}, for j = 1, 2. The second piecewise quadratic differential system is Z j = ẋ = −y + d j x + l j x 2 + m j x y + n j y 2 , y = x(1 − y), if (x, y) ∈ Σ V j ,(5) where the discontinuity straight line is Σ V = {(x, y) : x = 0}, with Σ V j = {(−1) j x < 0}, for j = 1, 2. We notice that, in general, (4) and (5) are discontinuous piecewise differential systems. The first main result (Theorem 1.1) provides a lower bound for the number of limit cycles of small amplitude in both situations. It is remarkable that in the second one we are using all the parameters for having a complete unfolding. For the continuous cases, labeled as (4 c ) and (5 c ) respectively, the local cyclicity problem is completely solved. See Theorems 1.2 and 1.3. Theorem 1.1. There are values of the parameters such that from the origin of systems (4) and (5) bifurcate 4 and 7 crossing limit cycles of small amplitude, respectively, multiplicities taken into account. As we will see in the proof of the above result, the limit cycles are obtained by studying which are the maximal orders of weak-foci together with the respective unfoldings. Both numbers provide lower bounds for the local cyclicity of the origin. For the study of the corresponding upper bounds, the cyclicity of the centers should be also studied. Which needs a more accurate analysis. This problem is completely solved in the next two results, getting the least upper bounds for the maximum number of crossing limit cycles of small amplitude that can bifurcate from the origin, multiplicities taken into account, that is, providing its local cyclicity. We remark the large difference between the number of limit cycles taking into account that both families have the same number of parameters. We also observe that the considered families have no sliding segment near the origin. As we will explain in Section 2, the pseudo-Hopf bifurcation does not take place. If we were interested in this kind of bifurcation, we could add an extra parameter in the first components of system (5) obtaining an extra limit cycle. We notice that the study of this phenomenon in system (4) breaks the chosen canonical form and also the invariant straight lines. Theorem 1.2. The piecewise differential equation (4) is continuous if and only if l 1 = l 2 =: l and d 1 = d 2 =: d. Therefore, it becomes ẋ = −y + d x + l x 2 + m j x y + n j y 2 , y = x(1 − y), if (x, y) ∈ Σ H j , (4 c ) being Σ H j = {(−1) j y < 0}, for j = 1, 2. Moreover, the local cyclicity of the origin of the above family is 3, multiplicities taken into account. Theorem 1.3. The piecewise differential equation (5) is continuous if and only if n 1 = n 2 =: n. Therefore, it writes as ẋ = −y + d j x + l j x 2 + m j x y + n y 2 , y = x(1 − y), if (x, y) ∈ Σ V j , (5 c ) being Σ V j = {(−1) j x < 0}, for j = 1, 2. Moreover, the local cyclicity of the origin of the above family is 2, multiplicities taken into account. After a detailed analysis of the number of limit cycles of small amplitude bifurcating from the origin in families (4) and (5), we finish providing an answer to the respective center-focus problems, firstly for the discontinuous case and secondly for the continuous one. ( H 1 ) m 1 = m 2 = 0; (H 2 ) l 1 − l 2 = m 1 − m 2 = l 1 + n 2 = l 1 + n 1 = 0; (H 3 ) l 1 + l 2 + 1 = m 1 − m 2 = n 1 + n 2 − 1 = 0. Theorem 1.5. For family (5) with d 1 = d 2 = 0, the origin is a center if, and only if, one of the next conditions holds: (V 1 ) l 1 − l 2 = m 2 + m 1 = n 1 − n 2 = 0; (V 2 ) l 1 − l 2 = n 2 + l 2 = n 1 + l 2 = 0; (V 3 ) l 1 − 2l 2 = m 2 = n 2 = n 1 + 2l 2 = 0; (V 4 ) 2l 1 − 2l 2 + 1 = m 2 = m 1 = n 2 = n 1 − 1 = 0; (V 5 ) 2l 1 − 2l 2 − 1 = m 2 = m 1 = n 1 = n 2 − 1 = 0; (V 6 ) l 1 − 2l 2 − 1 = m 2 = n 2 − 1 = n 1 + 2l 2 + 1 = 0; (V 7 ) 2l 1 − l 2 = m 1 = n 1 = n 2 + l 2 = 0; (V 8 ) 2l 1 − l 2 + 1 = m 1 = n 2 + l 2 = n 1 − 1 = 0. The next corollaries follow straightforwardly from the above results, considering the continuity conditions given in Theorem 1.2 when the corresponding family has an equilibrium point of weak-focus type at the origin. Corollary 1.6. For family (4 c ) with d = 0, the origin is a center if, and only if, one of the next conditions holds: (H c 1 ) m 1 = m 2 = 0; (H c 2 ) m 1 − m 2 = l + n 2 = l + n 1 = 0; (H c 3 ) 2l + 1 = m 1 − m 2 = n 1 + n 2 − 1 = 0. Corollary 1.7. For family (5 c ) with d 1 = d 2 = 0, the origin is a center if, and only if, one of the next conditions holds: (V c 1 ) l 1 − l 2 = m 2 + m 1 = 0; (V c 2 ) l 1 − l 2 = n + l 2 = n + l 2 = 0; (V c 3 ) l 1 = l 2 = m 2 = n = 0; (V c 6 ) l 1 + 1 = l 2 + 1 = m 2 = n − 1 = 0; (V c 7 ) l 1 = l 2 = m 1 = n = 0; (V c 8 ) l 1 + 1 = l 2 + 1 = m 1 = n − 1 = 0. The paper is structured as follows. In Section 2, we present the basic tools necessary to prove the results of this work. In Section 3, we provide sufficient conditions so that the presented families have a center at the origin. Next, in Section 4, we analyze the highest-order weak-foci equilibrium points together with the small amplitude limit cycle bifurcation and, as usual, the necessary conditions to have a center equilibrium point. The last section is devoted to showing which is the upper bound for the cyclicity when the families are continuous, finishing with the proofs of Theorems 1.2 and 1.3. The degenerate Hopf bifurcation The main results of this paper follow studying the return map near an equilibrium point of monodromic type located in the separation straight line. In fact, studying the composition of two half-return maps because the proofs are mainly based on considering two piecewise polynomial vector fields having a nondegenerate equilibrium point of centerfocus type. Hence, the analysis of both maps can be realized by computing the Taylor series of the solution with respect to the initial condition, working in polar coordinates. But, instead of using the composition of both maps, we will compute the difference map that is equivalent. We recall first how these Taylor series can be computed and then how they are used to study lower and upper bounds for the cyclicity. That is, the number of limit cycles of small amplitude bifurcating from the equilibrium point. The piecewise system (2) can be written in polar coordinates, (x, y) = (r cos θ, r sin θ), as dr ± dθ = n k=2 R ± k (θ)r k 1 + n k=2 Θ ± k (θ)r k−1 = ∞ k=2 S ± k (θ)r k ,(6) where R ± k (θ) = cos θ P ± k (cos θ, sin θ) + sin θ Q ± k (cos θ, sin θ), Θ ± k (θ) = cos θ Q ± k (cos θ, sin θ) − sin θ P ± k (cos θ, sin θ), being P ± k , Q ± k , and S ± k polynomials in sin θ and cos θ. We consider the solution of the initial value problems defined by (6) with r + (r 0 , 0) = r 0 and r − (r 0 , π) = r 0 , written in Taylor series with respect to r 0 , defined when |r 0 | ≪ 1, as r ± (θ, r 0 ) = r 0 + ∞ k=2 u + k (θ)r k 0 , if θ ∈ (0, π), r 0 + ∞ k=2 u − k (θ)r k 0 , if θ ∈ (π, 2π ). Hence, we can define the positive half-return map Π + (r 0 ) = r + (r 0 , π) and the negative half-return map Π − (r 0 ) = r − (r 0 , 2π). Instead of considering the composition of both maps we will define its equivalent displacement map, see for example [9], ∆(r 0 ) = Π − −1 (r 0 ) − Π + (r 0 ) = ∞ k=2 W k r k 0 ,(7) where, the function (Π − ) −1 (r 0 ) is the inverse of the negative half-return map (Π − ) (r 0 ), as it is illustrated in Figure 2. As usual in this kind of analysis, the first nonvanishing W k is called the kth-order Lyapunov quantity of the piecewise polynomial system (2). This approach was also used in [15,20,24]. It is well-known that, as the usual Lyapunov quantities for analytic vector fields, the coefficients W k are polynomials depending on the coefficients of the polynomial perturbations P ± k , Q ± k . Finally, we will say that the origin of (2) is a weak-focus of order ℓ when W j = 0, 1 ≤ j ≤ ℓ − 1 and W ℓ = 0. Moreover, the stability of the equilibrium point is given by the sign of the first nonzero Lyapunov quantity W ℓ . This first non-vanishing coefficient provides also the stability of the equilibrium. As usual, choosing adequately the perturbation parameters in P ± k , Q ± k we obtain limit cycles of small amplitude bifurcating, in this case, from the origin. This perturbation analysis is known as the degenerated Hopf bifurcation. The classical Hopf bifurcation in the study of analytic planar vector fields is characterized by the birth of a limit cycle from a weak-focus of first-order. More specifically, in the analytical context, the first nonvanishing coefficient has always an odd subscript and the limit cycle, which is of small amplitude, bifurcates from the origin changing adequately the sign of the trace of the Jacobian matrix of the corresponding system near the equilibrium point. See more details in [2,25]. This bifurcation, in the piecewise context, is associated with the study of the return map in a fold-fold type point. In this case, the first non-vanishing coefficient has always an even subscript. The limit cycle of small amplitude bifurcates from the origin changing also the stability of the origin. In this case, the size of the sliding segment takes the role of the trace in the analytic context. See more details of this phenomenon of codimension 1 in [22] or also in [18] for the codimension 2 case. Recently, this bifurcation is also known as pseudo-Hopf bifurcation, see [4]. When, as in our case, we fix the existence of an equilibrium of monodromic type in both upper and lower systems of (2), all the coefficients in r 0 appear in the Taylor development (7). All our families (4), (4 c ), (5), and (5 c ) have the origin as an equilibrium point. Hence, W 0 = 0 and we have no sliding (nor escaping) segment. It is easy to check that by adding the trace parameter in upper and lower systems (2) generically the first non-vanishing coefficient in (7) is W 1 . As in the analytic scenario when we have an equilibrium point of focus type. In fact, W 1 = 0 if and only if the sum of the traces (of the upper and lower systems) is zero. Clearly, when W 1 = 0 and W 2 = 0, the stability of the equilibrium point is given by the sign of W 2 . More concretely, the origin of the system is stable (resp. unstable) when W 2 < 0 (resp. W 2 > 0). Consequently, in the system, when W 1 is a small enough positive (resp. negative) real number, a small (resp. unstable) limit cycle bifurcates from the origin. In this case, it is important to remark that the equilibrium point (or equilibrium points) remains located at the origin. So, this bifurcation is also similar to the one previously denoted as Hopf bifurcation. From the above analysis, in the bifurcation of an analytic planar piecewise vector field, when we have a weak-focus of order k we get (generically) k limit cycles. See more details in [17]. This bifurcation problem with varying parameters and taking into account multiplicities is studied in [19]. As we will see, as our families are polynomial, the study of the complete unfolding is more intricate. We notice again that, as all our families (4), (4 c ), (5), and (5 c ) have no sliding, we will only get up to k − 1 limit cycles of small amplitude. Although for families (5 c ) and (5) this pseudo-Hopf bifurcation makes sense. Because the invariant straight line 1 − y = 0 remains unchanged when a constant term is added in the first components. The sufficient conditions for the center problem This section is devoted to proving that the families in Theorems 1.4 and 1.5 are centers. This is done in Propositions 3.3 and 3.4, respectively. We consider centers such that the period annulus is formed only by crossing periodic orbits. The key point is based, except by a special change of variables introduced in Definition 3.1, on the existence of three centers type. They are rigorously defined in Theorem 3.2. The first ones are of Darboux type because they have a piecewise first integral; the second ones have the usual timereversibility, and the third ones are also symmetric but have the identity as the half-return map. To simplify the reading, in the next definition and the main result we take Σ H = {y = 0} as the separation curve. The result and the definition can be easily generalized considering other separation curves. See all the different cases drawn in Figure 3. We notice that for the third class of systems in the above result also the complete vector field Z has the detailed symmetry. Proof of Theorem 3.2. We notice that from the piecewise differential system (2), it is easy to see that the origin is a monodromic point and this property will be used along the proof. (a) From the canonical form, as it is written the piecewise differential system (2), it is easy to check that the Taylor series of the first integrals should start as H ± (x, y) = x 2 + y 2 + · · · . Let x 2 < 0 < x 1 small enough real numbers such that (x j , 0) ∈ Σ H , j = 1, 2, and are connected by monodromic solutions of (2) in upper and lower halfplanes. Consider the function H(x 1 , x 2 ) = H ± (x 2 , 0) − H ± (x 1 , 0) x 2 − x 1 = n i=1 H i (x 1 , x 2 ), being H i homogeneous polynomials of degree i in (x 1 , x 2 ). We notice that we have only one function H because, by hypothesis, H ± (x, 0) = H ± (x, 0). Moreover, H 1 (x 1 , x 2 ) = x 1 + x 2 and, when (x 1 , 0) and (x 2 , 0) are on the same level curve of H + , or H − , we have H(x 1 , x 2 ) = 0. As, near the origin ∂ H/∂x 2 = 0, we can apply the Implicit Function Theorem to show the existence of a unique half-return map g : I → I, where I is an interval containing the origin. Moreover, g satisfies H(x, g(x)) = 0 and g(x) = −x + · · · . The proof follows from the uniqueness of the half-return map, because it is the same in both, upper and lower, regions. See Figure 3.(a). (b) The monodromy property together with the time-symmetry (8), allows us to use the classical result of analytic reversibility systems to prove this item. See more details in [23]. A drawing of this situation can be seen in Figure 3.(b). (c) The proof follows by the reversibility property (9) that satisfy the upper and lower solutions. Both are symmetric with respect to the y-axis, so the respective half-return maps are equal. More concretely, each point (x, 0) in Σ H is sent to the corresponding symmetric one (−x, 0). See this property in Figure 3.(c). The following results are direct consequences of the last theorem. Proof. The first family H 1 is a center using directly Theorem 3.2.(c). The second family H 2 satisfies that Z 1 = Z 2 . Hence it is, in fact, a quadratic vector field of Lotka-Volterra type following the classification in [28]. So there exists a first integral that coincides in both regions Σ H 1 and Σ H 2 . The proof finishes applying Theorem 3.2.(a). The proof that the last family H 3 has a center follows from Theorem 3.2.(b), proving that, after writing the systems in the usual polar coordinates (x, y) = (r cos θ, r sin θ) , the change r = 4R/(4 + A j (θ)R), with j = 1, 2 where A 1 (θ) =(3 − l 1 − 3n 2 ) sin θ + m 2 cos θ − (1 + l 1 − n 2 ) sin 3θ − m 2 cos 3θ, A 2 (θ) =(1 + l 1 + 3n 2 ) sin θ + m 2 cos θ + (1 + l 1 − n 2 ) sin 3θ − m 2 cos 3θ is a twin Σ H -transformation. We notice that both rational changes of coordinates of the above proof are known because they allow us to change a system with a homogeneous nonlinearity to an Abel differential equation, see for example [5]. Proposition 3.4. For each family V i , i = 1, . . . , 8, listed in Theorem 1.5, the corresponding piecewise system (5) has a center at the origin. Proof. As the separation line is Σ V = {x = 0}, we can apply Theorem 3.2 after changing the variables (x, y) by (y, x). The first family V 1 is time-reversible with respect to the change (x, y, t) → (−x, y, −t) and, from the comment above, we have a center applying Theorem 3.2.(b). The proof for the remaining families follows using Theorem 3.2.(a) and all the first integrals H ± , here denoted by H 1 , H 2 , will be of Darboux type and they will write as H j (x, y) = (f j,1 (x, y)) λ j,1 (f j,2 (x, y)) λ j,2 (f j,3 (x, y)) λ j,3 . Where λ j,3 can be zero when only two invariant algebraic curves are necessary in the center characterization. It is necessary to check that all of them are well defined in a neighborhood of the origin. We only provide the polynomials f j,1 , f j,2 , f j,3 and the exponents λ j,1 , λ j,2 , λ j,3 . Using the first remark of the proof, in addition of finding the first integrals we will check the condition H 1 (0, y) = H 2 (0, y), for all y, being H 1 and H 2 the first integrals defined in x > 0 and x < 0, respectively. • For the case V 2 we have H j (0, y) = (1 − y) l 2 (l 2 y + 1), for j = 1, 2. Being f j,1 = 1 − y, f j,2 = (−m j + (4l 2 2 + m 2 j + 4l 2 ) 1/2 )x/2 + l 2 y + 1, f j,3 = (−m j − (4l 2 2 + m 2 j + 4l 2 ) 1/2 )x/2 + l 2 y + 1, λ j,1 = l 2 , λ j,2 = (1 + m j (4l 2 2 + m 2 j + 4l 2 ) −1/2 )/2, λ j,3 = (1 − m j (4l 2 2 + m 2 j + 4l 2 ) −1/2 )/2. • For family V 3 we can take f 1,1 = f 2,1 = 1 − y, f 1,2 = (−m 1 + (16l 2 2 + m 2 1 + 8l 2 ) 1/2 )x/2 + 2l 2 y + 1, f 1,3 = (−m 1 − (16l 2 2 + m 2 1 + 8l 2 ) 1/2 )x/2 + 2l 2 y + 1, λ 1,1 = λ 2,1 = 2l 2 , λ 1,2 = (1 + 2m 1 (16l 2 2 + m 2 1 + 8l 2 ) −1/2 )/2, λ 1,3 = (1 − 2m 1 (16l 2 2 + m 2 1 + 8l 2 ) −1/2 )/2, f 2,2 = −(2l 2 + 1)l 2 x 2 + 2l 2 y + 1, λ 2,2 = 1. Here H j (0, y) = (1 − y) 2l 2 (2l 2 y + 1), for j = 1, 2. • For family V 4 , we have H j (0, y) = (1 − y) 2l 2 (2l 2 y + 1), for j = 1, 2, where f 1,1 = −f 2,1 = y − 1, f 1,2 = (2l 2 + 1)l 2 x 2 + (2l 2 y + 1)(y − 1), λ 1,1 = λ 2,1 − 1 = 2l 2 − 1, λ 1,2 = λ 2,2 = 1, f 2,2 = −(2l 2 + 1)l 2 x 2 + 2l 2 y + 1. • For case V 5 we have f 1,1 = f 2,1 = y − 1, f 1,2 = (2l 2 2 + 3l 2 + 1)x 2 − (2l 2 + 1)y − 1, λ 1,1 = λ 2,1 + 1 = 2l 2 + 1, λ 1,2 = λ 2,2 = 1, f 2,2 = (2l 2 2 + 3l 2 + 1)x 2 + ((2l 2 + 1)y + 1)(y − 1), with H j (0, y) = ((2l 2 + 1)y + 1) (y − 1) 2l 2 +1 , for j = 1, 2. • In family V 6 we have H j (0, y) = ((2l 2 + 1)y + 1) (y − 1) 2l 2 +1 , with j = 1, 2, and f 1,1 = f 2,1 = y − 1, f 1,2 = − m 1 + (16l 2 2 + m 2 1 + 24l 2 + 8) 1/2 x/2 + (2l 2 + 1)y + 1, f 1,3 = − m 1 − (16l 2 2 + m 2 1 + 24l 2 + 8) 1/2 x/2 + (2l 2 + 1)y + 1, λ 1,1 = λ 2,1 + 1 = 2l 2 + 1, λ 1,2 = [1 + m 1 (16l 2 2 + m 2 1 + 24l 2 + 8) −1/2 ]/2, λ 1,3 = [1 − m 1 (16l 2 2 + m 2 1 + 24l 2 + 8) −1/2 ]/2, f 2,2 = (2l 2 2 + 3l 2 + 1)x 2 + ((2l 2 + 1)y + 1)(y − 1), λ 2,2 = 1. • Family V 7 is equivalent to V 3 , just by interchanging the left and right differential systems. • For the last family V 8 we can take f 1,1 = f 2,1 = 1 − y, f 1,2 = −(l 2 + 1)l 2 x 2 /2 + (l 2 y + 1)(1 − y), λ 1,1 = λ 2,1 − 1 = l 2 − 1, λ 1,2 = 1, f 2,2 = −m 2 + (4l 2 2 + m 2 2 + 4l 2 ) 1/2 x/2 + l 2 y + 1, f 2,3 = −m 2 − (4l 2 2 + m 2 2 + 4l 2 ) 1/2 x/2 + l 2 y + 1, λ 2,2 = (1 + (4l 2 2 + m 2 2 + 4l 2 ) −1/2 m 2 )/2, λ 2,2 = (1 − (4l 2 2 + m 2 2 + 4l 2 ) −1/2 m 2 )/2, being H j (0, y) = (1 − y) l 2 (l 2 y + 1), for j = 1, 2. The Maximal Order of a Weak-Focus and the Bifurcation of Crossing Limit Cycles In this section, we will provide the conditions of the parameters such that systems (4 c ), (4), (5 c ) have the maximal order of a weak-focus located at the origin and also the unfolding of crossing limit cycles of small amplitude in each family. This is done in Propositions 4.1, 4.2 and 4.3. The complete study of system (5) is more intricate. Proposition 4.4 provides the maximal order of each weak-focus and some values of the parameters such that this maximality is attained. Finally, in Proposition 4.5 we get the complete unfolding of some of them. Consequently, the proof of Theorem 1.1 is finished. Although the proofs of Theorems 1.2 and 1.3 will be done in the next section, the explicit unfoldings follow from the following results. This section is structured in two subsections. The first contains all the results referred to the case with the x-axis as the separation straight line. The second is devoted to the result being the y-axis as the separation straight line. As we have explained in the introduction, in the following results we will always have one crossing limit cycle of small amplitude less than the order of each weak-focus. Because our canonical forms have no sliding segment. The Horizontal Case. Proposition 4.1. The maximal weak-focus order of the origin of the piecewise differential system (4 c ) is 4. This maximal property is obtained when the parameters are on T c = {d = 2l + n 1 + n 2 = m 1 − m 2 = 0; m 2 (n 1 − n 2 )(n 1 + n 2 − 1) = 0}. Additionally, the weak-foci on T c unfold 3 limit cycles of small amplitude bifurcating from the origin, multiplicities taken into account and perturbing inside family (4 c ). Proof. The first necessary condition to have a nondegenerate equilibrium point of centerfocus type at the origin of (4 c ) is d = 0. Because the trace and the determinant of the Jacobian matrix are zero and one, respectively. With the mechanism described in Section 2, straightforward computations provide the first Lyapunov quantities W n . In particular, W 1 = 0 because d = 0 and W 2 =2(m 1 − m 2 )/3, W 3 =πm 2 (2l + n 1 + n 2 )/8, W 4 =4m 2 (n 1 − n 2 )(6l + 4n 1 + 4n 2 − 1)/45.(10) The proof of the maximality follows checking that W 2 (T c ) = W 3 (T c ) = 0 and W 4 (T c ) = 4m 2 (n 1 − n 2 )(n 1 + n 2 − 1)/45 = 0 and that the solutions of the polynomial system {W 2 = W 3 = W 4 = 0} provide the centers detailed in Corollary 1.6, which are centers using Proposition 3.3 and the continuity condition. As the determinant of the Jacobian matrix of W 2 , W 3 with respect to (m 1 , n 2 ) on T c , det J = det(Jac [(W 2 , W 3 ), (m 1 , n 2 )] | T ⌋ ) = 2/3 0 0 1/8 π m 2 = πm 2 /12, is different from zero, we have two hyperbolic limit cycles bifurcating from the origin under the condition d = 0. The third limit cycle emerges from the origin in a similar way as the classical Hopf bifurcation being d small enough and different from zero. As we have explained previously. The unfolding taking into account the multiplicities can be proved using the results in [19]. Proposition 4.2. The maximal weak-focus order of the origin of the piecewise differential system (4) is 5. This maximality is obtained when the parameters are on T = {d 1 = d 2 = 2l 2 +3n 1 −n 2 = 2l 1 −n 1 +3n 2 = m 1 −m 2 = 0; m 2 (n 1 +n 2 −1)(n 2 −n 1 ) = 0}. Additionally, the weak-foci on T unfold 4 limit cycles of small amplitude bifurcating from the origin, multiplicities taken into account, and perturbing inside family (4). Proof. As the proof follows similarly to the proof of Proposition 4.1, we only detail the differences. For system (4), the origin is a nondegenerate weak-focus when d 1 = d 2 = 0. The first Lyapunov quantities are W 2 =2(m 1 − m 2 )/3, W 3 =πm 2 (l 1 + l 2 + n 2 + n 1 )/8, W 4 = − 2m 2 (l 1 + l 2 + 1)(3l 1 + l 2 + 4n 2 )/45, W 5 = − πm 2 (l 1 + l 2 + 1)(l 1 − l 2 ) 2 /1536, W 6 = − '2m 2 (l 2 + 2)(l 2 − 1)(l 1 + l 2 + 1)(l 1 − l 2 )/4725.(11) Straightforward computations show that over T we have W 2 = W 3 = W 4 = 0 and W 5 = πm 2 (n 1 + n 2 − 1)(n 1 − n 2 ) 2 /384 = 0. The maximality follows from the fact that W 2 6 ⊂ W 1 , W 2 , . . . , W 5 and that under the conditions W 2 = W 3 = W 4 = W 5 = 0 we have the centers detailed in Theorem 1.4, which are centers because of Proposition 3.3. The unfolding of limit cycles bifurcating from the origin follows also similarly to the previous proof. When d 1 = d 2 = 0 the determinant of the Jacobian matrix of W 2 , W 3 , W 4 with respect to (l 2 , m 1 , n 2 ) = 0 over T is det(J) = π(m 2 ) 2 (n 1 + n 2 − 1)/90, where J =        0 2 3 0 π 8 m 2 π 8 m 2 π 8 m 2 2 45 m 2 (21n 1 − 19n 2 − 1) 0 4 45 m 2 (9n 1 − 11n 2 + 1)        . Hence, we have three limit cycles of small amplitude and the fourth bifurcates taking d 2 = 0 and d 1 = 0 small enough but with an adequate sign. The unfolding taking into account the multiplicities is proved using [19]. Additionally, the weak-foci on F c unfold 2 limit cycles of small amplitude bifurcating from the origin, multiplicities taken into account and perturbing inside family (5 c ). Proof. The proof follows basically using the same steps as the proof of Proposition 4.1. Here for computing th Lyapunov quantities first we need to consider a rotation of angle −π/2 in order that the separation straight line be the x-axis. Once again we have that W 1 = 0, when d 1 = d 2 = 0. The first Lyapunov quantities are W 2 =4(l 1 − l 2 )/3, W 3 =π(m 1 + m 2 )(l 2 + n)/8.(12) When W 2 = W 3 = 0 we have one center at the origin as the ones listed in Corollary 1.7, but they are centers because of Proposition 3.4, assuming the continuity condition. Consequently, the property of maximality and the existence of the condition F c follow. Like in the previous two proofs, the complete unfolding also follows. Here the linearity condition of W 2 with respect to l 1 or l 2 provides the first limit cycle of small amplitude. The second, as above, taking d 2 = 0 and d 1 = 0 small enough. Proposition 4.4. The maximal weak-focus order of the origin of the piecewise differential system (5) is 8. In particular, there are at least four families exhibiting this maximality: F 1 ={d 1 = d 2 = m 1 = m 2 = 0, l 1 = −13/4, l 2 = −3/2, n 2 = −1/2}, F 2 ={d 1 = d 2 = m 1 = m 2 = 0, l 1 = 9/4, l 2 = 1/2, n 2 = 3/2}, F ± 3 ={d 1 = d 2 = m 1 = m 2 = 0, l 1 = (±1 + 6l 2 + f (l 2 ))/4, n 2 = 1/2 ± f (l 2 )/5, l 2 ∈ L},(13) where L = {−3/4, −1/4, −3/2} and f (l 2 ) = 20l 2 2 + 20l 2 + 10. Proof. The proof follows basically using the same steps as the previous proofs, but the computations are more intricate. As above, we will start assuming d 1 = d 2 = 0 to get W 1 = 0. Next, in order to apply the algorithm described in Section 2, as in the proof of Proposition 4.3, we need to do a rotation of angle −π/2 to compute the Lyapunov quantities in this case. As usual, the property of maximality will follow solving the algebraic system of equations S 7 = {W 2 = W 3 = W 4 = W 5 = W 6 = W 7 = 0},(14) checking that there exists at least one real solution such that W 8 = 0 and proving that all the solutions of S 8 = {W 2 = W 3 = W 4 = W 5 = W 6 = W 7 = W 8 = 0},(15) imply W k = 0 = 0 for k ≥ 9. This last step is a consequence of Proposition 3.4. Finally, we will prove the unfolding of 7 limit cycles described in the last statement, using, in this last step, the parameters d 1 , d 2 . Straightforward computations allow us to get the first Lyapunov quantities which are polynomials in the parameters space (l 1 , l 2 , m 1 , m 2 , n 1 , n 2 ). Because of the size, we only detail the first one which, using W 2 = 0, provides the condition n 1 = −2l 1 + 2l 2 + n 2 .(16) The direct application of the algorithm of Section 2 provides the coefficients of the displacement function (7) that write, some of them, as polynomials in π, before using that the previous should vanish. So being W j i polynomials with rational coefficients in (l 1 , l 2 , m 1 , m 2 , n 2 ). Using a computer algebra system we can see that W 8 . Moreover, we can write W 3 =(l 2 m 2 − l 1 m 1 + m 2 n 2 + m 1 n 2 + 2l 2 m 1 )/8, 3 2 − 240l 2 2 l 1 + 144l 2 2 n 2 + 192l 2 l 2 1 − 240l 2 l 1 n 2 − 8l 2 m 2 2 − 8l 2 m 2 m 1 + 60l 2 n 2 2 − 48l 3 1 + 96l 2 1 n 2 − 60l 1 n 2 2 − 8m 2 2 n 2 − 8m 2 m 1 n 2 − 48l 2 2 + 72l 2 l 1 − 36l 2 n 2 − 24l 2 1 + 36l 1 n 2 )/45, W 5 =(l 1 − l 2 )(22l 2 2 m 2 + 52l 2 2 m 1 + 67l 2 l 1 m 2 − 52l 2 l 1 m 1 + 45l 2 m 2 n 2 − 52l 2 1 m 2 + 13l 2 1 m 1 + 15l 1 m 2 n 2 + 10m 2 n 2 2 + 33l 2 m 2 + 26l 2 m 1 − 26l 1 m 2 − 13l 1 m 1 + 20m 2 n 2 )/384, W 4 =(96lW 6 =8(l 2 − l 1 )(15664l 4 2 − 41576l 3 2 l 1 + 41128l 3 2 n 2 + 40440l 2 2 l 2 1 − 79808l 2 2 l 1 n 2 + 41730l 2 2 n 2 2 − 17272l 2 l 3 1 + 50992l 2 l 2 1 n 2 − 51084l 2 l 1 n 2 2 + 20038l 2 n 3 2 + 2744l 4 1 − 10776l 3 1 n 2 + 15582l 2 1 n 2 2 − 11322l 1 n 3 2 + 3920n 4 2 − 10024l 3 2 + 17980l 2 2 l 1 − 19942l 2 2 n 2 − 10964l 2 l 2 1 + 22268l 2 l 1 n 2 − 13869l 2 n 2 2 + 2240l 3 1 − 6250l 2 1 n 2 + 7023l 1 n 2 2 − 3482n 3 2 + 1096l 2 2 − 1416l 2 l 1 + 1219l 2 n 2 + 434l 2 1 − 557l 1 n 2 + 444n 2 2 )/20475, W [1] 7 =m 1 (l 2 − l 1 ) n 2 l 2 (2l 2 + 1)(l 2 + 1) W 7,20 (l 2 ) + (2l 2 − l 1 ) W 7,23 (l 2 ) , W [0] 8 =(l 2 − l 1 ) n 2 W 8,14 (l 2 ) + (2l 2 − l 1 )(2l 2 − 2l 1 − 1) W 8,24 (l 2 ) , W [0] 10 = 1024 21049875 9 i=1 R i , where R 1 =l 2 , R 2 = l 2 − 2l 1 − 1, R 3 = 2l 2 − l 1 + 1, R 4 = l 2 − l 1 , R 5 = l 2 + 1, R 6 =2l 2 − l 1 , R 7 = 2l 2 − 2l 1 + 1, R 8 = 2l 2 − 2l 1 − 1, R 9 = l 2 − 2l 1 , and W 7,k and W 8,k are polynomials with rational coefficients of degree k. From the above computations, systems (14) and (15), using the condition (16), are now written as 10 to solve the center problem, it is useful to use it. Hence, after considering the equivalent system S 7 = { W 3 = W 4 = W 5 = W 6 = W [1] 7 = 0}, S 8 = { W 3 = W 4 = W 5 = W 6 = W [1] 7 = W [0] 8 = 0}.S 10 = { W 3 = W 4 = W 5 = W 6 = W [1] 7 = W [0] 8 = W [0] 10 = 0}, we obtain the families of the statement of Theorem 1.5. So, with Proposition 3.4 we have that all are center families and, consequently, the maximal weak-focus order is 8 and the first statement follows. The second part of the statement follows solving partially system S 7 . More concretely, solving S [1] 7 = { W 3 = W 4 = W 5 = W 6 = m 1 = 0} and obtaining weak-foci of order 8. In fact, we have that on each of them W i = 0, for i = 3, . . . , 7, and W 8 = 0. More concretely, W 8 (F 1 ) = 2/3, W 8 (F 2 ) = −2/3, W 8 (F ± 3 ) = − (4l 2 + 3) 2 (4l 2 + 1) 2 189000 232l 3 2 + 348l 2 2 + 222l 2 + 53 ± (52l 2 2 + 52l 2 + 17)f (l 2 ) . Proposition 4.5. The weak-foci F 1 and F 2 defined in (13) unfold 7 limit cycles of small amplitude bifurcating from the origin, multiplicities taken into account, when we perturb inside family (5). Proof. We will follow the same unfolding procedure as in the previous results assuming d 2 = 0. We will focus our attention only to the point F 1 , the other follows similarly. Using the linearity dependence on d 1 and n 1 of W 1 and W 2 defined in (16), we can restrict our analysis to the study of the transversality condition of the Taylor series of the next Lyapunov quantities near F 1 , with respect to the parameters (m 1 , m 2 , l 1 , l 2 , n 2 ). Taking the perturbation We notice that if u i = 0 we have a weak-focus of order 8. Moreover, with the Implicit Function Theorem we have new coordinates v 3 , . . . , v 7 , in the parameter space, such that W j (u) = v j , para j = 3, . . . , 6 and u 7 = v 7 . Hence the transversality condition is satisfied up to W 6 . The last step is the computation of the Taylor series of W 7 when v 3 = v 4 = v 5 = v 6 = 0. Straightforward computations provide W 7 (v 7 ) = − 80339 31104 v 3 7 + O(v 4 7 ). The unfolding is complete because the above first coefficient has an odd power in the remaining parameter v 7 . More details on the used technique can be seen in [16]. We remark that the complete unfolding study for the other families of weak-foci in Proposition 4.4 is more difficult because of the dependence on the parameter l 2 . But it can be seen that only linear developments are not enough. Proof of Theorem 1.5. This proof is a direct consequence of the proofs of Propositions 4.4 and 3.4, since every candidate to be a center is given nullifying the first eight Lyapunov quantities W i , for i = 1, . . . , 8 obtained in (17). The cyclicity problem in the continuous classes We finish the work by studying the maximum number of limit cycles of small amplitude that bifurcate from the origin in the continuity classes (4 c ) and (5 c ). That is obtaining its cyclicity and proving Theorems 1.2 and 1.3. Proof of Theorem 1.2. Using Proposition 4.1 we know that from the origin of system (4 c ) bifurcates at least 3 limit cycles of small amplitude. The multiplicity property follows like the previous results using [19]. The upper bound follows from Theorem 9, of Chapter 2, given in [25] because the ideal, I = W 2 , W 3 , W 4 , generated by the Lyapunov quantities, given in (10), is radical. The radicality proves that under the condition d = 0 we have at most two limit cycles. The third limit cycle appears, using d, as in a classical Hopf bifurcation. See more details in [2] or again [25]. Proof of Theorem 1.3. The proof follows analogously as the above proof using that the ideal generated by the Lyapunov quantities W 2 , W 3 defined in (12) is also radical. The above approach can not be used for studying the cyclicity of (4) and (5) because the ideal generated by the corresponding Lyapunov quantities it is not radical. Theorem 1 . 4 . 14For family (4) with d 1 = d 2 = 0, the origin is a center if, and only if, one of the next conditions holds: Figure 2 . 2The positive and negative half-return maps Π + and (Π − ) −1 , respectively. Definition 3. 1 . 1Let Ψ ± : R 2 → R 2 be bijective transformations and Σ H = {y = 0}. We say thatΨ(x, y) = Ψ + (x, y), if y > 0, Ψ − (x, y), if y < 0, is a twin Σ H -transformation when Ψ + (x, 0) = Ψ − (x, 0).Theorem 3.2. Let Z be a piecewise differential system of the form (2) with Σ H = {y = 0}. Then, applying a twin Σ H -transformation if necessary, we have a center at the origin in the following cases: (a) There exist first integralsH ± of Z ± satisfying H + (x, 0) = H − (x, 0). (b) Z is invariant with respect to the change (x, y, t) → (x, −y, −t).(8)(c) Z ± are invariant with respect to the change (x, y, t) → (−x, y, −t). Figure 3 . 3The three different center types detailed in Theorem 3.2 Proposition 3. 3 . 3For each family H i , i = 1, 2, 3, listed in Theorem 1.4, the corresponding piecewise system (4) has a center at the origin. Proof of Theorem 1 . 4 . 14The necessary conditions for having a center at the origin are d 1 = d 2 = 0 and the equation (11) obtained in the proof of Proposition 4.2. The sufficiency is provided by Proposition 3.3. 4.2. The Vertical Case. Proposition 4.3. The maximal weak-focus order of the origin of the piecewise differential system (5 c ) is 3. This maximality is obtained when the parameters are on F c = {d 1 = d 2 = l 1 − l 2 = 0; (m 1 + m 2 )(l 2 + n) = 0}. F 1 , 1e = [m 1 = e 1 , m 2 = e 2 , l 1 = −13/4 + e 3 , l 2 = −3/2 + e 4 , n 2 = −1/2 + e 5 ] and with the linear change of variables in the parameter space, 5 , e 5 = u 7 , we have that the Taylor series of the Lyapunov quantities write as W j (u) =u j + O(u 2 ), for j = 3, . . . , 3 − 12u 4 + O(u 2 ), W 8 (u) =2/3 + O(u). we haveW 3 =π W 3 , W 4 = W 4 , W 5 = π W 5 , W 6 = W 6 , W 7 = W[0] 7 + π W [1] 7 , W 8 = W [0] 8 + π W [1] 8 , W 9 = W [0] 9 + π W [1] 9 + π 2 W [2] 9 , W 10 = W [0] 10 + π W [1] 10 + π 2 W [2] 10 , W 11 = W [0] 11 + π W [1] 11 + π 2 W [2] 11 + π 3 W [3] 11 , ∈ W 3 . . . , W 6 , W ∈ W 3 , . . . , W 6 , W 11 ∈ W 3 . . . , W 6 , W[0] 7 [1] 8 [1] 7 , and W [i] 9 , W [0] 10 2 , W [1] 10 , W [2] 10 , W [i] [1] 7 , W [0] ∈ W 3 . . . , W 6 , W it is clear that W 10 = 0. But although it is not necessary to use WAs W [0] 10 2 [1] 7 , W [0] 8 [0] 10 2 = 0 on S 8 and so also W [0] [0] AcknowledgementsThis work has been realized thanks to the Brazilian CAPES Agency (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Finance Code 001), the Catalan AGAUR Agency (grant 2017 SGR 1617), the Spanish Ministerio de Ciéncia, Innovación y Universidades via the Agencia Nacional de Investigación (grants PID2019-104658GB-I00 and CEX2020-001084-M), and the European Union's Horizon 2020 research and innovation programme (grant Dynamics-H2020-MSCA-RISE-2017-777911). Nonsmooth modeling and simulation for switched circuits. V Acary, O Bonnefon, B Brogliato, Springer69DordrechtV. Acary, O. Bonnefon, and B. Brogliato. Nonsmooth modeling and simulation for switched circuits, volume 69. Dordrecht: Springer, 2011. Theory of bifurcations of dynamic systems on a plane. A A Andronov, E A Leontovich, I I Gordon, A G Maȋer, Halsted Press [A division of John Wiley & SonsNew York-Toronto, OntA. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maȋer. Theory of bifurcations of dynamic systems on a plane. Halsted Press [A division of John Wiley & Sons], New York-Toronto, Ont., 1973. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. N N Bautin, Am. Math. Soc., Transl. 10072Sbornik, n. Ser.N. N. Bautin. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Am. Math. Soc., Transl. 100, 19 p. (1954); translation from Mat. Sbornik, n. Ser. 30(72), 181-196 (1952)., 1952. The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems. J Castillo, J Llibre, F Verduzco, Nonlinear Dynam. 903J. Castillo, J. Llibre, and F. Verduzco. The pseudo-Hopf bifurcation for planar discontinuous piece- wise linear differential systems. Nonlinear Dynam., 90(3):1829-1840, 2017. The number of limit cycles of a certain second order autonumous system. L A Cherkas, Differencial'nye Uravnenija. 12L. A. Cherkas. The number of limit cycles of a certain second order autonumous system. Differen- cial'nye Uravnenija, 12(5):944-946, 960, 1976. Some tests for the absence or uniqueness of limit cycles. Differencial'nye Uravnenija. L A Cherkas, L I Zhilevich, 6L. A. Cherkas and L. I. Zhilevich. Some tests for the absence or uniqueness of limit cycles. Differen- cial'nye Uravnenija, 6:1170-1178, 1970. L A Cherkas, L I Zhilevich, The limit cycles of certain differential equations. Differencial'nye Uravnenija. 8L. A. Cherkas and L. I. Zhilevich. The limit cycles of certain differential equations. Differencial'nye Uravnenija, 8:1207-1213, 1972. Limit cycles for a quadratic system with an invariant straight line and some evolution of phase portraits. Qualitative theory of differential equations. B Coll, J Llibre, Colloq. Math. Soc. János Bolyai. 533rd ColloqB. Coll and J. Llibre. Limit cycles for a quadratic system with an invariant straight line and some evolution of phase portraits. Qualitative theory of differential equations, 3rd Colloq., Szeged/Hung. 1988, Colloq. Math. Soc. János Bolyai 53, 111-123 (1990)., 1990. The center problem for discontinuous Liénard differential equation. B Coll, R Prohens, A Gasull, Int. J. Bifurcation Chaos Appl. Sci. Eng. 99B. Coll, R. Prohens, and A. Gasull. The center problem for discontinuous Liénard differential equa- tion. Int. J. Bifurcation Chaos Appl. Sci. Eng., 9(9):1751-1761, 1999. Piecewise-smooth dynamical systems. M Bernardo, C J Budd, A R Champneys, P Kowalczyk, Theory and applications. 163SpringerM. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk. Piecewise-smooth dynamical systems. Theory and applications, volume 163. New York, NY: Springer, 2008. Differential equations with discontinuous righthand sides. A F Filippov, Mathematics and its Applications (Soviet Series). 18Kluwer Academic Publishers GroupTranslated from the RussianA. F. Filippov. Differential equations with discontinuous righthand sides, volume 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. Trans- lated from the Russian. Bifurcation sets of continuous piecewise linear systems with two zones. E Freire, E Ponce, F Rodrigo, F Torres, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2811E. Freire, E. Ponce, F. Rodrigo, and F. Torres. Bifurcation sets of continuous piecewise linear systems with two zones. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28(11):2073-2097, 1998. Limit cycles from a monodromic infinity in planar piecewise linear systems. E Freire, E Ponce, J Torregrosa, F Torres, J. Math. Anal. Appl. 4962124818E. Freire, E. Ponce, J. Torregrosa, and F. Torres. Limit cycles from a monodromic infinity in planar piecewise linear systems. J. Math. Anal. Appl., 496(2):23, 2021. Id/No 124818. The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. E Freire, E Ponce, F Torres, Publ. Mat., Barc. E. Freire, E. Ponce, and F. Torres. The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ. Mat., Barc., 2014:221-253, 2014. Center-focus problem for discontinuous planar differential equations. A Gasull, J Torregrosa, Int. J. Bifurcation Chaos Appl. Sci. Eng. 137A. Gasull and J. Torregrosa. Center-focus problem for discontinuous planar differential equations. Int. J. Bifurcation Chaos Appl. Sci. Eng., 13(7):1755-1765, 2003. Lower bounds for the local cyclicity for families of centers. J Giné, L F S Gouveia, J Torregrosa, J. Differ. Equations. 275J. Giné, L. F. S. Gouveia, and J. Torregrosa. Lower bounds for the local cyclicity for families of centers. J. Differ. Equations, 275:309-331, 2021. Local cyclicity in low degree planar piecewise polynomial vector fields. L F S Gouveia, J Torregrosa, Nonlinear Anal., Real World Appl. 60103278L. F. S. Gouveia and J. Torregrosa. Local cyclicity in low degree planar piecewise polynomial vector fields. Nonlinear Anal., Real World Appl., 60:19, 2021. Id/No 103278. Generic bifurcations of low codimension of planar Filippov systems. M Guardia, T M Seara, M A Teixeira, J. Differ. Equations. 2504M. Guardia, T. M. Seara, and M. A. Teixeira. Generic bifurcations of low codimension of planar Filippov systems. J. Differ. Equations, 250(4):1967-2023, 2011. The maximum number of zeros of functions with parameters and application to differential equations. M Han, J Yang, J. Nonlinear Mod. Anal. 31M. Han and J. Yang. The maximum number of zeros of functions with parameters and application to differential equations. J. Nonlinear Mod. Anal., 3(1):13-34, 2021. On Hopf bifurcation in non-smooth planar systems. M Han, W Zhang, J. Differential Equations. 2489M. Han and W. Zhang. On Hopf bifurcation in non-smooth planar systems. J. Differential Equations, 248(9):2399-2416, 2010. Bifurcation of limit cycles for a perturbed piecewise quadratic differential systems. G L Ji, C J Liu, P H Li, Acta Math. Sin. (Engl. Ser.). 383G. L. Ji, C. J. Liu, and P. H. Li. Bifurcation of limit cycles for a perturbed piecewise quadratic differential systems. Acta Math. Sin. (Engl. Ser.), 38(3):591-611, 2022. One-parameter bifurcations in planar Filippov systems. Y A Kuznetsov, S Rinaldi, A Gragnani, Int. J. Bifurcation Chaos Appl. Sci. Eng. 138Y. A. Kuznetsov, S. Rinaldi, and A. Gragnani. One-parameter bifurcations in planar Filippov sys- tems. Int. J. Bifurcation Chaos Appl. Sci. Eng., 13(8):2157-2188, 2003. Time-reversal symmetry in dynamical systems: a survey. J S W Lamb, J A G Roberts, Phys. D. 1121-2Time-reversal symmetry in dynamical systemsJ. S. W. Lamb and J. A. G. Roberts. Time-reversal symmetry in dynamical systems: a survey. Phys. D, 112(1-2):1-39, 1998. Time-reversal symmetry in dynamical systems (Coventry, 1996). Degenerate Hopf bifurcation in nonsmooth planar systems. F Liang, M Han, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22316F. Liang and M. Han. Degenerate Hopf bifurcation in nonsmooth planar systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22(3):1250057, 16, 2012. Bifurcation of planar vector fields and Hilbert's sixteenth problem. R Roussarie, Birkhäuser164BaselR. Roussarie. Bifurcation of planar vector fields and Hilbert's sixteenth problem, volume 164. Basel: Birkhäuser, 1998. The limit cycles of the equation u(x + 1)du = (−x + ax 2 + bxu + cu + du 2 )dx. Differencial'nye Uravnenija. G S Rychkov, 8G. S. Rychkov. The limit cycles of the equation u(x + 1)du = (−x + ax 2 + bxu + cu + du 2 )dx. Differencial'nye Uravnenija, 8:2257-2259, 1972. Theory of limit cycles. Y Q Ye, S L Cai, L S Chen, K C Huang, D J Luo, Z E Ma, E N Wang, M S Wang, X A Yang, Translations of Mathematical Monographs. Chi Y. Lo66American Mathematical Societysecond editionY. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M. S. Wang, and X. A. Yang. Theory of limit cycles, volume 66 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, second edition, 1986. Translated from the Chinese by Chi Y. Lo. Quadratic systems with center and their perturbations. H Żo, J. Differential Equations. 1092H.Żo ladek. Quadratic systems with center and their perturbations. J. Differential Equations, 109(2):223-273, 1994. Matemática Departamento De, SI-2000Rodovia Washington Luís. São Carlos235Universidade Federal de São CarlosKm. Brazil Email address: [email protected] de Matemática, Universidade Federal de São Carlos, Rodovia Wash- ington Luís, Km 235, Caixa Postal 676, SI-2000, São Carlos (SP), Brazil Email address: [email protected] Matemàtiques Departament De, 08193 Bellaterra. Barcelona (SpainUniversitat Autònoma de BarcelonaDepartament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellater- ra, Barcelona (Spain); . Barcelona Bellaterra, Spain) Email address: [email protected], Barcelona (Spain) Email address: [email protected]
[]
[ "The Mu3e Experiment", "The Mu3e Experiment" ]
[ "Gavin Hesketh [email protected] \nUniversity College London\n\n", "Sean Hughes \nUniversity of Liverpool\n\n", "Ann-Kathrin Perrevoort \nKarlsruhe Institute of Technology\n\n", "Nikolaos Rompotis \nUniversity of Liverpool\n\n" ]
[ "University College London\n", "University of Liverpool\n", "Karlsruhe Institute of Technology\n", "University of Liverpool\n" ]
[]
The Mu3e experiment at the Paul Scherrer Institut will search for the lepton-numberviolating decay µ + → e + e − e + , extending the sensitivity by four orders of magnitude compared to existing limits. This probe of new physics is complementary to the existing collider, dark matter and neutrino particle physics programmes, and part of a global programme investigating the charged lepton flavour sector. As well as the main µ + → e + e − e + search, Mu3e will also extend the sensitivity to low-mass dark photons, and additional flavour-violating decays involving long-lived or stable particles.
null
[ "https://arxiv.org/pdf/2204.00001v2.pdf" ]
247,922,616
2204.00001
c0c58f34f27713e2d210cf3279ae459f4c9713cb
The Mu3e Experiment 4 Apr 2022 Gavin Hesketh [email protected] University College London Sean Hughes University of Liverpool Ann-Kathrin Perrevoort Karlsruhe Institute of Technology Nikolaos Rompotis University of Liverpool The Mu3e Experiment 4 Apr 2022on behalf of the Mu3e Collaboration Submitted to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021) The Mu3e experiment at the Paul Scherrer Institut will search for the lepton-numberviolating decay µ + → e + e − e + , extending the sensitivity by four orders of magnitude compared to existing limits. This probe of new physics is complementary to the existing collider, dark matter and neutrino particle physics programmes, and part of a global programme investigating the charged lepton flavour sector. As well as the main µ + → e + e − e + search, Mu3e will also extend the sensitivity to low-mass dark photons, and additional flavour-violating decays involving long-lived or stable particles. Introduction Charged lepton flavour violating (CLFV) processes offer unique discovery potential for physics beyond the Standard Model (BSM), bringing sensitivity to new physics that is complementary to the existing collider, dark matter and neutrino particle physics programmes. The Mu3e experiment [1] at the Paul Scherrer Institut (PSI) is part of a global programme of experiments searching for the "golden channels" of CLFV in the muon sector: µ + → e + γ, µ + → e + e − e + , µ − N → e − N . This programme will bring a significant increase in sensitivity compared to previous searches, probing new physics mass scales up to 10 3 -10 4 TeV. Of the experiments which will start taking data in the coming years, Mu3e is the only one which will search for the decay µ + → e + e − e + (MEG-II [2,3] targets the µ + → e + γ signal, while Mu2e [4], COMET [5] and DeeMe [6] will search for µ − N → e − N ). Flavour violation is a known feature of the quark and neutrino sectors of the Standard Model. And while charged lepton flavour appears to be conserved, it is not protected by any known global symmetry and occurs at the 1-loop level via neutrino oscillation. In the Standard Model, it is suppressed to an unobservably small rate O(10 −50 ) [7,8] in those loops, where ∆m ij is the squared mass difference between the neutrino mass eigenstates i and j. However, significant enhancements to this rate are predicted by many BSM scenarios, and the observation of any CLFV signal would be the unambiguous observation of new physics. The nature of any CLFV signal will depend on the underlying physics. For example, in processes dominated by γ-penguin diagrams, the µ + → e + γ rate is expected to be higher than µ + → e + e − e + or µ − N → e − N . However, the reverse is true if Z or Hpenguin diagrams or tree level Z or lepto-quark models dominate. There is an extensive and growing literature exploring these different scenarios, for example [9], which can be parameterised with a general effective operator approach, as used in Section 2. However, the conclusion is clear: exploring all three "golden" muon channels is essential. The current best limit on µ + → e + e − e + was set by the SINDRUM collaboration [10], excluding a branching ratio over 1.0 × 10 −12 at 90% confidence level (CL). With a twophase approach, Mu3e aims to extend the sensitivity to 10 −16 . Phase-1, currently under construction, will utilise the πE5 beam-line at PSI to study up to 10 8 muon (µ + ) stops per second and reach a sensitivity of 2 × 10 −15 on the branching fraction. An upgraded detector for Phase-2 plans to make use of the High-Intensity Muon Beam upgrades at PSI to study 2 × 10 9 muon stops per second and reach the target sensitivity of 10 −16 on the branching fraction. Mu3e uses an innovative detector design, with extremely low material budgets in order to minimise multiple scattering for low energy electrons. It consists of four layers of HV-MAPS silicon pixel sensors thinned to 50 µm, and scintillating fibre and tile detectors providing sub-ns timing resolution. The detector is surrounded by a solenoid magnet providing a 1 T field; electrons from muon decays can pass through the detector several times as they follow helical paths in this field. This significantly extends the lever-arm for measurement, and hence the momentum resolution. As well as the main µ + → e + e − e + search, Mu3e can search for signatures of the form µ + → e + X, where X can decay to e + e − promptly, after travelling some distance, or escaping the detector before decaying. This covers a range of ALP, familon, majoron and Z models. This White-paper will briefly cover the expected sensitivity of the various searches possible at Mu3e, along with an overview of the experimental design. 2 The search for µ + → e + e − e + To study the effects of different BSM scenarios on the decay kinematics, and therefore detector acceptance, we will consider the general effective Lagrangian proposed by Kuno and Okada [11]: L µ→eee = − 4G F √ 2 m µ A R µ R σ µν e L F µν + m µ A L µ L σ µν e R F µν + g 1 (µ R e L ) (e R e L ) + g 2 (µ L e R ) (e L e R ) + g 3 (µ R γ µ e R ) (e R γ µ e R ) + g 4 (µ L γ µ e L ) (e L γ µ e L ) + g 5 (µ R γ µ e R ) (e L γ µ e L ) + g 6 (µ L γ µ e L ) (e R γ µ e R ) + H.c. (2.1) where the form factors A R,L describe tensor-type (dipole) couplings, which contribute to µ + → e + e − e + primarily through higher-order photon penguin diagrams; the other terms describe tree-level contact interactions with g 1,2 describing scalar-type and g 3−6 vector-type interactions. The expected constraints on these effective operators have been studied in detail in, for example, [12]. Figure 1 (from [12]) shows the allowed regions in two planes, defined by Wilson coefficients C V RR ee (equivalent to g 3 in eq. 2.1), C SLL ee (equivalent to g 1 in eq. 2.1), and C D L (equivalent to A R in eq. 2.1). It is worth noting that, for the purely leptonic contacttype interactions (Fig. 1a), Mu3e is perhaps unsurprisingly the most sensitive. The other channels (µ + → e + γ and µ − N → e − N ) are also "blind" to certain regions of the parameter space due to cancellations, while this is never the case for µ + → e + e − e + . When considering the dipole-type interaction (Fig. 1b), the different channels are more complementary. When considering the flavour dependnce in the contact-type interactions, the muon conversion experiments set more strungent limits than µ + → e + e − e + or µ + → e + γ. The studies in [12] highlight the degeneracy between different terms in the effective Lagrangian in 2.1, and the need for inputs from all three muon decay channels in order to resolve such degeneracy in the event of any observation of a signal. The general effective Lagrangian in Eq. 2.1 can also be used to study signal kinematics and provides an important input in the design of the Mu3e experiment. Two important considerations are the acceptance, defined as the fraction of µ + → e + e − e + decays where all decay products have a transverse momentum p T above some minimal value p T,min ; and the energy distribution of the highest energy decay product in µ + → e + e − e + decays. Both distributions are shown in Fig. 2 (taken from [1]) and illustrate that maximising sensitivity requires the ability to reconstruct electrons from half the muon mass down to as low as possible in p T . The other key consideration in the µ + → e + e − e + search is backgrounds, which fall into two categories: physics and combinatorics. The primary physics background is internal conversion: µ + → e + e − e + ν e ν µ , which can be separated from the signal by the presence of two neutrinos. Combinatorics arise from the coincidence of one or more standard Michel decays with an e − arising primarily from Bhabha scattering or radiative decay (µ + → ν µ e + ν e γ) with subsequent photon conversion in the material of the detector. The three tracks in such cases typically do not have a common origin, and are not coincident in time. All backgrounds can be controlled using vertexing and kinematic requirements, selecting three tracks consistent with e + e − e + from a common origin, and with the reconstructed vertex momentum < 4 MeV and mass consistent with the muon mass (103 MeV< m eee < 110 MeV). The combinatoric background can be further reduced by coincident timing requirements, with time resolutions below 100 ps required to obtain a two orders of magnitude reduction. These physics requirements drive the design of the Mu3e experiment, which combines an extremely low material budget tracking detector with scintillating fibres and tiles for timing measurements. The Phase-1 Mu3e design is briefly reviewed below, with full details in [1]. In order to obtain optimal momentum and vertexing resolution on low energy (< m µ /2) electrons, Mu3e consists of four layers of High-Voltage Monolithic Active Pixel Sensors (HV-MAPS) surrounding the muon stopping target. These sensors have a pixel size of 80 × 80 µm 2 , and are thinned to 50 µm giving a thickness of X/X 0 = 0.1% per layer when including the full assembly. Between the second and third pixel layers, the scintillating fibre detector provides the first time measurement. The fibre detector consists of three layers of 250 µm diameter fibres, providing a time resolution of 250 ps with an overall thickness of X/X 0 = 0.2%. The entire detector sits in a wide-bore (1 m diameter) superconducting solenoid magnet providing a 1 T field. In this multiple-scattering dominated regime, optimal momentum resolution is obtained for tracks which "recurl" in the magnetic field and pass through the detector a second time. To increase the acceptance for such tracks, upstream and downstream recurl stations, consisting of two further layers of pixel sensors surrounding a layer of plastic scintillator tiles are installed. Individual tiles measure 6.3 × 6.2 × 5mm 3 , and are mounted in seven modules of 416 tiles each to provide full azimuthal coverage in each recurl station. The size of the tiles is chosen to provide improved time resolution (< 50 ps) compared to the fibre detector, with a thickness which ensures high (> 99%) efficiency. The additional scattering in the tiles is not relevant, as only the pixel hits before reaching the tiles are used in track fitting. Figure 3 shows the layout of the Phase-1 Mu3e detector. The high rate of coincidences means that simply triggering on three coincident tracks does not provide a sufficient reduction in data rate to storage. Instead, Mu3e will operate with a continuous, triggerless readout, with a fast online track reconstruction used to select events containing three tracks consistent with a common vertex. The online reconstruction is based on time-slices ("frames") of the full detector readout, with 4-hit tracks reconstructed in each frame on a GPU farm. The full offline tracking and signal selection, based on longer recurling (6 or 8 hit) tracks to optimise resolution, is then carried out on the stored frames. Figure 4a shows the expected vertex mass distribution based on the full offline selection. Figure 4b shows the expected evolution of sensitivity with running time; existing limits will be superseded within days, and the target sensitivity reached with around 400 days of data taking with a muon stopping rate of 1 × 10 8 s −1 . . Mu3e Phase-2 To reach the final target sensitivity of 10 −16 on the branching fraction for µ + → e + e − e + , a higher rate of muon stops is required. The High Intensity Muon Beam (HiMB) currently under study at PSI would deliver a stopping rate of 2 × 10 9 s −1 , but is not expected to be available before 2028. In order to deal with this higher stopping rate and resultant higher occupancy and rate of coincident backgrounds, upgrades to the Mu3e timing detectors are necessary, as well as possible improvements to the pixel sensors to improve time resolution, and extensions to the detector stations to increase the acceptance. Such upgrades are currently under study. 3 The search for µ + → e + X In addition to searches for µ + → e + e − e + decays, Mu3e can also investigate lepton-flavorviolating decays of the type µ + → e + X, where X denotes a neutral light particle that escapes the experiment undetected. An example for such a particle is the familon which arises as a pseudo-Goldstone boson from an additional broken flavour symmetry [13]. The current strongest limits on the branching ratio of µ + → e + X are set by the experiment by Jodidio et al. at TRIUMF [14] for massless X with B(µ + → e + X) < 2.6 × 10 −6 at 90% CL, as well as the TWIST experiment [15] for 13 MeV < m X < 80 MeV with B(µ + → e + X) < 9 × 10 −6 at 90% CL on average. The characteristic signature of µ + → e + X decays is a mono-energetic positron whose energy is determined by the mass m X of the undetected particle X. These positrons would appear as a narrow peak on top of the smooth momentum spectrum of positrons from Standard Model muon decays. In contrast to the µ + → e + e − e + search, the final state contains only a single positron and would thus not pass event filtering on the online GPU farm. Therefore, a dedicated search strategy is required, and the analysis is performed on histograms which are filled with track fit information as part of the online track reconstruction. Since the full track reconstruction of all tracks in every event frame is performed online, histograms of the total momentum as well as the azimuthal and polar angle of the decay electrons and positrons can be recorded. This unprecedented dataset of the order of 10 15 µ + decays allows not only for µ + → e + X searches but also for studies of Standard Model µ + decays. As a drawback of this approach, event-by-event information is lost and the offline reprocessing of track reconstruction is not possible so that the optimum momentum resolution of recurling tracks cannot be achieved. The acceptance of the Mu3e detector determines the mass reach of the search. With a minimum p T (e) of about 10 MeV for the positron to be reconstructed, m X of at most 95 MeV can be studied. It is worth noting that complementary experiments sensitive to higher m X up to the muon mass are currently being discussed [16]. Low m X might be out of reach as well, as the characteristic edge of the momentum spectrum of Michel decays µ + → e + νν is currently used for calibration. Alternative calibration methods based on Bhabha and Mott scattering are under investigation. The sensitivity in Phase 1 of the Mu3e experiment is estimated in toy Monte Carlo . Expected limits on the branching fraction B(µ + → e + X) at 90 % CL for various masses m X . The momentum calibration is either obtained from the same spectrum leaving out a window around the µ + → e + X signal (red line), or from another process such as Bhabha or Mott scattering (black line). The current strongest observed limits by the TWIST experiment [15] are shown for comparison (blue line). TWIST results by courtesy of R. Bayes. studies based on simulated momentum spectra. Signal and background events are simulated with the Geant4-based simulation of the Mu3e experiment. Standard Model muon decays form the background dominated by the Michel decay µ + → e + νν. Additional effects like tracks from Bhabha scattering or tracks which recurl multiple times are also considered as background contributions. Signal decays are simulated as two-body decays. The tracks are reconstructed from 4 hits as in the online track reconstruction. The reconstructed momentum spectra for background events and µ + → e + X signal events with an exemplary mass of m X = 60 MeV are shown in Figure 5. In Figure 6, the expected sensitivity of the Phase 1 Mu3e experiment is shown for two different calibration approaches. In the first scenario, the calibration is derived from the Michel spectrum. In this approach, a momentum window around the the expected µ + → e + X signal is left out during the calibration step. In the second scenario, it is assumed that the calibration is derived in an alternative way for example with Bhabha or Mott scattering events. In the first scenario, the sensitivity is deteriorated when the left out momentum window covers the Michel edge, i.e. at low m X . In the first phase of the Mu3e experiment, µ + → e + X decays with branching ratios in the order of 10 −8 can be tested, an improvement in sensitivity by a factor of around 600 with respect to the results by the TWIST experiment [15]. In addition to increased statistics in Phase 2, this search could further profit from improvements in momentum resolution. The feasibility of online reconstruction of long 6 or 8 hit tracks in view of the Phase 2 upgrade is currently being investigated. The search for µ + → e + + long-lived particles The Mu3e dataset will allow searches for muon decays to light pseudoscalar particles that are long-lived and decay within the first silicon layer. An exploratory analysis documented in Ref. [17] has shown that such a final state in Mu3e has competitive sensitivity to other experiments. This analysis, however, has not used a detailed simulation of the Mu3e detector. In this section, this study is repeated using the latest Mu3e detector simulation and realistic track and vertex selection criteria. In Ref. [17] axion-like particles, a, are considered with couplings to the SM leptons α with α = e, µ, τ parameterized as: L a = 1 Λ ∂ µ a¯ α γ µ (g V αβ + g A αβ γ 5 ) β , (4.1) where Λ is an effective energy scale and g V , g A are the current structure matrices. From this interaction term, the decay width of the muon decay µ → ea and, hence, its branching ratio, is found to be proportional to m 3 µ /(16πΛ 2 µe ) where Λ eµ = Λ/ (g V eµ ) 2 + (g A eµ ) 2 with the proportionality factor being a function of m a and m e under the approximation that m µ m e . The current structure matrices are assumed that they are such that the axionlike particle a decays only to ee with a lifetime that is calculated from the width Γ(a → ee) and is found to be proportional to the particle masses and Λ ee = Λ/g A ee . Therefore, the free parameters for this model are (i) the mass of the axion-like particle m a , (ii) Λ eµ that controls the branching ratio of the decay, and (iii) Λ ee that controls the lifetime of a and hence the location of its decay vertex inside the detector. Muon decays to axion-like particles a are generated using 1 MadGraph5 aMC@NLO [18] and the resulting decay products are reconstructed using the Mu3e detector simulation. The same selection criteria for tracks and vertices are used as in the standard µ → eee signal search, described in Ref. [1]. In this particular decay, the muon is assumed to stop on the target and to decay backto-back to an electron and an axion-like particle a. The a particle will have a given boost and will decay depending on the assumed lifetime to two electrons. Due to the topology of the decay, the electron from the muon decay has a track that, if extrapolated, will meet the a particle's decay vertex. Therefore, for as long as there is no requirement for the reconstructed vertices to be close to the target surface, the Mu3e vertex reconstruction should be very efficient. The distance of the reconstructed vertex of the eee system from the actual a particle decay vertex for long tracks selected for this analysis is shown in Fig. 7a. This distribution is compatible with the vertex resolution performance for µ → eee decays. The efficiency on signal of various m a and lifetime values, as well as the efficiency for µ → eee decays for comparison, is shown in Fig. 7b. No requirement on the distance of the vertices from the target is applied for these results. The efficiencies shown in Fig. 7b are interpreted in the m a -Λ ee plane for different values of Λ eµ assuming the Mu3e Phase 1 expected muon decays on target as a 90% confidence (b) Figure 7. The distance between the reconstructed vertex of the e + e + e − system using the standard Mu3e experiment vertex reconstruction algorithm and the true vertex of the axion-like particle is shown in (a). This is for an axion-like particle of mass 40 MeV and lifetimes of 0.01 ns, 0.1 ns and 1 ns. The efficiency of the standard Mu3e experiment signal selection for axion-like particle masses in the range 10-90 MeV and for a range of lifetimes is shown in (b). level upper limit assuming no signal is found in a zero background experiment. This is shown in Fig. 4. On the same figure, exclusions from beam dump experiments and electron g − 2 reproduced from Ref. [17] are also shown. The whole available region of the m a -Λ ee parameter space between the existing constraints and the muon mass kinematic cutoff is excluded for Λ eµ < 5 × 10 13 GeV. To reduce combinatoric backgrounds, a requirement on the distance between the vertex and target my be required. The impact of a ealistic value of 3 mm (suggested in Ref. [19]) has been studied and the outcome is that such a requirement has a small effect on the sensitivity and does not impair the sensitivity of the analysis to the axion parameter space. 5 The search for e + e − -resonances in µ + → e + e − e + ν e ν µ In addition to µ + → e + e − e + searches and searches for long-lived particles, the dataset that will be recorded by the Mu3e experiment also allows for searches for resonances in µ + → e + e − e + ν e ν µ . One example that has been further studied are dark photon A decays into e + e − pairs. The dark photon is the messenger of a potential vector portal to the dark sector. It interacts with Standard Model particles via kinetic mixing with the photon and Z boson, i.e. via coupling to the electro-magnetic current. If the dark photon is light enough, it can be radiated in muon decays: µ + → A e + ν e ν µ . In the following, the Lagrangian from [20] is used with the parameter determining the strength of the kinetic mixing. In this study, the sensitivity of the Mu3e experiment to promptly decaying dark photons A → e + e − emitted in muon decays is estimated with a simplified detector simulation and without considering background from Bhabha scattering events. In the study presented in the following, the full Mu3e detector simulation and event reconstruction are used. Signal events are generated with MadGraph5 aMC@NLO 2.4.3 [18] up to dark photon masses of m A = 80 MeV. At larger masses, the parameter space in reach of Mu3e is already excluded by existing experimental limits. The simulated dark photons decay promptly to e + e − so that the same track and vertex reconstruction as for the µ + → e + e − e + search can be employed. The background is dominated by the rare muon decay µ + → e + e − e + ν e ν µ . As additional background sources, accidental combinations of Bhabha scattering events with Michel decays are considered which contribute on average by a factor of 800 less than the rare muon decay. Further types of accidental background are even less likely and therefore not considered in this study. Spectra of the reconstructed invariant e + e − -mass m ee for background and signal are shown in Figure 9. The signal distribution features a narrow peak on top of a broader distribution which stems from e + e − -combination of the positron from the muon decay and the electron from the A decay. This contribution can be reduced by selecting the e + e −combination with the lower m ee for lighter A , and the combination with the higher m ee for heavier A . The optimum transition point lies around 45 MeV. The sensitivity to prompt dark photon decays in muon decays in the Phase 1 Mu3e experiment is estimated with toy Monte Carlo studies. At low m A , branching fractions of 5 × 10 −9 at 90% CL can be investigated, while at higher m A , branching fraction of 3 × 10 −12 at 90% CL can be reached (see Figure 10a). The expected limits on the branching fraction are translated into limits on the kinetic mixing parameter . The reach in the − m A parameter of the Phase 1 Mu3e experiment is shown alongside observed limits is shown in Figure 10b. Conclusion The Mu3e experiment will search for the lepton-number-violating decay µ + → e + e − e + . Using a two-phase approach and innovative design, it will extend the sensitivity to the branching ratio for this process to 10 −16 , four orders of magnitude beyond current limits. In the context of the global programme covering µ + → e + e − e + , µ + → e + γ and µ − N → e − N , Mu3e provides unique and complementary sensitivity to flavour-violating new physics. The Mu3e experimental design also allows the search for a range of other processes, including dark photons, axion-like particles and long-lived particles. In each case, Mu3e can extend existing sensitivities in the accessible range of parameter space for such models. Figure 1 . 1Allowed regions (given at the scale m W ) in the 1a C V RR ee -C SLL ee plane, and the 1b C V RR ee -C D L plane. Existing (solid lines) and projected (dashed lines) are shown for µ + → e + γ (green), µ + → e + e − e + (red) and µ − N → e − N (blue). Figures from[12]. Figure 2 . 2A,g3,g4, no intf.) dipole&vector (A,g3,g4, constr. intf.) dipole&vector (A,g3,g4, destr. intf.) dipole&vector (A,g5,g6, no intf.) dipole&vector (A,g5,g6, constr. intf.) dipole&vector (A,g5,g6, destr. intf.) 2a the acceptance, defined as the fraction of µ + → e + e − e + decays where all decay products have p T greater than p T,min . 2b the energy distribution of the highest energy decay product in µ + → e + e − e + decays. Both are shown for the range of effective operators defined by 2.1. Figures from[1]. Figure 3 . 3The layout of the Phase-1 Mu3e detector. Figures from[1]. Figure 4 . 44a Simulation of the reconstructed vertex mass showing backgrounds and possible signal contributions. 4b the evolution of the Mu3e Phase 1 signal sensitivity with time. Figures from [1] Simulated µ + → e + X signal events with mX = 60 MeV. Figure 5 . 5Spectra of the reconstructed positron momentum of simulated background events from Standard Model muon decays (background) and µ + → e + X events with m X = 60 MeV. The tracks are reconstructed from 4 hits in the central detector like it is done on the GPU filter farm. Figure 6 6Figure 6. Expected limits on the branching fraction B(µ + → e + X) at 90 % CL for various masses m X . The momentum calibration is either obtained from the same spectrum leaving out a window around the µ + → e + X signal (red line), or from another process such as Bhabha or Mott scattering (black line). The current strongest observed limits by the TWIST experiment [15] are shown for comparison (blue line). TWIST results by courtesy of R. Bayes. Figure 8 . 8Upper limits in the two dimensional parameter space of the simple axion-like particle model discussed in the text. Simulated background events. Both combinations of e + e − are considered. Simulated A signal events. Both combinations of e + e − are considered. Simulated A events. The e + e − -combination with the lower invariant mass is shown. Simulated A signal events.The e + e −combination with the higher invariant mass is shown. Figure 9 . 9Spectra of the reconstructed invariant mass m ee of simulated background and dark photon signal events with m A = 20 MeV, 45 MeV and 70 MeV. limits on the branching fraction at 90% CL. limits on the kinetic mixing parameter at 90%CL. Adapted from[21]. Figure 10 . 10Expected sensitivity to prompt dark photon decays in µ + → e + e − e + ν e ν µ in the first phase of the Mu3e experiment. The authors would like to thank Andrea Thamm for providing help with the MadGraph5 aMC@NLO cards for this study. Technical design of the phase I Mu3e experiment. K Arndt, Mu3e collaboration10.1016/j.nima.2021.165679Nucl. Instrum. Meth. A. 10141656792009.11690Mu3e collaboration, K. Arndt et al., Technical design of the phase I Mu3e experiment, Nucl. Instrum. Meth. A 1014 (2021) 165679, [2009.11690]. Search for the lepton flavour violating decay µ + → e + γ with the full dataset of the MEG experiment. A M Baldini, MEG collaboration10.1140/epjc/s10052-016-4271-x1605.05081Eur. Phys. J. C. 76434MEG collaboration, A. M. Baldini et al., Search for the lepton flavour violating decay µ + → e + γ with the full dataset of the MEG experiment, Eur. Phys. J. C 76 (2016) 434, [1605.05081]. The design of the MEG II experiment. A M Baldini, MEG II collaboration10.1140/epjc/s10052-018-5845-61801.04688Eur. Phys. J. C. 78380MEG II collaboration, A. M. Baldini et al., The design of the MEG II experiment, Eur. Phys. J. C 78 (2018) 380, [1801.04688]. . L Bartoszek, Mu2e collaboration1501.05241Technical Design ReportMu2e collaboration, L. Bartoszek et al., Mu2e Technical Design Report, 1501.05241. COMET Phase-I Technical Design Report. R Abramishvili, COMET collaboration10.1093/ptep/ptz1251812.09018PTEP. 2020COMET collaboration, R. Abramishvili et al., COMET Phase-I Technical Design Report, PTEP 2020 (2020) 033C01, [1812.09018]. Status of the DeeMe Experiment, an Experimental Search for µ-e Conversion at J-PARC MLF. N Teshima, 10.22323/1.369.00821911.07143PoS. 201982N. Teshima, Status of the DeeMe Experiment, an Experimental Search for µ-e Conversion at J-PARC MLF, PoS NuFact2019 (2020) 082, [1911.07143]. The processes µ → eγ, µ → e − e − e + , neutrino' → neutrino γ in the weinberg-salam model with neutrino mixing. S Petcov, Soviet Journal of Nuclear Physics. 25340S. Petcov, The processes µ → eγ, µ → e − e − e + , neutrino' → neutrino γ in the weinberg-salam model with neutrino mixing, Soviet Journal of Nuclear Physics 25 (1977) 340. τ → µµµ at a rate of one out of 10 14 tau decays?. P Blackstone, M Fael, E Passemar, 10.1140/epjc/s10052-020-8059-71912.09862Eur. Phys. J. C. 80506P. Blackstone, M. Fael and E. Passemar, τ → µµµ at a rate of one out of 10 14 tau decays?, Eur. Phys. J. C 80 (2020) 506, [1912.09862]. Charged Lepton Flavour Violation: An Experimental and Theoretical Introduction. L Calibbi, G Signorelli, 10.1393/ncr/i2018-10144-01709.00294Riv. Nuovo Cim. 41L. Calibbi and G. Signorelli, Charged Lepton Flavour Violation: An Experimental and Theoretical Introduction, Riv. Nuovo Cim. 41 (2018) 71-174, [1709.00294]. Search for the Decay µ + → e + e + e −. U Bellgardt, SINDRUM collaboration10.1016/0550-3213(88)90462-2Nucl. Phys. B. 299SINDRUM collaboration, U. Bellgardt et al., Search for the Decay µ + → e + e + e − , Nucl. Phys. B 299 (1988) 1-6. Y Kuno, Y Okada, 10.1103/RevModPhys.73.151hep-ph/9909265Muon decay and physics beyond the standard model. 73Y. Kuno and Y. Okada, Muon decay and physics beyond the standard model, Rev. Mod. Phys. 73 (2001) 151-202, [hep-ph/9909265]. Renormalisation-group improved analysis of µ → e processes in a systematic effective-field-theory approach. A Crivellin, S Davidson, G M Pruna, A Signer, 10.1007/JHEP05(2017)1171702.03020JHEP. 05117A. Crivellin, S. Davidson, G. M. Pruna and A. Signer, Renormalisation-group improved analysis of µ → e processes in a systematic effective-field-theory approach, JHEP 05 (2017) 117, [1702.03020]. Axions and Family Symmetry Breaking. F Wilczek, 10.1103/PhysRevLett.49.1549Phys. Rev. Lett. 49F. Wilczek, Axions and Family Symmetry Breaking, Phys. Rev. Lett. 49 (1982) 1549-1552. Search for Right-Handed Currents in Muon Decay. A Jodidio, 10.1103/PhysRevD.34.1967,10.1103/PhysRevD.37.237Phys. Rev. 34A. Jodidio et al., Search for Right-Handed Currents in Muon Decay, Phys. Rev. D34 (1986) 1967. Search for two body muon decay signals. R Bayes, TWIST collaboration10.1103/PhysRevD.91.0520201409.0638Phys. Rev. 9152020TWIST collaboration, R. Bayes et al., Search for two body muon decay signals, Phys. Rev. D91 (2015) 052020, [1409.0638]. Muχe -A Search for Familons in Muon Decay Using HPGe Detectors. D Koltick, S Huang, F Bergin, J Chen, H Cao, 10D. Koltick, S. Huang, F. Bergin, J. Chen and H. Cao, Muχe -A Search for Familons in Muon Decay Using HPGe Detectors, 10, 2021, 2110.02164. Lepton flavor violation with displaced vertices. J Heeck, W Rodejohann, 10.1016/j.physletb.2017.11.0671710.02062Phys. Lett. B. 776J. Heeck and W. Rodejohann, Lepton flavor violation with displaced vertices, Phys. Lett. B 776 (2018) 385-390, [1710.02062]. The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. J Alwall, R Frederix, S Frixione, V Hirschi, F Maltoni, O Mattelaer, 10.1007/JHEP07(2014)0791405.0301JHEP. 0779J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07 (2014) 079, [1405.0301]. A.-K Perrevoort, 10.11588/heidok.00024585Sensitivity Studies on New Physics in the Mu3e Experiment and Development of Firmware for the Front-End of the Mu3e Pixel Detector. Ph.D. thesisA.-K. Perrevoort, Sensitivity Studies on New Physics in the Mu3e Experiment and Development of Firmware for the Front-End of the Mu3e Pixel Detector, Ph.D. thesis, U. Heidelberg (main), 2018. 10.11588/heidok.00024585. Projections for Dark Photon Searches at Mu3e. B Echenard, R Essig, Y.-M Zhong, 10.1007/JHEP01(2015)1131411.1770JHEP. 01113B. Echenard, R. Essig and Y.-M. Zhong, Projections for Dark Photon Searches at Mu3e, JHEP 01 (2015) 113, [1411.1770]. Dark Photon Search in the Mass Range Between 1.5 and 3.4 GeV/c 2. M Ablikim, BESIII collaboration10.1016/j.physletb.2017.09.067Phys. Lett. 7741705.04265BESIII collaboration, M. Ablikim et al., Dark Photon Search in the Mass Range Between 1.5 and 3.4 GeV/c 2 , Phys. Lett. B774 (2017) 252-257, [1705.04265].
[]
[ "Fourier Analysis of the Parametric Resonance of the Neutrino Oscillation in the Presence of Inhomogeneous Matter", "Fourier Analysis of the Parametric Resonance of the Neutrino Oscillation in the Presence of Inhomogeneous Matter" ]
[ "Joe Sato \nPhysics Department\nSaitama University\n255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan\n", "Masafumi Koike \nPhysics Department\nSaitama University\n255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan\n", "Toshihiko Ota ‡email:[email protected]§ \nInstitüt für Theoretische Physik und Astrophysik\nUniversität Würzburg\n97074WürzburgGermany\n", "Masako Saito [email protected] \nPhysics Department\nSaitama University\n255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan\n" ]
[ "Physics Department\nSaitama University\n255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan", "Physics Department\nSaitama University\n255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan", "Institüt für Theoretische Physik und Astrophysik\nUniversität Würzburg\n97074WürzburgGermany", "Physics Department\nSaitama University\n255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan" ]
[]
We study the parametric resonance of the neutrino oscillation through the matter whose density varies spatially. The Fourier analysis of the matter effect enables us to clarify the parametric resonance condition, which is summarized in a frequency matching between the neutrino oscillation and the spatial variation of the matter density. As a result, the n-th Fourier mode of a matter density profile modifies the energy spectrum of the νµ → νe appearance probability at around the n-th dip.
10.22323/1.074.0140
[ "https://arxiv.org/pdf/0810.3104v1.pdf" ]
18,153,673
0810.3104
865bf251f137ad7356958db493153de1fa52d85c
Fourier Analysis of the Parametric Resonance of the Neutrino Oscillation in the Presence of Inhomogeneous Matter 17 Oct 2008 Joe Sato Physics Department Saitama University 255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan Masafumi Koike Physics Department Saitama University 255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan Toshihiko Ota ‡email:[email protected]§ Institüt für Theoretische Physik und Astrophysik Universität Würzburg 97074WürzburgGermany Masako Saito [email protected] Physics Department Saitama University 255 Shimo-Okubo, Sakura-ku338-8570SaitamaJapan Fourier Analysis of the Parametric Resonance of the Neutrino Oscillation in the Presence of Inhomogeneous Matter 17 Oct 2008 We study the parametric resonance of the neutrino oscillation through the matter whose density varies spatially. The Fourier analysis of the matter effect enables us to clarify the parametric resonance condition, which is summarized in a frequency matching between the neutrino oscillation and the spatial variation of the matter density. As a result, the n-th Fourier mode of a matter density profile modifies the energy spectrum of the νµ → νe appearance probability at around the n-th dip. The oscillation probability of neutrinos passing through the interior of the Earth is strongly affected by the matter on the baseline. In such cases, the spatial variation of the matter density can make the probability quite different from that under the constant matter density [1,2,3]. The possibility of the parametric resonance of the neutrino oscillation with the matter density has been discussed [1]. We exploit the Fourier analysis of the matter density profile to study the effect of parametric resonance [3,4]. Generally, the parametric resonance of an oscillation emerges when an oscillation parameter changes with a resonant frequency, which is typically twice the natural frequency of the system. Our formulation based on the Fourier analysis gives a clear view of this frequency matching, which is essential to the parametric resonance. We carry out our analysis in a two-flavor framework. This simplification does not spoil our point as can be seen in Fig. 1, in which we compare the ν µ → ν e appearance probability in the three-and two-flavor calculation, taking the baseline length L = 12000 km. The three-flavor result assumes the density profile of the Preliminary Reference Earth Model (solid curve), while two-flavor result uses the constant density case (dotted curve) and the cosine variation around the average density (broken curve). Renormalized by the factor of sin 2 2θ 23 , the two-flavor result with the cosine profile shows the similar feature as the three-flavor one. The evolution equation of two flavors of neutrinos ν e and ν µ in the matter density ρ(x) is i d dx ν e (x) ν µ (x) = 1 2E δm 2 2 − cos 2θ sin 2θ sin 2θ cos 2θ + a(x) 0 0 0 ν e (x) ν µ (x) ,(1) where δm 2 , θ, and E are the mass-square difference, the mixing angle, and the energy of neutrinos, respectively. The matter effect a(x) is given by ξ = x/L, we obtain from Eq. (1) a(x) = 2 √ 2EG F N A Y e ρ(x), where G F is the Fermi constant, N A is the Avogadro number, and Y e is the proton-to-nucleon ratio. Defining ∆ ≡ δm 2 L/2E, ∆ m (x) ≡ a(x)L/2E, and z(x) = ν e (x) exp[ x 0 ds i∆ m (s)/2],z ′′ (ξ) + 1 4 ∆ m0 − ∆ cos 2θ 2 + ∆ 2 sin 2 2θ + 2(∆ m0 − ∆ cos 2θ) ∞ n=1 ∆ mn cos 2nπξ + ∞ n=1 ∆ mn cos 2nπξ 2 + 4nπi ∞ n=1 ∆ mn sin 2nπξ z(ξ) = 0 .(2) The parametric resonance can arise under the presence of the density profile. Suppose that only the n-th Fourier mode of the density profile is present so that ∆ m (ξ) = ∆ m0 + ∆ mn cos 2nπξ. The parametric resonance is expected when the frequency of the density profile 2nπ is twice the natural frequency of the system ω 0 ≡ (∆ m0 − ∆ cos 2θ) 2 + ∆ 2 sin 2 2θ − 1 2 ∞ n=1 ∆ 2 mn /2 , i.e. ω 0 = 2nπ/2 = nπ. This frequency matching condition amounts to E = E (res) n ≡ δm 2 L 2 1 ∆ m0 cos 2θ ± 4n 2 π 2 − ∆ 2 m0 sin 2 2θ − 1 2 ∞ n=1 ∆ 2 mn .(3) We show in Fig. 2 the ν µ → ν e oscillation probability for various values (A) of ρ 1 and (B) of ρ 2 . Figure 2 indicates that the n-th Fourier mode of the matter density modifies the oscillation probability around the n-th dip: ω 0 = nπ, which corresponds to the parametric resonance energy Eq. (3). This modification is understood by the parametric resonance as shown elsewhere [5]. In summary, we studied the parametric resonance of the neutrino oscillation through the matter whose density varies spatially, analyzing the matter effect by the Fourier expansion. We have shown that the n-th Fourier mode of the density profile modifies the oscillation probability around the parametric resonance energy E (res) n defined in Eq. (3). This condition is understood in terms of the frequency matching between the natural frequency of the system and that of the density profile. Figure 1 :Figure 2 : 12Oscillation probability from ν µ to ν e for the baseline length of 12000 km. The solid curve (A) is evaluated by three-flavor analysis with the matter density profile taken from the PREM. The dotted (B) and broken (C) curves are results of the two flavor calculation with (B) constant density with ρ 0 = 7.58 g/cm 3 , and (C) cosine profile ρ(x) = ρ 0 + ρ 1 cos 2πx/L with ρ 1 = −4.32 g/cm 3 . Values of other parameters taken are described in the figure. The ν µ → ν e oscillation probabilities with ρ n = 0, −0.5, and −1.0 g/cm 2 , where (A) n = 1 and (B) n = 2. The former includes ρ 1 = −5.0 g/cm 2 case also. Values of other parameters are shown in the graphs. and introducing an Fourier expansion ∆ m (ξ) = n=0 ∆ mn cos 2nπξ with * email: [email protected] † email: [email protected] ‡ email: [email protected] § email: [email protected]∞ . V K Ermilova, Kr. Soob. Fiz. 526Short Notices of the Lebedev InstituteV. K. Ermilova et al., Kr. Soob. Fiz. [Short Notices of the Lebedev Institute] 5 (1986) 26; . E K Akhmedov, Sov. J. Nucl. Phys. 47301Yad. Fiz.E. K. Akhmedov, Sov. J. Nucl. Phys. 47 (1988) 301 [Yad. Fiz. 47 (1988) 475]; . P I Krastev, A Y Smirnov, Phys. Lett. B. 226341P. I. Krastev and A. Y. Smirnov, Phys. Lett. B 226 (1989) 341; . Q Y Liu, A Y Smirnov, Nucl. Phys. B. 524505Q. Y. Liu and A. Y. Smirnov, Nucl. Phys. B 524 (1998) 505; . Q Y Liu, S P Mikheyev, A Y Smirnov, Phys. Lett. B. 440319Q. Y. Liu, S. P. Mikheyev and A. Y. Smirnov, Phys. Lett. B 440 (1998) 319; . E K Akhmedov, Nucl. Phys. B. 53825E. K. Akhmedov, Nucl. Phys. B 538 (1999) 25; . E K Akhmedov, A Dighe, P Lipari, A Y Smirnov, Nucl. Phys. B. 5423E. K. Akhmedov, A. Dighe, P. Lipari and A. Y. Smirnov, Nucl. Phys. B 542 (1999) 3; . E K Akhmedov, M Maltoni, A Y Smirnov, Phys. Rev. Lett. 95211801E. K. Akhmedov, M. Maltoni and A. Y. Smirnov, Phys. Rev. Lett. 95 (2005) 211801. . S T Petcov, Phys. Lett. B. 434321S. T. Petcov, Phys. Lett. B 434 (1998) 321; . M V Chizhov, S T Petcov, Phys. Rev. Lett. 831096M. V. Chizhov and S. T. Petcov, Phys. Rev. Lett. 83 (1999) 1096; . M V Chizhov, S T Petcov, Phys. Rev. D. 6373003M. V. Chizhov and S. T. Petcov, Phys. Rev. D 63 (2001) 073003. . T Ota, J Sato, Phys. Rev. D. 6393004T. Ota and J. Sato, Phys. Rev. D 63 (2001) 093004; . T Ota, J Sato, Phys. Rev. D. 6753003T. Ota and J. Sato, Phys. Rev. D 67 (2003) 053003. . M Koike, J Sato, Mod. Phys. Lett. A. 141297M. Koike and J. Sato, Mod. Phys. Lett. A 14 (1999) 1297. . M Koike, in preparationM. Koike et al., in preparation.
[]
[ "Global dynamics of coupled standard maps", "Global dynamics of coupled standard maps" ]
[ "T Manos \nCenter for Research and Applications of Nonlinear Systems (CRANS)\nDepartment of Mathematics\nUniversity of Patras\nGR-26500Greece\n\nObservatoire Astronomique de Marseille-Provence (OAMP)\n2 Place Le Verrier, Cédex 04F-13248MarseilleFrance\n", "Ch Skokos \nAstronomie et Systèmes Dynamiques\nIMCCE, Observatoire de Paris\n77 Av. Denfert-Rochereau, F-75014ParisFrance\n", "T Bountis [email protected]@imcce.fr \nCenter for Research and Applications of Nonlinear Systems (CRANS)\nDepartment of Mathematics\nUniversity of Patras\nGR-26500Greece\n" ]
[ "Center for Research and Applications of Nonlinear Systems (CRANS)\nDepartment of Mathematics\nUniversity of Patras\nGR-26500Greece", "Observatoire Astronomique de Marseille-Provence (OAMP)\n2 Place Le Verrier, Cédex 04F-13248MarseilleFrance", "Astronomie et Systèmes Dynamiques\nIMCCE, Observatoire de Paris\n77 Av. Denfert-Rochereau, F-75014ParisFrance", "Center for Research and Applications of Nonlinear Systems (CRANS)\nDepartment of Mathematics\nUniversity of Patras\nGR-26500Greece" ]
[]
Understanding the dynamics of multi-dimensional conservative dynamical systems (Hamiltonian flows or symplectic maps) is a fundamental issue of nonlinear science. The Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems [1], is an ideal tool for this purpose. In the present paper we make a first step towards the dynamical study of multi-dimensional maps, by obtaining some interesting results for a 4-dimensional (4D) symplectic map consisting of N = 2 coupled standard maps [2]. In particular, using the new GALI3 and GALI4 indices, we compute the percentages of regular and chaotic motion of the map equally reliably but much faster than previously used indices, like GALI2 (known in the literature as SALI).
10.1007/978-3-540-75826-6_38
[ "https://arxiv.org/pdf/0801.3923v1.pdf" ]
18,256,316
0801.3923
fee100eddaea6730a9e2b5e0f656fd71855cd09c
Global dynamics of coupled standard maps 25 Jan 2008 T Manos Center for Research and Applications of Nonlinear Systems (CRANS) Department of Mathematics University of Patras GR-26500Greece Observatoire Astronomique de Marseille-Provence (OAMP) 2 Place Le Verrier, Cédex 04F-13248MarseilleFrance Ch Skokos Astronomie et Systèmes Dynamiques IMCCE, Observatoire de Paris 77 Av. Denfert-Rochereau, F-75014ParisFrance T Bountis [email protected]@imcce.fr Center for Research and Applications of Nonlinear Systems (CRANS) Department of Mathematics University of Patras GR-26500Greece Global dynamics of coupled standard maps 25 Jan 2008 Understanding the dynamics of multi-dimensional conservative dynamical systems (Hamiltonian flows or symplectic maps) is a fundamental issue of nonlinear science. The Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems [1], is an ideal tool for this purpose. In the present paper we make a first step towards the dynamical study of multi-dimensional maps, by obtaining some interesting results for a 4-dimensional (4D) symplectic map consisting of N = 2 coupled standard maps [2]. In particular, using the new GALI3 and GALI4 indices, we compute the percentages of regular and chaotic motion of the map equally reliably but much faster than previously used indices, like GALI2 (known in the literature as SALI). Summary. Understanding the dynamics of multi-dimensional conservative dynamical systems (Hamiltonian flows or symplectic maps) is a fundamental issue of nonlinear science. The Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems [1], is an ideal tool for this purpose. In the present paper we make a first step towards the dynamical study of multi-dimensional maps, by obtaining some interesting results for a 4-dimensional (4D) symplectic map consisting of N = 2 coupled standard maps [2]. In particular, using the new GALI3 and GALI4 indices, we compute the percentages of regular and chaotic motion of the map equally reliably but much faster than previously used indices, like GALI2 (known in the literature as SALI). Definition and behavior of GALI Let us first briefly recall the definition of GALI and its behavior for regular and chaotic motion, adjusting the results obtained in [1] to symplectic maps. Considering of a 2N -dimensional map, we follow the evolution of an orbit (using the equations of the map) together with k initially linearly independent deviation vectors of this orbit − → ν 1 , − → ν 2 , ..., − → ν k with 2 ≤ k ≤ 2N (using the equations of the tangent map). The Generalized Alignment Index of order k is defined as the norm of the wedge or exterior product of the k unit deviation vectors: GALI k (i) = ν 1 (i) ∧ν 2 (i) ∧ ... ∧ν k (i)(1) and corresponds to the volume of the generalized parallelepiped, whose edges are these k vectors. We note that the hat (ˆ) over a vector denotes that it is of unit magnitude and that i is the discrete time. In the case of a chaotic orbit all deviation vectors tend to become linearly dependent, aligning in the direction of the eigenvector which corresponds to the maximal Lyapunov exponent and GALI k tends to zero following an exponential law ∼ e −[(σ1−σ2)+(σ1−σ3)+...+(σ1−σ k )]i , where σ 1 , . . . , σ k are approx- imations of the first k largest Lyapunov exponents. In the case of regular motion on the other hand, all deviation vectors tend to fall on the N -dimensional tangent space of the torus on which the motion lies. Thus, if we start with k ≤ N general deviation vectors they will remain linearly independent on the N -dimensional tangent space of the torus, since there is no particular reason for them to become aligned. As a consequence GALI k remains practically constant for k ≤ N . On the other hand, GALI k tends to zero for k > N , since some deviation vectors will eventually become linearly dependent, following a particular power law, i. e. GALI k (i) ∼ i 2(N −k) . Dynamical study of a 4D standard map As a model for our study we consider the 4D symplectic map: x ′ 1 = x 1 + x ′ 2 x ′ 2 = x 2 + K 2π sin(2πx 1 ) − B 2π sin[2π(x 3 − x 1 )] x ′ 3 = x 3 + x ′ 4 x ′ 4 = x 4 + K 2π sin(2πx 3 ) − B 2π sin[2π(x 1 − x 3 )] (mod 1), (2) which consists of two coupled standard maps [2] and is a typical nonlinear system, in which regions of chaotic and regular dynamics are found to coexist. In our study we fix the parameters of the map (2) to K = 0.5 and B = 0.05. In Fig. 1, we show the behavior of GALIs for two different orbits: a regular orbit R with initial conditions (x 1 , x 2 , x 3 , x 4 ) = (0.55, 0.10, 0.54, 0.01) (Fig. 1a), and a chaotic orbit C with initial conditions (x 1 , x 2 , x 3 , x 4 ) = (0.55, 0.10, 0.005, 0.01) (Fig. 1b). The positive Lyapunov exponents of orbit C were found to be σ 1 ≈ 0.070, σ 2 ≈ 0.008. From the results of Fig. 1 we see that the evolution of GALIs is described very well by the theoretically obtained approximations presented in Sect. 1. Let us now turn our attention to the study of the global dynamics of map (2). From the results Fig. 1 we conclude that in the case of 4D maps, GALI 2 has different behavior for regular and chaotic orbits. In particular, GALI 2 tends exponentially to zero for chaotic orbits (GALI 2 ∼ e −(σ1−σ2)i ) while it fluctuates around non-zero values for regular orbits. This difference in the behavior of the index can be used to obtain a clear distinction between regular and chaotic orbits. Let us illustrate this by following up to i = 4000 iterations, all orbits whose initial conditions lie on a 2-dimensional grid of 500 × 500 equally spaced points on the subspace x 3 = 0.54, x 4 = 0.01, of the 4-dimensional phase space of the map (2), attributing to each grid point a color according to the value of GALI 2 at the end of the evolution. If GALI 2 of an orbit becomes less than 10 −10 for i < 4000 the evolution of the orbit is stopped, its GALI 2 value is registered and the orbit is characterized as chaotic. The outcome of this experiment is presented in the left panel of Fig. 2. But also GALI 4 can be used for discriminating regular and chaotic motion. From the theoretical predictions for the evolution of GALI 4 , we see that after i = 1000 iterations the value of GALI 4 of a regular orbit should become of the order of 10 −16 , since GALI 4 ∼ t −4 , although the results of Fig. 1 show that more iterations are needed for this threshold to be reached, due to an initial transient time where GALI 4 does not decrease significantly. On the other hand, for a chaotic orbit GALI 4 has already reached extremely small values at i = 1000 due to its exponential decay (GALI 4 ∼ e −4σ1i ). Thus, the global dynamics of the system can be revealed as follows: we follow the evolution of the same orbits as in the case of GALI 2 and register for each orbit the value of GALI 4 after i = 1000 iterations. All orbits having values of GALI 4 significantly smaller than 10 −16 are characterized as chaotic, while all others are considered as non-chaotic. In the right panel of Fig. 2 we present the outcome of this procedure. From the results of Fig. 2, we see that both procedures, using GALI 2 or GALI 4 as a chaos indicator, give the same result for the global dynamics of the system, since in both cases 16% of the orbits are characterized as chaotic. These orbits correspond to the black colored areas n both panels of Fig. 2. One important difference between the two procedures is their computational efficiency. Even though GALI 4 requires the computation of four deviation vectors, instead of only two that are needed for the evaluation of GALI 2 , using GALI 4 we were able to get a clear dynamical 'chart', not only for less iterations of the map (1000 instead of 4000 needed for GALI 2 ), but also in less CPU time. In particular, for the computation of the data of the left panel of Fig. 2 (using GALI 2 ) we needed 1 hour of CPU time on an Athlon 64bit, 3.2GHz PC, while for the data of the left panel of the same figure (using GALI 4 ) only 14 minutes of CPU time were needed. Using the above-described method, both for GALI 2 and GALI 4 , we were able to compute very fast and accurately the percentages of regular motion for several values of parameter B. In Fig. 3 we plot the percentage of regular orbits for B ∈ [0, 2.5] where B varies with a step δB = 0.1. We see that the two curves practically coincide, but using GALI 2 we needed almost four times more CPU time. So, it becomes evident that a well-tailored application of GALI k , with 2 < k, can significantly diminish the CPU time required for the detailed 'charting' of phase space regions, compared with that for GALI 2 . Fig. 1 . 1The evolution of GALI k , k = 2, 3, 4, with respect to the number of iterations i for a) the regular orbit R and b) the chaotic orbit C. The plotted lines correspond to functions proportional to n −2 , n −4 in a) and to e −(σ 1 −σ 2 )i , e −2σ 1 i , e −4σ 1 i for σ1 = 0.070, σ2 = 0.008 in b). Fig. 2 . 2Regions of different values of the GALI2 (left panel) and GALI4 (right panel) for a grid of 500 × 500 initial conditions on the subspace x3 = 0.54, x4 = 0.01 of map (2) for K = 0.5 and B = 0.05. Fig. 3 . 3Percentages of regular orbits on the subspace x3 = 0.54, x4 = 0.01 of map (2) for K = 0.5, as a function of the parameter B ∈ [0, 2.5]. AcknowledgmentsT. Manos was supported by the "Karatheodory" graduate student fellowship No B395 of the Univ. of Patras, the program "Pythagoras II" and the Marie . Ch, T Skokos, Ch Bountis, Antonopoulos, Physica D. 231Ch. Skokos, T. Bountis and Ch. Antonopoulos, Physica D, 231, 30, (2007). . H Kantz, P Grassberger, J. Phys. A: Math. Gen. 21127H. Kantz and P. Grassberger, J. Phys. A: Math. Gen, 21 L127, (1988).
[]
[ "ELLIPSOIDAL BGK MODEL NEAR A GLOBAL MAXWELLIAN", "ELLIPSOIDAL BGK MODEL NEAR A GLOBAL MAXWELLIAN" ]
[ "Seok-Bae Yun " ]
[]
[]
The BGK model has been widely used in place of the Boltzmann equation because of the qualitatively satisfactory results it provides at relatively low computational cost. There is, however, a major drawback to the BGK model: The hydrodynamic limit at the Navier-Stokes level is not correct. One evidence is that the Prandtl number computed using the BGK model does not agree with what is derived from the Boltzmann equation. To overcome this problem, Holway[21]introduced the ellipsoidal BGK model where the local Maxwellian is replaced by a non-isotropic Gaussian. In this paper, we prove the existence of classical solutions of the ES-BGK model when the initial data is a small perturbation of the global Maxwellian. The key observation is that the degeneracy of the ellipsoidal BGK model is comparable to that of the original BGK model or the Boltzmann equation in the range −1/2 < ν < 1.
10.1137/130932399
[ "https://arxiv.org/pdf/1507.03277v1.pdf" ]
11,129,282
1507.03277
3caf747b62d9a0f5168fbc1ec2707129e9163e48
ELLIPSOIDAL BGK MODEL NEAR A GLOBAL MAXWELLIAN 12 Jul 2015 Seok-Bae Yun ELLIPSOIDAL BGK MODEL NEAR A GLOBAL MAXWELLIAN 12 Jul 2015 The BGK model has been widely used in place of the Boltzmann equation because of the qualitatively satisfactory results it provides at relatively low computational cost. There is, however, a major drawback to the BGK model: The hydrodynamic limit at the Navier-Stokes level is not correct. One evidence is that the Prandtl number computed using the BGK model does not agree with what is derived from the Boltzmann equation. To overcome this problem, Holway[21]introduced the ellipsoidal BGK model where the local Maxwellian is replaced by a non-isotropic Gaussian. In this paper, we prove the existence of classical solutions of the ES-BGK model when the initial data is a small perturbation of the global Maxwellian. The key observation is that the degeneracy of the ellipsoidal BGK model is comparable to that of the original BGK model or the Boltzmann equation in the range −1/2 < ν < 1. Introduction The dynamics of a non-ionized monatomic rarefied gas is governed by the Boltzmann equation. But the complex structure of the collision operator has long been a major obstacle for theoretical and computational investigation of the Boltzmann equation. To overcome this difficulty, Bhatnagar et al. [6], and independently Walender [40], introduced a model equation called the BGK model, where the collision operator is replaced by a relaxation operator. Since then, it has been widely used in place of the Boltzmann equation for various computational experiments, since this model provides very satisfactory results at relatively low computational cost compared to the Boltzmann equation. But the BGK model has a major drawback. Hydrodynamic limit at the Navier Stokes level is not satisfactory in that the Prandtl number -defined as the ratio between the viscovity and the thermal conductivity -computed using the BGK model is incorrect: The Prandlt number for the Navier-Stokes equation is around 0.7, but the computation using the BGK model yields 1. To resolve this problem, Holway suggested a variant of the BGK model, called the ellipsoidal BGK model (ES-BGK model) [21]: ∂ t F + v · ∇ x F = A ν M ν (F ) − F , F (x, v, 0) = F 0 (x, v). (1.1) F (x, v, t) denotes the velocity distribution function representing the number density on the phase space point (x, v) in T 3 x × R 3 v at time t ∈ R + . A ν is the collision frequency whose explicit form will be given later. The non-isotropic Gaussian M ν (F ) in the r.h.s of (1.1) is defined as follows: First, we define the macroscopic density ρ, bulk velocity U , temperature T and the stress tensor Θ by ρ(x, t) = R 3 F (x, v, t)dv, ρ(x, t)U (x, t) = R 3 F (x, v, t)vdv, 3ρ(x, t)T (x, t) = R 3 F (x, v, t)|v − U (x, t)| 2 dv, ρ(x, t)Θ(x, t) = R 3 F (x, v, t)(v − U ) ⊗ (v − U )dv, and introduce the temperature tensor T ν as a linear combination of T and Θ: T ν =   (1 − ν)T + νΘ 11 νΘ 12 νΘ 13 νΘ 21 (1 − ν)T + νΘ 22 νΘ 23 νΘ 31 νΘ 32 (1 − ν)T + νΘ 33   = (1 − ν)T Id + νΘ. The non-isotropic Gaussian M ν is now defined as follows: M ν (F ) = ρ det(2πT ν ) exp − 1 2 (v − U ) ⊤ T −1 ν (v − U ) . We note that the temperature is recovered as the trace of T ν : 3T = Θ 11 + Θ 22 + Θ 33 = trΘ = trT ν . The collision frequency A ν takes the following explicit form: A ν = ρ T 1 − ν , − 1 2 < ν < 1. The free parameter ν is introduced to derive the correct Prandtl number. The restriction on the range of ν is imposed to guarantee that the temperature tensor T ν remains positive definite. (See [2]). Prandtl number computed via the Chapman-Enskog expansion using the ES-BGK model is given by P r = 1/(1 − ν) (See [2,10,21,34]). The two most important cases in the range −1/2 < ν < 1 are ν = 0 and ν = (P r − 1)/P r ≈ −3/7: When ν = 0, (1.1) reduces to the classical BGK model, whereas ν = (P r − 1)/P r corresponds to the ES-BGK model with the correct Prandtl number. The relaxation operator of the ES-BGK model satisfies the following cancelation property [1,2]: R 3 M ν (F ) − F   1 v |v| 2   dv = 0, which implies the conservation of mass, momentum and energy: T 3 x ×R 3 v F (x, v, t)dxdv = T 3 x ×R 3 v F 0 (x, v)dxdv, T 3 x ×R 3 v F (x, v, t)vdxdv = T 3 x ×R 3 v F 0 (x, v)vdxdv, T 3 x ×R 3 v F (x, v, t)|v| 2 dxdv = T 3 x ×R 3 v F 0 (x, v)|v| 2 dxdv. (1.2) Entropy dissipation property was established recently in [2]: d dt T 3 x ×R 3 v F (t) log F (t)dxdv ≤ 0. It is important to note that, as in the case of the original BGK model or the Boltzmann equation, the only possible equilibrium state for (1.1) is the local Maxwellian: M (F ) = ρ (2πT ) 3/2 e − |v−U | 2 2T . To see this, let's assume that M ν (f ) = f . We then recall the definition of Θ and T ν to see that R 3 F (v)(v − U ) ⊗ (v − U )dv = ρΘ, M ν (F )(v)(v − U ) ⊗ (v − U )dv = ρT ν . Therefore, upon multiplying (v − U ) ⊗ (v − U ) to both sides of M ν (F ) = F and integrating with respect to v, we have ρT ν = ρΘ. In view of the definition of T ν , this leads to (1 − ν)T Id + νΘ = Θ. Thus, Θ = T Id, and we see, from the definition of T ν , that T ν = T Id for 3 × 3 identity matrix Id. This gives M ν (F ) = ρ det(2πT Id) exp − 1 2 (v − U ) ⊤ {T Id} −1 (v − U ) = ρ (2πT ) 3/2 e − |v−U | 2 2T = M (F ). That is, M ν (F ) reduces to the usual local Maxwellian M (F ). In this paper, we study the existence of classical solutions of (1.1) and their asymptotic behavior when the initial data is a small perturbation of the normalized global Maxwellian: µ(v) = 1 (2π) 3 e − |v| 2 2 . (1.3) We define the perturbation f around µ by the relation: : F (x, v, t) = µ+ √ µf (x, v, t) and, ac- cordingly, F 0 (x, v) = µ+ √ µf 0 (x, v∂ t f + v · ∇ x f = L ν f + Γ(f ), f (x, v, 0) = f 0 (x, v). (1.4) where L ν denotes the linearized relaxation operator and Γ(f ) is the nonlinear part. In section 2, we verify that L ν can be represented as a ν-perturbation of the linearized relaxation operator of the original BGK model: L ν f = (P 0 f − f ) + νP 1 f + νP 2 f. (1.5) Here, P 0 denotes the macroscopic projection operartor on the the linear space generated by { √ µ, v √ µ, |v| 2 √ µ}. P 1 and P 2 are operators related to the burnett functions, which play a crucial role in the hydrodynamic limit of the Boltzmann equation at the Navier-Stokes level. (See [3]). In general, the coercivity estimate of the linearized collision or relaxation operators for spatially inhomogeneous collisional kinetic equations are degenerate, and the major difficulty in obtaining the global existence in the perturbative regime lies in removing the degeneracy to recover the full coercivity [18,19,20]. When the spatial variable lies in T 3 , the usual recipe is the use of the Poincare inequality together with a system of macroscopic equations and the conservation laws (See, for example, [19,20]). In the whole space, where the Poincare inequality is not available, additional consideration has to be made to compensate the still lingering degeneracy [12,13,14,23,36,38]. Therefore, it is very important to capture the degenerate coercivity estimate of the linearized relaxation operator first. In our case, it is not clear whether the presence of the additional terms P 1 and P 2 make the linearized relxation operator more degenerate or not. In Theorem 2.8, we show that, for −1/2 < ν < 1, the degenerate coercive estimate of L ν f is comparable to that of L 0 f = (P 0 − I)f , for which the usual energy method is well-established (See Theorem 2.8): L ν f, f L 2 v ≤ −C ν {I − P 0 }f 2 L 2 v , (−1/2 < ν < 1) , for some constant C ν > 0. This indicates that the dissipative property of the linearized relaxation operator for the ES-BGK model is essentially same as that of the BGK model or Boltzmann equation. On the other hand, since the ES-BGK model is obtained by replacing the temperature function T by the temperature tensor T ν in the classical BGK model, additional difficulties related to T ν , which was not observed in the classical BGK model arise. First, in each step of the iteration scheme designed to obtain the local in time existence of the solution, we need to check that the temperature tensor remains strictly positive definite, which is established in Proposition 3.1 as: C −1 ν2 T (x, t) Id ≤ T −1 ν (x, t) ≤ C −1 ν1 T (x, t) Id. where C ν1 = min{1 − ν, 1 + 2ν} and C ν1 = max{1 − ν, 1 + 2ν}. This also shows why the restriction of the range of the free parameter in the interval (−1/2, 1) is crucial: It is only in this range that the temperature tensor is comparable to T , and, therefore, the nonisotropic Gaussian is comparable to the local Maxwellian. Secondly, due to the presence of the free parameter ν in the definition of the temperature field T ν , it is a priori not clear whether the nonlinear perturbation Γ(f ) can be estimated uniformly with respect to ν near ν = 0 because the the inverse of the temperature tensor T −1 ν may have problematic terms involving 1/ν. Such a singularity at ν = 0 is undesirable considering that the case ν = 0 corresponds to the classical BGK model. The above equivalence estimate guarantees that such singularity never shows up when −1/2 < ν < 1. The mathematical theory for the BGK model has a rather short history. The first rigorous existence result can be traced back to Ukai [37], where he considered stationary problem for 1 dimensional BGK model in a periodic bounded domain. Perthame established the existence of weak solutions of the BGK model with constant collision frequency in [27] assuming only the finite mass, momentum, energy and entropy. See also [7]. The uniqueness was considered in a more stringent functional space involving the pointwise decay in velocity [28]. Mischler considered similar problems in the whole space in [25]. Extension to L p was carried out in [42]. Issautier established regularity estimates for the BGK model and proved the convergence of a Monte-Carlo type scheme to the regular distribution function in [22]. The convergence property of a semi-Lagrangian scheme for the BGK model was studied in [29]. In near Maxwellian regime, Bellouquid [5] obtained the global well posedness in the whole space using Ukai's spectral analysis argument [36]. In the periodic case, Chan employed the energy method developed by Liu et al. [24] to establish the global in time classical solution near global Maxwellians [11]. The convergence rate to the equilibrium was not known in this work, which was derived in [41]. For fluid dynamic limit of the BGK model, see [30,31]. The ES-BGK model has attracted only limited attention until very recently since it was not clear whether the entropy dissipation property holds for this model. It was proved in the affirmative, at least at the formal level, in [2], which revived the interest on this model. To our knowledge, no existence result has been established for the ellipsoidal BGK. For numerical test for the ES-BGK model, we refer to [1,15,16,26,43]. For general review of the mathematical and physical theory of the Boltzmann equation and the BGK model, see [8,9,17,32,33,35,39]. Before proceeding further, we define some notations. • When there is no risk of confusion, we use generic constants C. Their value may change from line to line but does not depend on important parameters. • We define the index set i < j by i<j a ij = a 12 + a 23 + a 31 . • e i (i = 1, 2, 3) denote the standard coordinate unit vectors in R 3 . • 0 n denotes n-dimensional zero vector. • I(m, n; a, b) denotes a (m + n) × (m + n) diagonal matrix whose first m diagonal elements are a and following n diagonal elements are b. • ·, · L 2 v and ·, · L 2 x,v denote the standard L 2 inner product on R 3 v and T 3 x × R 3 v respectively: f, g L 2 v = R 3 f (v)g(v)dv, f, g L 2 x,v = T 3 ×R 3 f (x, v)g(x, v)dxdv. • · L 2 v and · L 2 x,v denote the standard L 2 inner norms on R 3 v and T 3 x × R 3 v respectively: f 2 L 2 v = R 3 |f (v)| 2 dv 1 2 , f 2 L 2 x,v = T 3 ×R 3 |f (x, v)| 2 dxdv 1 2 . • We employ the following notations for the multi-indices and differential operators: α = [α 0 , α 1 , α 2 , α 3 ], β = [β 1 , β 2 , β 3 ], and ∂ α β = ∂ α0 t ∂ α1 x1 ∂ α2 x2 ∂ α3 x3 ∂ β1 v1 ∂ β2 v2 ∂ β3 v3 . For simplicity, when only the spatial derivatives are involved, we write ∂ α x = ∂ α1 x1 ∂ α2 x2 ∂ α3 x3 . 1.1. Main results. We now state our main result. We first define the high order energy functional E(f (t)): E f (t) = 1 2 |α|+|β|≤N ∂ α β f (t) 2 L 2 x,v + |α|+|β|≤N t 0 ∂ α β f (s) 2 L 2 x,v ds. (1) The distribution function is non-negative for all t ≥ 0: F = µ + √ µf ≥ 0, and satisfies the conservation laws (2.5). (2) The high order energy functional E f (t) is uniformly bounded: E f (t) ≤ CE f 0 . (3) The distribution function converges to the global equilibrium exponentially fast: |α|+|β|≤N ∂ α β f (t) L 2 x,v ≤ Ce −C ′ t for some constant C and C ′ . (4) Iff denotes another solution corresponding to initial datef 0 satisfying the same assumptions, then we have the following uniform L 2 -stability estimate: f (t) −f (t) L 2 x,v ≤ C f 0 −f 0 L 2 x,v . This paper is organized as follows: In section 2, we consider the derivation of the linearized ES-BGK equation and the main result is stated. We also derive the coercive estimate and determine the kernel of L ν . In section 3, various estimates on the macroscopic field are established and, based on this, the local in time existence is obtained. In section 4, the nonlinear energy estimate is derived, which readily leads to the global existence and the asymptotic behavior. Linearization In this section, we consider the linearzation of the ES-BGK model around the global Maxwellian (1.3). For some technical reason, we define G ν as follows: G ν = 1 − ν 3 3ρT + ρ|U | 2 2 Id + ν ρΘ + ρU ⊗ U 2 − ρ 2 Id. Due to the symmetry of G, we can view G as an element in R 6 : {G 11 , G 22 , G 33 , G 12 , G 23 , G 31 } . We also define J ν to be the Jacobian matrix for the change of variable (ρ, U, T ν ) → (ρ, ρU, G): J ν ≡ ∂(ρ, ρU, G ν ) ∂(ρ, U, T ν ) . Lemma 2.1. (1) J ν is given by                 1 0 0 0 0 0 0 0 0 0 U 1 ρ 0 0 0 0 0 0 0 0 U 2 0 ρ 0 0 0 0 0 0 0 U 3 0 0 ρ 0 0 0 0 0 0 A ν 11 1+2ν 3 ρU 1 1−ν 3 ρU 2 1−ν 3 ρU 3 1 2 ρ 0 0 0 0 0 A ν 22 1−ν 3 ρU 1 1+2ν 3 ρU 2 1−ν 3 ρU 3 0 1 2 ρ 0 0 0 0 A ν 33 1−ν 3 ρU 1 1−ν 3 ρU 2 1+2ν 3 ρU 3 0 0 1 2 ρ 0 0 0 A ν 12 ν 2 ρU 2 ν 2 ρU 1 0 0 0 0 1 2 ρ 0 0 A ν 23 0 ν 2 ρU 3 ν 2 ρU 2 0 0 0 0 1 2 ρ 0 A ν 31 ν 2 ρU 3 0 ν 2 ρU 1 0 0 0 0 0 1 2 ρ                 , where A ν ij is defined as A ν ii = 1 2 T ii + (1 − ν)|U | 2 + 3νU 2 i 3 − 1 , A ν ij = ν 2 (T ij + U i U j ) . (2) J −1 ν is given by                    1 0 0 0 0 0 0 0 0 0 − U1 ρ 1 ρ 0 0 0 0 0 0 0 0 − U2 ρ 0 1 ρ 0 0 0 0 0 0 0 − U3 ρ 0 0 1 ρ 0 0 0 0 0 0 A 11 − 2(1+2ν) 3 U1 ρ − 2(1−ν) 3 U2 ρ − 2(1−ν) 3 U3 ρ 2 ρ 0 0 0 0 0 A 22 − 2(1−ν) 3 U1 ρ − 2(1+2ν) 3 U2 ρ − 2(1−ν) 3 U3 ρ 0 2 ρ 0 0 0 0 A 33 − 2(1−ν) 3 U1 ρ − 2(1−ν) 3 U2 ρ − 2(1+2ν) 3 U3 ρ 0 0 2 ρ 0 0 0 A 12 − U2 ρ −ν U1 ρ 0 0 0 0 2 ρ 0 0 A 23 0 −ν U3 ρ −ν U2 ρ 0 0 0 0 2 ρ 0 A 31 −ν U3 ρ 0 −ν U1 ρ 0 0 0 0 0 2 ρ                    , where A ν ij is defined as A ii = − 1 ρ T ii + (1 − ν)|U | 2 + 3νU 2 i 3 − 1 , A ij = ν ρ (−T ij + U i U j ). (3) When F = µ, J ν and J −1 ν reduce to the following simpler form: J ν | F =µ = I(4, 6; 1, 1/2) and J −1 ν F =µ = I(4, 6; 1, 2) . For the definition of I(m, n; a, b), see the notation at the end of the introduction. Proof. The proof is straightforward but very tedious. We omit the proof. Lemma 2.2. We have (1) Derivatives for det T ν : (1 ≤ i, j ≤ 3, i = j) ∂ det T ν ∂ρ F =µ = 0, ∂ det T ν ∂U i F =µ = 0, ∂ det T ν ∂T ii F =µ = 1, ∂ det T ν ∂T ij F =µ = 0. (2) Derivatives for M ν : (1 ≤ i, j ≤ 3, i = j) ∂M ν ∂ρ F =µ = µ(v), ∂M ν ∂U i F =µ = v i µ(v), ∂M ν ∂T ii F =µ = v 2 i − 1 2 µ(v), ∂M ν ∂T ij F =µ = v i v j µ(v). Proof. (1) A straightforward calculation leads to the following explicit form of the determinant of T ν : det T ν = T 11 T 22 T 33 − T 2 23 T 11 − T 2 31 T 22 − T 2 12 T 33 . (2.1) Then (1) follows from explicit calculations using ∂T ij ∂ρ = 0, ∂T ij ∂U = 0, ∂T ij ∂T ℓk = 1 (i = ℓ, j = k) 0 (otherwise) , and T ij µ = δ ij . (2) We only consider ∂M ∂Tij . Other terms can be obtained similarly. We first observe that ∂M ν ∂T ij = − 1 2 1 | det T | ∂ det T ∂T ij + 1 2 (v − U )T −1 ∂T ∂T ij T −1 (v − U ) M ν , When i = j = 1, we have ∂M ν ∂T 11 F =µ =    − 1 2 + 1 2 v ⊤   1 0 0 0 0 0 0 0 0   v    M ν = v 2 1 − 1 2 µ. ∂Mν ∂T22 , ∂Mν ∂T33 can be obtained in the same manner. In the case i = j, we observe that ∂M ν ∂T 12 F =µ =    0 + 1 2 v ⊤   0 1 0 1 0 0 0 0 0   v    µ = v 1 v 2 µ. Similarly, we have ∂M ∂T 23 F =µ = v 2 v 3 µ and ∂M ∂T 31 F =µ = v 3 v 1 µ. Now, we are ready to prove the main theorem of this section, which basically says that the linearized relaxation operator is composed of ν-perturbation of the projection on the macroscopic kernel and nonlinear terms. Theorem 2.3. Let F = µ + √ µf . Then the ellipsoidal Gaussian M ν (F ) can be expanded around µ as follows: M ν (F ) = µ + (P ν f ) √ µ + 1≤i,j≤3 1 0 D 2 (ρ θ ,ρ θ U θ ,G θ ) M(θ) ij (1 − θ) 2 dθ f, e i L 2 v f, e j L 2 v , Here, P ν is given by a ν-perturbation of the usual macroscopic projection P 0 : P ν f ≡ P 0 f + ν(P 1 f + P 2 f ), where P 0 f = f √ µdv √ µ + f v √ µdv · v √ µ + f |v| 2 − 3 √ 6 √ µdv |v| 2 − 3 √ 6 √ µ, P 1 f = 3 i=1 f 3v 2 i − |v| 2 3 √ 2 √ µdv 3v 2 i − |v| 2 3 √ 2 √ µ, P 2 f = i<j f v i v j √ µdv v i v j √ µ. and M ν (θ) denotes M ν (θ) = ρ θ det(2πT θ ) exp − 1 2 (v − U θ ) ⊤ T −1 θ (v − U θ ) , where the transitional macroscopic fields ρ θ , U θ , G θ and T θ are defined by ρ θ = θρ + (1 − θ), ρ θ U θ = θρU, and G θ = θG, and T θ =   (1 − ν)T θ + νΘ θ11 νΘ θ12 νΘ θ13 νΘ θ21 (1 − ν)T θ + νΘ θ22 νΘ θ23 νΘ θ31 νΘ θ32 (1 − ν)T θ + νΘ θ33   . Proof. We define g(θ) as g(θ) = M θ(ρ, ρU, G) + (1 − θ) 1, 0 3 , 0 6 = M (ρ θ , ρ θ U θ , G θ ) . Note that g(θ) represents the transition from the global Maxwellian µ(v) to the ellipsoidal Gaussian M ν (F ). Then we have from the Taylor's theorem g(1) = g(0) + g ′ (0) + 1 0 g ′′ (θ)(1 − θ) 2 dθ. (2.2) The first term in the right hand side is the global Maxwellian: g(0) = µ. We now consider the second and the third terms: (i) g ′ (0): We observe from Lemma 2.2 that D (ρ,U,Tν ) M ν (0) = ∂M ν ∂ρ , ∂M ν ∂U , ∂M ν ∂T ν F =µ = 1, v, v 2 1 − 1 2 , v 2 2 − 1 2 , v 2 3 − 1 2 , v 1 v 2 , v 2 v 3 , v 3 v 1 µ(v). Then, using the identities in Lemma 2.1 and Lemma 2.2, g ′ (0) can be represented as g ′ (0) = d dθ M θ(ρ, ρU, G) + (1 − θ)(1, 0, 0, 0, 0 6 ) θ=0 = (ρ − 1, ρU, G) ⊤ · J −1 θ D (ρ θ ,U θ ,T θ ) M θ=0 = (ρ − 1, ρU, G) ⊤ · J −1 × 1, v, v 2 1 − 1 2 , v 2 3 − 1 2 , v 2 3 − 1 2 , v 1 v 2 , v 2 v 3 , v 3 v 1 µ = f √ µdv µ + f v √ µdv · vµ + 2 3 i=1 G ii v 2 i − 1 2 µ + 2 i<j G ij v i v j µ. Here J θ denotes ∂(ρ θ U θ ,G θ ) ∂(ρ θ U θ ,T θ ) and we used J 0 = J . (ii) g ′′ (θ): By an explicit computation, we find g ′′ (θ) = d 2 M dθ 2 (θ(ρ − 1, ρU, G) + (1 − θ)(1, 0, 0 6 )) = (ρ − 1, ρU, G) ⊤ D 2 ρ θ ,ρ θ U θ ,G θ ) M(θ) (ρ − 1, ρU, G). (iii) We claim that g ′ (0) = P ν f √ µ. Note that it is enough to establish 2 3 i=1 G ii v 2 i − 1 2 √ µ = f |v| 2 − 3 √ 6 √ µdv |v| 2 − 3 √ 6 √ µ + 3 i=1 f 3v 2 i − |v| 2 3 √ 2 √ µdv 3v 2 i − |v| 2 3 √ 2 √ µ. We first observe that G ii (i = 1, 2, 3) can be decomposed as G ii = 1 − ν 3 R 3 f |v| 2 2 dv + ν R 3 f v 2 i 2 dv − R 3 1 2 f dv = R 3 f |v| 2 − 3 6 + ν 3v 2 i − |v| 2 6 √ µdv, so that 2 3 i=1 R 3 f G ii v 2 i − 1 2 √ µ = 2 3 i=1 R 3 f |v| 2 − 3 6 + ν 3v 2 i − |v| 2 6 √ µdv v 2 i − 1 2 √ µ = 2 3 i=1 R 3 f |v| 2 − 3 6 √ µdv v 2 i − 1 2 √ µ + 2ν 3 i=1 R 3 f 3v 2 i − |v| 2 6 √ µdv v 2 i − 1 2 √ µ + A + B. We compute A as A = 2 R 3 f |v| 2 − 3 6 √ µdv 3 i=1 v 2 i − 1 2 √ µ = 2 R 3 f |v| 2 − 3 6 √ µdv |v| 2 − 3 2 √ µ = R 3 f |v| 2 − 3 √ 6 √ µdv |v| 2 − 3 √ 6 √ µ. For B, we observe that v 2 i − 1 2 = 3v 2 i − |v| 2 6 + |v| 2 − 3 6 , and 3 i=1 3v 2 i − |v| 2 6 = 0, to derive B = 2 3 i=1 R 3 f 3v 2 i − |v| 2 6 √ µdv 3v 2 i − |v| 2 6 + |v| 2 − 3 6 √ µ = 2 3 i=1 R 3 f 3v 2 i − |v| 2 6 √ µdv 3v 2 i − |v| 2 3 √ µ + 2 3 i=1 R 3 f 3v 2 i − |v| 2 6 √ µdv |v| 2 − 3 6 √ µ = 3 i=1 R 3 f 3v 2 i − |v| 2 3 √ 2 √ µdv 3v 2 i − |v| 2 3 √ 2 √ µ + 2 R 3 f 3 i=1 3v 2 i − |v| 2 6 =0 √ µdv |v| 2 − 3 6 √ µ = 3 i=1 R 3 f 3v 2 i − |v| 2 3 √ 2 √ µdv 3v 2 i − |v| 2 3 √ 2 √ µ. Plugging (i), (ii), (iii) into 2.2, we obtained the desired result. We now consider the linearization of the collision frequency. Lemma 2.4. The collision frequency A ν can be linearized around the normalized global Maxwellian as follows: A ν = 1 1 − ν + 1 1 − ν A p , where A p = 1 0 J −1 θ (T θ , 0 3 , 1/3ρ θ Id)(1 − θ)dθ · (ρ − 1, ρU, G). Proof. We expand A ν by the Taylor's theorem. Then the second term reads 1 0 D (ρ θ ,ρ θ U θ ,G θ ) (ρ θ T θ )(1 − θ)dθ · (ρ − 1, ρU, G). Note that D (ρ θ ,ρ θ U θ ,G θ ) = J −1 θ D (ρ θ ,U θ ,T θ ) , to see A p = 1 0 J −1 θ D (ρ θ ,U θ ,T θ ) (ρ θ T θ )(1 − θ)dθ · (ρ − 1, ρU, G) = 1 0 J −1 θ (T θ , 0 3 , 1/3ρ θ Id)(1 − θ)dθ · (ρ − 1, ρU, G). Instead of writing down D 2 (ρ θ ,ρ θ U θ ,G θ ) explicitly, we introduce generic notations which considerably simplify the argument. We first observe that D 2 (ρ θ ,ρ θ U θ ,G θ ) M(θ) = J −1 θ D (ρ θ ,U θ ,T θ ) J −1 θ D (ρ θ ,U θ ,T θ ) M(θ) . We then invoke Lemma 2.1 to conclude the following lemma. Lemma 2.5. There exist generic polynomials P M i,j , R M i,j such that (ρ − 1, ρU, G) ⊤ D 2 (ρ θ ,ρ θ U θ ,G θ ) M(θ) (ρ − 1, ρU, G) = i,j P M i,j (ρ θ , v − U θ , U θ , T −1 θ , ν) R M i,j (ρ θ , det T θ ) exp − 1 2 (v − U θ ) ⊤ T −1 θ (v − U θ ) f, e i f, e j , where P M i,j (x 1 , . . . , x n ) and R M i,j (x 1 , . . . , x n ) satisfy the following structural assumptions (H M ): • (H Mν 1) P Mν i,j is a polynomial such that P i,j (0, 0, . . . , 0) = 0. • (H Mν 2) R Mν i,j is a monomial. Lemma 2.6. There exist generic polynomials P Aν i , R Aν i such that J −1 θ (T θ , 0 3 , 1/3ρ θ Id) · (ρ − 1, ρU, G) = i P Aν i (ρ θ , U θ , T νθ , ν) R A ν i (ρ θ ) f, e i , where P A ν i,j (x 1 , . . . , x n ) and R A ν i,j (x 1 , . . . , x n ) satisfy the following structural assumptions (H Aν ). • (H Aν 1) P Aν i is a polynomial such that P i,j (0, 0, . . . , 0) = 0. • (H Aν 2) R Aν i is a monomial. Note that P M ij , R M ij ,Q Mν ij (θ) = 1 √ µ 1 0 P Mν i,j (ρ θ , v − U θ , T −1 θ , ν) R M ij (ρ θ , det T νθ ) exp − 1 2 (v − U θ ) ⊤ T −1 νθ (v − U θ ) (1 − θ) 2 dθ and Q Aν i (θ) = 1 0 P Aν i (ρ θ , U θ , T θ , ν) R Aν i (ρ θ ) (1 − θ)dθ. Then the relaxation operator and the collision frequency can be expressed in a more succinct form: M ν (F ) − F = P ν f − f √ µ + Q Mν ij f, e i L 2 v f, e i L 2 v , and A ν = 1 1 − ν + 1 1 − ν Q ν i f, e i L 2 v . We summarize the result in the following proposition. Proposition 2.1. The relaxation operator can be linearized around the normalized global Maxwellian µ as follows A ν M ν (F ) − F = 1 1 − ν 1 + i Q Aν i f, e i (P ν f − f ) + i,j Q Mν ij f, e i f, e j √ µ. We now substitute the standard perturbation F = µ + √ µf into (1.1) and apply proposition 2.1 to obtain the perturbed ES-BGK model: ∂ t f + v · ∇ x f = L ν f + Γ(f ), f (x, v, 0) = f 0 (x, v), (2.3) where f 0 (x, v) = F0−µ √ µ . The linearized relaxation operator L ν and the nonlinear perturbation Γ(f ) are defined as follows: L ν f = 1 1 − ν P ν f − f , (2.4) and Γ(f ) = 1 1 − ν i Q A i f, e i P ν f − f + 1 1 − ν 1≤i,j≤3 Q M i,j f, e i L 2 v f, e j L 2 v + 1 1 − ν 1≤i,j≤3 Q A ν i Q M j,k f, e i L 2 v f, e j L 2 v f, e k L 2 v ≡ Γ 1 (f, f ) + Γ 2 (f, f ) + Γ 3 (f, f, f ). The conservation laws in (1.2) now take the following form: T 3 x ×R 3 v f (x, v, t) √ µ dxdv = T 3 x ×R 3 v f 0 (x, v) √ µ dxdv, T 3 x ×R 3 v f (x, v, t)v √ µ dxdv = T 3 x ×R 3 v f 0 (x, v)v √ µ dxdv, T 3 x ×R 3 v f (x, v, t)|v| 2 √ µ dxdv = T 3 x ×R 3 v f 0 (x, v)|v| 2 √ µ dxdv. Therefore, if initial data shares the same mass, momentum and energy with µ, the conservation laws read T 3 x ×R 3 v f (x, v, t) √ µ dxdv = 0, T 3 x ×R 3 v f (x, v, t)v √ µ dxdv = 0, T 3 x ×R 3 v f (x, v, t)|v| 2 √ µ dxdv = 0. (2.5) 2.1. Analysis of the linearized relaxation operator. We now study the dissipative mechanism of the linearized operator. We start with the following technical lemma. Lemma 2.7. P 0 , P 1 and P 2 satisfies the following properties: (1) P 0 , P 2 and P 2 are orthonormal projections: P 2 0 = P 0 , P 2 1 = P 1 , P 2 2 = P 2 . (2) P 0 , P 1 and P 3 are mutually orthogonal in the following sense: P 0 P 1 = P 1 P 0 = P 0 P 2 = P 2 P 0 = P 1 P 2 = P 2 P 1 = 0. Proof. (1) The first and third identities P 2 0 = P 0 and P 2 2 = P 2 follow from the fact that { √ µ, v √ µ, |v| 2 √ µ} and {v 1 v 2 √ µ, v 2 v 3 √ µ, v 3 v 1 √ µ} form orthonormal bases respectively. To show P 2 1 = P 1 , we first observe that (3v 2 i − |v| 2 ) √ µ, (3v 2 i − |v| 2 ) √ µ L 2 v = 12, (1 = 1, 2, 3) (3v 2 i − |v| 2 ) √ µ, (3v 2 j − |v| 2 ) √ µ L 2 v = −6 (i = j). Using this, we have for c i (v) = (3v 2 i − |v| 2 )/3 √ 2 P 2 1 f = P 1 f, c 1 L 2 v c 1 + f, c 2 L 2 v c 2 + f, c 3 L 2 v c 3 = 1 3 {2 f, c 1 L 2 v − f, c 2 L 2 v − f, c 3 L 2 v }c 1 + 1 3 {− f, c 1 L 2 v + 2 f, c 2 L 2 v − f, c 3 L 2 v }c 1 + 1 3 {− f, c 1 L 2 v − f, c 2 L 2 v + 2 f, c 3 L 2 v }c 1 = f, 2c 1 − c 2 − c 3 3 L 2 v c 1 + f, −c 1 + 2c 2 − c 3 3 L 2 v c 2 + f, −c 1 − c 2 + 2c 3 3 L 2 v c 3 = f, c 1 L 2 v c 1 + f, c 2 L 2 v c 2 + f, c 3 L 2 v c 3 = P 1 f. In the last line, we used c 1 + c 2 + c 3 = 0. (2) Straightforward calculations gives √ µ, (3v 2 i − |v| 2 ) √ µ L 2 v = v ℓ √ µ, (3v 2 i − |v| 2 ) √ µ L 2 v = (|v| 2 − 3) √ µ, (3v 2 i − |v| 2 ) √ µ L 2 v = 0, and v i v j √ µ, (3v 2 k − |v| 2 ) √ µ L 2 v = 0. This implies (2). We now prove the main theorem of this section. Note that that the estimate is uniform with respect to ν. Theorem 2.8. For − 1 2 < ν < 1, we have L ν f, f L 2 v ≤ − min 1, 1 − |ν| 1 − ν (I − P 0 )f 2 L 2 v . Proof. From the definition of L ν , we have (1 − ν) L ν f, f L 2 v = P ν f − f, f L 2 v = P 0 f − f + ν(P 1 + P 2 )f, f L 2 v = − (I − P 0 )f 2 L 2 v + ν (P 1 + P 2 )f, f L 2 v . (2.6) We recall from Lemma 2.7 that (P 1 + P 2 ) ⊥ P 0 , which gives (P 1 + P 2 )f, f L 2 v = (P 1 + P 2 )(I − P 0 )f, (I − P 0 )f L 2 v = (P 1 + P 2 )(I − P 0 )f 2 L 2 v . (2.7) We then observe from Lemma 2.7 that P 1 + P 2 is a projection operator: (P 1 + P 2 ) 2 = P 2 1 + P 1 P 2 + P 2 P 1 + P 2 2 = P 1 + P 2 , which leads to (P 1 + P 2 )(I − P 0 )f L 2 v ≤ (I − P 0 )f 2 L 2 v . (2.8) Therefore, we have from (2.6) -(2.7) (1 − ν) L ν f, f L 2 v ≤ − min{(1 − ν), (1 − |ν|)} (I − P 0 )f 2 L 2 v . Since (1 − ν) > 0, this completes the proof. Corollary 2.1. For −1/2 < ν < 1, the kernel of the linearized relaxation operator is given by Ker{L ν } = Ker{L 0 } = span{ √ µ, v √ µ, |v| 2 √ µ}. Estimates on the macroscopic field 3.1. Estimates on the macroscopic field. To control the nonlinear perturbation Γ(f ) in the energy norm, we first need to establish various estimates for macroscopic quantities. Throughout this section, C ν > 0 means that C ν is strictly positive for all −1/2 < ν < 1. Lemma 3.1. Let E(t) be sufficiently small, then there exists a positive constant C > 0 and C ν > 0 such that (1) |ρ(x, t) − 1| ≤ C E(t), (2) |U i (x, t)| ≤ C E(t), (1 ≤ i ≤ 3) (3) |T νii (x, t) − 1| ≤ C ν E(t), (1 ≤ i ≤ 3) (4) |T ij (x, t)| ≤ νC E(t), (1 ≤ i < j ≤ 3) Proof. (1) We have from Hölder inequality |ρ(x, t) − 1| = R 3 f √ µdv ≤ f L 2 x ≤ E(t). (2) Using the lower bound estimate of ρ, Hölder inequality and R 3 µvdv = 0, we see that |U i | = 1 ρ R 3 f v i √ µdv ≤ f L 2 x,v 1 − E(t) ≤ E(t) 1 − E(t) ≤ C E(t). (3) For the upper bound of T ii , we compute as follows: T ii = (1 − ν)T + νΘ ii = (1 − ν) 3 1 ρ R 3 (µ + √ µf )|v| 2 dv − |U | 2 + ν 1 ρ R 3 (µ + √ µf )v 2 i dv − U 2 i ≤ (1 − ν) 3ρ 3 + R 3 f |v| 2 √ µdv + ν ρ 1 + R 3 f v 2 i √ µdv ≤ 1 − ν 3 3 + √ 15 f L 2 x,v ρ + ν 1 + √ 3 f L 2 x,v ρ ≤ 1 + C ν f L 2 x,v ρ ≤ 1 + C ν E(t) 1 − √ E(t) . Therefore, T ii − 1 ≤ C E(t) 1 − E(t) ≤ C E(t). (3.1) Using the lower bound estimate for ρ and U i , we estimate the lower bound similarly as T ii = (1 − ν)T + νθ ii = (1 − ν) 3ρ R 3 (µ + √ µf )|v| 2 dv − |U | 2 + ν ρ R 3 (µ + √ µf )v 2 i dv − U 2 i = (1 − ν) 3ρ 3 + R 3 f |v| 2 √ µdv − |U | 2 + ν ρ 1 + R 3 f v 2 i √ µdv − U 2 i ≥ 1 − ν 3ρ 3 − √ 15 f L 2 x,v − CE(t) + ν ρ 1 − √ 3 f L 2 x,v − CE(t) ≥ 1 − C ν f L 2 x,v − CE(t) ρ ≥ 1 − C ν E(t) − CE(t) 1 + √ E(t) ≥ 1 − C ν E(t) 1 + √ E(t) . Hence we have T ii − 1 ≥ −C ν √ E(t) 1 + E(t) ≥ −C ν E(t). (3.2) (3.1) and (3.2) give the desired result for T ii (i = 1, 2, 3). (4) T ij can be estimated similarly as |T ij | ≤ ν ρ R 3 f v i v j √ µdv + ν|U i ||U j | ≤ ν f L 2 x,v 1 − E(t) + νCE(t) ≤ ν E(t) 1 − E(t) + νCE(t) ≤ νC E(t). Lemma 3.2. Suppose E(t) is sufficiently small. Then there exists a positive constant C |α| > 0 and C |α|,ν > 0 such that (1) |∂ α ρ(x, t)| ≤ E(t), (2) |∂ α U (x, t)| ≤ C |α| E(t), (3) |∂ α T ij (x, t)| ≤ C |α|,ν E(t). Proof. (1) Since ∂ α µ = 0, we have |∂ α ρ| = ∂ α R 3 µ + f √ µdv = |∂ α f | √ µdv ≤ ∂ α f L 2 x,v ≤ E(t). (2) A straightforward computation using U = 1 ρ f v √ µdv and the chain rule gives to |∂ α U | ≤ C |α| ρ 2|α|   |α1|≤N R 3 |∂ α1 f ||v| √ µdv     1 + |α2|≤N |∂ α2 ρ|   |α| . Then the use of Hölder inequality and the estimate (1) leads to |∂ α U | ≤ C |α| E(t) 1 + E(t) |α| 1 − E(t) 2|α| ≤ C |α| E(t). (3) Recall T ij = 1−ν 3ρ f |v| 2 √ µdv − |U | 2 + ν ρ f v 2 i √ µdv − U 2 i . Therefore, by the same argument as in (2) above, we have |∂ α T ij | ≤ C |α|,ν ρ 2|α|   |α1|≤N R 3 |∂ α1 f ||v| 2 √ µdv     1 + |α2|≤N |∂ α2 ρ|   |α| + C |α|,ν E(t) ≤ C |α|,ν E(t) 1 + E(t) |α| 1 − E(t) 2|α| + C |α|,ν E(t) ≤ C |α|,ν E(t). Lemma 3.3. Let E(t) be sufficiently small. Then, we have positive constants C > 0 and C ν > 0 independent of θ such that (1) |ρ θ (x, t) − 1| ≤ C E(t), (2) |U θ (x, t)| ≤ C E(t), (3) T θii (x, t) − 1 ≤ C ν E(t), (i = 1, 2, 3), (4) T θij (x, t) ≤ C ν E(t), (i < j). Proof. (1) By Lemma 3.1 and the definition of ρ θ , we have |ρ θ − 1| = θ|ρ − 1| ≤ θ E(t) ≤ E(t). (2) follows directly from (1), Lemma 3.1 and the definition of U θ : U θ = θ ρ θ ρU. (3) We divide the case into i = j and i = j. When i = j, we have from the definition of G θ that for i = 1, 2, 3: 1 − ν 3 ρ θ (Θ θ11 + Θ θ22 + Θ θ33 ) + ρ θ |U θ | 2 2 + ν ρ θ Θ θii + ρU 2 θi 2 − ρ θ 2 = θ 1 − ν 3 ρ(Θ 11 + Θ 22 + Θ 33 ) + ρ|U | 2 2 + ν ρΘ ii + ρU 2 i 2 − ρ 2 ,(3. 3) Summing over i = 1, 2, 3, we obtain ρ θ (Θ θ11 + Θ θ22 + Θ θ33 ) 2 = θ ρ(Θ 11 + Θ 22 + Θ 33 ) 2 + ρ θ |U θ | 2 − θρ|U | 2 2 + 3 2 (ρ θ − ρθ).ν ρ θ Θ θii + ρ θ U 2 θi 2 = − 1 − ν 3 θ ρ(Θ 11 + Θ 22 + Θ 33 ) + |U | 2 2 − 1 − ν 2 (ρ θ − θρ) + θ 1 − ν 3 ρ(Θ 11 + Θ 22 + Θ 33 ) + |U | 2 2 + ν ρΘ ii + U 2 i 2 + ρ θ − θρ 2 = νθ ρΘ ii + ρU 2 i 2 + ν 2 (ρ θ − θρ). (3.5) In view of (3.4) and (3.5), we see that T θii = θ 1 − ν 3 ρ(Θ 11 + Θ 22 + Θ 33 ) ρ θ + 1 − ν 3 ρ θ |U θ | 2 − θρ|U | 2 ρ θ + (1 − ν) ρ θ − θρ ρ θ + νθ ρΘ ii + ρU 2 i ρ θ + ν ρ θ − θρ ρ θ = θ 1 − ν 3 ρ(Θ 11 + Θ 22 + Θ 33 ) ρ θ + νθ ρΘ ii + ρU 2 i ρ θ + 1 − ν 3 ρ θ |U θ | 2 − θρ|U | 2 ρ θ + ρ θ − θρ ρ θ = ρ ρ θ T ii + 1 − ν 3 ρ θ |U θ | 2 − θρ|U | 2 ρ θ + ρ θ − θρ ρ θ . (3.6) Therefore, applying Lemma 3.1, Lemma 3.2 and the estimate (1) and (2) above, we find that T θii ≤ θ(1 + C ν E(t)) 2 1 − E(t) + 1 − ν 3 (1 + E(t))E(t) 1 − E(t) + 1 + E(t) − θ(1 − √ E(t)) 1 − E(t) ≤ 1 + Cθ E(t) + C E(t) 1 − E(t) . This leads to T θii − 1 ≤ Cθ √ E(t) + C E(t) 1 − E(t) ≤ C E(t). (3.7) Lower bound estimate for T θii can be derived analogously as T θii − 1 ≥ −C √ E(t).|T θii − 1| ≤ C √ E(t). The case for i = j is simpler. We first observe from the definition of G θij that ν ρ θ Θ θij + ρU θi U θj 2 = θν ρΘ ij + ρU i U j 2 , Hence we have T θij = ρ ρ θ θ(Θ ij + U i U j ) − U θi U θj . (3.9) Then we can proceed similarly to obtain the desired result. (1) |∂ α ρ θ (x, t)| ≤ E(t), (2) |∂ α U θ (x, t)| ≤ C |α| E(t), (3) |∂ α T θ (x, t)| ≤ C |α| E(t). for some positive constant C |α| . Proof. The proof is almost identical to Lemma 3.2. We omit the proof. Lemma 3.5. Let E(t) be sufficiently small. Then determinant of the temperature tensor T ν satisfies the following estimates: (1) |∂ α {det T ν }|, |∂ α {det T νθ }| ≤ C E(t), (2) | det T ν |, | det T νθ | ≥ 1 2 , for a positive constant C independent of ν. Proof. We recall the explicit formula for det T ν derived in the proof of Lemma 2.2: det T ν = T 11 T 22 T 33 − T 2 23 T 11 − T 2 31 T 22 − T 2 12 T 33 , det T θ = T θ11 T θ22 T θ33 − T 2 θ23 T θ11 − T 2 θ31 T θ22 − T 2 θ12 T θ33 . Then (1) follow from the direct application of the estimates on the derivatives of the macroscopic fields in the preceding lemmas. To prove (2), we recall from Lemma 3.2 and Lemma 3.3 that T ii = 1 + o(E(t)) (i = 1, 2, 3), T ij = o (E(t)) (i = j). which leads to det T ν , det T θν = {1 + o(E(t))} 3 − 1 + 3 {o(E(t))} 2 {1 + o(E(t))} ≥ 1 − o(E(t)) ≥ 1 2 for sufficiently small E(t). 3.2. Uniform estimate on the temperature tensor. Recall that the nonlinear perturbation Γ(f ) contains inverse of the temperature tensor T −1 νθ : Q M ij = 1 √ µ i,j P M ij (ρ, v − U θ , U θ , T −1 θ , ν) R M ij (ρ θ , det T θ ) exp − 1 2 (v − U θ ) ⊤ T −1 θ (v − U θ )) . Now, since T ν (and T θ ) contains ν, rough estimates of its inverse may involve factors inversely proportional to ν in it, which make it impossible to derive estimates uniform around ν = 0. This is a serious problem considering that the ν = 0 corresponds to the classical BGK model. In what follows, we will carefully investigate the temperature tensor T ν and show that the seemingly problematic 1/ν factor actually does not cause any harm. The key observation is that T ν is essentially equivalent to the temperature T under our assumptions on ν. Proposition 3.1. Let −1/2 < ν < 1. Define constant C ν1 and C ν2 by C ν1 = min{1 − ν, 1 + 2ν}, C ν2 = max{1 − ν, 1 + 2ν}. Then the temperature tensor is comparable to the temperature in the following sense: C ν1 T (x, t)Id ≤ T ν (x, t) ≤ C ν2 T (x, t)Id. Furthermore, if E(f (t)) be sufficiently small, then T ν is invertible and C −1 ν2 T (x, t) Id ≤ T −1 ν (x, t) ≤ C −1 ν1 T (x, t) Id. Proof. (1) We first observe from the definition of T ν that ρT ν =   (1 − ν)ρT + νρΘ 11 νρΘ 12 νρΘ 13 νρΘ 21 (1 − ν)T + νρΘ 22 νρΘ 23 νΘ 31 νρΘ 32 (1 − ν)T + νρΘ 33   = (1 − ν)ρT Id + νρΘ = (1 − ν) 3 R 3 F (x, v, t)|v − U | 2 dvId + ν R 3 F (x, v, t)(v − U ) ⊗ (v − U )dv. Then a direct computation using k ⊤ {(v − U ) ⊗ (v − U )}k = {(v − U ) · k} 2 shows that for any k in R 3 k ⊤ {ρT ν }k = (1 − ν) 3 R 3 F (x, v, t)|v − U | 2 dv |k| 2 + ν R 3 F (x, v, t) (v − U ) · k 2 dv. We split the estimate into the following two cases. When 0 ≤ ν < 1, we have k ⊤ {ρT ν }k ≥ (1 − ν) 3 |k| 2 R 3 F (x, v, t)|v − U | 2 dv. In the case − 1 2 ≤ ν < 0, we apply Cauchy-Schwartz inequality to the second term to get k ⊤ {ρT ν }k ≥ (1 − ν) 3 R 3 F (x, v, t)|v − U | 2 dv |k| 2 + ν R 3 F (x, v, t)|v − U | 2 |k| 2 dv = (1 + 2ν) 3 |k| 2 R 3 F (x, v, t)|v − U | 2 dv. Therefore, we have (3.10) or equivalently, k ⊤ {ρT ν }k ≥ 1 3 min{1 − ν, 1 + 2ν}|k| 2 R 3 F (x, v, t)|v − U | 2 dv,k ⊤ T ν k ≥ min{1 − ν, 1 + 2ν}|k| 2 T, (3.11) We then apply Lemma 3.1 to compute T (x, t) = 1 ρ R 3 F (x, v, t)|v − U | 2 dv = 1 ρ R 3 F (x, v, t)|v| 2 dv − ρ|U | 2 = 1 1 − E(t) R 3 {µ + √ µf } |v| 2 v − CE(t) ≥ 1 1 − E(t) R 3 µ|v| 2 dv − f L ∞ x,v R 3 √ µ|v| 2 dv − CE(t) ≥ 3 − C E(t) 1 − E(t) ≥ 3 − C E(t) (3.12) for some generic constant C. From (3.11) and (3.12), we conclude that for any fixed −1/2 < ν < 1 and for sufficiently small E(t), T ν is invertible and T −1 ν ≤ 1 min{1 − ν, 1 + 2ν} T −1 Id. The proof for the upper bound is similar. Lemma 3.6. Let −1/2 < ν < 1. Suppose E(f (t)) be sufficiently small. Then there exists a positive constant C ν < ∞ such that X ⊤ {T −1 ν }Y ≤ C ν X 2 + Y 2 }, for X, Y in R 3 . Proof. By Proposition 3.1, T ν is invertible under the assumption of the lemma. Moreover, Since T ν is symmetric, T −1 ν also is symmetric. Therefore, we can compute X ⊤ T −1 ν Y = 1 2 (X + Y ) ⊤ T −1 ν (X + Y ) − X ⊤ T −1 ν X − Y ⊤ T −1 ν Y ≤ 1 2 (X + Y ) ⊤ T −1 ν (X + Y ) + 1 2 X ⊤ T −1 ν X + 1 2 Y ⊤ T −1 ν Y ≤ C min {1 − ν, 1 + 2ν} X 2 + Y 2 . for any two vectors X and Y in R 3 . Similar result holds for T θ : Lemma 3.7. Let −1/2 < ν < 1. Suppose E(f (t) ) is sufficiently small. Then T θ is invertible, and there exists a positive constant C ν < ∞ such that X ⊤ {T −1 θ }Y ≤ C ν X 2 + Y 2 }. Proof. In view of (3.6) and (3.9), we can write ρT ν as ρ θ T θ = θρT + 1 − ν 3 (ρ θ |U θ | 2 − θρ|U | 2 ) + ρ θ − θρ Id + νθ(ρ U ⊗ U − ρD) − ν(ρ θ U θ ⊗ U θ − ρ θ D θ ), so that k ⊤ {ρ θ T θ } k = θk {ρT } k + (ρ θ − θρ)|k| 2 + 1 − ν 3 ρ θ |U θ | 2 − θρ|U | 2 |k| 2 + νθ ρ(U · k) 2 − ρk ⊤ Dk − ν ρ θ (U θ · k) 2 − ρ θ k ⊤ D θ k , for k ∈ R 3 . D and D θ denote the diagonal matrix with diagonal elements U 2 1 , U 2 2 , U 2 3 and U 2 θ1 , U 2 θ2 , U 2 θ3 respectively: D =   U 2 1 0 0 0 U 2 2 0 0 0 U 2 3   , D θ =   U 2 1θ 0 0 0 U 2 2θ 0 0 0 U 2 3θ   . Then, employing Lemma 4.1 and 4.3, we obtain k ⊤ {ρ θ T θ }k ≥ 1 2 min{1 + 2ν, 1 − ν}|k| 2 + (1 − θ)|k| 2 + O(E(t))|k| 2 ≥ 1 3 min{1 + 2ν, 1 − ν}|k| 2 , k ∈ R 3 , for sufficiently small E(t). By virtue of Lemma 3.3 (1) k ⊤ {T θ }k ≥ 1 4 min{1 + 2ν, 1 − ν}|k| 2 , The rest of the proof is similar to the proof of Lemma 3.6. Lemma 3.8. Let −1/2 < ν < 1. Suppose E(f (t)) is sufficiently small. Then there exists a positive constant C ν,α < ∞ such that (1) X ⊤ {∂ α T −1 ν }Y ≤ C ν,α X 2 + Y 2 }, (2) X ⊤ {∂ α T −1 νθ }Y ≤ C ν,α X 2 + Y 2 }, for X, Y in R 3 . Proof. We have proved in Lemma 3.6 that T ν is strictly positive definite for −1/2 < ν < 1 when E(t) is sufficiently small. Therefore, T ν is invertible. Now, applying ∂ to T ν T −1 ν = I, we see that ∂T ν T −1 ν + T ν ∂ T −1 ν = 0, and thus, ∂{T −1 ν } = T −1 ν {∂T ν } T −1 ν . (3.13) Then the case |α| = 1 follows directly from this identity and Lemma 3.6 and Lemma 3.4. For general case, we recall ∂ α T ν = |β|+|γ|=|α| ∂ β T ν ∂ γ T −1 ν (3.14) and use the induction argument. The proof for T θ is almost identical. We omit it. 3.3. Local existence. We first estimate the nonlinear term Γ(f ). Note that, in contrast to the Boltzmann equation, we need to use the estimates on the macroscopic fields established in the previous section to control Γ(f ) in the energy norm. Lemma 3.9. The bilinear perturbation Γ satisfies the following estimates: (1) ∂ α β Γ(f )gdv ≤ C |α1|+|α2| ≤|α| ∂ α1 f L 2 x,v ∂ α2 f L 2 v h L 2 v +C |α1|+|α2|≤|α| |β2|≤|β| ∂ α1 f L 2 x,v ∂ α2 β2 f L 2 v ∂ α3 f L 2 v h L 2 v , +C |α1|+|α2|+|α3| ≤|α| ∂ α1 f L 2 x,v ∂ α2 f L 2 v ∂ α3 f L 2 v h L 2 v ,(2)Γ 1,2 (f, g)f dv + Γ 1,2 (g, f )f dv ≤ C sup x g L 2 x,v f 2 L 2 x,v . Γ 3 (f, g, h)f dv + Γ 3 (g, f, h)f dv + Γ 3 (g, h, f )f dv ≤ C sup x g L 2 v sup x h L 2 v f 2 L 2 x,v . (3) Γ 1,2 (f, g)h + Γ 1,2 (g, h)h L 2 x,v ≤ C sup x,v |h| sup x f L 2 v g L 2 x,v , Γ 3 (f, g, h)r + Γ 3 (g, f, h)r + Γ 3 (g, h, f )r L 2 x,v ≤ C sup x,v |h| sup x f L 2 v sup x g L 2 v h L 2 x,v . Proof. Recall that the Γ consists of Γ 1 , Γ 2 and Γ 3 . We prove this lemma only for Γ 2 , because the proof for the remaining parts are similar. Utilizing macroscopic estimates established in the previous section, we find that there exists a polynomial P α,β , which is generically defined, such that ∂ α β M ν (ρ θ , U θ , T νθ ) = C α,β |P α,β (ν, ∂ρ θ , ∂U θ , ∂T νθ )| exp − 1 2 (v − U θ ) ⊤ T −1 νθ (v − U θ ) ≤ C α,β P (v) exp − 1 − o E f (t) |v| 2 2 + o E f (t) ≤ C α,β,ε exp − 1 − o(E(t)) 1 2 − ε |v| 2 + o E f (t) , where ∂ denotes any of ∂ m n such that m ≤ |α| and n ≤ |β|. Therefore, there exists a positive number a depending on α, β and ν such that 1 √ µ ∂ α β Q M ≤ C α,β exp −a|v| 2 (3.15) (1) By (3.15) and Hölder inequality, we see R 3 |∂ α β Γ 2 (f )g|dv ≤ |α1|+|α2|+|α3| =|α| R 3 ∂ α β Q M ∂ α1 f, e i ∂ α2 f, e j gdv ≤ C |α1|+|α2|+|α3| =|α3| R 3 exp −a|v| 2 ∂ α1 f, e i ∂ α2 f, e j gdv ≤ C |α1|+|α2|+|α3| =|α| R 3 exp −|v| 2 gdv ∂ α1 f L 2 v ∂ α2 f L 2 v ≤ C |α1|+|α2|+|α3| =|α| exp −a|v| 2 L 2 v g L 2 v ∂ α1 f L 2 v ∂ α2 f L 2 v ≤ C |α1|+|α2|+|α3| =|α| ∂ α1 f L 2 v ∂ α2 f L 2 v g L 2 v . (2) can be estimated similarly as Γ 2 (f, g)f dxdv ≤ C R 3 f L 2 v g L 2 v R 3 exp −a|v| 2 f dv dx ≤ C R 3 f L 2 v g L 2 v f L 2 v dx ≤ C sup x g L 2 v f 2 L 2 x,v . (3) For Φ ∈ L 2 x,v , we have Γ 2 (f, g)r, Φ ≤ C R 3 f L 2 v g L 2 v rΦ L 2 v dx ≤ C sup x,v |r| R 3 f L 2 v g L 2 v Φ L 2 v dx ≤ C sup x,v |r| R 3 f 2 L 2 v g 2 L 2 v dx 1 2 Φ L 2 x,v ≤ C sup x,v |r| sup x f L 2 v g L 2 x,v Φ L 2 x,v . Therefore, the duality argument gives Γ 2 (f, g)r L 2 x,v ≤ C sup x,v |r| sup x f L 2 v g L 2 x,v . From the estimates in Lemma 3.9, the following local existence theorem can be proved by standard arguments (See, e.g [19,41]). Theorem 3.10. Let ν be a fixed constant such that −1/2 ≤ ν < 1. Let F 0 = g 0 + √ µf 0 ≥ 0 and f 0 satisfies the conservation laws (2.5). Then there exists M 0 > 0, T * > 0, such that if T * ≤ M0 2 and E(f 0 ) < M0 2 , there is a unique solution f (x, v, t) to the ES-BGK model (2.3) such that (1) The high order energy E f (t) is continuous in [0, T * ) and uniformly bounded: sup 0≤t≤T * E f (t) ≤ M 0 . (2) The distribution function remains positive in [0, T * ): F (x, v, t) = µ + √ µf (x, v, t) ≥ 0. (3) The conservation laws (2.5) hold for all [0, T * ]. Proof. We consider the following iteration scheme. ∂ t F n+1 + v · ∇ x F n+1 = ρ n T n 1 − ν M ν (F n ) − F n+1 , (3.16) where M(F n ) is defined by M ν (F n ) = ρ n det(2πT ) exp 1 2 (v − U n ) {T n ν } −1 (v − U n ) . ρ n , U n and T n ν denote the local density, bulk velocity and the temperature tensor associated with F n = µ + √ µf n . With estimates on the nonlinear perturbation in Lemma 3.9, it is standard to prove the local existence (See [19,41]). The only thing to be careful about is whether the temperature tensor T n ν remains strictly positive definite for each n, so that the iteration scheme is well-defined in each step. But this follows directly from Proposition 3.1 and Lemma 3.6 -3.8. Global Existence Now, having all the necessary estimates at hand, the global existence can be established using standard arguments (See [19,41]). We sketch the proof in this section. First, we need to recover the degeneracy of the linearized relaxation operator to obtain the full coercivity. For this, we define a(x, t) = R 3 f √ µdv, b i (x, t) = R 3 f v i √ µdv (i = 1, 2, 3), c(x, t) = R 3 f |v| 2 √ µdv. We also define a macroscopic projection P as follows: P f = a(x, t) √ µ + i b i (x, t)v i √ µ + c(x, t)|v| 2 √ µ. Note that P is not identical to P 0 but equivalent. Since L ν {P f } = 0 for −1/2 < ν < 1 by Corollary 2.1, we can split the linearized ES-BGK model (2.3) into the macroscopic part and the microscopic part as follows: {∂ t + v · ∇ x }{P f } = −{∂ t + v · ∇ x }{(I − P )f } + L{(I − P )f } + Γ(f ). We then expand the l.h.s and r.h.s with respect to the following basis (1 ≤ i, j ≤ 3): √ µ, v i √ µ, v i v j √ µ, v 2 i √ µ, v i |v| 2 √ µ ,(4.1) and compare coefficients on both sides to obtain the following micro-macro system [19]: ∂ t a = ℓ a + h a , ∂ t b i + ∂ xi a = ℓ abi + h abi , ∂ xi b j + ∂ xj b i = ℓ ij + h ij (i = j) (4.2) ∂ xi b i + ∂ t c = ℓ bci + h bci , ∂ xi c = ℓ ci + h ci , for i, j = 1, 2, 3. Then, by carefully studying this system, we find that the macroscopic part can be controlled by the macroscopic part as follows (See [19]): |α|≤N ∂ α a L 2 x + ∂ α b L 2 x + ∂ α c L 2 x ≤ C |α|≤N −1 ∂ α (ℓ ν + h ν ) L 2 x . (4.3) We slightly abused the notation on the r.h.s for the simplicity of presentation. On the other hand, we can bound ℓ ν and h ν by the energy norm of f as |α|≤N −1 ∂ α (ℓ ν + h ν ) L 2 x ≤ C ν |α|≤N (I − P )∂ α f L 2 x,v + C ν M 0 |α|≤N ∂ α f L 2 x,v . Combining this with (4.3), we see that |α|≤N ∂ α P f 2 L 2 x,v ≤ 1 C |α|≤N ∂ α a 2 L 2 x,v + ∂ α b 2 L 2 x,v + ∂ α c 2 L 2 x,v ≤ C |α|≤N ∂ α (I − P )f 2 L 2 x,v + C √ M 0 |α|≤N ∂ α f 2 L 2 x,v , which implies |α|≤N P ∂ α f 2 L 2 x,v ≤ C |α|≤N (I − P )∂ α f 2 L 2 x,v . (4.4) Therefore, Proposition 2.8 together with (4.4) and the equivalence of P 0 and P imply the coercivity estimate for L ν : There exists δ ν = δ(ν) > 0 such that |α|≤N L ν ∂ α f, ∂ α f ≤ −δ ν |α|≤N ∂ α f (t) L 2 x,v , (4.5) when f is sufficiently small in the energy norm. We are now ready to derive the nonlinear energy estimates which enables us to extend the local solution into the global one. Let f be the smooth local in time solution constructed in Theorem 3.10. Taking ∂ α on both sides of (2.3), we obtain ∂ t ∂ α f + v · ∇ x ∂ α f = L∂ α f + ∂ α Γ(f ). We then take inner product with ∂ α f d dt ∂ α f 2 L 2 x,v ≤ L∂ α f, ∂ α f L 2 x,v + ∂ α Γ(f ), ∂ α f L 2 x,v , and apply the coercivity estimates (4.5) together with the nonlinear estimates in Lemma 3.9 to derive E α 0 : 1 2 d dt ∂ α f 2 L 2 x,v + δ ν |α|≤N ∂ α f 2 L 2 x,v ≤ C E(f (t))D(f (t)), where D(f (t)) denotes D(f (t)) = |α|+|β|≤N ∂ α β f (t) 2 L 2 x,v . We now turn to the general case involving the derivatives in the velocity variables. Applying ∂ α β to (2.3), we get ∂ t + v · ∇ x + ν 0 ∂ α β f = ∂ β1 v · ∇ x ∂ α β−β1 f + ∂ β P ∂ α f + ∂ α β Γ(f, f ). We multiply ∂ α β f , integrate over R 3 x × R 3 v and apply Hölder inequality with Lemma 3.9 to see E α β : 1 2 d dt ∂ α β f 2 L 2 x,v + ν 0 ∂ α β f 2 L 2 x,v ≤ C i ∂ α+ei β−ei f L 2 x,v ∂ α β f L 2 x,v + C ∂ α f L 2 x,v ∂ α β f L 2 x,v + C Ef ((t) )D(f (t)). Then, we split the first two terms in the r.h.s using Young's inequality and gather relevant terms together to obtain E α β : 1 2 d dt ∂ α β f 2 L 2 x,v + ν 0 2 ∂ α β f 2 L 2 x,v ≤ C ε i ∂ α+ei β−ei f 2 L 2 x,v + C ε ∂ α f 2 L 2 x,v + CE(f (t) )D(f (t)). Now, we observe that the r.h.s of |β|=m+1 E α β can be controlled by the good terms of C m |β|=m E α β + C m α E α if C m is sufficiently large. By good terms, we mean the production terms on the l.h.s. Therefore, we can find constants C m and δ m inductively such that |α|+|β|≤N, |β|≤m C m d dt ∂ α β f 2 L 2 x,v + δ m ∂ α β f 2 L 2 x,v ≤ C N E(f (t) )D(f (t)). From this energy estimate, the existence of global solutions follows from the standard continuity argument. Remaining part of the Theorem 1.1 can be established in the exactly same manner as in the classical BGK case [41]. This completes the proof. Acknowledgement The author would like to thank Prof. Yan Guo and Prof. Kazuo Aoki and Prof. Tai Theorem 1. 1 . 1Let −1/2 < ν < 1 and N ≥ 4. Let F 0 = µ + √ µf 0 ≥ 0 and suppose f 0 satisfies (2.5). Then there exist positive constants δ ν and C = C(N, ν), such that if E(f 0 ) < δ ν , then there exists a unique global solution f to (1.4) such that generically. They may change line after line during the argument. But explicit form is not important as long as we keep in mind the structural assumptions H M and H Aν . To simplify the notation further, we define Q M ij and Q Aν i as Lemma 3 . 4 . 34Let E(t) be sufficiently small. Then we have -Ping Liu for fruitful discussions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2014R1A1A1006432) ). Then, after linearization around the global Maxwellian,the ES-BGK model takes the following form (See Section 2 for precise definition of each term) Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. P Andries, J.-F Bourgat, P Le Tallec, B Perthame, Comput. Methods Appl. Mech. Engrg. 19131P. Andries, J.-F. Bourgat, P. Le Tallec, B. Perthame, Numerical comparison between the Boltz- mann and ES-BGK models for rarefied gases, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 31, 3369-3390. The Gaussian-BGK model of Boltzmann equation with small Prandtl number. P Andries, P Le Tallec, J.-P Perlat, B Perthame, Eur. J. Mech. B Fluids. 196P. Andries, P. Le Tallec, J.-P. Perlat, B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B Fluids 19 (2000), no. 6, 813-830. Fluid dynamic limites of kinetic equations. I. Formal derivations. C Bardos, F Golse, D Levermore, J. Stat. Phys. 631-2C. Bardos, F. Golse, D. Levermore, Fluid dynamic limites of kinetic equations. I. Formal derivations, J. Stat. Phys. 63 (1991), no. 1-2, 323-344. Fluid dynamic limites of kinetic equations. II. Convergence proofs for the Boltzmann equation. C Bardos, F Golse, D Levermore, Comm. Pure. Appl. Math. 46C. Bardos, F. Golse, D. Levermore, Fluid dynamic limites of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure. Appl. Math 46 (1993), 667-753. Global existence and large-time behavior for BGK model for a gas with non-constant cross section. A Bellouquid, Transport Theory Statist. Phys. 322A. Bellouquid, Global existence and large-time behavior for BGK model for a gas with non-constant cross section, Transport Theory Statist. Phys. 32 (2003) no. 2, 157-185. Krook A model for collision processes in gases. Small amplitude process in charged and neutral one-component systems. P L Bhatnagar, E P Gross, M , Physical Revies. 94P.L. Bhatnagar, E. P. Gross, M. Krook A model for collision processes in gases. Small amplitude process in charged and neutral one-component systems, Physical Revies, 94 (1954), 511-525. A BGK model for small Prandtl number in the Navier-Stokes approximation. F Bouchut, B Perthame, J. Stat. Phys. 711-2F. Bouchut, B. Perthame, A BGK model for small Prandtl number in the Navier-Stokes approxima- tion, J. Stat. Phys. 71 (1993), no. 1-2, 191-207. The Boltzmann Equation and Its Application. C Cercignani, Springer-VerlagC. Cercignani, The Boltzmann Equation and Its Application, Springer-Verlag, 1988. The Mathematical Theory of Dilute Gases. C Cercignani, R Illner, M Pulvirenti, Springer-VerlagC. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases. Springer-Verlag, 1994. C Chapman, T G Cowling, The mathematical theory of non-uniform gases. Cambridge University PressC. Chapman, T. G. Cowling, The mathematical theory of non-uniform gases, Cambridge University Press, 1970. An energy method for the BGK model. W M Chan, City University of Hong KongM. Phil thesisW. M. Chan, An energy method for the BGK model, M. Phil thesis, City University of Hong Kong, 2007. R Duan, hypocoercivity of the linearized dissipative kinetic equations. R. Duan, hypocoercivity of the linearized dissipative kinetic equations, Nonlinearity, 24 (2011), no. 8, 2165-2189 Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space. R Duan, R Strain, Comm. Pure. Appl. Math. 64R. Duan, R. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure. Appl. Math. 64 (2011), no2, 375-413. Strain Optimal time decay of the Vlasov-Poinsson-Boltzmann system in R 3 , Arch. Rational. R Duan, R , Mech. Anal. 1991R. Duan R. Strain Optimal time decay of the Vlasov-Poinsson-Boltzmann system in R 3 , Arch. Ra- tional. Mech. Anal. 199 (2010), no.1, 291-328. An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation. F Filbet, S Jin, J. Sci. Comput. 462F. Filbet, S. Jin, An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput. 46 (2011), no.2, 204-224. Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls. M A Galli, R Torczynski, Phys. Fluids. 2330601M.A. Galli, R. Torczynski, Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls, Phys. Fluids, 23 (2011) 030601 The Cauchy Problmes in Kinetic Theory. R Glassey, SIAMR. Glassey, The Cauchy Problmes in Kinetic Theory, SIAM 1996. The Boltzmann equation in the whole space. Y Guo, Indiana Univ. Math. J. 534Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53 (2004). no.4, 1081-1094 The Vlasov-Maxwell-Boltzmann system near Maxwellians. Y Guo, Invent. Math. 1533Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math. 153 (2003) no.3, 593-630 The Vlasov-Poisson-Boltzmann system near Maxwellians. Y Guo, Comm. Pure. Appl. Math. 559Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure. Appl. Math., 55 (2002) no.9, 1104-1135. Kinetic theory of schock structure using and ellipsoidal distribution function. L H Holway, Proc. Fourth Internat. Sympos., Univ. Toronto, 1964). Fourth Internat. Sympos., Univ. Toronto, 1964)New YorkAcademic PressIL. H. Holway, Kinetic theory of schock structure using and ellipsoidal distribution function, Rarefied Gas Dynamics, Vol. I (Proc. Fourth Internat. Sympos., Univ. Toronto, 1964), Academic Press, New York, (1966), pp. 193-215. Convergence of a weighted particle method for solving the Boltzmann (B.G,K.) equaiton. D Issautier, Siam Journal on Numerical Analysis. 336D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (B.G,K.) equaiton, Siam Journal on Numerical Analysis, 33, no 6 (1996), 2099-2199. The Boltzmann equation and thirteen moments. S Kawashima, Japan J. Appl. Math. bf. 7S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math. bf 7 (1990), 301-320. Energy method for Boltzmann equation. T.-P Liu, T Yang, T S.-H Yu, Phys. D. 1883-4T.-P. Liu, T. Yang, T. S.-H. Yu, Energy method for Boltzmann equation, Phys. D 188 (2004), no. 3-4, 178-192. Uniqueness for the BGK-equation in R n and the rate of convergence for a semi-discrete scheme. S Mischler, Differential integral Equations. 95S. Mischler, Uniqueness for the BGK-equation in R n and the rate of convergence for a semi-discrete scheme, Differential integral Equations 9 (1996), no.5, 1119-1138. Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number. L Mieussens, H Struchtrup, Phys. Fluids. 168L. Mieussens, H. Struchtrup, Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number, Phys. Fluids 16 (2004), no.8. 2797-2813 Global existence to the BGK model of Boltzmann equation. B Perthame, J. Differential Equations. 821B. Perthame, Global existence to the BGK model of Boltzmann equation J. Differential Equations. 82 (1989), no.1, 191-205. Weighted L ∞ bounds and uniqueness for the Boltzmann BGK model. B Perthame, M Pulvirenti, Arch. Rational Mech. Anal. 1253B. Perthame, M. Pulvirenti, Weighted L ∞ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal. 125 (1993), no. 3, 289-295. Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation. G Russo, P Santagati, S , - B Yun, SIAM J. Numer. Anal. 50311111135G. Russo, P. Santagati, S,-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation, SIAM J. Numer. Anal. 50 (2012), no. 3, 11111135. From the BGK model to the Navier-Stokes equations. L Saint-Raymond, Ann. Sci. Ecole Norm. Sup. 362L. Saint-Raymond, From the BGK model to the Navier-Stokes equations, Ann. Sci. Ecole Norm. Sup 36 (2003), no.2, 271-317. Discrete time Navier-Stokes limit for the BGK Bltzmann equation. L Saint-Raymond, Comm. Partial Differential Equations. 271-2L. Saint-Raymond, Discrete time Navier-Stokes limit for the BGK Bltzmann equation, Comm. Partial Differential Equations 27 (2002), no. 1-2, 149-184. Y Sone, Kinetic Theory and Fluid Mechanics. BostonBirkhäuserY. Sone, Kinetic Theory and Fluid Mechanics, Boston: Birkhäuser, 2002. Y Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications. BostonBrikhäuserY. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Boston: Brikhäuser, 2006. The BGK-model with velocity-dependent collision frequency. H Struchtrup, Contin. Mech. Thermodyn. 91H. Struchtrup, The BGK-model with velocity-dependent collision frequency, Contin. Mech. Thermo- dyn. 9 (1997), no.1 , 23-31. Mesoscopic transport equaitons for rarefied gas flows: Approximation methods in kinetic theory. H Struchtrup, SpringerH. Struchtrup, Mesoscopic transport equaitons for rarefied gas flows: Approximation methods in kinetic theory, Springer. 2005. On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation. S Ukai, Proc. Japan Acad., Ser. A53. Japan Acad., Ser. A53S. Ukai, On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation, Proc. Japan Acad., Ser. A53, 179-184 (1974) Stationary solutions of the BGK model equation on a finite interval with large boundary data. S Ukai, Transport theory Statist. Phys. 21S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data, Transport theory Statist. Phys. 21 (1992) no.4-6. Mathematical Theory of Boltzmann equation. S Ukai, T Yang, Lecture Notes Series. 8Liu Bie Ju Center for Math. Sci, City University of Hong KongS. Ukai, T. Yang, Mathematical Theory of Boltzmann equation, Lecture Notes Series. no. 8, Liu Bie Ju Center for Math. Sci, City University of Hong Kong, 2006. A Review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics. C Villani, I. North-Holland. AmsterdamC. Villani, A Review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics. Vol. I. North-Holland. Amsterdam, 2002, 71-305 P Walender, On the temperature jump in a rarefied gas. Ark, FysP. Walender, On the temperature jump in a rarefied gas, Ark, Fys. 7 (1954), 507-553. Cauchy problem for the Boltzmann-BGK model near a global Maxwellian. S.-B Yun, J. Math. Phy. 511224S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian, J. Math. Phy. 51 (2010), no. 12, 123514, 24pp. L p solutions to the Cauchy problem of the BGK equation. X Zhang, S Hu, J. Math. Phys. 481117X. Zhang, S. Hu, L p solutions to the Cauchy problem of the BGK equation, J. Math. Phys. 48 (2007) no.11, 113304, 17pp. Ellipsoidal statistical Bhatnagar-Gross-Krook model with velocity dependent collision frequency. Y Zheng, H Struchtrup, 127103, 17pp. Department of Mathematics. 17Sungkyunkwan UniversityRepublic of Korea E-mail address: [email protected]. Zheng, H. Struchtrup, Ellipsoidal statistical Bhatnagar-Gross-Krook model with velocity depen- dent collision frequency, Phys. Fluids 17 (2005), 127103, 17pp. Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea E-mail address: [email protected]
[]
[ "Photoinduced magnetic bound state in itinerant correlated electron system with spin-state degree of freedom", "Photoinduced magnetic bound state in itinerant correlated electron system with spin-state degree of freedom" ]
[ "Yu Kanamori \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n", "Jun Ohara \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n\nCore Research for Evolutional Science and Technology (CREST)\n980-8578SendaiJapan\n", "Sumio Ishihara \nDepartment of Physics\nTohoku University\n980-8578SendaiJapan\n\nCore Research for Evolutional Science and Technology (CREST)\n980-8578SendaiJapan\n" ]
[ "Department of Physics\nTohoku University\n980-8578SendaiJapan", "Department of Physics\nTohoku University\n980-8578SendaiJapan", "Core Research for Evolutional Science and Technology (CREST)\n980-8578SendaiJapan", "Department of Physics\nTohoku University\n980-8578SendaiJapan", "Core Research for Evolutional Science and Technology (CREST)\n980-8578SendaiJapan" ]
[]
Photo-excited state in correlated electron system with spin-state degree of freedom is studied. We start from the two-orbital extended Hubbard model where energy difference between the two orbitals is introduced. Photo-excited metastable state is examined based on the effective model Hamiltonian derived by the two-orbital Hubbard model. Spin-state change is induced by photoirradiation in the low-spin band insulator near the phase boundary. High-spin state is stabilized by creating a ferromagnetic bound state with photo-doped hole carriers. An optical absorption occurs between the bonding and antibonding orbitals inside of the bound state. Time-evolution for photo-excited states is simulated in the time-dependent mean-field scheme. Pair-annihilations of the photo-doped electron and hole generate the high-spin state in a low-spin band insulator. We propose that this process is directly observed by the time-resolved photoemission experiments.
10.1103/physrevb.86.045137
[ "https://arxiv.org/pdf/1204.1847v1.pdf" ]
118,495,979
1204.1847
5f777ca170702ec924868dc8637b124ae5469458
Photoinduced magnetic bound state in itinerant correlated electron system with spin-state degree of freedom (Dated: May 10, 2014) Yu Kanamori Department of Physics Tohoku University 980-8578SendaiJapan Jun Ohara Department of Physics Tohoku University 980-8578SendaiJapan Core Research for Evolutional Science and Technology (CREST) 980-8578SendaiJapan Sumio Ishihara Department of Physics Tohoku University 980-8578SendaiJapan Core Research for Evolutional Science and Technology (CREST) 980-8578SendaiJapan Photoinduced magnetic bound state in itinerant correlated electron system with spin-state degree of freedom (Dated: May 10, 2014)arXiv:1204.1847v1 [cond-mat.str-el] 9 Apr 2012numbers: 7820Ls7110-w7820Bh7847J- Photo-excited state in correlated electron system with spin-state degree of freedom is studied. We start from the two-orbital extended Hubbard model where energy difference between the two orbitals is introduced. Photo-excited metastable state is examined based on the effective model Hamiltonian derived by the two-orbital Hubbard model. Spin-state change is induced by photoirradiation in the low-spin band insulator near the phase boundary. High-spin state is stabilized by creating a ferromagnetic bound state with photo-doped hole carriers. An optical absorption occurs between the bonding and antibonding orbitals inside of the bound state. Time-evolution for photo-excited states is simulated in the time-dependent mean-field scheme. Pair-annihilations of the photo-doped electron and hole generate the high-spin state in a low-spin band insulator. We propose that this process is directly observed by the time-resolved photoemission experiments. I. INTRODUCTION Optical properties and photo-induced phenomena in solids are one of the attractive themes in recent solid state physics. In particular, correlated electron system is one of the main targets for photo-induced exotic phenomena. Because of strong electron-electron interaction and multidegrees of freedom, e.g. spin, charge, orbital and so on, a number of electronic and structural phases are realized under a subtle balance of interactions. 1 By irradiation of intensive laser pulse into one of the phases, a system is transferred into different phase transiently or permanently. This is termed photo-induced phase transition (PIPT) phenomena. 2 Nowadays, a number of experimental and theoretical studies have been done in PIPT phenomena in transition-metal oxides, 3-7 low-dimensional organic salts, [8][9][10][11] and others. Among the multi-degrees of freedom, spin-state degree of freedom has attracted much attention from view point of optical manipulation of magnetism. In a certain magnetic ion, different magnitude of the spin angular momentum is realized by changing external fields, such as temperature, pressure and photons. This is termed the spinstate transition and is caused by a competition between the crystalline field splitting and the Hund's coupling. A well known example of the photo-induced spin state change is seen in the so-called spin cross-over complexes, such as Prussian blue analogue complex. [12][13][14][15] Here, photons bring about a charge transfer from the neighboring Fe ions to Co ions, associated with the spin-state change in Co ions from the low-spin (LS) state to the high-spin (HS) one. A main mechanism of the cooperative spinstate transition in a series of materials is supposed to be the elastic interaction; [16][17][18][19] a local volume change of a metal-ligand cluster propagates over a crystal lattice. Another material where photo-induced spinstate change is realized is the perovskite cobaltites R 1−x A x CoO 3 (R: a rear-earth ion, A: an alkaline-earth ion) and their families. [20][21][22] In an undoped compound LaCoO 3 , 23,24 a formal valence of a Co ion is 3+ with a d 6 electron configuration. There are possible three spin states: the LS state with the (t 2g ) 6 (e g ) 0 configuration, the intermediate-spin (IS) state with (t 2g ) 5 (e g ) 1 , and the HS state with (t 2g ) 4 (e g ) 2 . It is supposed from the electric resistivity and the magnetic susceptibility measurements that the LS band insulator in low temperatures are changed into the HS or IS metallic state with increasing the temperature (T ). [25][26][27][28] By substitution of R by A, corresponding to hole doping into the non-magnetic insulating ground state, a system shows ferromagnetic metallic behavior. [29][30][31][32] One key point to understand the electronic and magnetic properties in cobaltites are strong correlation between electron conduction and magnetism, i.e. charge and spin degrees of freedom of electrons. 33 Optical irradiation and manipulation in perovskite cobaltites and related materials have been examined by the ultrafast optical pump-probe measurements. 34,35 Recently, detailed experiments and analyses have been done in so-called A-site ordered perovskite-type RBaCo 2 O 6.δ crystals by Okimoto and co-workers. 34 After pump pulse is introduced into the LS insulator, a metallic state, which is different from the high-temperature metallic state, is observed in the optical conductivity spectra. This photo-induced state strongly depends on the R species, which is supposed to control a ratio of electron correlation and band width. These experiments suggest that strong correlation between electronic and magnetic states remains even in the photo excited state, and tell us that the photo-irradiation phenomena in the cobalt oxides should be reexamined from different viewpoint from the photo-induced spin-state change in spin-cross over complexes. In this paper, photo-induced spin-state change in correlated electron systems is studied theoretically. From the two-orbital Hubbard model, the effective Hamilto-nian for the photo-excited state is derived. The photoexcited metastable state is obtained through analyses of the effective Hamiltonian by using the exact diagonalization method. By irradiation of photons into the LS band insulator near the phase boundary, the HS state is induced. It is found that the HS state is stabilized by forming a bound state with a photo-doped hole. This bound state brings about a characteristic peak structure in the optical spectra in the photo-excited state. A time evolution after photo-irradiation is examined in the timedependent mean-field scheme. A creation of the HS state is caused by a pair annihilation of photo-doped electron and hole. This mechanism is able to be confirmed by the time-resolved photoemission spectroscopy experiments. In Sect. II, the model Hamiltonian and the effective model for the photo-excited states are introduced. In Sect. III, numerical results of the electronic states before and after photo-irradiation are presented. In Sect. IV, the time-dependence of the photo-excited states are shown. Section V is devoted to discussion and concluding remarks. A brief report for the previous studies in the photo-induced metastable state was published in Ref. 36. II. MODEL A. two-orbital Hubbard model We start from the two-orbital Hubbard model as a minimal model to examine the photo-induced spin-state change. Two orbitals, termed A and B corresponding to the e g and t 2g orbitals in a Co ion, respectively, are introduced in each site in a lattice. The crystalline field splitting between A and B is represented by ∆ = ε A − ε B > 0 where ε A and ε B are the level energies of the A and B orbitals, respectively. The model Hamiltonian is given as H = H U + H t ,(1) where we define the on-site term, H U = ∆ iσ c † iAσ c iAσ + U iγ n iγ↑ n iγ↓ + U ′ iσσ ′ n iAσ n iBσ ′ + J iσσ ′ c † iAσ c † iBσ ′ c iAσ ′ c iBσ + I iγ c † iγ↑ c † iγ↓ c iγ↓ c iγ↑ ,(2) and the inter-site term, H t = − ij γσ t γ c † iγσ c jγσ + H.c. .(3) Here, c † iγσ is the electron creation operator at site i with orbital γ(= A, B) and spin σ(=↑, ↓). We define the number operator n iγσ = c † iγσ c iγσ and a subscript = (B, A). The intra-orbital Coulomb interaction U , the inter-orbital Coulomb interaction U ′ , the pair-hopping I, and the Hund's coupling J are introduced. The electron transfer integrals between the nearest-neighboring (NN) sites are set to be diagonal with respect to the orbital. We assume a relation t B < t A by considering the transfer-integrals in perovskite oxides, and we take t A = 1 as a unit of energy. Let us consider the local electronic structure, in which two electrons occupy each site, and the electron transfers are set to be zero. The following LS state with S = 0 and the HS state with S = 1 are the possible ground states (see Figs. 1(a) and (b)). The eigen function and the eigen energy for the LS state are given as lowest-energy state inside of the energy surface, where density of the photo-excited electron-hole pairs is fixed. A schematic picture is shown in Fig. 2. Bold curves represent the adiabatic energy surfaces before and after photoirradiation as functions of a number of the HS sites. Photons excite a system from the lowest-energy surface to the higher-energy surface. Through several kinds of relaxation processes, the system settles down in the lowest energy state in the higher energy surface. Instead of timedependent simulations for the photo-excited dynamics, we examine the lowest-energy state inside of the energy surface where a number of the electron-hole pair is one in a N -site system. This state is termed the photo-induced metastable state, from now on. We derive the two effective Hamiltonians, where numbers of the electron-hole pairs are zero and one in a N -site system. The effective Hamiltonians are derived by the perturbational processes from the two-orbital Hubbard model in Eq. (1). The inter-site transfer term, H t , is treated as the perturbation term. As for the effective Hamiltonian before the photo-irradiation, the HS and LS states, defined in Eq. (4) and Eqs. (7)- (9), respectively, are adopted as the basis states. Other local-states, where two electrons occupy each site, have higher energies of the order of ∆, J, and U than the LS and HS sates. By considering all of the second-order perturbational processes, the Hamiltonian is given as H 0 = H U + J HH ij (S i · S j − 1) P H i P H j − J LL ij P L i P L j − J HL ij P L i P H j + P H i P L j − J ++ ij I − i I − j (S i · S j − 1) + (S i · S j − 1) I + i I + j + J +− ij I − j (S i · S j + 1) I + i + I − i (S i · S j + 1) I + j .(10) Here, S i is the spin operator defined by S i = (1/2) γσσ ′ c † iγσ σ σσ ′ c iγσ ′ with the Pauli matrices σ, and P L i and P H i are the projection operators for the LS and HS state defined by P L i = |ψ Li ψ Li | ,(11) and P H i = l=(+,0,−) |ψ Hli ψ Hli | ,(12) respectively. The operators I + i and I − i change the spin state as I + i = |ψ H0i ψ Li | ,(13) and I − i = |ψ Li ψ H0i | .(14) The prefactors in each term in Eq. (10) are the exchange constants defined by J HH = t 2 A + t 2 B U + J ,(15)J LL = 4f 2 B f 2 A t 2 A + t 2 B 2U ′ + 2∆ J − U − J ,(16)J HL = t 2 A + t 2 B f 2 B U ′ + ∆ J − ∆ + f 2 A U ′ + ∆ J + ∆ ,(17)J ++ = 2t A t B f B f A 1 U + J + 1 2U ′ − U − J + 2∆ J ,(18)J +− = 2t A t B f 2 B U ′ + ∆ J − ∆ + f 2 A U ′ + ∆ J + ∆ ,(19) where we define ∆ J = √ ∆ 2 + J 2 . The effective Hamiltonian after photo-irradiation is derived in the same way. As the unperturbed states, in addition to the LS and HS states, we introduce the states where numbers of electrons in a site are one or three (see Figs. 1(c) and (d)). These local states are termed the hole state and the electron state, respectively. The wave functions are given as |ψ hσ = c † Aσ |0 ,(20) and |ψ eσ = c † Aσ c † B↑ c † B↓ |0 ,(21) respectively. The eigen energies are E e = ∆+U +2U ′ −J for the electron state and E h = 0 for the hole state. We assume that a number of both the electron state and the hole state is one in a N -site cluster. The calculated effective Hamiltonian is classified by the electronic states in the NN sites as H 1 = H 0 + H eh + H e + H h .(22) The first term H 0 corresponds to H 0 in Eq. (10), where both the electron and hole states are not concerned in the interactions. The second term is for the interactions between the electron state and the hole state. The third and fourth terms describe the interactions between the electron state and LS or HS, and the interactions between the hole state and LS or HS, respectively. Explicit forms for the Hamiltonian are given in Appendix. The ground state before photoirradiation and the photo-induced metastable state are obtained in the effective Hamiltonians in Eqs. (10) and (22), respectively, which are analyzed by the exact-diagonalization method based on the Lanczos algorithm. Time evolutions in the photo-induced dynamics are calculated in the two-orbital Hubbard model in Eq. (1). III. ELECTRONIC STATES BEFORE AND AFTER PHOTOIRRADIATION A. Ground State Electronic structure in the ground state is examined by analyzing the effective Hamiltonian H 0 in a finite size cluster system. Several physical quantities are plotted in Fig. 3 as a function of the Hund's coupling J at ∆/t A = 10. We introduce the number density of the HS states which are estimated from the electron number in the orbital A defined by n HS = 1 N i n iA ,(23) the spin correlation function, S(q) = 1 2N 2 ij e −iq·(ri−rj ) S i · S j ,(24)I(q) = 4 N 2 ij e −iq·(ri−rj ) I z i I z j .(25) Here, we define the spin-state operator as a projection operator by I z i = 1 2 m=±1,0 |ψ Hmi ψ Hmi | − |ψ Li ψ Li | ,(26) which takes 1/2 and −1/2 for the HS and LS states, respectively. With increasing J, three different phases appear in Fig. 3. In a region for small J, both n HS and S(q) are zero, and I(0, 0) is almost one. On the other hand, in a region of large J, n HS and I(0, 0) are one, and S(π, π) is the largest. Two phases are identified as the LS band insulator and the HS antiferromagnetic Mott insulator. Between the two, there is an intermediate phase where n HS = 0.5, and I(π, π) is one. These data imply that the HS and LS states are aligned alternately. This phase is termed the spin-state ordered phase. 37 This alternate ordering of the HS and LS states is caused by the fourth term in the right hand side in Eq. (10); J LH given in Eq. (17) represents the attractive interaction between the LS and HS states. Numerical data in several J and ∆ are summarized in a phase diagram shown in Fig. 4, where the phase boundaries in the ground state and those in the photoinduced metastable state are plotted in one figure. The detailed results in the metastable state will be presented in Sec. III B. Here, we identify the LS (HS) phase as a state, where electron numbers of the A orbital is smaller (larger) than 0.3. We confirm that the size dependence of the phase boundaries is of the order of 0.01t A . The LS and HS phases appear in regions of large ∆ and large J, respectively. The spin-state ordered phase appears between the two phases. B. Photo-induced Metastable State Several physical quantities in the photo-induced metastable state are presented in Fig. 5 as a function of the Hund's coupling J at ∆/t A = 10. The number density of HS states is estimated from a number of electrons in the orbital A defined by n HS = 1 N − 2 i n iA − 1 ,(27) where the electron and hole states are subtracted in a denominator. It is shown in the Fig. 5(a) that n HS in the photo-induced metastable state is finite between 3.25 < J/t A < 3.30 where n HS in the ground state is zero. Different value between the two states implies that one HS state is generated in the N -site cluster. This phase in the photo-induced metastable state is distinct from the spin-state ordered phase observed in the ground state; the spin-state correlation functions at any q are not remarkable, and a weak spin correlation at q = (0, 0) is observed. Detail properties of this phase are introduced latter. The phase diagram in the photo-excited metastable state is presented in Fig. 4, together with that in the ground state. The phase boundary between the LS and LS-HS mixed phases shifts to a region of the LS phase. There is a parameter region where the LS phase in the ground state is changed into the LS-HS mixed phase in the photo-excited metastable state. That is to say, the photo-irradiation induces the HS state in the LS phase at vicinity of the phase boundary. We note that the spinstate change also occurs from the HS phase to the mixed phase. Now we examine the electronic structure in the photoinduced HS state in more detail. We introduce the electronic-state distribution function defined by g M (n) = z −1 n j∈nN N i P M i+j P h i ,(28) where j∈nN N implies a summation of j connecting the n-th NN sites of i, and z n is a number of the n-th NN sites. The operator P M i (M = L, H, e) is the projection operator for the M -state at site i. The operators for the LS and HS states are define in Eqs. (11) and (12), respectively, and those for the electron and hole states are defined as P e i = σ |ψ eσi ψ eσi | ,(29) and P h i = σ |ψ hσi ψ hσi | ,(30) respectively. This function, g M (n), describes distribution of the local electronic states at the n-th NN sites from the photo-induced hole state, as shown in Fig. 6(a). Numerical results of the distribution functions in a twodimensional cluster are shown in Fig. 6(b). Parameter values are chosen to be J = 3.3t A and ∆ = 3.3t A in which the HS state is induced by photo-irradiation. A characteristic feature is shown in the HS distribution function; g H (n) is nearly 0.25 at n = 1 and zero at n ≥ 2. This implies a local bound state between the HS state and a photo-doped hole state. The size dependence of g H (n) is checked in the one-dimensional clusters, and results are shown in the inset of Fig. 6 in this bound state is monitored by the correlation function defined by i =j S j · S i P H j P h i which represents the spin correlation between the hole and HS states. Calculated value is about 0.5 which implies a ferromagnetic spin correlation. Figure 6 also shows that g e (n) monotonically increases with n. This is due to the kinetic-energy gain of the photo-excited electron. A schematic electronic structure in the photo-induced metastable state is presented in Fig. 7. Here we discuss a mechanism of the ferromagnetic HShole bound state. In the ground state, the energy difference per site between the LS state and the HS state is given by ∆E HS−LS ≡ E HS − E LS = (U ′ + ∆ − J) − (U + ∆ − √ ∆ 2 + I 2 ) in the local limit. Let us consider a situation that photo-excited electrons and holes are introduced in the LS phase at vicinity of the phase boundary, and these photo-carriers move in the system. Magnitude of the exchange process between the hole state and the LS state is given by f 2 B t B (see Eq. (A2)). Thus, when the electron and hole states move around the LS background without generation of the HS state, the kinetic energy of the hole state is −zf 2 B t B where z is a number of the NN sites. On the other hand, when the ferromagnetic HShole bound state is generated, one electron occupies the bonding orbital in the bound state and this energy gain is −t A as shown in Fig. 7. Since the electron states move in the same ways in both the two cases, energy differ- ence are ∆E BS = (−t A ) − (−zf 2 B t B ) . When this energy gain due to the bound state, |∆E BS |, overcomes the energy cost for the HS generation, ∆E HS−LS , the HS-hole bound state is realized. Above consideration for the energy balance is confirmed in the band width dependence of the phase di-agram. In Fig.8, the phase diagrams in the ground state and the photo-induced metastable state are plotted as functions of a ratio of the band widths for the A and B bands, i.e. t B /t A . In a region of t B /t A < 0.1, there is a phase space where the LS phase in the ground state is changed into the HS-LS mixed phase in the photo-induced metastable state. With increasing t B /t A , this phase space is shrunken and disappears. This is explained from above consideration where stability of the photo-induced HS state is controlled by a factor ∆E BS = (−t A ) − (−zf 2 B t B ) . This tendency for stability of the HS state is similar to the previous results in the chemical doping. 33 C. Optical Spectra In this subsection, we show the optical spectra in the photo-induced metastable state. The optical absorption spectra are defined by α αβ (ω) = − 1 N π Im ψ 0 | j α 1 ω − H ef f + E 0 + iη j β |ψ 0 ,(31) where H ef f is taken to be H 0 in Eq. (10) and H 1 in Eq. (22) for the ground state and the photo-induced metastable state, respectively, |ψ 0 and E 0 are the corresponding lowest energy state and energy, respectively, and α(β) represents a Cartesian coordinate. We introduce the current operator j α = i iγσ t γ c † iγσ c i+αγσ − H.c. ,(32) which is defined in the restricted Hilbert space in each effective Hamiltonian. A damping constant is introduced as η. The optical spectra are calculated by the exact diagonalization method based on the recursion procedure. Two dimensional finite-size clusters with the periodic boundary condition is adopted. The absorption spectra in the photo-induced metastable state, where the HS state is induced by photo-irradiation, are shown in Fig. 9. The system size is taken to be N = 8 and 10. Characteristic two peaks appear in the spectra at ω = 2.1t A and ω = 2.8t A in the case of N = 8. These are termed the peaks B and D, and their energies are denoted ω B and ω D , respectively. In order to assign these peaks, we calculate the bond correlation function in the excited states given as B (l,m) (ω n ) = − ij γσ ψ(ω n )| × P m i P l j c † iγσ c jγσ P l i P m j + H.c. |ψ(ω n ) .(33) where |ψ(ω n ) is the eigen function of the Hamiltonian H 1 corresponding to the final state of the n-th optical absorption peak, and ω n is its eigen energy. and eigen energies are obtained by using the conjugate gradient method. This function measures the bond correlation between the l and m local electronic states in the photo-excited state. Numerical results of this correlation function together with the optical absorption spectra are presented in Fig. 9(a) Results in different size cluster of N = 10 are shown in Fig. 9(b). Numerical values of B (h,HS) and B (e,LS) are almost same with the values in N = 8. Energy of the peak B is almost unchanged, but that of the peak D decreases with increasing N . These size dependences are consistent with the assignments that the peak B is attributed to the local excitation, and the peak D is related to the kinetic motion of the photo-excited electron. We further examine the size dependence of the peak positions in the one-dimensional clusters, and observe that the energy of the peak D decreases with the system size. 36 This peak is interpreted as a Drude-like component in the thermodynamic limit. The optical absorption spectra in the photo-induced metastable state, where HS is not induced, are presented in Fig. 10 (see bold lines). Two peak structure is observed in N = 10 and are almost overlapped around ω = 3.7t A in N = 8. Results are well reproduced by the following hard-core two-fermion model defined by H HC = −t A ij (a † i a j + H.c.) − t B ij (b † i b j + H.c.) − t ex ij (a † i b † j b i a j + H.c.),(34) where a i and b i are the spin-less fermion operators at site i and describe annihilations of the electron state and the hole state, respectively. We take a condition of a † i b † i = 0. The first and second terms represent kinetic motions of the electron and hole states in the LS phase, respectively, and the third term represents an exchange of the electron and hole states. This model is derived in the limiting case of ∆ >> I as follows. From Eqs. (4)-(6), we have |ψ L = c † B↑ c † B↓ |0 which is set to be a vacuum, 0 , in this model. The electron and hole states are defined from this vacuum as |ẽ = a † i 0 and |h = b † i 0 , respectively. The matrix elements for the exchange of the electron (hole) and LS states, corresponding to the first (second) term in Eq. (34), are given by (A2) and (A12). The exchange of the electron and hole states, corresponding to the last term in Eq. (34), are given in the matrix elements in Eqs. (A6) and (A9). We confirm numerically that this contribution to the optical spectra is much smaller than other terms, and set to be a small constant t ex = 10 −10 t A in the numerical calculation. In this effective model, the current operator along an α direction is given by t A f 2 B ∼ t A (t B f 2 B ∼ t B ) from Eqs.j α HC = it A i (a † i a i+α − H.c.) − it B i (b † i b i+α − H.c.),(35) The optical absorption spectra obtained in the HC model are shown by broken lines in Fig. 10. One pair of the a and b fermions is introduced in the N -site clusters. Spectra in the effective Hamiltonian are well reproduced by the HC model. Size dependences of the peak positions are examined in detail in this model. Two dimensional clusters with N = 8, 10, 4 × 4, 6 × 6, 8 × 8, 10 × 10, 12 × 12 with the periodic boundary condition are adopted. Peak energies for the lowest three peaks are plotted in the inset of Fig. 10 as functions of N −1 . Energies tend to be zero in the thermodynamic limit. We interpret that these peaks originate from the metallic behaviors of photo-doped electron and hole. IV. TIME-DEPENDENCE OF PHOTO-EXCITED STATE In this section, we show the real-time evolution of photo-exited state calculated in the mean-field scheme, 38,39 and reveal a mechanism of the photoinduced HS state. A. Formulation Time evolution of the photo-excited state is analyzed. A mean-field type decoupling is applied into the Coulomb and exchange interaction terms in the two-orbital Hub-bard model in Eq. (1) as follows, H MF = iγσ n iγσ U n iγσ − U ′ σ ′ n iγσ ′ − J n iγσ − U iγ n iγ↑ n iγ↓ + U ′ iσσ ′ n iAσ n iBσ ′ + J iσ n iAσ n iBσ − ij γσ t γ c † iγσ c jγσ + H.c. + ∆ i n ia ,(36) where · · · implies the average calculated by the timedependent mean-field wave function, and a subscriptσ is defined byσ = (↑, ↓) for σ = (↓, ↑). We note that the pair-hopping interaction in the Hamiltonian and the Fock terms are not taken into account. This is essential to reproduce the electronic states in the case of t A = t B = 0. The initial electronic wave function before the photoexcitation is obtained by solving the self-consistent equations. The photo-irradiation is simulated by excitations of electrons from the highest occupied levels to the lowest unoccupied ones at time τ = 0 with conserving the z-component of the total spin-angular momentum and the total momentum. The time evolution of the wave function is calculated in the time-dependent mean-field scheme. The time-dependent Shrödinger equation for the ν-th level, |φ ν (τ ) , is given as |φ ν (τ ) = P exp −i τ 0 dτ ′ H MF (τ ′ ) |φ ν (0) ,(37) where H MF (τ ) is the time-dependent Hamiltonian given in Eq. (36), and P is the time-ordering operator. The wave function at time τ + dτ , where dτ is short time distance, is calculated from the wave function at time τ by expanding the exponential factor as |φ ν (τ + dτ ) = µ ϕ µ (τ )|φ ν (τ ) e −iεµ(τ )dτ |ϕ µ (τ ) ,(38) where |ϕ µ (τ ) is the eigen state of H MF (τ ) with the eigen energy ε µ (τ ). In the numerical calculation, we take dτ t A = 10 −3 ∼ 10 −4 , and check that the total energy is conserved within the order of 10 −2 percent. Phase diagram in the ground state is presented in Fig. 11. Phase boundaries are determined by the HS density. We also plot the results obtained by the exact diagonalization method applied to the effective Hamiltonian shown in Fig. 4. Two results are qualitatively similar with each other, although the LS phase in the present calculation shifts to a low J region. B. Numerical Results Time evolution of the photo-excited electronic state is examined in a two dimensional N = 10 × 10 site cluster with the periodic boundary condition. A number of photon in the cluster is chosen to be N ph = 10, which are introduced into the LS phase at a vicinity of the boundary (J = 3.1t A , ∆ = 10t A ). We monitor numbers of the HS state, the photo-doped electron state, and the photodoped hole state, by the following physical quantities, N HS = iσ N HS iσ , N e = iσ N e iσ , and N h = iσ N h iσ with N e iσ = n iAσ (1 − n iAσ ) n iBσ n iBσ ,(39)N h iσ = (1 − n iAσ )(1 − n iAσ ) n iBσ (1 − n iBσ ),(40) and N HS iσ = n iAσ (1 − n iAσ ) n iBσ (1 − n iBσ ),(41) respectively. It is noticed that these are defined in products of the mean-field number density in each orbital and spin, instead of the projection operators such as P e i (Eq. (29)), P h i (Eq. (30)) and P H i (Eq. (12)), which cannot be calculated directly in the mean-field scheme. Time dependence of these numbers are plotted in Fig. 12 N e and N h decrease monotonically. That is, changes in the three numbers are correlated with each other. This result is interpreted that the HS states are created by annihilation of the photo-induced electron and hole states. Let us consider a situation where the electron and hole states adjoin, as shown in Fig. 12(b). When an electron in the A orbital transfers to the hole state, a LS-HS pair is generated. This pair is also generated by the electron transfer in the B orbital. This is termed the electron-hole pair annihilation process, from now on. Snapshots for the local electronic states are shown in Fig. 13. At τ = 10t −1 A , three numbers are almost homogeneous. At τ = 100t −1 A , distributions for the electron states start to be inhomogeneous and a vertical shape domain appears in N e i↑ . At τ = 500t −1 A , the electron states in this vertical-shape domain begin to be localized. It is shown that, at the sites where N e i↑ is large, N h i↑ and N HS are also large. These data support the mechanism of the HS creation due to the electron-hole annihilation process. After the HS states are generated, N h i↑ and N e ↑ at the same sites still remain to be large. This observation is not contradict to the electron-hole annihilation, but is due to the definitions of N h i↑ and N e i↑ [see Eqs. (40) and (39)]; these are represented in the products of the meanfield number density, instead of the projection operators. We have numerically confirmed that, as N e i and N h i increase, N HS i increases, and then N e i and N h i decrease. Relation between the numerical results of the real-space snapshots and the HS-hole bound state introduced in the previous section will be discussed in Sect. V. The electron-hole pair annihilation processes are also examined by the time dependent density of state (DOS). We define DOS as A(ω) = A e (ω) + A h (ω),(42) with the electron part A e (ω) = ν δ[ω − ε ν (τ )] c † ν c ν ,(43) and the hole part A h (ω) = ν δ[ω − ε ν (τ )] c ν c † ν .(44) The operator c † ν is the creation operator obtained by diagonalizing the Hamiltonian at time τ , ε ν (τ ) is the corresponding mean-field energy, and · · · implies the average in terms of the wave function of |ψ(τ ) . In the numerical calculation, the delta functions in Eqs. A , the original gap is almost filled out by in-gap states. These data are consistent with the localization of the electron and hole states observed in the snapshots in Fig. 13. Based on these results, we consider the energy balance in the electron-hole pair annihilation processes. On-site mean-field energies of the LS, HS, electron, and hole states are given by E LS MF = U , E HS MF = U ′ + ∆ − J, E e MF = U + 2U ′ + ∆ − J, and E h MF = 0, respectively. When one electron-hole pair is changed into one LS state and one HS state, the on-site energy is changed as ∆E eh→LH ≡ E LS MF + E HS MF − E e MF + E h MF = −U ′ which is negative, i.e. energy loss. This energy is compensated by the kinetic energy of the hole and electron states, which are not concerned in the pair-annihilation processes. This is confirmed in DOS at τ t A = 900 (see Fig. 14(d)); the hole part of DOS in the B-orbital band distributes not only to the top of the band, but also down to the middle of the band. This indicates increasing of the kinetic energy of holes with time. In the last part of this section, we examine, on the time-evolution of the photo-induced HS generation, roles of the relativistic spin-orbit (SO) interaction which breaks the spin angular-momentum conservation. Here we mimic the SO interaction in the 3d orbitals as follows, H SO = iξ i c † ia↑ c ib↓ + c † ia↓ c ib↑ − c † ib↑ c ia↓ − c † ib↓ c ia↑ ,(45) with the SO interaction constant ξ. It is demonstrated that, when this interaction acts on the LS state, the HS state is created as follows, H SO c † ib↑ c † ib↓ |0 = −iξ c † ia↑ c † ib↑ − c † ia↓ c † ib↓ |0 . (46) Numerical results of the time evolutions of N HS and N h are presented in Fig. 15, where the SO interaction constant is taken to be ξ = 0 and 0.3t A . Before τ = 200t −1 A , the SO interaction effects are not seen in N HS . However, beyond τ = 200t −1 A , N HS starts to decrease in the case of a finite ξ. The observed reduction of the HS state in the case of finite ξ is due to the transition from the HS to LS states through the SO interaction, as shown schematically in Fig. 15(b). This result indicate that roles of the SO interaction on the spin-state transition is destructive rather than constructive. V. DISCUSSION AND CONCLUSION In this section, we remark i) a connection between the calculated results in the photo-excited metastable state and the time dependent simulation in the photo-excited state, which are presented in Sects. III and IV, respectively, and ii) implications of the present theoretical results to the recent experiments. In Sect. IV, we show in the time-dependent simulation that a pair annihilation of photo-doped electron and hole generates a HS state. In this scheme, here we discuss a stability of the HS state and a role of the photo-doped hole. A correlation between the local HS state and the hole state around the local HS state is examined numerically. We introduce the number density of the local HS state at time τ , N HS iσ (τ ), defined in Eq. (41), and the number density of the hole state around the site i at time τ ′ denoted as ρ iσ (τ ′ ) ≡ ′ j N h jσ (τ ′ ). We define N h iσ (τ ′ ) in Eq. (40), and introduce a symbol ′ j which implies a summation for the NN sites of i. Data sets of ρ iσ (τ ) and N HS iσ (τ +∆τ ) with ∆τ = 10t −1 A are obtained in 100 times simulations with different initial states. Numerical results are shown in Fig. 16. A positive correlation between the two quantities is seen in this figure. In particular, in a region of high density of hole, (ρ iσ (τ ) 0.1), a number of data for small N HS iσ (τ + ∆τ ) is a few. On the other hand, in a region of low density of hole, (ρ iσ (τ ) 0.1), value of N HS iσ (τ +∆τ ) distributes. These results are interpreted that in the case that the hole density around the photo-induced HS state is low, a probability of the survival HS state number is randomly distributed. On the other hand, in the case of high hole density around the HS state, the HS density increases with increasing the hole density. These relations between the photo-doped hole and the HS state are consistent with the results in (color online) Correlation between the number density of the local HS state at time τ , N HS iσ (τ + ∆τ ), and the number density of the hole state around the site i at time τ ′ , ρiσ(τ ). We take ∆τ = 10/tA. Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, J = 3.1tA, ∆ = 10tA, tB = 0.05tA, N ph = 10, and ∆τ = 10t −1 A . Sect. III, where the HS state is stabilized by forming the HS-hole bound state. Next we compare the present calculated results with the experimental data reported in Ref. 34. As introduced previously, key points in the optical pump-probe experiments in RBaCo 2 O 6.δ are a) a photo-induced metallic state is different from the high-temperature metallic state, and b) this photo-induced state strongly depends on the R species. From the calculated results, we propose that the observed metallic state is attributed to the HShole bound state. Experimental spectral weight induced by the photon pumping is interpreted to be the dipole transition inside of the bound state. We have checked by the exact diagonalization method in a small size cluster that a clear bound state between thermal hole carriers and the HS states is not stabilized in finite temperatures. 36 It is well known that, in the perovskite crystal, the electron transfer intensity is systematically controlled by the R species through a changing of a Co-O-Co bond angle. The smaller the ionic radius of the R ion is, the smaller the e g band width is. With increasing the ionic radius from Tb to Sm, the photo-induced metallic state is remarkably seen in the experimental optical conductivity spectra. These data correspond to the calculated results in the phase diagram in Fig. 4. Increasing of the transfer integral of the A band, being equivalent to decreasing of ∆/t A , is indicated by the arrow in this phase diagram. A system is transferred from the phase, where the spinstate is not changed by photo-excitation, to the phase, where the HS state is induced by photo-irradiation. This consistency between the theory and the experiments is additional evidence of existence of the photo-induced HShole bound state. In conclusion, we study the photo-induced spin-state change in a correlated electron system. The photoinduced metastable state is examined in the effective Hamiltonian which is derived by the two-orbital Hubbard model. By photo-irradiation into the LS phase near the phase boundary with the mixed phase, the HS state is induced and is stabilized by forming a bound state with a photo-doped hole state. The optical transition inside of this bound state appears. Time dependent simulation for the photo-excited state is also performed on the twoorbital Hubbard model in the time-dependent mean-field scheme. A pair annihilation of the photo-doped electron and hole states generates the HS state. This process reflects on the time-dependent DOS. The present results propose a new state of the photo-excited matter in correlated electron system with multi-degrees of freedom. 0 0 −J eL −f 2 B t A 0 0 0 0 −f 2 B t A −J eL          . (A12) where J eH = t 2 B 4J + t 2 A U + U ′ + 2J + t 2 A f 2 B ∆ + U − U ′ + J − ∆ J + t 2 A g 2 B ∆ + U − U ′ + J + ∆ J ,(A13) and J eL = t 2 A f 2 B g 2 B 2∆ J + t 2 A f 2 A ∆ + 2U ′ + J + ∆ J + f 2 A t 2 B 2 3 ∆ − U + U ′ + ∆ J − J + f 2 A t 2 B 2 1 ∆ − U + U ′ + ∆ J + J . (A14) The basis set is {|ψ e↓ , ψ H+1 , |ψ e↑ , ψ 0 , |ψ H+1 , ψ e↓ , |ψ H0 , ψ e↑ , |ψ e↑ , ψ L , |ψ L , ψ e↑ }. † Present address: Department of Physics, Hokkaido University, Sapporo 060-0810, Japan. FIG. 1 : 1(color online) Local electronic configurations. γ = (A, B) for γ FIG. 2 : 2(color online) Adiabatic energy surfaces before and after photo-irradiation. Horizontal axis represents a number of the HS state. . 3: (color online) (a) The number density of the HS state nHS, (b) the spin correlation function S(q), and (c) the spin-state correlation function I(q) in the ground state. The parameter values are chosen to be U = 4J, U ′ = 2J, ∆ = 10tA, and tB = 0.05tA. A two dimensional cluster of the N = 8 sites with the periodic boundary condition is adopted. and the spin-state correlation function defined by FIG. 4 : 4(color online) Phase diagram in the plane of the crystalline field splitting ∆ and the Hund's coupling J. Broken and bold lines represent the phase boundaries in the ground state and in the photo-induced metastable state, respectively. Abbreviations, HS, LS, and H/L represent the HS phase, the LS phase, and the HS-LS mixed phase, respectively. A vertical dotted line represents the parameter region where the data in Figs. 3 and 5 are calculated. Parameter values are chosen to be U = 4J, U ′ = 2J, and tB = 0.05tA. A two dimensional cluster of the N = 8 sites with the periodic boundary condition is adopted. . 5: (color online) (a) The number density of the HS states nHS, (b) the spin correlation function S(q), and (c) the spin-state correlation function I(q) in the photo-excited metastable state. In comparison, results of nHS in the ground state are also plotted in (a). Parameter values are chosen to be U = 4J, U ′ = 2J, ∆ = 10tA, and tB = 0.05tA. A two dimensional cluster of the N = 8 with the periodic boundary condition sites is adopted. (b). The HS distribution is located at the NN sites of the hole state, and almost no size dependence is seen in the results. Different numerical values of g H (n) in one and two dimensional clusters, i.e. 0.25 and 0.5, are attributed to difference of z n . Spin structure online) (a) A schematic definition of the distribution function gM (n). (b) Distribution function of the electronic states as functions of a distance from the hole state. Bold, broken and dotted lines represent the distribution functions for the HS, LS and electron states, respectively. Two dimensional N = 10 site cluster with the periodic boundary condition is adopted. The inset shows distribution functions for the HS state in one dimensional N =6, 8, and 10 site clusters with the periodic boundary condition. Parameter values are chosen to be J = 3.3tA, U = 4J, U ′ = 2J, ∆ = 10tA, and tB = 0.05tA. FIG. 7 : 7(color online) (left) A schematic picture of the HShole bound state. (right) Energy levels inside of the HS-hole bound state. . 8: (color online) Phase diagrams in the ground state and the photo-excited metastable state in the plane of tB/tA and J/tA. A shaded area implies a parameter region where the HS state is induced by photo-irradiation. Two dimensional N = 8 site clusters with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, and ∆ = 10tA. online) Optical absorption spectra in the photo-induced metastable state, where the HS is induced. Bond correlation function B (l,m) (ω) is also plotted. Red broken lines and blue dotted lines are for (l, m) = (h, HS) and (l, m) = (e, LS), respectively. Cluster size is N = 8 in (a), and N = 10 in (b). Parameter values are chosen to be J = 3.3tA, U = 4J, U ′ = 2J, ∆ = 10.0tA, tB = 0.05tA, and η = 0.2tA. , where we set (l, m) = (h, HS) and (e, LS). In the ground state, i.e. ω = 0, B (h,HS) (ω = 0) ∼ −1 and B (e,LS) (ω = 0) ∼ −3. These values are consistent with the picture presented in Fig. 7, where a photo-doped hole forms a bound state with HS, and a photo-doped electron is located in a bottom of the A-orbital band. In the excited state corresponding to the peak B, B (e,LS) (ω B ) ≃ B (e,LS) (0) and B (h,HS) (ω B ) ∼ 1 > B (h,HS) (0). This value of B (h,HS) (ω B ) is interpreted that an electron occupies the antibonding orbital in the HS-hole bound state, and the peak B is assigned as an excitation between the bonding and antibonding orbitals inside of the bound state. As for the peak D, B (e,LS) (ω D ) > B (e,LS) . 10: (color online) Optical absorption spectra in the photo-induced metastable state, where HS state is not induced. Broken lines represent the spectra obtained by the HC fermion model. Cluster sizes are N = 8 and 10. Inset shows size dependences of the peak energies in the optical absorption spectra calculated in the HC fermion model. Parameter values are chosen to be J = 3.1tA, U = 4J, U ′ = 2J, ∆ = 10.0tA, tB = 0.05tA, and η = 0.2tA in the original model, and tB = 0.05tA, tex = 10 −10 tA, η = 0.2tA in the HC fermion model. B (h,HS) (ω D ) ≃ B (h,HS) (0) which imply that a change in the photo-doped electron motion is concerned in this peak. 11: (color online) Phase diagram in the ground state obtained by the mean-field approximation. Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. As a comparison, phase boundaries obtained by the exact diagonalization method on the effective Hamiltonian (N = 8) are plotted by broken lines. Abbreviations, HS, LS, and H/L, represent the HS phase, the LS phase, and the HS-LS mixed phase, respectively. Parameter values are chosen to be U = 4J, U ′ = 2J, and tB = 0.05tA. (a). Except for the early time below τ t A = 10, where all three are almost constant, N HS increases, and . 12: (color online) (a) Time evolutions of numbers of the HS state, the electron state and the hole state. (b) A schematic picture of the electron-hole pair annihilation processes. Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, J = 3.1tA, ∆ = 10tA, tB = 0.05tA, and N ph = 10. FIG . 13: (color online) Snapshots of the electron state, the hole state and the HS state. Time is chosen to be 10t −1 A , 100t −1 A , 500t −1 A and 900t −1 A . Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, J = 3.1tA, ∆ = 10tA, tB = 0.05tA, and N ph = 10. (43) and (44) are replaced by the Lorentz function with a damping con-stant η = 0.1t A .Numerical results of the time-dependent DOS are shown inFig. 14. At τ = 10t −1 A , an energy gap exists between the narrow B band and the wide A band. Tiny online) Density of states. The electron part and the hole part of DOS are represented by pink solid and blue solid lines, respectively. Time is taken to be 10t −1 A , 100t −1 A , 500t −1 A , and 900t −1 A . Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, J = 3.1tA, ∆ = 10tA, tB = 0.05tA, η = 0.1tA, and N ph = 10. weights of the hole and electron parts of DOS are observed in the top of the B band and the bottom of the A band, respectively. At time τ = 500t −1 A , the top of the B band and the bottom of the A band start to separate from the main bands. Finally, at τ = 900t −1 15: (color online) (a) Numbers of the HS state, the electron state and the hole state in the model with the spinorbit interaction. Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, J = 3.1tA, ∆ = 10tA, tB = 0.05tA, and N ph = 10. Ten data sets with different initial values for the time evolutions are averaged. (b) A schematic picture of the transition between the HS state and the LS state due to the SO interaction. FIG. 16: (color online) Correlation between the number density of the local HS state at time τ , N HS iσ (τ + ∆τ ), and the number density of the hole state around the site i at time τ ′ , ρiσ(τ ). We take ∆τ = 10/tA. Two dimensional N = 10 × 10 site cluster with the periodic boundary condition is adopted. Parameter values are chosen to be U = 4J, U ′ = 2J, J = 3.1tA, ∆ = 10tA, tB = 0.05tA, N ph = 10, and ∆τ = 10t −1 A . AcknowledgmentsAuthors would like to thank H. Matsueda, Y. Inoue, Y. Okimoto, S. Koshihara, S. Iwai and T. Arima for their valuable discussions. This work was supported by KAKENHI from MEXT, Optical Science of Dynamically Correlated Electrons (DYCE), Tohoku University "Evolution" program, and Grand Challenges in Next-Generation Integrated Nanoscience. YK is supported by the global COE program "Weaving Science Web beyond Particle-Matter Hierarchy" of MEXT, Japan. Parts of the numerical calculations have been performed in the supercomputing systems in ISSP, University of Tokyo, and Kyoto University.In this Appendix, explicit formulae of the effective Hamiltonian for the photo-induced metastable state are presented. Matrix elements in terms of the electronic states in NN sites are shown. These are classified by the electron number, n, and the z-component of the total spin-angular momentum, S z , as H (n,Sz) . The wave functions in the two sites are denoted as |ψ i , ψ j . In the following notation, each term in the Hamiltonian in Eq.andThe basis set is {|ψ e↑ , ψ h↑ |ψ h↑ , ψ e↑ }.(4) (n = 4, S z = 0)where J eh± = J eh1 ± J eh2 andThe basis set is {|ψ e↓ , ψ h↑ , |ψ e↑ , ψ h↓ , |ψ h↑ , ψ e↓ , |ψ h↓ , ψ e↑ }.(5) (n = 5, S z = 3/2)The basis set is {|ψ e↑ , ψ H+1 , |ψ H+1 , ψ e↑ }.(6) (n = 5, S z = 1/2) . S Maekawa, T Tohyama, S E Barnes, S Ishihara, W Koshibae, G Khliullin, Physics of Transition Metal Oxides. Springer VerlagS. Maekawa, T. Tohyama, S. E. Barnes, S. Ishihara, W. Koshibae, and G. Khliullin, Physics of Transition Metal Oxides (Springer Verlag, Berlin, 2004). . K Nasu, Photo Induced Phase Transition (World Scientific. and references thereinK. Nasu, Photo Induced Phase Transition (World Scien- tific, Singapore, 2004), and references therein. . M Fiebig, K Miyano, Y Tomioka, Y Tokura, Science. 2801925M. Fiebig, K. Miyano, Y. Tomioka, and Y. Tokura, Science 280, 1925 (1998). . A Cavalleri, Cs Tóth, C W Siders, J A Squier, F Ráksi, P Forget, J C Kieffer, Phys. Rev. Lett. 87237401A. Cavalleri, Cs. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, Phys. Rev. Lett. 87, 237401 (2001). . H Okamoto, T Miyagoe, K Kobayashi, H Uemura, H Nishioka, H Matsuzaki, A Sawa, Y Tokura, Phys. Rev. B. 83125102H. Okamoto, T. Miyagoe, K. Kobayashi, H. Uemura, H. Nishioka, H. Matsuzaki, A. Sawa, and Y. Tokura, Phys. Rev. B 83, 125102 (2011). . H Matsueda, S Ishihara, J. Phys. Soc. Jpn. 7683703H. Matsueda and S. Ishihara, J. Phys. Soc. Jpn. 76, 083703 (2007). . Y Kanamori, H Matsueda, S Ishihara, Phys. Rev. Lett. 103115101Phys. Rev. BY. Kanamori, H. Matsueda and S. Ishihara, Phys. Rev. Lett. 103, 267401 (2009). ibid., Phys. Rev. B 82, 115101 (2010). . S Iwai, S Tanaka, K Fujinuma, H Kishida, H Okamoto, Y Tokura, Phys. Rev. Lett. 8857402S. Iwai, S. Tanaka, K. Fujinuma, H. Kishida, H. Okamoto, and Y. Tokura, Phys. Rev. Lett. 88, 057402 (2002). . M Chollet, L Guerin, N Uchida, S Fukaya, H Shimoda, T Ishikawa, K Matsuda, T Hasegawa, A Ota, H Yamochi, G Saito, R Tazaki, S Adachi, S Koshihara, Science. 30786M. Chollet, L. Guerin, N. Uchida, S. Fukaya, H. Shi- moda, T. Ishikawa, K. Matsuda, T. Hasegawa, A. Ota, H. Yamochi, G. Saito, R. Tazaki, S. Adachi, and S. Koshi- hara, Science 307, 86 (2005). . N Tajima, J Fujisawa, N Naka, T Ishihara, R Kato, Y Nishio, K Kajita, J. Phys. Soc. Jpn. 74511N. Tajima, J. Fujisawa, N. Naka, T. Ishihara, R. Kato, Y. Nishio, and K. Kajita, J. Phys. Soc. Jpn. 74, 511 (2005). . K Yonemitsu, K Nasu, J. Phys. Soc. Jpn. 7511008K. Yonemitsu and K. Nasu, J. Phys. Soc. Jpn. 75, 011008 (2006). . O Sato, T Iyoda, A Fukushima, K Hashimoto, Sci- ence 272 704O. Sato, T. Iyoda, A. Fukushima, and K. Hashimoto, Sci- ence 272 704 (1996). . A Bleuzen, C Lomenech, V Escax, F Villain, F Varret, C C Dit Moulin, M Verdaguer, J. Am. Chem. Soc. 1226648A. Bleuzen, C. Lomenech, V. Escax, F. Villain, F. Varret, C. C. dit Moulin, and M. Verdaguer, J. Am. Chem. Soc. 122 6648 (2000). . V Escax, A Bleuzen, C C Dit Moulin, F Villain, A Goujon, F Varret, M Verdaguer, J. Am. Chem. Soc. 12312536V. Escax, A. Bleuzen, C. C. dit Moulin, F. Villain, A. Gou- jon, F. Varret, and M. Verdaguer, J. Am. Chem. Soc. 123 12536 (2001). . O Sato, J. Photoch. Photobio. C: Photochem. Rev. 5203O. Sato, J. Photoch. Photobio. C: Photochem. Rev. 5 203 (2004). . N Willenbacher, H Spiering, J. Phys. C. 211423N. Willenbacher and H. Spiering, J. Phys. C 21, 1423 (1988). . A L Tchougréeff, M B Darkhovskii, Int. J. Quantum Chem. 57903A. L. Tchougréeff and M. B. Darkhovskii, Int. J. Quantum Chem. 57, 903 (1996). . M Nishino, K Boukheddaden, Y Konishi, S Miyashita, Phys. Rev. Lett. 98247203M. Nishino, K. Boukheddaden, Y. Konishi, and S. Miyashita, Phys. Rev. Lett. 98, 247203 (2007). . S Miyashita, P A Rikvold, T Mori, Y Konishi, M Nishino, H Tokoro, Phys. Rev. B. 8064414S. Miyashita, P. A. Rikvold, T. Mori, Y. Konishi, M. Nishino and H. Tokoro, Phys. Rev. B 80, 064414 (2009). . C Frontera, J L García-Muñoz, A Llobet, M A G Aranda, Phys. Rev. B. 65180405C. Frontera, J. L. García-Muñoz, A. Llobet, and M. A. G. Aranda, Phys. Rev. B 65, 180405(R) (2002). . S Tsubouchi, T Kyomen, M Itoh, P Ganguly, M Oguni, Y Shimojo, Y Morii, Y Ishii, Phys. Rev. B. 6652418S. Tsubouchi, T. Kyomen, M. Itoh, P. Ganguly, M. Oguni, Y. Shimojo, Y. Morii, and Y. Ishii, Phys. Rev. B 66, 052418 (2002). . Y Okimoto, X Peng, M Tamura, T Morita, K Onda, T Ishikawa, S Koshihara, N Todoroki, T Kyomen, M Itoh, Phys. Rev. Lett. 10327402Y. Okimoto, X. Peng, M. Tamura, T. Morita, K. Onda, T. Ishikawa, S. Koshihara, N. Todoroki, T. Kyomen, and M. Itoh, Phys. Rev. Lett. 103, 027402 (2009). . K Asai, P Gehring, H Chou, G Shirane, Phys. Rev. B. 4010982K. Asai, P. Gehring, H. Chou, and G. Shirane, Phys. Rev. B 40, 10982 (1989). . Y Tokura, Y Okimoto, S Yamaguchi, H Taniguchi, T Kimura, H Takagi, Phys. Rev. B. 581699Y. Tokura, Y. Okimoto, S. Yamaguchi, H. Taniguchi, T. Kimura, and H. Takagi, Phys. Rev. B 58, 1699(R) (1998). . R R Heikes, R C Miller, R Mazelsky, Physica. 301600R. R. Heikes, R. C. Miller, and R. Mazelsky, Physica 30, 1600 (1964). . P M Raccah, J B Goodenough, Phys. Rev. 155932P. M. Raccah, and J. B. Goodenough, Phys. Rev. 155, 932 (1967). . S Yamaguchi, Y Okimoto, Y Tokura, Phys. Rev. B. 5411022S. Yamaguchi, Y. Okimoto, and Y. Tokura, Phys. Rev. B 54, 11022(R) (1996). . T Saitoh, T Mizokawa, A Fujimori, M Abbate, Y Takeda, M Takano, Phys. Rev. B. 554257T. Saitoh, T. Mizokawa, A. Fujimori, M. Abbate, Y. Takeda, and M. Takano, Phys. Rev. B 55, 4257 (1997). . C N R Rao, Om Parkash, D Bahadur, P Ganguly, S Nagabhushana, J. Solid State Chem. 22353C. N. R. Rao, Om Parkash, D. Bahadur, P. Ganguly, and S. Nagabhushana, J. Solid State Chem. 22, 353 (1977). . M A Señarís-Rodríguez, J B Goodenough, J. Solid State Chem. 118323M. A. Señarís-Rodríguez and J. B. Goodenough, J. Solid State Chem. 118, 323 (1995). . M Itoh, I Natori, J. Phys. Soc. Jpn. 64970M. Itoh and I. Natori, J. Phys. Soc. Jpn. 64, 970 (1995). . K Tsutsui, J Inoue, S Maekawa, Phys. Rev. B. 594549K. Tsutsui, J. Inoue, and S. Maekawa, Phys. Rev. B 59, 4549 (1999). . R Suzuki, T Watanabe, S Ishihara, Phys. Rev. B. 8054410R. Suzuki, T. Watanabe, and S. Ishihara, Phys. Rev. B 80, 054410 (2009). . Y Okimoto, T Miyata, M S Endo, M Kurashima, K Onda, T Ishikawa, S Koshihara, M Lorenc, E Collet, H Cailleau, T Arima, Phys. Rev. B. 84121102Y. Okimoto, T. Miyata, M. S. Endo, M. Kurashima, K. Onda, T. Ishikawa, S. Koshihara, M. Lorenc, E. Collet, H. Cailleau, and T. Arima, Phys. Rev. B 84, 121102(R) (2011). Meeting abstracts of the Physical Society of. S Iwai, S Tomimoto, Y Okimoto, J P He, Y Kaneko, H Okamoto, Y Tokura, Japan. 592-4673S. Iwai, S. Tomimoto, Y. Okimoto, J. P. He, Y. Kaneko, H. Okamoto, and Y. Tokura, Meeting abstracts of the Physical Society of Japan, 59(2-4), 673. . Y Kanamori, H Matsueda, S Ishihara, Phys. Rev. Lett. 107167403Y. Kanamori, H. Matsueda, and S. Ishihara, Phys. Rev. Lett. 107, 167403 (2011). . D I Khomskii, U Löw, Phys. Rev. B. 69184401D. I. Khomskii and U. Löw, Phys. Rev. B 69, 184401 (2004). . A D Mclachlan, M A Ball, Rev. Mod. Phys. 36844A. D. McLachlan and M. A. Ball, Rev. Mod. Phys. 36, 844 (1964). . A Terai, Y Ono, Prog. Theor. Phys. Suppl. 113177A. Terai and Y. Ono, Prog. Theor. Phys. Suppl. 113, 177 (1993).
[]
[ "On Certain Divisibility Property of Polynomials", "On Certain Divisibility Property of Polynomials" ]
[ "Luis F Cáceres ", "José A Vélez Marulanda " ]
[]
[]
We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover, we give a characterization for D-rings that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations.
null
[ "https://arxiv.org/pdf/1010.5952v1.pdf" ]
117,974,321
1010.5952
6f3d89124111c99267b044849b35d57b0c2f9fb3
On Certain Divisibility Property of Polynomials 28 Oct 2010 October 29, 2010 Luis F Cáceres José A Vélez Marulanda On Certain Divisibility Property of Polynomials 28 Oct 2010 October 29, 2010 We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover, we give a characterization for D-rings that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations. Introduction Assume that f (x) and g(x) are polynomials with integer coefficients. Assume also that for all k ∈ Z such that g(k) = 0 we have g(k)|f (k) in Z. Can we say that g(x)|f (x) in Z[x]? Certainly this is not always true. For instance, if we consider p any prime, f (x) = x p − x and g(x) = p, using Fermat's little theorem we see that for all k ∈ Z, g(k)|f (k), but g( x) ∤ f (x) in Z[x] although f (x) g(x) ∈ Q[x] . This motivates the following definition introduced by H. Gunji & D. L. MacQuillan in [1]. Definition 1.1. Let R be an integral domain. R is a D-ring if given polynomials f (x) and g(x) with coefficients in R such that for almost all k ∈ R, g(k)|f (k) then f (x) g(x) ∈ K[x], where K is the field of fractions of R. Fields cannot be D-rings: if F is a field, trivially k|1 for all nonzero element k ∈ F but 1 x ∈ F [x]. From now on, we always assume all the rings are commutative with identity. Let R be a ring. For any polynomial f (x) ∈ R[x] denote S(f ) the set of all nonzero prime ideals p of R such that the congruence f (x) ≡ 0 mod p is solvable in R, i.e. there exists k ∈ R such that f (k) ∈ p. In particular, if c ∈ R, S(c) is precisely the set of prime ideals of R that contain c. Proposition 1.2. Let R be an integral domain and R × the multiplicative group of R . The following properties are equivalent: (i) R is a D-ring. (ii) Every polynomial over R which satisfies f (k) ∈ R × for almost all k ∈ R must be a constant. We review the definition of algebraic integers. Let R be a subring of a ring L. An element α ∈ L is integral over R if there exists a monic polynomial f (x) ∈ R[x] such that f (α) = 0. In particular, when R = Z, the element α is said to be an algebraic integer in L. It is well-known that the set B consisting of all the elements that are integral over R is a ring which is called the integral closure of R in L. In particular, if R = Z and L is a field containing Z, the integral closure of Z in L is called the ring of integers of L, and we denote this ring by O L . For example, let d be a square-free integer and consider Q( √ d) = {a+b √ d : a, b ∈ Q}, then ring of integers in Q( √ d) is O Q( √ d) = Z[ω] = {a + bω : a, b ∈ Z} where ω = √ d, if d ≡ 2, 3 mod 4 1+ √ d 2 , if d ≡ 1 mod 4 We say that an integral domain R is integrally closed if R is equal to its integral closure in its field of fractions. For example, the integral closure of Z in Q is itself, which implies that Z is integrally closed. It is stated in [1, Cor 1, pg 293] that the following result is a direct consequence of Proposition 1.2. We provide in the following section we prove that Z is a D-ring (see Lemma 2.2). Then we can see that Proposition 1.3 is actually a consequence of the following result. Proposition 1.4. Let R be an integrally closed domain and K be its ring of fractions. Let K ⊆ L be a finite Galois extension of fields and let C be the integral closure of R in L. If R is a D-ring then C is also a D-ring. In the following section we provide an elementary proof of Proposition 1.4 which proves Proposition 1.3 without using Proposition 1.2. Alternative Characterizations of D-rings (x) in R[x] such that for k ∈ R (g(k) = 0 ⇒ g(k)|f (k)) then either f (x) = 0 or deg g ≤ deg f . Proof. (⇒) Assume that R is a D-ring. Let g(x), f (x) ∈ R[x] such that for all k ∈ R with (g(k) = 0 ⇒ g(k)|f (k)). Since g(x) has finitely many zeros, g(k)|f (k) for almost all k ∈ R. Since R is a D-ring, f (x) g(x) ∈ K[x]. Then there exists p(x) ∈ K[x] such that f (x) = p(x)g(x). Suppose f (x) = 0, then deg f = deg(pg) = deg p + deg g ≥ deg g. (⇐) Let g(x), f (x) ∈ R[x] such that for almost all k ∈ R, g(k)|f (k). Let A = {k 1 , . . . , k n } be a finite subset of R such that g(k)|f (k) for all k ∈ R − A. Let k 1 , . . . , k s ∈ A such that g(k i ) = 0 for i = 1, . . . , s and let β = g(k 1 ) · · · g(k s ). If s = 0, let β = 1. Consequently, for all k ∈ R (g(k) = 0 ⇒ g(k)|βf (k)). By hypothesis, βf (x) = 0 or deg g ≤ deg βf . If βf (x) = 0, then f (x) = 0, which trivially implies f (x) g(x) ∈ K[x] . Suppose deg g ≤ deg βf and let g(x) = a n x n + · · · + a 0 . By the division algorithm there exist q(x), r(x) ∈ K[x] and s ∈ Z + such that a s n βf (x) = g(x)q(x)+r(x), with r(x) = 0 or deg r < deg g. Let α = a s n β and suppose that deg r < deg g. Therefore for all k ∈ R with g(k) = 0 we have both g(k)|αf (k) and g(k)|g(k)q(k), implying g(k)|r(k). Using the hypothesis for the polynomials g(x) and r(x) we obtain r(x) = 0 or deg r ≥ deg g. Therefore r(x) = 0 and hence αf ( x) = g(x)q(x). It follows that f (x) g(x) = α −1 q(x) ∈ K[x] . In others words, R is a D-ring. Lemma 2.2. Let g(x), f (x) ∈ Z[x] such that for k ∈ Z, (g(k) = 0 ⇒ g(k)|f (k)). Then f (x) = 0 or deg g ≤ deg f . Consequently, Z is a D-ring. Proof. Let g(x) = a n x n + · · · + a 1 x + a 0 and f ( x) = b m x m + · · · + b 1 x + b 0 be polynomials in Z[x] such that f (x) = 0 and deg g = n > deg f . Without loss of generality, assume a n , b m > 0. Then we can find k ∈ Z large enough such that g(k) = 0 and a n k n + · · · + a 1 k + a 0 > b m k m + · · · + b 1 k + b 0 which implies that g(k) ∤ f (k). This proves the first conclusion of Let R, K, L and C as in the hypothesis of Proposition 2.3. Assume that p(x) = α n x n + · · · + a 1 x + a 0 ∈ L[x] and σ ∈ Gal(L/K) are arbitrary and let (iv) N L/K (p)(a) ∈ R for all a ∈ R. p σ (x) = σ(α n )x n + · · · σ(α 1 )x + σ(α 0 ). Observe that if p(x) = r(x)s(x) with r(x), s(x) ∈ L[x] then p σ (x) = r σ (x)s σ (x). Moreover, if a ∈ R Proof. Statements (i) and (ii) follow directly from the definition of N L/K (p)(x). Let q(x) = N L/K (p)(x) (note that q(x) ∈ C[x]). Let τ ∈ Gal(L/K) be fixed but arbitrary. Then τ • σ ∈ Gal(L/K) and (p τ ) σ (x) = p τ •σ (x) for all σ ∈ Gal(L/K). Note that τ induces a permutation of the finite group Gal(L/K). Then q τ (x) = τ •σ∈Gal(L/K) p τ •σ (x) = q(x) which implies q(x) ∈ R[x] proving (iii). Note that (iv) is a direct consequence of (iii). Proof of Proposition 1.4. Let f (x), g(x) ∈ C[x] be such that for all k ∈ C (g(k) = 0 ⇒ g(k)|f (k)). Consider F (x) = N L/K (f )(x) and G(x) = N L/K (g)(x). By Lemma 2.4, F (x), G(x) ∈ R[x]. Let b ∈ R such that G(b) = 0. Then g(b) = 0 which implies g(b)|f (b). Therefore σ(g(b) )|σ(f (b)) for all σ ∈ Gal(L/K). Using properties of divisibility together with Lemma 2.4 we obtain G(a) =   σ∈Gal(L/K) σ(g(a))   |   σ∈Gal(L/K) σ(f (a)))   = F (a) . We have proved that for every b ∈ R, (G(b) = 0 ⇒ G(b)|F (b)). Using Proposition 2.1, we obtain that either deg G ≤ deg F or F (x) = 0. Using Lemma 2.4 we obtain deg g ≤ deg f or f (x) = 0. It follows from Proposition 2.1 that C is a D-ring. It is clear that Proposition 1.3 follows directly from Proposition 1.4 together with Lemma 2.2. In the following, we give a characterization of D-rings that are also unique factorization domains. Let R be a unique factorization domain. For all p(x) ∈ R[x] we denote by C(p(x)) the content of p(x), i.e. the greatest common divisor of the coefficients of p(x). Remember that p(x) ∈ R[x] is said to be primitive if C(p(x)) is a unit. Gauss' lemma states that the product of two primitive polynomials over a unique factorization domain is also primitive. This implies that if f (x), g(x) and h(x) are polynomials in R[x] with g(x) primitive and mh(x) = f (x)g(x) for some m ∈ R, there exists a polynomial q(x) ∈ R[x] such that f (x) = mq(x). Proposition 2.5. Let R be a unique factorization domain and let K its field of fractions. R is a D-ring if and only if given f (x), g(x) ∈ R[x] with g(x) nonconstant and primitive such that for all k ∈ R, (g(k) = 0 ⇒ g(k)|f (k)), then g(x)|f (x) in R[x]. Proof. (⇒). Let f (x), g(x) ∈ R[x] with g(x) nonconstant primitive such that for all k ∈ R, g(k) = 0 ⇒ g(k)|f (k). It is clear that for almost all k ∈ R, g(k)|f (k). Since R is a D-ring we have that f (x) g(x) = p(x) ∈ K[x] . Let p(x) = rn sn x n + rn−1 sn−1 x n−1 + · · · + r1 s1 x + r0 s0 , where r i , s i ∈ R, with s i = 0 for all i = 0, . . . , n. Let m = s 0 s 1 · · · s n , therefore mp(x) ∈ R[x]. Take h(x) = mp(x). We have mf (x) = mp(x)g(x) = h(x)g(x), with g(x) primitive. Then there exists q(x) ∈ R[x] such that h(x) = mq(x), and so mf (x) = mq(x)g(x). Therefore f (x) = q(x)g(x), with q(x) ∈ R[x]; i.e. g(x)|f (x) in R[x]. (⇐). Let f (x), g(x) ∈ R[x] such that for almost all k ∈ R we have that g(k)|f (k). Let A = {k 1 , . . . , k n } be a finite subset of R such that g(k)|f (k) for all k ∈ R − A. Let k 1 , . . . , k s ∈ A such that g(k i ) = 0 for i = 1, . . . , s and let β = g(k 1 ) · · · g(k s ). If s = 0, let β = 1. Then for all k ∈ R such that g(k) = 0 we have g(k)|βf (k). We can write g(x) = αh(x) where h(x) is primitive with deg h = deg g ≥ 1 and α is the content of g(x). Let k ∈ R such that h(k) = 0. Therefore g(k) = 0 and g(k)|βf (k); but h(k)|g(k), so h(k)|βf (k). By hypothesis, h(x)|βf (x) in R[x]. Hence, there exists p(x) ∈ R[x] such that βf (x) = p(x)h(x) and so αβf (x) = p(x)(αh(x)) = p(x)g(x). Therefore f (x) = (αβ) −1 p(x)g(x) where (αβ) −1 p(x) ∈ K[x], i.e. f (x) g(x) ∈ K[x]. Hence, R is a D-ring. The following result provides a positive answer to our original question about divisibility of polynomials with integer coefficients. Theorem 2.6. If given f (x), g(x) ∈ Z[x] with g(x) primitive, deg g(x) ≥ 1 and for all k ∈ Z, (g(k) = 0 ⇒ g(k)|f (k)), then g(x)|f (x) in Z[x]. Proof. Since Z is a unique factorization domain, the result follows from Proposition 2.5 together with Lemma 2.2. Example 2.7. For all n ≥ 0 consider p n (x) and q n (x) defined as follows. p 0 (x) = 1, p 1 (x) = x, p n+1 (x) = 2xp n (x) − p n−1 (x), q 0 (x) = 0, q 1 (x) = 1, q n+1 (x) = 2xq n (x) − q n−1 (x). n p n (x) q n (x) 0 1 0 1 x 1 2 2x 2 − 1 2x 3 4x 3 − 3x 4x 2 − 1 4 8x 4 − 8x 2 + 1 8x 3 − 4x 5 16x 5 − 20x 3 + 5x 16x 4 − 12x 2 + 1 6 32x 6 − 48x 4 + 18x 2 − 1 32x 5 − 32x 3 + 6x 7 64x 7 − 112x 5 + 56x 3 − 7x 64x 6 − 80x 4 + 24x 2 − 1 8 128x 8 − 256x 6 + 160x 4 − 32x 2 + 1 128x 7 − 192x 5 + 80x 3 − 8x Table 1: Polynomials p n (x) and q n (x) for n = 0, . . . , 8 In [3] is proved the following congruence for all a ∈ Z − {0, −1} and n ≥ 1. q 2n (a) ≡ 0 mod p n (a), This implies that for all a ∈ Z with p n (a) = 0 then p n (a)|q 2n (a). Can we deduce p n (x)|q 2n (x) as polynomials? We can check this for example when n = 4: note that q 8 (x) = 128x 7 − 192x 5 + 80x 3 − 8x = 8x(2x 2 − 1)(8x 4 − 8x 2 + 1) and p 4 (x) = 8x 4 − 8x 2 + 1 which clearly shows p 4 (x)|q 8 (x). It is straightforward to prove that for all n ≥ 1 the polynomials p n (x) are primitive with deg p n ≥ 1. It follows from Proposition 2.5 that for all n ≥ 1, p n (x)|q 2n (x). Proposition 1 . 3 . 13Let Q ⊆ L be a finite Galois extension of fields. Then the ring of algebraic integers O L in L is a D-ring. Proposition 2 . 1 . 21Let R be an integral domain and let K be its field of fractions. R is a D-ring if and only if for given polynomials f (x) and g Lemma 2.2 by contradiction. The second conclusion follows from Proposition 2.1 Proposition 2.3. Let R be an integral domain and K be its field of fractions. Assume that K ⊆ L is a finite Galois extension of fields and let C be the integral closure of R in L. Then σ(C) = C for all σ ∈ Gal(L/K). Moreover, if R is integrally closed, then R = {b ∈ C : σ(b) = b, for all σ ∈ Gal(L/K)}. For a proof of Proposition 2.3 see for example [2, Prop 2.19]. then p σ (a) = σ(p(a)). It follows from Proposition 2.3 that p τ (x) = p(x) for all τ ∈ Gal(L/K) if and only if p(x) ∈ R[x]. Let N L/K (p)(x) = σ∈Gal(L/K) p σ (x). It follows that for all a ∈ R, N L/K (p)(a) = σ∈Gal(L/K) p σ (a) = σ∈Gal(L/K) σ(p(a)). Lemma 2 . 4 . 24Let R, K, L and C as in the hypothesis of Proposition 2.3 and let p(x) ∈ C[x] be arbitrary. (i) deg N L/K (p) = |Gal(L/K)| deg p. (ii) N L/K (p)(x) = 0 if and only if p(x) = 0. ( iii ) iiiFor any nonconstant polynomial f (x) ∈ R[x], the set S(f ) is nonempty. As a consequence of Proposition 1.2, the ring Z[W ] where W = {1/p : p is prime and p ≡ 1 mod 4 or p = 2} is not a D-ring using the nonconstant polynomial f (x) = x 2 + 1 and the fact that for all k ∈ Z[W ], f (k) ∈ Z[W ] × (see [1, Example 1, pg 293]).(iv) For any nonconstant polynomial f (x) ∈ R[x] and any non-zero c ∈ R, the set S(f ) − S(c) is infinite. Proof. See [1, Prop 1, pg 290]. . H Gunji, D L Macquillan, On Rings with Certain Divisibility Property, Michigan Math. J. 22Gunji H. & MacQuillan D.L. On Rings with Certain Divisibility Property, Michigan Math. J., 22 (1975), 289-299. An Invitation to Arithmetic Geometry. D Lorenzini, Graduate Studies in Mathematics. 9American Mathematical SocietyLorenzini, D. An Invitation to Arithmetic Geometry, Graduate Stud- ies in Mathematics, Volume 9, American Mathematical Society, 1996. Proof of Recursive Unsolvability of Hilbert's Tenth Problem. Y V Matiyasevich, J Jones, American Math. Monthly. 8Matiyasevich Y.V. & Jones J.P. Proof of Recursive Unsolvability of Hilbert's Tenth Problem, American Math. Monthly, 8 (1991), 689- 709.
[]
[ "The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach *", "The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach *" ]
[ "Samuel Herrmann ", "Etienne Tanré " ]
[]
[]
Under some weak conditions, the first-passage time of the Brownian motion to a continuous curved boundary is an almost surely finite stopping time. Its probability density function (pdf) is explicitly known only in few particular cases. Several mathematical studies proposed to approximate the pdf in a quite general framework or even to simulate this hitting time using a discrete time approximation of the Brownian motion. The authors study a new algorithm which permits to simulate the first-passage time using an iterating procedure. The convergence rate presented in this paper suggests that the method is very efficient.
10.1137/151006172
[ "https://arxiv.org/pdf/1501.07060v2.pdf" ]
36,547,243
1501.07060
8af661e50b7e55ab326f1089087543280b2d66cb
The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach * January 29, 2015 Samuel Herrmann Etienne Tanré The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach * January 29, 2015and phrases: first-passage timeBrownian motionpotential the- oryrandomized algorithm 2010 AMS subject classifications: primary 65C05; secondary 65N7560G40 Under some weak conditions, the first-passage time of the Brownian motion to a continuous curved boundary is an almost surely finite stopping time. Its probability density function (pdf) is explicitly known only in few particular cases. Several mathematical studies proposed to approximate the pdf in a quite general framework or even to simulate this hitting time using a discrete time approximation of the Brownian motion. The authors study a new algorithm which permits to simulate the first-passage time using an iterating procedure. The convergence rate presented in this paper suggests that the method is very efficient. Introduction Modeling biological or physical systems often requires to handle with onedimensional diffusion processes. The marginal probability distribution of such processes, at a fixed time, permits to describe quite precisely the model. Nevertheless, in many applications, this information is insufficient and the description of the whole paths becomes crucial. This is namely the case for variety problems related to neuronal sciences, financial derivatives with barriers, ruin probability of an insurance fund, optimal stopping problems,... In these frameworks, the main task is the description of the first passage time densities for time-dependent boundaries (for level-crossing problems see [1,2]). For instance, let us focus our attention on a simple interpretation of the neural transmission. When a neuron is stimulated by pressure, heat, light, or chemical information, its membrane voltage changes as time elapses and, as soon as it reaches a constant threshold, the depolarization phenomenon occurs and the voltage is reset to a resting potential. The family of integrate-and-fire spiking neuron models is based on this simple interpretation. The firing time therefore corresponds to the first-passage time of the membrane potential, represented by a stochastic mean-reverting process (usually the Ornstein-Uhlenbeck process) to the neural threshold (for introduction of noise in neuron systems see Part I Chapter 5 in [9], for the integrate-and-fire model see Chapter 10 in [8]). Our main motivation is to emphasize an algorithmic approach in order to approximate the first-passage time of the Brownian motion to curved boundaries. Using simple time transformations will also permit to apply the results to the Ornstein-Uhlenbeck process (see Section 3). In order to describe approximations of the first-passage time of the Brownian motion, we assume that this stopping time is almost surely finite. In this way, we introduce particular conditions for this property to be satisfied. Let us consider a continuous function ϕ : R + → R satisfying the following hypothesis: ϕ(0) > 0 and lim sup t→∞ ϕ(t) √ 2t log log t < 1. We then define the hitting time τ ϕ = inf{t > 0 : B t = ϕ(t)} (0.1) where (B t , t ≥ 0) stands for a standard one-dimensional Brownian motion. Under (H1), the a.s. finiteness of τ ϕ is an obvious consequence of the law of the iterated logarithm (see e.g. [11,Th.9.23 p.112]). It is quite difficult to obtain precise informations about this stopping time in general situations. Durbin [6,7] proposed to approximate the first-passage distribution p(t) dt = P(τ ϕ ∈ dt) of the Brownian motion as follows: p can be represented by a serie expansion p(t) = k j=1 (−1) j−1 q j (t) + (−1) k r k (t), k ≥ 1, where q j for 1 ≤ j ≤ k and r k are defined by multiple integrals depending on the boundary ϕ. The approximation simply consists in truncating the series. Let us note that the first term corresponds in fact to the tangent approximation of Strassen [15] and Daniels [4]. The convergence of the series and the error bounds can be precised if the curved boundary is wholly concave or wholly convex. For particular cases, the probability density function can be computed explicitly. Lerche [12] used the method of images in order to obtain explicit expressions of the p.d.f. p. However, in practice, for general boundaries, the expression emphasized by Durbin does not permit to simulate easily the hitting time. One way to approximate τ ϕ and especially to compute the probability for the hitting time to be smaller than some given T > 0, is to use a time discretization of the Brownian motion on [0, T ]. The time interval is then split into n small intervals of the kind [(k − 1)T /n, kT /n], with 1 ≤ k ≤ n. At each endpoint kT /n, the event B kT /n < ϕ(kT /n) has to be tested and at the same time we need to know if, given the Brownian values at the endpoints, the Brownian paths on the small intervals (the conditional distribution corresponds therefore to the Brownian bridge one) hit the curved boundary. The probability not to hit the boundary on a small interval for the Brownian bridge can be approximated [10]. This method can become onerous if the observed time interval [0, T ] is large. Let us also note that the time-splitting can be replaced by a space-splitting namely for the first time the Brownian motion exits from a given interval [3]. The aim of this study is to present a new method of approximation of τ ϕ by some families (τ ϕ ) ≥0 which satisfy exact simulation and such that τ ϕ converges toward τ ϕ in distribution as tends to 0. Stopping such sequence at a small level = 0 induces the error term in the approximation. Two different families of sequences will be emphasized and the associated convergence rates are estimated. The first algorithm developed in Section 1 concerns increasing curved boundaries and the second one, Section 2, permits to deal with quite general boundaries provided that its derivative is bounded. In the last section, we present different examples in order to illustrate the algorithm efficiency. First-passage time to non-decreasing boundaries Let us assume that the boundary ϕ satisfies (H1) and that the following additional conditions hold ϕ : R + → R is a non-decreasing C 1 -continuous function,(H2)2ϕ (t) √ 1 + t ≤ 1, ∀t ≥ 0.(H3) We introduce the algorithm associated to the hitting time τ ϕ defined by (0.1). Algorithm (A1). Let > 0 be a small parameter and (G n ) n≥0 a sequence of independent standard Gaussian distributed random variables. Initialization: T 0 = 0, T 1 = 0, T 2 = (ϕ(0)/G 0 ) 2 and N = 1. While ϕ(T 2 ) − ϕ(T 1 ) > do: (T 0 , T 1 , T 2 ) ← T 1 , T 2 , T 2 + (ϕ(T 2 ) − ϕ(T 1 )) 2 /G 2 N N ← N + 1. (1.1) Outcome: τ ϕ ← T 2 and N . Let us just note that Algorithm (A1) is very simple to use since each step only requires one Gaussian distributed random variable. Moreover it is a approximation of the first-passage time: Theorem 1.1. 1. Let us assume that the boundary function ϕ satisfies (H1), (H2) and (H3) then the random variable τ ϕ defined in Algorithm (A1) converges in distribution towards τ ϕ defined by (0.1) as tends to zero. More precisely F (t − ) − 3 √ √ 2π ≤ F (t) ≤ F (t), for any t ≥ ,(1. 2) where F (resp. F ) is the cumulative distribution function of τ ϕ (resp. τ ϕ ). 2. There exists a constant C > 0 such that the random number of iterations N defined in Algorithm (A1) satisfies: E[N ε ] ≤ C | log |. (1.3) The parameter describes the precision of the approximation. The number of steps in the Algorithm (A1) is very small (even smaller than usual results obtained for algorithms based on random walks on spheres, which are close to Algorithm (A1), see [14]) : in fact the constant appearing in (1.3) can be explicitly computed: for any constant 0 < κ < 1/2, there exists 0 (κ) > 0 such that (1.3) is satisfied as soon as < 0 , with the particular constant C = 1 mκ , m = log(4) + 2 √ 2 √ π µ and µ = ∞ 0 (log |x|) e −x 2 dx. (1.4) The proof of Theorem 1.1 is based on a main argument developed in the following proposition: each step of Algorithm (A1) has to be related to a particular part of the Brownian paths before hitting the boundary. Proposition 1.2. Let (B t , t ≥ 0) be a standard one-dimensional Brownian motion. We define the following sequence of stopping times: s 0 = T 0 = 0 and for any n ≥ 1: s n := inf t ≥ 0 : B t+Tn−1 = ϕ(T n−1 ) and T n := s 1 + . . . + s n , (1.5) where the function ϕ satisfies (H1), (H2) and (H3). Then the following properties hold: 1. (T n ) n≥0 is a non-decreasing sequence which almost surely converges towards τ ϕ . 2. Let n ≥ 1, then the probability distribution of s n+1 given the σ-algebra F n := σ{s 1 . . . , s n } is identical as (ϕ(T n ) − ϕ(T n−1 )) 2 /G 2 n where (G n ) n≥0 is a sequence of independent standard Gaussian random variables. Moreover s 1 (d) = (ϕ(0)/G 0 ) 2 . 3. Let M := inf{n ≥ 1 : ϕ(T n ) − B Tn ≤ }, then T M and τ ϕ , defined in Algorithm (A1), are identically distributed, so are M and N . Let us note that the mean of each random variable s n defined by (1.5) is infinite since E[G −2 ] = +∞ where G is a standard Gaussian variable. Proposition 1.2 suggests that the first-passage time can be obtained as a sum of positive random variables of infinite average, we easily deduce E[τ ϕ ] = +∞. In the particular case of increasing boundaries ϕ, the sum has infinitely many terms. Proof of Proposition 1.2. Step 1. By construction, the sequence (T n ) n≥0 is non-decreasing and nonnegative: it converges almost surely to T ∞ . Since ϕ is a non-decreasing boundary, T n ≤ τ ϕ for any n ≥ 0. In particular T ∞ is less than τ ϕ which is a finite stopping time due to the law of the iterated logarithm, see (H1) followed by discussion. Consequently, the random variable B T∞ is well defined. Since ϕ is non-decreasing, we get B Tn = ϕ(T n−1 ) for any n ≥ 1. Taking the large n limit leads to B T∞ = ϕ(T ∞ ), the Brownian paths and the function ϕ being continuous. We deduce that T ∞ = τ ϕ . Step 2. Let us first consider the stopping time s 1 . Using the reflection principle of the Brownian paths and a scaling property, we obtain: P(s 1 > t) = P sup 0≤u≤t B u < ϕ(0) = P(|B t | < ϕ(0)) = P(B 2 1 < ϕ(0) 2 /t) = P(ϕ(0) 2 /G 2 0 > t), t ≥ 0. The general n-th case can be proven using similar arguments combined with the Markov property of the Brownian motion: P(s n+1 > t|F n ) = P sup Tn≤u≤Tn+t B u < ϕ(T n ) F n = P sup 0≤u≤t B u+Tn − B Tn < ϕ(T n ) − ϕ(T n−1 ) F n = P sup 0≤u≤tB u < ϕ(T n ) − ϕ(T n−1 ) F n , whereB is a Brownian motion independent of F n . Step 3. Using the results developed in Step 2, we observe that (s n ) n∧M and the sequence of values T 2 , defined in Algorithm (A1), have the same distribution. It is therefore obvious that T M and τ ϕ are identically distributed. Indeed the stopping time can be rewritten as follows: M = inf{n ≥ 1 : ϕ(T n ) − ϕ(T n−1 ) ≤ }. (1.6) Proof of Theorem 1.1. Step 1. Let us recall that T n is defined by (1.5). By Proposition 1.2, T n ≤ τ ϕ for any n ≥ 0 and in particular T M ≤ τ ϕ . Hence P(T M ≤ t) ≥ P(τ ϕ ≤ t), ∀t ≥ 0. Since τ ϕ has the same distribution as T M , we obtain F (t) ≥ F (t), ∀t ≥ 0, (1.7) where F and F are the associated cumulative distribution functions. Let us now prove the second bound in (1.2). For t ≥ , F (t − ) = P(τ ϕ ≤ t − ) = P(T M ≤ t − ) ≤ P(T M ≤ t − , τ ϕ > t) + P(τ ϕ ≤ t) ≤ P(|T M − τ ϕ | > ) + F (t). (1.8) Combining the Markov property of the Brownian motion and the reflection principle leads to P := P(|T M − τ ϕ | > ) ≤ 1 − P sup 0≤u≤ B T M +u ≥ sup 0≤u≤ ϕ(T M + u) ≤ 1 − P sup 0≤u≤ B T M +u − B T M ≥ sup 0≤u≤ ϕ(T M + u) − ϕ(T M ) + ≤ 1 − P sup 0≤u≤ B T M +u − B T M ≥ ϕ(T M + ) − ϕ(T M ) + ≤ 1 − P sup 0≤u≤ B u ≥ sup T M ≤θ≤T M + ϕ (θ) + ≤ 1 − P |B | ≥ sup T M ≤θ≤T M + ϕ (θ) + . Using Hypothesis (H3) and straightforward computations permits to obtain P(|T M − τ ϕ | > ) ≤ 1 − P(|B | ≥ 3 /2) ≤ 3 2π . (1.9) The lower bound in (1.2) holds due to both (1.8) and (1.9). Step 2. Let us now focus our attention to the efficiency of this algorithm. We need to estimate the number of steps which depends on the small parameter . Using the third result presented in Proposition 1.2 on one hand and (1.6) on the other hand, we obtain P(N > n) = P(M > n) = P(ϕ(T 1 ) − ϕ(T 0 ) > , . . . , ϕ(T n ) − ϕ(T n−1 ) > ). Hypothesis (H3) implies P(N > n) ≤ P(s 1 > 2 , . . . , s n > 2 ). (1.10) Step 2.1. Let us first estimate the previous upper-bound. We introduce a sequence of independent standard Gaussian random variables (G n ) n≥0 and define X n = log(4G 2 n ), Ξ n = n k=0 X k and Z n = n k=0 Ξ k . (1.11) Let us define Π(n, ) := P(s n > 2 ). By Proposition 1.2, we know that the random variables s n+1 are related to G n and therefore Π(1, ) = P(2 G 2 0 < ϕ(0) 2 ) = P log(4G 2 0 ) < − log( ) + log(2) + 2 log ϕ(0) = P Z 0 < − log( ) + log(2) + 2 log ϕ(0) . Let us prove that, for n ≥ 1, we have the general formula: Π(n, ) ≤ P Z n−1 < − log( ) + (2n − 1) log(2) + (2n) log ϕ(0) . (1.12) By Proposition 1.2, we have for n ≥ 2, Π(n, ) = P (ϕ(T n−1 ) − ϕ(T n−2 )) 2 > 2 G 2 n−1 . (1.13) Since ϕ is a non decreasing function satisfying Hypothesis (H3), the following upper-bound holds for n ≥ 2: ϕ(T n−1 ) − ϕ(T n−2 ) ≤ T n−1 − T n−2 2 1 + T n−2 ≤ s n−1 2 √ 1 + s n−2 . (1.14) Hence for n = 2, (1.13) and (1.14) imply Π(2, ) ≤ P s 2 1 2 2 > 2 G 2 1 = P (2G 2 1 )(2G 2 0 ) 2 < ϕ(0) 4 = P(2X 0 + X 1 < − log( ) + 3 log(2) + 4 log ϕ(0)) = P(Z 1 < − log( ) + 3 log(2) + 4 log ϕ(0)). Using the lower-bound 1+s n−1 ≥ s n−1 and similar arguments as those developed previously, the general case is expressed as follows: Π(n, ) ≤ P s 2 n−1 2 2 (1 + s n−2 ) > 2 G 2 n−1 ≤ P s 3 n−2 2 2 2 4 s 2 n−3 > 2 G 2 n−1 G 4 n−2 ≤ P s n−1 2 s n−2 1 > 22 2 2 4 . . . 2 2(n−2) G 2 n−1 G 4 n−2 . . . G 2(n−2) 2 ≤ P s 2 1 2 2 G 2 1 n−1 1 s n−2 1 > 22 2 2 4 . . . 2 2(n−2) G 2 n−1 G 4 n−2 . . . G 2(n−2) 2 ≤ P ϕ(0) 2n > 22 2 2 4 . . . 2 2(n−1) G 2 n−1 G 4 n−2 . . . G 2n 0 ≤ P Z n−1 < − log( ) + (2n − 1) log(2) + (2n) log ϕ(0) . Step 2.2. By (1.10) and the arguments developed in Step 2.1, we obtain P(N > n) ≤ P(s n > 2 ) ≤ P(Z n−1 − EZ n−1 < η( , n) − EZ n−1 ), where η( , n) := − log( ) + (2n − 1) log(2) + (2n) log ϕ(0). Let us observe that, for any n ≥ 0, m := E[X n ] = log(4) + 2 √ 2 √ π µ > 0 where µ is defined by (1.4). Hence E[Z n ] = n k=0 E[Ξ n ] = n k=0 k j=0 E[X j ] = m n k=0 (k + 1) = m(n + 1)(n + 2) 2 . Thus, for n large enough, η( , n) − EZ n−1 < 0. Introducing d n := |mn(n + 1)/2 − η( , n)|, we observe that, for any 0 < κ < 1/2 there exists ℵ(κ, ) ∈ N such that d n > mn 2 (1/2 − κ) for n sufficiently large that is n ≥ ℵ(κ, ). After straightforward computations, we can choose ℵ(κ, ) := | log(2 )| mκ + 1 2κ − log(2ϕ(0)) mκ + 1. (1.15) Markov's inequality leads to P(N > n) ≤ P(|Z n−1 − E[Z n−1 ]| > d n ) ≤ E[(Z n−1 − E[Z n−1 ]) 4 ] d 4 n . (1.16) Let us note that X j := X j − m are i.i.d. random variables with finite moments of any order. We denote m k := E[X k j ]. Therefore we obtain Z n−1 := E[(Z n−1 − E[Z n−1 ]) 4 ] = E n−1 k=0 k j=0 X j 4 = E n−1 j=0 (n − j)X j 4 = n−1 j=0 (n − j) 4 m 4 + 2 0≤j<k≤n−1 (n − j) 2 (n − k) 2 m 2 2 ≤ m 4 30 n(n + 1)(6n 3 + 9n 2 + n − 1) + m 2 2 36 n 2 (n + 1) 2 (2n + 1) 2 . (1.17) Hence, there exist a constant C 0 > 0 such that E[(Z n−1 − E[Z n−1 ]) 4 ] ≤ C 0 n 6 . Combining the previous inequality with (1.15) and (1.16) leads to P(N > n) ≤ C 0 m 4 (1/2 − κ) 4 1 n 2 , for n ≥ ℵ(κ, ). Consequently, the following upper-bound holds E[N ] = n≥0 P(N > n) ≤ ℵ(κ, ) + C 0 m 4 (1/2 − κ) 4 n≥ℵ(κ, ) 1 n 2 . In order to conclude, it suffices to note that ℵ(κ, ) → ∞ as → 0, the second term in the previous inequality therefore becomes small as → 0: the leading term is finally ℵ(κ, ) which is equivalent to | log(2 )|/(mκ) by (1.15). First-passage time to boundaries with bounded derivative The algorithm presented in Section 1 is simple to achieve (it only requires independent Gaussian random variables) and efficient: the averaged number of steps is of the order | log | where stands for the small parameter appearing in the rejection sampling (see Theorem 1.1). In order to apply Algorithm (A1) the curved boundary, the Brownian motion is going to hit, has to satisfies suitable conditions: (H1), (H2) and (H3). Asking for the monotonicity of the function ϕ is quite restrictive, that's why we present an extension of the algorithm which is of course less efficient (even if the average number of steps is still very small) but which permits to deal with more general boundaries. Let us introduce the following assumption: there exist two constants ρ + > 0 and ρ − > 0 such that ϕ : R + → R is a C 1 -continuous function satisfying sup t≥0 ϕ (t) ≤ ρ + and inf t≥0 ϕ (t) ≥ −ρ − .(H4) For such boundaries, we present an algorithm which permits for any K ∈ R + to approximate the hitting time τ K ϕ = τ ϕ ∧ K, where τ ϕ is defined in (0.1). Let us introduce some notations: the inverse Gaussian distribution of parameters µ > 0 and λ > 0 will be denoted by I(µ, λ) and is defined by its the probability distribution function: f (x) = λ 2πx 3 exp − λ(x − µ) 2 2µ 2 x 1 {x≥0} . Algorithm (A2). Let > 0 be a small parameter and r > ρ − where ρ − is defined in (H4). Initialization: (T, H) = (0, ϕ(0)) and N ,K = 0. While H > and T < K, simulateĜ an inverse Gaussian random variable with distribution I(H/r, H 2 ) and do:    H ← ϕ(T +Ĝ) − ϕ(T ) + rĜ, T ←Ĝ + T, N ,K ← N ,K + 1. (2.1) Outcome: τ ,K ϕ ← T ∧ K and N ,K . Algorithm (A2) is quite simple, it only requires the simulation of inverse Gaussian distributed random variables. Let us recall the following scaling property: ifĜ ∼ I(H/r, H 2 ) then HĜ/r ∼ I(1, rH). Moreover (rĜ−H) 2 G is Chi-squared distributed with one degree of freedom (the square of a standard Gaussian random variable). In order to simulate an inverse Gaussian random variable, we suggest to use the algorithm introduced by Michael, Schucany and Haas (see [13] or [5, p. 149]). Let us now state the efficiency of Algorithm (A2). The inverse Gaussian distribution does not permit to argue in a similar way as in Section 1. That's why we are going to use the general potential theory in order to upper-bound the averaged number of steps. This kind of arguments was already introduced in convergence results associated to the Random Walk on Spheres algorithm which permits to approximate the solution of the Dirichlet problem, see for instance [14]. Theorem 2.1. 1. Let us assume that the boundary function ϕ satisfies (H4) then the random variable τ ,K ϕ defined in Algorithm (A2) converges in distribution towards τ K ϕ = τ ϕ ∧ K where τ ϕ is defined by (0.1) as tends to zero. More precisely F ,K (t − ) − (1 + ρ) 2 π ≤ F K (t) ≤ F ,K (t), for any t ≥ , (2.2) where F K (resp. F ,K ) is the cumulative distribution function of τ K ϕ (resp. τ ,K ϕ ). 2. There exist positive constants a, b, κ 0 , κ 1 and 0 such that: for any ρ + ≤ κ 0 and any (K, r) satisfying (r + κ 0 )K ≤ κ 1 , the random number of iterations N ,K defined in Algorithm (A2) satisfies the following upper bound E[N ε,K ] ≤ (a + br)| log |, ∀ ≤ 0 . Proposition 2.2. Let (B t , t ≥ 0) be a standard one-dimensional Brownian motion. We introduce the following stopping times: s 0 = T K 0 = 0 and for any n ≥ 1: s n := inf t ≥ 0 : B t+T K n−1 = ϕ(T K n−1 ) − rt and T K n := (s 1 + . . . + s n ) ∧ K, (2.5) where the boundary ϕ satisfies (H4). Then the following properties hold: 1. (T K n ) n≥0 is a non-decreasing sequence which almost surely converges towards τ K ϕ . 2. On the event {s 1 +· · ·+s n < K}, the probability distribution of s n+1 given the σ-algebra F n := σ{T K 1 , . . . , T K n } is the inverse Gaussian distribution I(H n /r, H 2 n ) with Here the monotonicity property is just replaced by (H4) which permits easily to prove that T K n ≤ τ K ϕ . Let us now focus our attention to the second part of the statement. Due to the definition of s n+1 and since {T K n < K}, we get B T K n = ϕ(T K n−1 ) − rs n . Hence, we have H n := ϕ(T K n ) − ϕ(T K n−1 ) + rs n .s n+1 = inf{t ≥ 0 : B t+T K n − B T K n = ϕ(T K n ) − B T K n − rt} = inf{t ≥ 0 : W t = H n − rt}, where W t = B t+T K n − B T K n is a standard Brownian motion independent of F n and the F n adapted r.v. H n is defined by (2.6). The distribution of s n+1 corresponds to the distribution of the first passage time of the standard Brownian motion with drift at the constant level H n . The probability distribution is well known (see, for instance [11, p. 197]): P(s n+1 ∈ dt|F n ) = H n √ 2πt 3 exp − (H n − rt) 2 2t dt, we can consequently identify the inverse Gaussian distribution I(H n /r, H 2 n ). Proof of Theorem 2.1. Step 1. We can prove the convergence in distribution of τ ,K ϕ towards τ K ϕ using similar arguments as those presented in the proof of Theorem 1.1. The upperbound in (2.2) is an adaptation of (1.7) which requires that T K n ≤ τ K ϕ and that τ ,K ϕ and T M K are identically distributed. These conditions are satisfied, see Proposition 2.2. For the lower-bound in (2.2), we obtain F ,K (t − ) ≤ P(|T M K − τ K ϕ | > ) + F ,K (t), see (1.8) for the details. Let us note that |τ M K − τ K ϕ | > leads to the condition τ M K < K. Hence M K = M . Using the Markov property, the following bound holds: P(|T M K − τ K ϕ | > ) ≤ 1 − P sup 0≤u≤ B u ≥ + sup 0≤u≤ ϕ(T M + u) − ϕ(T M ) . Here (B t , t ≥ 0) stands for a standard Brownian motion independent of T M . Combining Hypothesis (H4) and the reflection principle of the Brownian motion leads to P(|T M − τ K ϕ | > ) ≤ 1 − P(|B | ≥ (1 + ρ + )) ≤ (1 + ρ + ) 2 π , and consequently to the lower bound (2.2). Step 2. Let us now focus our attention to the averaged number of steps in Algorithm (A2), denoted by N ,K . A rough description of the method: we aim to construct a Markov chain and to describe the associated potential. The classical potential theory then permits to obtain the announced bound. We introduce the Markov chain R n := (T n , H n ) for n ≥ 0. We recall that T n = s 1 + . . . + s n is defined by (2.5) and H n by (2.6). The stopping time M K defined in Proposition 2.2 can also be interpreted as the first time the Markov chain (R n , n ≥ 0) goes out of the domain E := [0, K]×] , +∞]. Let us consider the function f (x, y) = log(y), defined on E, and denote by P the infinitesimal generator associated to the Markov chain (R n ) n≥0 . By Proposition 2.2 and for any (t, h) ∈ E, we obtain P f (t, h) = E log(ϕ(t +Ĝ) − ϕ(t) + rĜ) , whereĜ is an inverse Gaussian distributed random variable with the following density function: p(x) = h √ 2πx 3 exp − (h − rx) 2 2x , x ≥ 0. By (H4), ϕ(t +Ĝ) − ϕ(t) ≤ ρ +Ĝ , we get P f (t, h) − f (t, h) ≤ log 1 + ρ + r + E log rĜ h . (2.7) Let us find now an explicit upper bound of P f − f . Using first the change of variables u = rx/h and secondly u → 1/u, we get E log rĜ h = ∞ 0 log rx h h √ 2πx 3 exp − (h − rx) 2 2x dx = hr 2π ∞ 0 log(u) u 3/2 exp − hr(1 − u) 2 2u du = hr 2π ∞ 1 (1 − u) log(u) u 3/2 exp − hr(1 − u) 2 2u du. (2.8) It is then obvious that E log rĜ h < 0. Let us now give a more precise upperbound. We set α = hr, then (2.8) emphasizes that E log rĜ h only depends Figure 1: Monte Carlo approximation of the function ψ on the parameter α, this dependence being continuous. Let us therefore denote this function ψ(α) (see Figure 1 below representing ψ obtained with the Monte-Carlo method sample size: 10 000). Simple computations lead to ψ(α) := E log rĜ h = − α 2π ∞ 0 u log(1 + u) (1 + u) 3/2 exp − αu 2 2(1 + u) du (2.9) ≤ − α 2π ∞ 0 u log(1 + u) (1 + u) 3/2 exp − αu 2 du ≤ − 1 √ 2π ∞ 0 w log(1 + w/α) (α + w) 3/2 exp − w 2 dw ≤ − 1 √ 2π ∞ 1/2 w log(1 + w/α) (α + w) 3/2 exp − w 2 dw. Using the inequality (α + w) ≤ (1 + 2α)w, we get ψ(α) ≤ − log(1 + (2α) −1 ) (1 + 2α) 3/2 √ 2π ∞ 1/2 1 √ w exp − w 2 dw ≤ − log(1 + (2α) −1 ) (1 + 2α) 3/2 P(G ≥ 1/2), where G is a standard gaussian r.v. and so P(G ≥ 1/2) ≈ 0.3085 We deduce from the previous upper-bound that lim α→0 + ψ(α) = −∞. Moreover the right hand side is a non decreasing function with respect to the variable α. Hence ψ(α) ≤ − log(3/2) 3 √ 3 P(G ≥ 1/2) ≈ −0.0241, for α ≤ 1. (2.10) Let us observe what happens for large values of the variable α. The Laplace method implies that ψ(α) ∼ − 1 2α as α → ∞. Let us prove now that there exists a constant c > 0 such that ψ(α) ≤ − c α , for any α ≥ 1. (2.11) For α ≥ 1, we get ψ(α) ≤ − α 2π ∞ 0 u log(1 + u) (1 + u) 3/2 exp − αu 2 2 du ≤ − α 2π 1 0 u log(1 + u) (1 + u) 3/2 exp − αu 2 2 du. Due to the convexity property of the logarithm function (log(1 + u) ≥ log(2)u) and the Cauchy-Schwarz inequality, we obtain ψ(α) ≤ − log(2) α2 3/2 1 2 E[G 2 ] − E[G 2 1 {G≥ √ α} ] ≤ − log(2) α2 3/2 1 2 − E[G 4 ] P(G ≥ √ α) ≤ − log(2) α2 3/2 1 2 − √ 3 2 e −α ≤ − log(2) α2 5/2 (1 − √ 3e −1 ), for α ≥ 1. We deduce that ψ(α) ≤ −c/α with c ≈ 0.0445 when α ≥ 1. Combining both inequalities (2.10) and (2.11) leads to the existence of a constant c > 0 such that ψ(α) ≤ −c 1 α ∧ 1 . (2.12) By (2.7), the following upper-bound holds: for f (x, y) = log(y), P f (t, h) − f (t, h) ≤ log 1 + ρ + r − c 1 hr ∧ 1 ≤ ρ + r − c 1 hr ∧ 1 , h ≥ 0, t ≥ 0. (2.13) Due to the definition of ρ + , we know that h ≤ ϕ(0) ∨ (r + ρ + )t ≤ ϕ(0) ∨ (r + ρ + )K, where ϕ is the boundary the process has to hit. In other words, there exist two constants κ 0 > 0 and κ 1 > 0 such that for any ρ + ≤ κ 0 and any (K, r) satisfying (r + κ 0 )K ≤ κ 1 the following bound holds ρ + ≤ c 2 1 h ∧ r . Hence: P f (t, h) − f (t, h) ≤ − c 2r 1 ϕ(0) ∧ κ 1 ∧ r =: −R −1 (r). We deduce that the function g(t, h) defined by g(t, h) = R(r) (f (t, h) − log ) satisfies g(t, h) ≥ 0 for any (t, h) ∈ E and P g(t, h) − g(t, h) ≤ −1 on E. The potential theory therefore implies: E[N ,K ] ≤ g(0, ϕ(0)) ≤ R(r)(log(ϕ(0)) − log( )). We finally deduce the existence of a > 0 and b > 0 such that E[N ε,K ] ≤ (a + br)| log | for small enough. For the particular case of a non increasing boundary function it suffices to vanish ρ + in (2.13) and to apply the same arguments of the potential theory in order to get (2.4) 3 Examples and numerics. In this section, we present three different examples which nicely illustrate the efficiency of these new algorithms (A1) and (A2). 3.1 Brownian hitting time of ϕ(t) = √ 1 + αt Let us first consider an application of Theorem 1.1. We observe that ϕ(t) = √ 1 + αt is an increasing function satisfying (H1), (H2) and (H3) for α ∈ [0, 1]. Consequently Algorithm (A1) converges and permits to obtain an approximation of the hitting time τ ϕ . In the figures, we present the link between the averaged number of steps and which characterizes the approximation error size. The first figure (resp. the second one) concerns: α = 1 (resp. α = 0.01), = 0.5 n (n is represented on the horizontal axis) and the number of simulation in order to estimate the averaged number of steps is 10 000. Let us now present the approximate distribution of the hitting time. Brownian hitting time of ϕ(t) = α + β cos(ωt) Let us now consider the first time the Brownian motion hits the periodic boundary ϕ(t) = α + β cos(ωt). Since the boundary is not an increasing function, we shall use Algorithm (A2). Theorem 2.1 ensures that the algorithm converges. Let us therefore use the Monte-Carlo method in order to estimate precisely the average number of steps. As explained in the previous section, the simulation procedure permits to approximate the stopping time τ ϕ ∧K for some given fixed time K. Figure 4 illustrates the approximation τ ϕ by τ ,K ϕ , where the parameters are fixed at α = 3.5, β = 3 and ω = π/2. The maximal time are K = 20 on one hand and K = 100 on the other hand and the error rate is given by = 0.5 n , for 1 ≤ n ≤ 10. A sample of 10E8 paths has been simulated to approximate the mean. We know that the mean number of steps is a decreasing function of and an increasing function of K. Figure 5 gives the evolution of the mean number of steps as a function of the truncation K. In practice, we obtained easily an impressively accurate approximation of τ ϕ . 3.3 The first time the Ornstein Uhlenbeck process hits the boundary ϕ(t) = α + β cos(ωt) The last example concerns the one-dimensional Ornstein-Uhlenbeck process defined as the unique solution of the following stochastic differential equation: dX t = dB t − λX t dt, X 0 = x 0 , (3.1) where (B t , t ≥ 0) is the standard Brownian motion. The aim is to approximate the first passage time through the curved boundary ϕ(t) = α + β cos(ωt) where ϕ(0) > x 0 . Since the Ornstein-Uhlenbeck process can be represented as a timechanged Brownian motion, the question is directly related to the main results of this study. Indeed the solution of (3.1) is given by X t = e −λt x 0 + t 0 e λs dB s , t ≥ 0. Using Levy's theorem, (X t , t ≥ 0) has the same distribution as (Y t , t ≥ 0) defined by Y t := e −λt x 0 + W u(t) , t ≥ 0, with u(t) := 1 2λ (e 2λt − 1) and W a standard Brownian motion. We deduce that T ϕ := inf{t ≥ 0 : X t = ϕ(t)} has the same distribution aŝ Consequently, in order to simulate the Ornstein-Uhlenbeck hitting time T ϕ ∧ K for some K, we simply use Algorithm (A2) and propose an approximation of the Brownian hitting time τ ψ ∧K withK := u(K) = (e 2λK − 1)/(2λ). Let us note that a straightforward computation leads to the following upperbound: |ψ (t)| ≤ λα + λβ + ωβ √ 1 + 2λt ≤ λα + λβ + ωβ, t ≥ 0. In other words, the continuous curve ψ satisfies Hypothesis (H4): Algorithm (A2) therefore converges and Theorem 2.1 can be applied. In the following numerical experiences, we will choose r = 0.5 + λα + λβ + ωβ. Figures 6 and 7 concern the following choice of parameters: x 0 = 0, α = 2, β = 1, ω = π/5, λ = 0.5. We have chosen K = 5 for Figure 6 and K = 10 for Figure 7. In both cases, the first figure represents the average number of steps as a function of n where the approximation parameter is chosen as 0.5 n , for n = 1, · · · , 10. The average has been estimated using 5.10E6 simulations. The second figure represents the distribution of T ϕ ∧ K for n = 10. We observe that the change of timeK = (e 2λK − 1)/(2λ) increases very fast with K and the number becomes quite large when K increases. Note however that the number of random variables we have to simulate keeps relatively small in comparaison with the use of a classical stopped Euler scheme usually used to approximate T ϕ . For non increasing functions ϕ: there exists two positive constants a and 0 such that E[N ε,K ] ≤ ar 2 K| log |, ∀ ≤ 0 . (2.4) This theorem is based on the following intermediate statement which is a modification of Proposition 1.2. Let M := inf{n ≥ 1 : ϕ(T K n )−B T K n ≤ }, M K := inf{n ≥ 1, T K n = K}, M K = M ∧ M K . then T M K and τ ,K ϕ , defined in Algorithm (A2), are identically distributed, so are M K and N ,K . Proof of Proposition 2.2. The first and the third part of the proof are left to the reader. They need similar arguments as those presented in Proposition 1.2. Figure 2 : 2E(N ): mean number of steps for = 0.5 n as a function of n. The boundary is ϕ(t) = √ 1 + αt. Figure 3 : 3Empirical distribution of the approximate first hitting time of the boundary ϕ(t) = √ 1 + αt. . n = 10, K = 20. Figure 4 : 4Approximation of τ ϕ with ϕ(t) = 3.5 + 3 cos(πt/2) Figure 5 : 5E(N ,K ϕ ) as a function of K. T ϕ := inf t ≥ 0 : e −λt x 0 + W u(t) = ϕ(t) = inf u −1 (s) ≥ 0 : W s = ϕ(u −1 (s))e λu −1 (s) − x 0 = u −1 (τ ψ ), where τ ψ := inf{t ≥ 0 : W t = ψ(t)}, ψ(t) := √ 1 + 2λt ϕ log(1 + 2λt) 2λ − x 0 . Figure 6 : 6First hitting time of ϕ(t) = α + β cos(ωt) by an Ornstein Uhlenbeck process solution of (3.1) (α = 2, β = 1, ω = π/5, λ = 0.5, K = 5.)(a) E(N 1/2 n ,K ) versus n (b) Distribution of T K, ϕ ( = 1/2 10 ). Figure 7 : 7First hitting time of ϕ(t) = α + β cos(ωt) by an Ornstein Uhlenbeck process solution of (3.1) (α = 2, β = 1, ω = π/5, λ = 0.5, K = 10). A survey of recent progress on level-crossing problems for random processes. J Abrahams, Communications and Networks. IanF. Blake and H.Vincent PoorNew YorkSpringerJ. Abrahams. A survey of recent progress on level-crossing problems for random processes. In IanF. Blake and H.Vincent Poor, editors, Communi- cations and Networks, pages 6-25. Springer New York, 1986. Level-crossing problems for random processes. I F Blake, W C Lindsey, IEEE Trans. Information Theory, IT. 19I. F. Blake and W. C. Lindsey. Level-crossing problems for random pro- cesses. IEEE Trans. Information Theory, IT-19:295-315, 1973. Simulation of Brownian motion at first-passage times. Z A Burq, O D Jones, Math. Comput. Simulation. 771Z. A. Burq and O. D. Jones. Simulation of Brownian motion at first-passage times. Math. Comput. Simulation, 77(1):64-71, 2008. The maximum size of a closed epidemic. H E Daniels, Advances in Appl. Probability. 6H. E. Daniels. The maximum size of a closed epidemic. Advances in Appl. Probability, 6:607-621, 1974. Nonuniform random variate generation. L Devroye, Springer-VerlagNew YorkL. Devroye. Nonuniform random variate generation. Springer-Verlag, New York, 1986. The first-passage density of a continuous Gaussian process to a general boundary. J Durbin, J. Appl. Probab. 221J. Durbin. The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Probab., 22(1):99-122, 1985. The first-passage density of the Brownian motion process to a curved boundary. J Durbin, J. Appl. Probab. 292With an appendix by D. WilliamsJ. Durbin. The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Probab., 29(2):291-304, 1992. With an appendix by D. Williams. Mathematical foundations of neuroscience. G B Ermentrout, D H Terman, Interdisciplinary Applied Mathematics. 35SpringerG. B. Ermentrout and D. H. Terman. Mathematical foundations of neu- roscience, volume 35 of Interdisciplinary Applied Mathematics. Springer, New York, 2010. Spiking neuron models. W Gerstner, W M Kistler, Cambridge University PressCambridgeSingle neurons, populations, plasticityW. Gerstner and W. M. Kistler. Spiking neuron models. Cambridge Uni- versity Press, Cambridge, 2002. Single neurons, populations, plasticity. Euler schemes and half-space approximation for the simulation of diffusion in a domain. E Gobet, ESAIM Probab. Statist. 5E. Gobet. Euler schemes and half-space approximation for the simulation of diffusion in a domain. ESAIM Probab. Statist., 5:261-297 (electronic), 2001. Brownian motion and stochastic calculus. I Karatzas, S E Shreve, Graduate Texts in Mathematics. 113Springer-Verlagsecond editionI. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991. Boundary crossing of Brownian motion. H R Lerche, Lecture Notes in Statistics. 40Springer-VerlagIts relation to the law of the iterated logarithm and to sequential analysisH. R. Lerche. Boundary crossing of Brownian motion, volume 40 of Lecture Notes in Statistics. Springer-Verlag, Berlin, 1986. Its relation to the law of the iterated logarithm and to sequential analysis. Generating random variates using transformations with multiple roots. J R Michael, W R Schucany, R W Haas, The American Statistician. 302J. R. Michael, W. R. Schucany, and R. W. Haas. Generating random vari- ates using transformations with multiple roots. The American Statistician, 30(2):88-90, 1976. Some continuous Monte Carlo methods for the Dirichlet problem. M E Muller, Ann. Math. Statist. 27M. E. Muller. Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Statist., 27:569-589, 1956. Almost sure behavior of sums of independent random variables and martingales. V Strassen, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability. Fifth Berkeley Sympos. Math. Statist. and ProbabilityBerkeley, Calif; Berkeley, CalifUniv. California PressIIContributions to Probability Theory. Part 1V. Strassen. Almost sure behavior of sums of independent random variables and martingales. In Proc. Fifth Berkeley Sympos. Math. Statist. and Prob- ability (Berkeley, Calif., 1965/66), pages Vol. II: Contributions to Proba- bility Theory, Part 1, pp. 315-343. Univ. California Press, Berkeley, Calif., 1967.
[]
[ "Salient Object Detection via Augmented Hypotheses", "Salient Object Detection via Augmented Hypotheses" ]
[ "Tam V Nguyen [email protected] \nDepartment for Technology, Innovation and Enterprise\nSingapore Polytechnic\n", "Jose Sepulveda [email protected] \nDepartment for Technology, Innovation and Enterprise\nSingapore Polytechnic\n" ]
[ "Department for Technology, Innovation and Enterprise\nSingapore Polytechnic", "Department for Technology, Innovation and Enterprise\nSingapore Polytechnic" ]
[]
In this paper, we propose using augmented hypotheses which consider objectness, foreground and compactness for salient object detection. Our algorithm consists of four basic steps. First, our method generates the objectness map via objectness hypotheses. Based on the objectness map, we estimate the foreground margin and compute the corresponding foreground map which prefers the foreground objects. From the objectness map and the foreground map, the compactness map is formed to favor the compact objects. We then derive a saliency measure that produces a pixelaccurate saliency map which uniformly covers the objects of interest and consistently separates foreand background. We finally evaluate the proposed framework on two challenging datasets, MSRA-1000 and iCoSeg. Our extensive experimental results show that our method outperforms state-ofthe-art approaches.
null
[ "https://arxiv.org/pdf/1505.07930v1.pdf" ]
6,505,016
1505.07930
88b471a3129c9b9ec3eb30adca08881d76deb1b5
Salient Object Detection via Augmented Hypotheses Tam V Nguyen [email protected] Department for Technology, Innovation and Enterprise Singapore Polytechnic Jose Sepulveda [email protected] Department for Technology, Innovation and Enterprise Singapore Polytechnic Salient Object Detection via Augmented Hypotheses In this paper, we propose using augmented hypotheses which consider objectness, foreground and compactness for salient object detection. Our algorithm consists of four basic steps. First, our method generates the objectness map via objectness hypotheses. Based on the objectness map, we estimate the foreground margin and compute the corresponding foreground map which prefers the foreground objects. From the objectness map and the foreground map, the compactness map is formed to favor the compact objects. We then derive a saliency measure that produces a pixelaccurate saliency map which uniformly covers the objects of interest and consistently separates foreand background. We finally evaluate the proposed framework on two challenging datasets, MSRA-1000 and iCoSeg. Our extensive experimental results show that our method outperforms state-ofthe-art approaches. Introduction The ultimate goal of salient object detection is to search for salient objects which draw human attention on the image. The research has shown that computational models simulating low-level stimuli-driven attention [Koch and Ullman, 1985;Itti et al., 1998] are quite successful and represent useful tools in many practical scenarios, including image resizing [Achanta et al., 2009], attention retargeting [Nguyen et al., 2013a], dynamic captioning [Nguyen et al., 2013b], image classification and action recognition [Nguyen et al., 2015]. The existing methods can be classified into biologically-inspired and computationally-oriented approaches. On the one hand, works belonging to the first class [Itti et al., 1998;Cheng et al., 2011] are generally based on the model proposed by Koch and Ullman [Koch and Ullman, 1985], in which the low-level stage processes features such as color, orientation of edges, or direction of movement. One example of this model is the work by Itti et al. [Itti et al., 1998], which use a Difference of Gaussians approach to evaluate those features. However, the resulting saliency maps are generally blurry, and often overemphasize small, purely local features, which renders this approach less useful for applications such as segmentation, detection, etc [Cheng et al., 2011]. On the other hand, computational methods relate to typical applications in computer vision and graphics. For example, frequency space methods [Hou and Zhang, 2007] determine saliency based on spectral residual of the Fourier transform of an image. The resulting saliency maps exhibit undesirable blurriness and tend to highlight object boundaries rather than its entire area. Since human vision is sensitive to color, different approaches use local or global analysis of color contrast. Local methods estimate the saliency of a particular image region based on immediate image neighborhoods, e.g., based on dissimilarities at the pixel-level [Ma and Zhang, 2003] or histogram analysis [Cheng et al., 2011]. While such approaches are able to produce less blurry saliency maps, they are agnostic of global relations and structures, and they may also be more sensitive to high frequency content like image edges and noise. In a global manner, [Achanta et al., 2009] achieves globally consistent results by computing color dissimilarities to the mean image color. Murray et al. : Saliency maps computed by our proposed AH method (t) and state-of-the-art methods (a-r), salient region detection (AC [Achanta et al., 2008]), attention based on information maximization (AIM [Bruce and Tsotsos, 2005]), context-aware (CA [Goferman et al., 2010]), frequency-tuned (FT [Achanta et al., 2009]), graph based saliency (GB [Harel et al., 2006]), global components (GC [Cheng et al., 2013]), global uniqueness (GU [Cheng et al., 2013]), global contrast saliency (HC and RC [Cheng et al., 2011]), spatial temporal cues (LC [Zhai and Shah, 2006]), visual attention measurement (IT [Itti et al., 1998]), maximum symmetric surround (MSS ), fuzzy growing (MZ [Ma and Zhang, 2003]), saliency filters (SF [Perazzi et al., 2012]), induction model (SIM [Murray et al., 2011]), spectral residual (SR [Hou and Zhang, 2007]), saliency using natural statistics (SUN [Zhang et al., 2008]), and the objectness map (s). Our result (t) focuses on the main salient object as shown in ground truth (u). ray et al., 2011] introduced an efficient model of color appearance, which contains a principled selection of parameters as well as an innate spatial pooling mechanism. There also exist different patch-based methods which estimate dissimilarity between image patches [Goferman et al., 2010;Perazzi et al., 2012]. While these algorithms are more consistent in terms of global image structures, they suffer from the involved combinatorial complexity, hence they are applicable only to relatively low resolution images, or they need to operate in spaces of reduced image dimensionality [Bruce and Tsotsos, 2005], resulting in loss of salient details. Despite many recent improvements, the difficult question is still whether "the salient object is a real object". That question bridges the problem of salient object detection into the traditional object detection research. In the latter object detection problem, the efficient sliding window object detection while keeping the computational cost feasible is very important. Therefore, there exist numerous objectness hypothesis generation methods proposing a small number (e.g. 1,000) of category-independent hypotheses, that are expected to cover all objects in an image [Lampert et al., 2008;Alexe et al., 2012;Uijlings et al., 2013;Cheng et al., 2014]. Objectness hypothesis is usually represented as a value which reflects how likely an image window covers an object of any category. Lampert et al. [Lampert et al., 2008] introduced a branch-and-bound scheme for detection. However, it can only be used to speed up classifiers that users can provide a good bound on highest score. Alexe et al. [Alexe et al., 2012] proposed a cue integration approach to get better prediction performance more efficiently. Uijlings et al. [Uijlings et al., 2013] proposed a selective search approach to get higher prediction performance. However, these methods are time-consuming, taking 3 seconds for one image. Recently, Cheng et al. presented a simple and fast objectness measure by using binarized normed gradients features which compute the objectness of each image window at any scale and aspect ratio only requires a few bit operations. This method can be run 1,000+ times faster than popular alternatives. In this work, we investigate applying objectness to the problem of salient object detection. We utilize the object hypotheses from the objectness hypothesis generation augmented with foreground and compactness constraint in order to produce a fast and high quality salient object detector. The exemplary object hypotheses and our saliency prediction are shown in the second and the third row of Figure 1, respectively. As we demonstrate in our experimental evaluation, each of our individual measures already performs close to or even better than some existing approaches, and our combined method currently achieves the best ranking results on two public datasets provided by [Achanta et al., 2009;Batra et al., 2010]. Figure 2 shows the comparison of our saliency map to other baselines in literature. The main contributions of this work can be summarized as follows. • We conduct the comprehensive study on how the objectness hypotheses affect the salient object detection. • We propose the foreground map and compactness map, derived from the objectness map, which can cover both global and local information of the saliency object. • Unlike other works in the literature, we evaluate our proposed method on two challenging datasets in order to know the impact of our work in different settings. Methodology In this section, we describe the details of our augmented hypotheses (AH), and we show how the objectness measures as well as the saliency assignment can be efficiently computed. Figure 3 illustrates the overview of our processing steps. Objectness Map In this work, we extract object hypotheses from the input image to form the objectness map. We assume that the salient objects attract more object hypotheses than other parts in the Figure 3: Illustration of the main phases of our algorithm. The object hypotheses are generated from the input image. The objectness map is later formed by accumulating all hypotheses. The foreground map is then created from the difference between the pixel's color and the background color obtained following the estimated margins. We then oversegment the image into superpixels and compute the compactness map based on the spatial distribution of superpixels. Finally, a saliency value is assigned to each pixel. image. As aforementioned, the objectness hypothesis generators propose a small number n p (e.g. 1,000) of categoryindependent hypotheses, that are expected to cover all objects in an image. Each hypothesis P i has coordinate (l i , t i , r i , b i ), where l i , t i areP i (x, y) = 1 if t i ≤ x ≤ b i and l i ≤ y ≤ r i 0 otherwise . (1) The objectness map is constructed by accumulating all object hypotheses: OB(x, y) = np i=1 P i (x, y).(2) The objectness map is later rescaled into the range [0. .1]. We observe that the objectness map discourages the object parts locating close to the image boundary. Thus we extend the original image by embedding an image border with the size is 10% of the original image's size. The addition image border is filled with the mean color of the original image. We perform the hypothesis extraction and compute the objectness map similar to the aforementioned steps. The final objectness map is cropped to the size of the original image. Figure 4 demonstrates the effect of our image extension and the shrinkage of the objectness map. Foreground Map The salient object tends to be distinctive from its surrounding context. Thus, we aim to model the background which can facilitate the object localization. In particular, the foreground map is computed by finding the difference between the color of the original image and the background image. In order to model the background, we first localize the salient object by the margin shown as the red rectangle in Fig 3d. To this end, we compute the accumulate objectness level by four directions n r , namely, top, bottom, left, and right. For each direction, the accumulated objectness level is bounded by a threshold θ. To boost this process, we utilize the integral image [Viola and Jones, 2001] computed from the objectness map. Finally, there are n r , 4 in this work, corresponding rectangles surrounding the salient object. Each bounding rectangle r i is represented by its mean color µ ri . The foreground value computed for each pixel (x, y) is computed as follows, F G(x, y) = nr i=1 I(x, y) − µ ri ,(3) where I(x, y) is the color vector of the pixel (x, y). Compactness Map The foreground map prefers the color of the salient object of the foreground. Unfortunately, it also favors the similar color appearing in the background. We observe that though the colors belonging to the background will be distributed over the entire image exhibiting a high spatial variance, the foreground objects are generally more compact [Perazzi et al., 2012]. Therefore, we compute the compactness map in order to remove the noise from the background. First, we compute the centroid of interest (x c , y c ) = y) ), where the objectnessforeground value OF (x, y) = OB(x, y) × F G(x, y). Intuitively, the pixel close to the centroid of interest tends to be more salient, whereas the farther pixels tend to be less salient. In addition, the saliency value of a certain pixel reduces if the path between the centroid and that pixel contains many low saliency values. The naive method is to compute the path from the centroid of interest to other pixels. However, it is time-consuming to perform this task in the pixel-level. Therefore, we transform it to superpixel-level. The image is oversegmented into superpixels, and the OF value of a superpixel Algorithm 1 Superpixel compactness computation is computed as the average OF values of all containing pixels. The over-segmented image can be formulated as a graph G = (V, E), where V is the list of vertices (superpixels) and E is the list of edges connecting the neighboring superpixels. ( (x,y) x×OF (x,y) (x,y) OF (x,y) , (x,y) y×OF (x,y) (x,y) OF (x,1: l = {v c }. 2: c = 0 ∈ R nsp . 3: t = ∅ 4: while l = ∅ do 5: for each vertex v i in l do 6: for each edge (v i , v j ) do 7: if c(v j ) < c(v i ) × OF (v j ) then 8: c(v j ) ← c(v i ) × OF (v j The procedure to compute the compactness values of superpixels is summarized in Algorithm 1. Denote v c as the superpixel containing the centroid of interest. The algorithm transfers the OF value from the v c to all other superpixels. The procedure performs a sequence of relaxation steps, namely assigning the compactness value c(v j ) of superpixel v j by the square root of its neighboring superpixel's compactness value and its own OF value. Our algorithm only relaxes edges from vertices v j for which c(v j ) has recently changed, since other vertices cannot lead to correct relaxations. Additionally, the algorithm may be terminated early when no recent changes exist. Finally, the compactness value CN is computed as: CN (x, y) = c(sp(x, y)),(4) where sp(x, y) returns the index of the superpixel containing pixel (x, y). Saliency Assignment We normalize the objectness map OB, foreground map F G, and compactness map CN to the range [0..1]. We assume that all measures are independent, and hence we combine these terms as follows to compute a saliency value S for each pixel: S(x, y) = OB(x, y) × F G(x, y) × CN (x, y).(5) The resulting pixel-level saliency map may have an arbitrary scale. In the final step, we rescale the saliency values within [0..1] and to contain at least 10% saliency pixels. Implementation Settings We apply the state-of-the-art objectness detection technique, i.e., binarized normed gradients (BING) , to produce a set of candidate object windows. Our selection of BING is two-fold. First, BING extractor has a weak training from the simple feature, e.g., binarized normed gradients. Therefore, it is useful comparing to bottom-up edge extractor. Second, the BING extractor is able to run 10 times faster than real-time, i.e., 300 frames per second (fps). BING hypothesis generator is trained with VOC2007 dataset [Everingham et al., 2010] same as in . In order to compute the foreground map, θ is set as 0.1 and we convert the color channels from RGB to Lab color space as suggested in [Achanta et al., 2009;Perazzi et al., 2012]. Regarding the image over-segmentation, we use SLIC [Achanta et al., 2012] for the superpixel segmentation. We set the number of superpixels as 100 as a trade-off between the fine oversegmentation and the processing time. Evaluation Datasets and Evaluation Metrics We evaluate and compare the performances of our algorithm against previous baseline algorithms on two representative benchmark datasets: the MSRA 1000 salient object dataset [Achanta et al., 2009] and the Interactive cosegmentation Dataset (iCoSeg) [Batra et al., 2010]. The MSRA-1000 dataset contains 1,000 images with the pixel-wise ground truth provided by [Achanta et al., 2009]. Note that each image in this dataset contains a salient object. Meanwhile, the iCoSeg contains 643 images with single or multiple objects in a single image. The first evaluation compares the precision and recall rates. High recall can be achieved at the expense of reducing the precision and vice-versa so it is important to evaluate both measures together. In the first setting, we compare binary masks for every threshold in the range [0..255]. In the second setting, we use the image dependent adaptive threshold proposed by [Achanta et al., 2009], defined as twice the mean saliency of the image: T a = 2 W × H (x,y) S(x, y).(6) In addition to precision and recall we compute their weighted harmonic mean measure or F − measure, which is defined as: F β = (1 + β 2 ) × P recision × Recall β 2 × P recision + Recall .(7) As in previous methods [Achanta et al., 2009;Cheng et al., 2013;Perazzi et al., 2012], we use β 2 = 0.3. For the second evaluation, we follow Perazzi et al. [Perazzi et al., 2012] to evaluate the mean absolute error (MAE) between a continuous saliency map S and the binary ground truth G for all image pixels (x, y), defined as: M AE = 1 W × H (x,y) |S(x, y) − G(x, y)|.(8) Performance on MSRA1000 dataset Following [Achanta et al., 2009;Perazzi et al., 2012;Cheng et al., 2013], we first evaluate our methods using a precision/recall curve which is shown in Figure 5. Our work [Achanta et al., 2009] with pixel accuracy saliency region annotation: (a) the average precision recall curve by segmenting saliency maps using fixed thresholds, (b) the average precision recall by adaptive thresholding (using the same method as in FT [Achanta et al., 2009], SF [Perazzi et al., 2012, GC [Cheng et al., 2013], etc.), (c) the mean absolute error of the different saliency methods to ground truth mask. Please check Figure 2 for the references to the publications in which the baseline methods are presented. Figure 6: Visual comparison of saliency maps on iCoSeg dataset. We compare our method (AH) to other 10 alternative methods. Our results are close to ground truth and focus on the main salient objects. reaches the highest precision/recall rate over all baselines. As a result, our method also obtains the best performance in terms of F-measure. We also evaluate the individual components in our system, namely, objectness map (OB), foreground map (FG), and compactness map (CN). They generally achieve the acceptable performance which is comparable to other baselines. The performance of the objectness map itself is outperformed by our proposed augmented hypotheses. In this work, our novelty is that we adopt and augment the conventional hypotheses by adding two key features: foregroundness and compactness to detect salient objects. When fusing them together, our unified system achieves the stateof-the-art performance in every single evaluation metric. As discussed in the SF [Perazzi et al., 2012] and GC [Cheng et al., 2013], neither the precision nor recall measure considers the true negative counts. These measures favor methods which successfully assign saliency to salient pixels but fail to detect non-salient regions over methods that suc- cessfully do the opposite. Instead, they suggested that MAE is a better metric than precision recall analysis for this problem. As shown in Figure 5c, our work outperforms the stateof-the-art performance [Cheng et al., 2013] Performance on iCoSeg dataset The iCoSeg dataset is "less popular" in the sense that some baselines do not even release detection results and sourcecode. We only reproduced 10 methods on iCoSeg thanks to their existing source-code. The visual comparison of saliency maps generated from our method and different baselines are demonstrated in Figure 6. Our results are close to ground truth and focus on the main salient objects. We first evaluate our methods using a precision/recall curve which is shown in Figure 7a, b. Our method outperforms all other baselines in both two settings, namely fixed threshold and adaptive threshold. As shown in Figure 7c, our method achieves the best performance in terms of MAE. Our work outperforms other methods by a large margin, 25%. Computational Efficiency It is also worth investigating the computational efficiency of different methods. In Table 1, we compare the average running time of our approach to the currently best performing methods on the benchmark images. We compare the performance of our method in terms of speed with methods with most competitive accuracy (GC [Cheng et al., 2013], SF [Perazzi et al., 2012]). The average time of each method is measured on a PC with Intel i7 3.3 GHz CPU and 8GB RAM. Performance of all the methods compared in this table are based on implementations in C++ and MATLAB. The CA method the slowest one because it requires an exhaustive nearestneighbor search among patches. Meanwhile, our method is able to run in a real-time manner. Our procedure spends most of the computation time on generating the objectness map (about 35%) and forming the compactness map (about 50%). From the experimental results, we find that our algorithm is effective and computationally efficient. Conclusion and Future Work In this paper, we have presented a novel method, augmented hypotheses (AH), which adopts the object hypotheses in order to rapidly detect salient objects. To this end, three maps are derived from object hypotheses: superimposed hypotheses form an objectness map, a foreground map is computed from deviations in color from the background, and a compactness map emerges from propagating saliency labels in the oversegmented image. These three maps are fused together to detect salient objects with sharp boundaries. Experimental results on two challenging datasets show that our results are 24% -25% better than the previous best results (compared against 10+ methods in two different datasets), in terms of mean absolute error while also being faster. For future work, we aim to investigate more sophisticated techniques for objectness measures and integrate more cues, i.e., depth [Lang et al., 2012] and audio [Chen et al., 2014] information. Also, we would like to study the impact of salient object detection into the object hypothesis process. Figure 1 : 1From top to bottom: original images, the objectness hypotheses, results of our saliency computation, and ground truth labeling. For a better viewing, only 40 object hypotheses are displayed in each image. Figure 4 : 4From left to right: the original image, the object hypotheses and the corresponding objectness map, the extended object hypotheses and the corresponding objectness map. end while 16: return compactness values c of superpixels. Figure 5 : 5Statistical comparison with 18 saliency detection methods using all the 1000 images from MSRA-1000 dataset Figure 7 : 7Statistical comparison with 10 saliency detection methods using all the 643 images from iCoSeg benchmark[Batra et al., 2010] with pixel accuracy saliency region annotation: (a) the average precision recall curve by segmenting saliency maps using fixed thresholds, (b) the average precision recall by adaptive thresholding (using the same method as in FT[Achanta et al., 2009], GC[Cheng et al., 2013], etc.), (c) the mean absolute error of the different saliency methods to ground truth mask. the coordinate of the top left point, whereas r i , b i are the coordinate of the bottom right point. Here, we formulate each hypothesis P i ∈ R H×W , where H and W are the height and the width of the input image I, respectively. The value of each element P i (x, y) is defined as: by 24%. One may argue that a simple boosting of saliency values similar as in [Perazzi et al., 2012] results would improve it. However, a boosting of saliency values could easily result in the boosting of low saliency values related to background that we also aim to avoid. Table 1 : 1Comparison of running times in the MSRA 1000 benchmark[Achanta et al., 2009].Method CA RC SF GC Ours Time (s) 51.2 0.14 0.15 0.09 0.07 Code Matlab C++ C++ C++ C++ AcknowledgmentsThis work was supported by Singapore Ministry of Education under research Grants MOE2012-TIF-2-G-016 and MOE2014-TIF-1-G-007. Saliency detection using maximum symmetric surround. Süsstrunk Achanta, Radhakrishna Achanta, Sabine Süsstrunk, ; Achanta, International Conference of Computer Vision Systems. CVPRReferences [Achanta and Süsstrunk, 2010] Radhakrishna Achanta and Sabine Süsstrunk. Saliency detection using maximum symmetric surround. In ICIP, pages 2653-2656, 2010. [Achanta et al., 2008] Radhakrishna Achanta, Francisco J. Estrada, Patricia Wils, and Sabine Süsstrunk. Salient re- gion detection and segmentation. In International Confer- ence of Computer Vision Systems, pages 66-75, 2008. [Achanta et al., 2009] Radhakrishna Achanta, Sheila S. Hemami, Francisco J. Estrada, and Sabine Süsstrunk. Frequency-tuned salient region detection. In CVPR, pages 1597-1604, 2009. SLIC superpixels compared to state-of-the-art superpixel methods. T-PAMI. Achanta, Measuring the objectness of image windows. T-PAMI. 34CVPRAchanta et al., 2012] Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aurélien Lucchi, Pascal Fua, and Sabine Süsstrunk. SLIC superpixels compared to state-of-the-art superpixel methods. T-PAMI, 34(11):2274-2282, 2012. [Alexe et al., 2012] Bogdan Alexe, Thomas Deselaers, and Vittorio Ferrari. Measuring the objectness of image win- dows. T-PAMI, 34(11):2189-2202, 2012. [Batra et al., 2010] Dhruv Batra, Adarsh Kowdle, Devi Parikh, Jiebo Luo, and Tsuhan Chen. icoseg: Interac- tive co-segmentation with intelligent scribble guidance. In CVPR, pages 3169-3176, 2010. Hierarchical matching with side information for image classification. Tsotsos ; Neil Bruce, John Bruce, ; Tsotsos, Chen, NIPS. 24CVPR[Bruce and Tsotsos, 2005] Neil Bruce and John Tsotsos. Saliency based on information maximization. In NIPS, 2005. [Chen et al., 2012] Qiang Chen, Zheng Song, Yang Hua, ZhongYang Huang, and Shuicheng Yan. Hierarchical matching with side information for image classification. In CVPR, pages 3426-3433, 2012. [Chen et al., 2014] Yanxiang Chen, Tam V. Nguyen, Mo- han S. Kankanhalli, Jun Yuan, Shuicheng Yan, and Meng Wang. Audio matters in visual attention. T-CSVT, 24(11):1992-2003, 2014. [Cheng et al., 2011] Ming-Ming Cheng, Guo-Xin Zhang, Niloy J. Mitra, Xiaolei Huang, and Shi-Min Hu. Global contrast based salient region detection. In CVPR, pages 409-416, 2011. Vibhav Vineet, and Nigel Crook. Efficient salient region detection with soft image abstraction. Cheng, CVPR. Ming-Ming Cheng, Ziming Zhang, Wen-Yan Lin, and Philip H. S. Torr.CVPRCheng et al., 2013] Ming-Ming Cheng, Jonathan Warrell, Wen-Yan Lin, Shuai Zheng, Vibhav Vineet, and Nigel Crook. Efficient salient region detection with soft image abstraction. In CVPR, pages 1529-1536, 2013. [Cheng et al., 2014] Ming-Ming Cheng, Ziming Zhang, Wen-Yan Lin, and Philip H. S. Torr. BING: Binarized normed gradients for objectness estimation at 300fps. In CVPR, 2014. The pascal visual object classes (VOC) challenge. [ Everingham, Stas Goferman, Lihi Zelnik-Manor, and Ayellet Tal. 88NIPS[Everingham et al., 2010] Mark Everingham, Luc Van Gool, Christopher Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (VOC) challenge. IJCV, 88(2):303-338, 2010. [Goferman et al., 2010] Stas Goferman, Lihi Zelnik-Manor, and Ayellet Tal. Context-aware saliency detection. In CVPR, pages 2376-2383, 2010. [Harel et al., 2006] Jonathan Harel, Christof Koch, and Pietro Perona. Graph-based visual saliency. In NIPS, pages 545-552, 2006. Saliency detection: A spectral residual approach. Zhang ; Xiaodi Hou, Liqing Zhang, CVPR. and Zhang, 2007] Xiaodi Hou and Liqing Zhang. Saliency detection: A spectral residual approach. In CVPR, 2007. Shifts in selective visual attention: towards the underlying neural circuitry. CVPR. 20Beyond sliding windows: Object localization by efficient subwindow searchet al., 1998] Laurent Itti, Christof Koch, and Ernst Niebur. A model of saliency-based visual attention for rapid scene analysis. T-PAMI, 20(11):1254-1259, 1998. [Koch and Ullman, 1985] C Koch and S Ullman. Shifts in selective visual attention: towards the underlying neural circuitry. Hum Neurobiol, 1985. [Lampert et al., 2008] Christoph H. Lampert, Matthew B. Blaschko, and Thomas Hofmann. Beyond sliding win- dows: Object localization by efficient subwindow search. In CVPR, 2008. STAP: Spatial-temporal attention-aware pooling for action recognition. T-CSVT. Sun: A bayesian framework for saliency using natural statistics. Lingyun Zhang, Matthew H. Tong, Tim K. Marks, Honghao Shan, and Garrison W. CottrellKoen van de Sande, Theo Gevers, and Arnold SmeuldersZhai and Shah15Journal of Visionet al., 2012] Congyan Lang, Tam V. Nguyen, Har- ish Katti, Karthik Yadati, Mohan S. Kankanhalli, and Shuicheng Yan. Depth matters: Influence of depth cues on visual saliency. In ECCV, pages 101-115, 2012. [Ma and Zhang, 2003] Yu-Fei Ma and HongJiang Zhang. Contrast-based image attention analysis by using fuzzy growing. In ACM MM, pages 374-381, 2003. [Murray et al., 2011] Naila Murray, Maria Vanrell, Xavier Otazu, and C. Alejandro Párraga. Saliency estimation us- ing a non-parametric low-level vision model. In CVPR, pages 433-440, 2011. [Nguyen et al., 2013a] Tam V. Nguyen, Bingbing Ni, Hairong Liu, Wei Xia, Jiebo Luo, Mohan Kankanhalli, and Shuicheng Yan. Image re-attentionizing. Multimedia, IEEE Transactions on, 15(8):1910-1919, 2013. [Nguyen et al., 2013b] Tam V. Nguyen, Mengdi Xu, Guangyu Gao, Mohan Kankanhalli, Qi Tian, and Shuicheng Yan. Static saliency vs. dynamic saliency: a comparative study. In ACM MM, pages 987-996, 2013. [Nguyen et al., 2015] Tam V. Nguyen, Zheng Song, and Shuicheng Yan. STAP: Spatial-temporal attention-aware pooling for action recognition. T-CSVT, 2015. [Perazzi et al., 2012] Federico Perazzi, Philipp Krähenbühl, Yael Pritch, and Alexander Hornung. Saliency filters: Contrast based filtering for salient region detection. In CVPR, pages 733-740, 2012. [Uijlings et al., 2013] Jasper Uijlings, Koen van de Sande, Theo Gevers, and Arnold Smeulders. Selective search for object recognition. IJCV, 104(2):154-171, 2013. [Viola and Jones, 2001] Paul A. Viola and Michael J. Jones. Robust real-time face detection. In ICCV, page 747, 2001. [Zhai and Shah, 2006] Yun Zhai and Mubarak Shah. Visual attention detection in video sequences using spatiotempo- ral cues. In ACM MM, pages 815-824, 2006. [Zhang et al., 2008] Lingyun Zhang, Matthew H. Tong, Tim K. Marks, Honghao Shan, and Garrison W. Cottrell. Sun: A bayesian framework for saliency using natural statistics. Journal of Vision, 8(7), 2008.
[]
[ "Kibble-Zurek scaling in the Yang-Lee edge singularity", "Kibble-Zurek scaling in the Yang-Lee edge singularity" ]
[ "Shuai Yin \nDepartment of physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n\nInstitute for Advanced Study\nTsinghua University\n100084BeijingP. R. China\n", "Guang-Yao Huang \nDepartment of Electronics\nKey Laboratory for the Physics and Chemistry of Nanodevices\nPeking University\n100871BeijingP. R. China\n", "Chung-Yu Lo \nDepartment of physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n", "Pochung Chen \nDepartment of physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n" ]
[ "Department of physics\nNational Tsing Hua University\n30013HsinchuTaiwan", "Institute for Advanced Study\nTsinghua University\n100084BeijingP. R. China", "Department of Electronics\nKey Laboratory for the Physics and Chemistry of Nanodevices\nPeking University\n100871BeijingP. R. China", "Department of physics\nNational Tsing Hua University\n30013HsinchuTaiwan", "Department of physics\nNational Tsing Hua University\n30013HsinchuTaiwan" ]
[]
We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLE-Ses) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0 + 1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0 + 1)-dimensional and the (1 + 1)-dimensional YLESes.The Kibble-Zurek mechanism [1, 2] describes universal scaling behavior in the driven critical dynamics in a variety of systems, ranging from classical to quantum phase transitions[3,4]. It separates the whole driven process into three stages: two adiabatic stages and one impulse stage. In the adiabatic region, the relaxation rate is larger than the transition rate and the system evolves along the instantaneous equilibrium state; while in the impulse stage, the relaxation rate is smaller than the transition rate because of the critical slowing down, and thus the system falls out of equilibrium essentially. Furthermore, the Kibble-Zurek scaling (KZS) [1, 2] shows that only the equilibrium critical exponents are needed to characterise the dynamic scaling behavior. All these exponents belong to one set, which is determined by the renormalization group flow near the critical point[3][4][5]. According to the KZS, the external driving will induce an effective correlation length, which divides the system into different domains. The domain walls will form topological defects, whose number can be scaled by the driving rate[3,4]. Additionally, for a finite-size system, the system size also becomes an scaling variable[6][7][8]. The KZS has been verified numerically and experimentally in both classical and quantum phase transitions [9-26].On the other hand, Yang and Lee[27,28]paved the way to understand phase transitions by analysing the zeros of the partition function in the complex plane of a symmetry-breaking field. It was shown that singular behaviors exist not only at the critical point with a vanishing symmetry-breaking field but also near the edge of the Lee-Yang zeros, where the applied symmetrybreaking field is purely imaginary[29]. The latter case is often referred to as the Yang-Lee edge singularity (YLES) and can be cast to a critical theory characterized by the Landau-Ginzburg action of a scalar field with an imaginary cubic coupling[30]. Although the YLES occurs in the complex parameter space, its critical properties can be detected in experiments[31].While there are many exotic scaling behaviors in YLES such as the divergence of the order parameter and negative correlation-length exponent in low dimensions[30], to the best of our knowledge, however, the nonequilibrium properties of the quantum YLES has rarely been investigated. Furthermore, the quantum YLES provides a prototype to study a class of dissipative phase transitions, which is characterised by the spontaneous paritytime symmetry breaking[32]. Different from usual quantum phase transitions which occur by tuning a parameter in the Hermitian Hamiltonian, dissipative phase transitions are induced by changing the strength of the dissipation[33]. Recently driven-dissipative open quantum systems have attracted much attention as they offer a promising route of quantum computations or state engineering[32][33][34][35][36]. These call for an investigation on the nonequilibrum behavior near their phase transitions. Some questions then arise: How to describe the driven dynamics across such phase transitions which exhibit YLES? Is the Kibble-Zurek mechanism still applicable? If the answer is yes, is there any new ingredient in such KZS?To answer these questions, we study the driven dynamics across the critical point of YLESes in a finitesize quantum Ising chain with an imaginary symmetrybreaking field[37]. We confirm that the KZS is applicable but there are some features which are quite different from the KZS in ordinary phase transitions. In particular, we show that while the mechanism leading to topological defects breaks down, for small size system the driven dy-
10.1103/physrevlett.118.065701
[ "https://arxiv.org/pdf/1609.06567v2.pdf" ]
43,621,765
1609.06567
942c671a34093147967943c0a5ca1afd9c6f6827
Kibble-Zurek scaling in the Yang-Lee edge singularity 21 Sep 2016 (Dated: September 22, 2016) Shuai Yin Department of physics National Tsing Hua University 30013HsinchuTaiwan Institute for Advanced Study Tsinghua University 100084BeijingP. R. China Guang-Yao Huang Department of Electronics Key Laboratory for the Physics and Chemistry of Nanodevices Peking University 100871BeijingP. R. China Chung-Yu Lo Department of physics National Tsing Hua University 30013HsinchuTaiwan Pochung Chen Department of physics National Tsing Hua University 30013HsinchuTaiwan Kibble-Zurek scaling in the Yang-Lee edge singularity 21 Sep 2016 (Dated: September 22, 2016)numbers: 0367Mn6460De6460Ht6470Tg We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLE-Ses) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0 + 1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0 + 1)-dimensional and the (1 + 1)-dimensional YLESes.The Kibble-Zurek mechanism [1, 2] describes universal scaling behavior in the driven critical dynamics in a variety of systems, ranging from classical to quantum phase transitions[3,4]. It separates the whole driven process into three stages: two adiabatic stages and one impulse stage. In the adiabatic region, the relaxation rate is larger than the transition rate and the system evolves along the instantaneous equilibrium state; while in the impulse stage, the relaxation rate is smaller than the transition rate because of the critical slowing down, and thus the system falls out of equilibrium essentially. Furthermore, the Kibble-Zurek scaling (KZS) [1, 2] shows that only the equilibrium critical exponents are needed to characterise the dynamic scaling behavior. All these exponents belong to one set, which is determined by the renormalization group flow near the critical point[3][4][5]. According to the KZS, the external driving will induce an effective correlation length, which divides the system into different domains. The domain walls will form topological defects, whose number can be scaled by the driving rate[3,4]. Additionally, for a finite-size system, the system size also becomes an scaling variable[6][7][8]. The KZS has been verified numerically and experimentally in both classical and quantum phase transitions [9-26].On the other hand, Yang and Lee[27,28]paved the way to understand phase transitions by analysing the zeros of the partition function in the complex plane of a symmetry-breaking field. It was shown that singular behaviors exist not only at the critical point with a vanishing symmetry-breaking field but also near the edge of the Lee-Yang zeros, where the applied symmetrybreaking field is purely imaginary[29]. The latter case is often referred to as the Yang-Lee edge singularity (YLES) and can be cast to a critical theory characterized by the Landau-Ginzburg action of a scalar field with an imaginary cubic coupling[30]. Although the YLES occurs in the complex parameter space, its critical properties can be detected in experiments[31].While there are many exotic scaling behaviors in YLES such as the divergence of the order parameter and negative correlation-length exponent in low dimensions[30], to the best of our knowledge, however, the nonequilibrium properties of the quantum YLES has rarely been investigated. Furthermore, the quantum YLES provides a prototype to study a class of dissipative phase transitions, which is characterised by the spontaneous paritytime symmetry breaking[32]. Different from usual quantum phase transitions which occur by tuning a parameter in the Hermitian Hamiltonian, dissipative phase transitions are induced by changing the strength of the dissipation[33]. Recently driven-dissipative open quantum systems have attracted much attention as they offer a promising route of quantum computations or state engineering[32][33][34][35][36]. These call for an investigation on the nonequilibrum behavior near their phase transitions. Some questions then arise: How to describe the driven dynamics across such phase transitions which exhibit YLES? Is the Kibble-Zurek mechanism still applicable? If the answer is yes, is there any new ingredient in such KZS?To answer these questions, we study the driven dynamics across the critical point of YLESes in a finitesize quantum Ising chain with an imaginary symmetrybreaking field[37]. We confirm that the KZS is applicable but there are some features which are quite different from the KZS in ordinary phase transitions. In particular, we show that while the mechanism leading to topological defects breaks down, for small size system the driven dy- We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLE-Ses) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0 + 1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0 + 1)-dimensional and the (1 + 1)-dimensional YLESes. The Kibble-Zurek mechanism [1,2] describes universal scaling behavior in the driven critical dynamics in a variety of systems, ranging from classical to quantum phase transitions [3,4]. It separates the whole driven process into three stages: two adiabatic stages and one impulse stage. In the adiabatic region, the relaxation rate is larger than the transition rate and the system evolves along the instantaneous equilibrium state; while in the impulse stage, the relaxation rate is smaller than the transition rate because of the critical slowing down, and thus the system falls out of equilibrium essentially. Furthermore, the Kibble-Zurek scaling (KZS) [1,2] shows that only the equilibrium critical exponents are needed to characterise the dynamic scaling behavior. All these exponents belong to one set, which is determined by the renormalization group flow near the critical point [3][4][5]. According to the KZS, the external driving will induce an effective correlation length, which divides the system into different domains. The domain walls will form topological defects, whose number can be scaled by the driving rate [3,4]. Additionally, for a finite-size system, the system size also becomes an scaling variable [6][7][8]. The KZS has been verified numerically and experimentally in both classical and quantum phase transitions [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. On the other hand, Yang and Lee [27,28] paved the way to understand phase transitions by analysing the zeros of the partition function in the complex plane of a symmetry-breaking field. It was shown that singular behaviors exist not only at the critical point with a vanishing symmetry-breaking field but also near the edge of the Lee-Yang zeros, where the applied symmetrybreaking field is purely imaginary [29]. The latter case is often referred to as the Yang-Lee edge singularity (YLES) and can be cast to a critical theory characterized by the Landau-Ginzburg action of a scalar field with an imaginary cubic coupling [30]. Although the YLES occurs in the complex parameter space, its critical properties can be detected in experiments [31]. While there are many exotic scaling behaviors in YLES such as the divergence of the order parameter and negative correlation-length exponent in low dimensions [30], to the best of our knowledge, however, the nonequilibrium properties of the quantum YLES has rarely been investigated. Furthermore, the quantum YLES provides a prototype to study a class of dissipative phase transitions, which is characterised by the spontaneous paritytime symmetry breaking [32]. Different from usual quantum phase transitions which occur by tuning a parameter in the Hermitian Hamiltonian, dissipative phase transitions are induced by changing the strength of the dissipation [33]. Recently driven-dissipative open quantum systems have attracted much attention as they offer a promising route of quantum computations or state engineering [32][33][34][35][36]. These call for an investigation on the nonequilibrum behavior near their phase transitions. Some questions then arise: How to describe the driven dynamics across such phase transitions which exhibit YLES? Is the Kibble-Zurek mechanism still applicable? If the answer is yes, is there any new ingredient in such KZS? To answer these questions, we study the driven dynamics across the critical point of YLESes in a finitesize quantum Ising chain with an imaginary symmetrybreaking field [37]. We confirm that the KZS is applicable but there are some features which are quite different from the KZS in ordinary phase transitions. In particular, we show that while the mechanism leading to topological defects breaks down, for small size system the driven dy-namics is still described by the KZS with (0 + 1)D critical exponents. For the medium size system, however, the driven dynamics can be described by the KZS with both (0 + 1)D and (1 + 1)D critical exponents. We also demonstrate that the same results hold for the cases of changing the longitudinal-field and changing transverse-field. Static properties of the YLES -We begin our study with the quantum Ising chain in an imaginary longitudinal field [37]. The Hamiltonian reads H = − L n=1 σ z n σ z n+1 − λ L n=1 σ x n − ih L n=1 σ z n ,(1) where σ z n and σ x n are the Pauli matrices in z and x direction, respectively, at site n, λ is the transverse-field, h is longitudinal-field, and L is the lattice size. The critical point of the ordinary ferromagnetic-paramagnetic phase transition is λ c = 1 and h = 0 [38] while there are critical points for the YLES at (λ L YL , h L YL ) when λ > λ c [30]. Although H is non-Hermitian, it has been demonstrated that the appearance of the YLES corresponds to the vanishment of the energy gap [39,40], similar to the ordinary quantum critical phenomena occurring in the Hermitian system [38]. One can also define an order parameter: [37,39]. For a fixed λ(λ > λ c ), when h < h L YL , the real part of model (1) dominates and the energy spectra are real [40]. Since these spectra are adiabatically connected with those for h = 0, the system is in a paramagnetic phase with M = 0. When h > h L YL , the dissipative part in model (1) plays significant roles. As a result, energy spectra become conjugate pairs [39,40]. It has been shown that the latter phase is a ferromagnetic phase with M = 0 [29]. Moreover, it has been demonstrated that the equilibrium singular behaviors near the critical point of the YLES can be described by the usual critical exponents [30]. For instance, β 0 = β 1 = 1, ν 0 = −1, δ 0 = −2, ν 1 = −5/2, δ 1 = −6 and the dynamic exponents z 0 = z 1 = 1 [30,37,39]. (The subscript indicates the space dimension.) Different from usual phase transitions which only occur in the thermodynamic limit, the YLES in model (1) can appear even at finite sizes [29,39]. The YLES near h L YL in model (1) belongs to the (0+1)D universality class, while the YLES near h ∞ YL belongs to the (1 + 1)D universality class [30]. Furthermore, it has been shown that [39] M ≡ Re[ Ψ * |σ z |Ψ / Ψ * |Ψ ]h L YL − h ∞ YL = C(λ)L − β 1 δ 1 ν 1 ,(2) in which C(λ) is a dimensionless function. Two deductions thus can be obtained as sketched in Fig. 1: (i) there must be an overlap critical region in which both the (0 + 1)D YLES and the (1 + 1)D YLES play indispensable roles; (ii) the critical region for the (0 + 1)D YLES must shrink as L increases, and when L → ∞, this region becomes a point. KZS for small-size systems-We first study the KZS for model (1) with a small size system, whose critical For a small size system, the evolution will cross the critical region (Green) described by the (0 + 1)D theory of the finite-size YLES; while for the system with a medium lattice-size, the evolution will cross the critical region (Blue) described by the (1 + 1)D theory of the infinite-size YLES and the overlap between these two critical regions. We show that the driven dynamics in the overlap region can be described by the KZS according to both the (0 + 1)D and (1 + 1)D critical theories of YLESes. region is the green region in Fig. 1. The critical properties in this region are described by the (0 + 1)D critical theory. We consider the case for changing h as h = h 0 + R h t, while λ(> λ c ) is fixed. Since h 0 is chosen to be far away from the YLES, it is irrelevant. Similar to the KZS in ordinary phase transitions [3,4], when |h− h L YL | > R β0δ0/ν0r0 h (r 0 = z 0 + β 0 δ 0 /ν 0 ), the relaxation rate, |h − h L YL | ν0z0/β0δ0 , is larger than the transition rate, R h /|h − h L YL |, and the evolution is in the adiabatic stage; and when |h − h L YL | < R β0δ0/ν0r0 h , the relaxation rate is smaller than the transition rate and the evolution is in the impulse stage [9][10][11][12]. Therefore, the dynamics of M near h L YL should still satisfy the KZS [5,19] M (h − h L YL , R h ) = R β 0 ν 0 r 0 h f a [(h − h L YL )R − β 0 δ 0 ν 0 r 0 h ],(3) in which f a is an analytical scaling function (similar definitions will always be implied). Equation (3) is applicable when the impulse region is embedded in the critical region of the (0 + 1)D critical point [41]. Otherwise, the information, which is not controlled by the (0 + 1)D critical theory, can be brought into the driven dynamics. Moreover, although we focus on the small-size system, there is no finite-size correction in Eq. (3), since L is irrelevant in this (0 + 1)D YLES. Additionally, for the driven dynamics in ordinary phase transitions, topological defects emerge after impulse stage since the whole lattice of the system is divided by the driven-induced length scale; in the (0 + 1)D YLES, however, the topological defects are not well-defined, since the lattice size can be microscopically small. KZS in the overlap region-For medium sizes, the critical regions for the (0 + 1)D and (1 + 1)D overlap with each other, as shown in Fig. 1. In this overlap region, besides Eq. (3), the dynamic scaling should satisfy the (1+1)D KZS with finite-size corrections being considered. Similar to the usual finite-size KZS [6,7], the scaling form of M reads (4) is applicable in the (1 + 1)D critical region (Blue region in Fig. 1). Since both f a and f b are analytical functions for any finite L and R h , one expects that in the overlap region, both Eq. (3) and Eq. (4) should be applicable. However, it seems quite unreasonable that one critical phenomenon can be explicated by two sets of scaling theories with disparate exponents. So, there must be some latent scaling properties for f a and f b in the overlap region. M (h−h ∞ YL , R h , L) = R β 1 ν 1 r 1 h f b [(h−h ∞ YL )R − β 1 δ 1 ν 1 r 1 h , L −1 R − 1 r 1 h ], (4) in which r 1 = z 1 + β 1 δ 1 /ν 1 . Equation To explore these hidden scaling properties, we start Thus, Eq (6) provides a constraint, which make the explicit scaling variable L in f b (f c ) behave like a dimensionless parameter and Eq. (3) is restored. from f b . When R h → 0 and h → h L YL , M diverges as M ∼ (h−h L YL ) 1/M (h−h L YL , R h , L) = R β 1 ν 1 r 1 h f c [(h−h L YL )R − β 1 δ 1 ν 1 r 1 h , L −1 R − 1 r 1 h ].(5) As L → ∞, the driven dynamics must be described by the (1 + 1)D KZS theory, i.e., Eq. (4) with L → ∞. However, at first glance, by taking L → ∞ in Eq. (3), one obtains M (h, R h ) = R β0/ν0r0 h f a [(h − h ∞ YL )R −β0δ0/ν0r0 h ]. This is apparently incorrect. The reason is that the critical region of the (0 + 1)D critical point shrinks to a point as L → ∞. So, for any finite driving rate, the impulse region of the (0 + 1)D KZS is broader than the (0 + 1)D critical region. As a consequence, Eq. (3) is not applicable anymore when L → ∞. KZS for changing λ-Besides changing h, one can also change λ to cross the critical point of the YLES. It has been proved that there is only one relevant direction in the parameter space of the YLES [30,39]. Therefore, the KZS for changing λ is exactly the same as that for changing h. This feature is different from the KZS in ordinary classical and quantum phase transitions, in which the relevant exponents of the KZS are usually different for changing different parameters. Numerical results-To verify the scaling theory, we solve directly the Schrödinger equation of the Hamiltonian (1) by using the finite difference method with periodic boundary condition. The time interval is chosen as 5 × 10 −5 . Smaller intervals have been checked to produce no appreciable changes. For small sizes, Fig. 2 shows the evolution of M under the external driving h = R h t + h 0 with a fixed λ (λ > λ c ). First, we find that the divergence of M at h L YL is rounded by the external driving. Second, in Figs. 2(b), after rescaling M and h with R h by using the (0 + 1)D exponents, we find that the rescaled curves match with each other in the vicinity of h L YL , confirming Eq. (3). For medium sizes, we compare the driven dynamics for fixed L and fixed LR Fig. 3(b2), we find that after the peaks, the collapse in Fig. 3(b2) is much better than that in Fig. 3(a2). The reason is that the critical region for the (1 + 1)D YLES is broader than the critical region for the (0 + 1)D YLES, as shown in Fig. (1). Moreover, comparing Figs. 2 and 3 (a), we find that the collapse region for the rescaled curves becomes smaller as the lattice size increases. This indicates that the regime, in which Eq. (3) is applicable, shrinks for larger L. To investigate the relation between Eqs. (3) and (4), we extract the order parameters at h L YL for various R h . First, Fig. 4(a), plotted on the double-logarithmic scale, shows that for different lattice sizes, the curves of M ver- sus R h are almost parallel straight lines, whose slopes are between −0.334 and −0.324, agrees with the theoretical value of β 0 /ν 0 r 0 = −1/3. Second, we plot in Fig. 4(b) the rescaled order parameter as the function of the rescaled lattice size with the (1 + 1)D critical exponents as input. The rescaled curves collapse onto one single curve. Thus, Eq. (5) is confirmed and this rescaled curve is just the scaling function f c (0, B). Third, as shown in Fig. 4(b), by plotting the rescaled curve in double-logarithmic scale, one finds that f c (0, B) itself is a power function, whose exponent is fitted to be −0.731. This exponent is close to the theoretical value of r 1 (β 0 δ 0 /ν 0 r 0 −β 1 δ 1 /ν 1 r 1 ) ≃ −0.7333, confirming Eq. (6). In addition, we also numerically study the driven dynamics for changing λ [40]. We confirm that the driven dynamics in the overlap critical region in the L −1 − λ plane can be described by both Eqs. (3) and (4) with h and h L YL (h ∞ YL ) being replaced by λ and λ L YL (λ ∞ YL ), respectively, however, the exponents therein keep unchanged. Summary and Discussion-In summary, we have studied the driven dynamics in the YLES. For the (0 + 1)D YLES, we have shown that although the topological defects is not well-defined, the KZS can still be applied to describe the driven dynamics. Additionally, in the overlap critical regions between the (0 + 1)D and (1 + 1)D YLES, we have found that the driven critical dynamics can be described by the KZS according to both the (0+1)D and (1+1)D critical theories, although their critical exponents are different. We have also explored the relation between dynamic scaling functions of the KZS in the (0 + 1)D and (1 + 1)D theory. Besides the experiments in condensed matter [31], recent experiments in cold atom physics also provides powerful approach to measure the Lee-Yang zeros [42,43]. These experiments can possibly be extended to manipulate the driven dynamics near the YLES. Our studies on the KZS in the overlap critical region will also shed some light on the investigation of the KZS in other complex systems. For example, the twodimensional quantum Ising model at finite temperatures exhibits both classical and quantum phase transition [38], and its phase diagram is similar to Fig. 1. The quantum critical regime overlaps with the classical critical region. By noting that the KZS has been verified in both the quantum criticality at finite temperature regions [7] and the classical criticality, one expects that the driven dynamics in the quantum-classical overlap critical region should exhibit similar properties. SUPPLEMENTAL MATERIAL I. The Yang-Lee edge singularity at finite size Energy spectra-To illustrate the Yang-Lee edge singularity (YLES) for finite-size systems [1], we show the lowest two eigenvalues for model (1). In Fig. 5, one finds that for fixed λ, when h < h L YL , the spectra are real; when h > h L YL , the spectra form conjugate pairs; and exactly at h L YL , the gap between the lowest two eigenvalues vanishes. For comparison, in Fig. 6, one finds that for fixed h, when λ > λ L YL , the spectra are real; when λ < λ L YL , the spectra form conjugate pairs; and at λ L YL , the gap vanishes. 34. 32. 30. 28. 26. 24. 10. 5. 0. 5. 10. 30. 25. 6. 4. 2. 0. 2. 4. 6. ReE ImE Estimation of the critical point of the YLES -As shown in Fig. 7, we estimate the critical point of the YLES by determining the position at which the order parameter diverges [1,2]. Different from usual phase transitions, in which M vanishes at the critical point, here, M tends to infinity when h (λ) tends to h L YL (λ L YL ) from the side of the ferromagnetic phase; while M = 0 when h (λ) tends to h L YL (λ L YL ) from the side of the paramagnetic phase. In Table I, we list values of h L YL and λ L YL with fixed λ and h, respectively, for different L. II. The KZS for changing the transverse-field λ In the main text, we discuss the main results for changing λ. Here, we demonstrate the details. We consider the driving λ = λ 0 + R λ t with λ 0 being far from the YLES. In Figs. 9(a1) and 9(a2), L is fixed and the (0 + 1)D critical exponents are employed to calculate the rescaled variables. After rescaling M and (λ − λ L YL ) with R λ , we find that the rescaled curves match with each other in the vicinity of λ L YL , confirming Eq. (3) with h and R h being replaced by λ and R λ , respectively. In Figs. 9(b1) and 9(b2), LR 1/r1 λ is fixed and the rescaled values are calculated according to the (1 + 1)D theory. We find that the rescaled curves collapse onto each other according to Eq. (4) with h and R h being replaced by λ and R λ , respectively. Thus, similar to the case of changing h, we conclude that the critical dynamics under changing λ near the critical point of the YLES of model (1) can be described by the KZS for both (0 + 1)D and (1 + 1)D. Comparing Fig. 9(a2) with Fig. 9(b2), we find that on the left-hand side of the peaks, the collapse in (b2) is much better than that in (a2). The reason is that the critical region for the infinite-size YLES is broader than the critical region for the finite-size YLES, similar to the case of changing h. Then we study the relation between Eqs. (3) and (4) for changing λ. Similar to the procedure of changing h, we extract the order parameters at λ L YL for R λ . Similar to Fig. 4(a), Fig. 10(a) shows that for different lattice sizes, the curves of M versus R λ are almost parallel lines on a double-logarithmic scale. By linearly fitting, we find that the slopes are between −0.327 and −0.318, close to the corresponding theoretical value for changing h. Similar to Fig. 4(b), we plot in Fig. 10(b) the rescaled order parameters as the function of the rescaled lattice sizes with the (1 + 1)D exponents for changing h as input. We find that the rescaled curves collapse onto each other, confirming Eq. (5) with h and R h being replaced by λ and R λ , respectively. Moreover, linearly fitting the rescaled curve, one finds that the slope is about −0.713, agree with the theoretical value of r 1 (β 0 δ 0 /ν 0 r 0 − β 1 δ 1 /ν 1 r 1 ) ≃ −0.7333, confirming Eq. (6) with h and R h being replaced by λ and R λ , respectively. PACS numbers: 03.67.Mn, 64.60.De, 64.60.Ht, 64.70.Tg online) Critical regions near critical points of YLESes. Critical points of finite-size YLESes link up into a critical curve (Solid green curve), which ends at the critical point of the infinite-size YLES h ∞ YL . The red arrow indicates the direction of changing h in KZS. Comparing Eq. (5) with Eq. (3), one finds that the scaling function f c (A, B) satisfy f c (A, B) = (B −r1 ) FIG. 2 . 2(Color online) Under increasing h with fixed λ = 5, the curves of M versus h − h 2 YL (h 2 YL = 2.933353) for fixed L = 2 in (a) match with each other in (b) when M and h−h 2 YL are rescaled by the (0 + 1)D exponents. The arrow points the direction of changing h. h0 is chosen as h0 ≡ h 2 YL − 1. . In Figs. 3(a1) and 3(a2), L is fixed and the (0 + 1)D critical exponents are employed to calculate the rescaled variables. Similar to Fig. 2, after rescaling M and h with R h , we find that the rescaled curves match with each other in the vicinity of h L YL , confirming Eq. (3). In contrast, in Figs. 3(b1) and 3(b2), LR 1/r1 h is fixed and the rescaled values are calculated according to the (1 + 1)D theory. We find that the rescaled curves collapse onto each other according to Eq. (4). Thus, we conclude that the driven critical dynamics near the critical point of YLES can be described by the KZS in both (0 + 1)D and (1 + 1)D. Comparing Fig. 3(a2) with FIG. 3 . 3(Color online) Under increasing h with fixed λ = 5, the curves of M versus h − h 10 YL (h 10 YL = 2.309176) for fixed L = 10 in (a1) match with each other in (a2) when M and h−h 10 YL are rescaled by the (0+1)D exponents; for comparison, the curves of M versus h − h ∞ YL for fixed LR b1) match with each other in (b2) when M and h − h ∞ YL are rescaled by the (1 + 1)D exponents. The arrows point the directions of changing h. FIG. 4 . 4(Color online) Under changing h with fixed λ = 5, (a) M at h L YL versus R h for different lattice sizes; and (b) the collapse of curves of the rescaled M at λ L YL versus the rescaled L. Double-logrithmaic scales are used in both (a) and (b). FIG. 5 . 5(Color online) The lowest two levels for λ = 5 and L = 8. The gap vanishes at h 8 YL = 2.320787. FIG. 6 . 6(Color online) The lowest two levels for h = 2.292475 and L = 8. The gap vanishes at λ 8 YL = 4.962513. FiniteFIG. 7 . 7-size scaling of the critical point of the YLES -Here we show the finite-size scaling of the critical point of the finite-size YLES [1]. For fixed λ (= 5), we plot in Fig 8(a) the difference between h L YL and h ∞ YL (= 2.292475) as a (Color online) (a) Estimations of h L YL for fixed λ; (b) Estimation of λ L YL for fixed h. function of L. Power fitting shows that this curve satisfies (h L YL − h ∞ YL ) ∝ L −2.366 , approximately agree with Eq. (2) in which β 1 δ 1 /ν 1 = 12/5. For comparison, with fixed h (= 2.292475), we plot in Fig 8(b) the difference between λ L YL and λ ∞ YL (= 5) as a function of L. Power fitting shows that this curve satisfies (λ L YL − λ ∞ YL ) ∝ L −2.370 . This indicates that λ has the same critical dimension with h [1]. Here we only consider the leading term. The higher order corrections have been discussed in Ref. 1. FIG . 8. (Color online) (a) Fitting of (h L YL − h ∞ YL ) versus L for fixed λ. (b) Fitting of (λ L YL − λ ∞ YL ) versus L for fixed h. FIG. 9 . 9(Color online) Under changing λ with fixed h = 2.292475, the curves of M versus (λ − λ 10 YL ) (λ 10 YL = 4.977885) for fixed L = 10 in (a1) match with each other in (a2) when M and (λ − λ 10 YL ) are rescaled by the (0 + 1)D exponents; for comparison, the curves of M versus (λ − λ ∞ YL ) for fixed LR 1/r 1 λ = 2.001689 in (b1) match with each other in (b2) when M and (λ − λ ∞ YL ) are rescaled by the (1 + 1)D exponents. The arrows point the directions of changing λ. . 10. (Color online) Under changing λ with fixed h = 2.292475, (a) M at λ L YL versus R λ for different lattice size; and (b) the collapse of curves of the rescaled M at λ L YL versus the rescaled L. Double-logrithmaic scales are used in both (a) and (b). δ0 according to the (0+1)D static scaling theory [40]. However, h L YL is not an explicit variable in f b . To expose this divergence, we substitute Eq. (2) into Eq. (4). After reassembling the scaling variables, we obtain TABLE I . IThe critical points of the YLES for various lattice sizesL h L YL for λ = 5 λ L YL for h = 2.292475 8 2.320787 4.962513 9 2.313911 4.971616 10 2.309176 4.977885 11 2.305794 4.982363 12 2.303305 4.985659 13 2.301426 4.988147 14 2.299978 4.990065 We wish to thank Fan Zhong, Zhong Wang and Zhongbo Yan for their helpful discussions. We acknowledge the support by Ministry of Science and Technology (MOST) of Taiwan through Grant No. 104-2628-M-007-005-MY3. We also acknowledge the support from the National Center for Theoretical Science (NCTS) of Taiwan. . T Kibble, J Phys. A. 91387T. Kibble, J Phys. A 9, 1387 (1976). . W H Zurek, Nature. 317505W. H. Zurek, Nature (London) 317, 505 (1985). . J Dziarmaga, Adv. Phys. 591063J. Dziarmaga, Adv. Phys. 59, 1063 (2010). . A Polkovnikov, K Sengupta, A Silva, M Vengalattore, Rev. Mod. Phys. 83863A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat- tore, Rev. Mod. Phys. 83, 863 (2011). . S Gong, F Zhong, X Huang, S Fan, New J. Phys. 1243036S. Gong, F. Zhong, X. Huang, and S. Fan, New J. Phys. 12, 043036 (2010). . C De Grandi, A Polkovnikov, A W Sandvik, Phys. Rev. B. 84224303C. De Grandi, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 84, 224303 (2011). . S Yin, P Mai, F Zhong, Phys. Rev. B. 8994108S. Yin, P. Mai, and F. Zhong, Phys. Rev. B 89, 094108 (2014). . Y Huang, S Yin, B Feng, F Zhong, Phys. Rev. B. 90134108Y. Huang, S. Yin, B. Feng, and F. Zhong, Phys. Rev. B 90 134108 (2014). . W H Zurek, U Dorner, P Zoller, Phys. Rev. Lett. 95105701W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005). . J Dziarmaga, Phys. Rev. Lett. 95245701J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005). . A Polkovnikov, Phys. Rev. B. 72161201A. Polkovnikov, Phys. Rev. B 72, 161201(R) (2005). . N D Antunes, P Gandra, R J Rivers, Phys. Rev. D. 73125003N. D. Antunes, P. Gandra, and R. J. Rivers, Phys. Rev. D 73, 125003 (2006). . D Sen, K Sengupta, S Mondal, Phys. Rev. Lett. 10116806D. Sen, K. Sengupta, and S. Mondal, Phys. Rev. Lett. 101, 016806 (2008). . S Deng, G Ortiz, L Viola, Europhys. Lett. 8467008S. Deng, G. Ortiz, and L. Viola, Europhys. Lett. 84, 67008 (2008). . B Damski, W H Zurek, Phys. Rev. Lett. 104160404B. Damski and W. H. Zurek, Phys. Rev. Lett. 104, 160404 (2010). . C De Grandi, A Polkovnikov, A W Sandvik, Phys. Rev. B. 84224303C. De Grandi, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 84, 224303 (2011). . M Kolodrubetz, B K Clark, D A Huse, Phys. Rev. Lett. 10915701M. Kolodrubetz, B. K. Clark, and D. A. Huse, Phys. Rev. Lett. 109, 015701 (2012). . M Kolodrubetz, D Pekker, B K Clark, K Sengupta, Phys. Rev. B. 85100505M. Kolodrubetz, D. Pekker, B. K. Clark, and K. Sen- gupta, Phys. Rev. B 85, 100505(R) (2012). . A Chandran, A Erez, S S Gubser, S L Sondhi, Phys. Rev. B. 8664304A. Chandran, A. Erez, S. S. Gubser, and S. L. Sondhi, Phys. Rev. B 86, 064304 (2012). . Q Hu, S Yin, F Zhong, Phys. Rev. B. 91184109Q. Hu, S. Yin, and F. Zhong, Phys. Rev. B 91, 184109 (2015). . S Ulm, J Roßnagel, G Jacob, C Degünther, S Dawkins, U Poschinger, R Nigmatullin, A Retzker, M Plenio, F Schmidt-Kaler, K Singer, Nat. Commun. 42290S. Ulm, J. Roßnagel, G. Jacob, C. Degünther, S. Dawkins, U. Poschinger, R. Nigmatullin, A. Retzker, M. Plenio, F. Schmidt-Kaler, and K. Singer, Nat. Commun. 4, 2290 (2013). . K Pyka, J Keller, H L Partner, R Nigmatullin, T Burgermeister, D M Meier, K Kuhlmann, A Retzker, M B Plenio, W H Zurek, A Campo, T E Mehlstäubler, Nat. Commun. 42291K. Pyka, J. Keller, H. L. Partner, R. Nigmatullin, T. Burgermeister, D. M. Meier, K. Kuhlmann, A. Retzker, M. B. Plenio, W. H. Zurek, A. del Campo, and T. E. Mehlstäubler, Nat. Commun. 4, 2291 (2013). . N Navon, A L Gaunt, R P Smith, Z Hadzibabic, Science. 347167N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Science 347, 167 (2015). . L W Clark, L Feng, C Chin, arXiv:1605.01023L. W. Clark, L. Feng, C. Chin, arXiv: 1605.01023. . C Liu, A Polkovnikov, A Sandvik, Phys. Rev. Lett. 114147203C. Liu, A. Polkovnikov, and A. Sandvik, Phys. Rev. Lett. 114, 147203 (2015). . C Liu, A Polkovnikov, A Sandvik, A Young, Phys. Rev. E. 9222128C. Liu, A. Polkovnikov, A. Sandvik, and A. Young, Phys. Rev. E 92, 022128 (2015). . C Yang, T Lee, Phys. Rev. 87404C. Yang and T. Lee, Phys. Rev. 87, 404 (1952). . T Lee, C Yang, Phys. Rev. 87410T. Lee and C. Yang, Phys. Rev. 87, 410 (1952). . P Kortman, R Griffiths, Phys. Rev. Lett. 271439P. Kortman and R. Griffiths, Phys. Rev. Lett. 27, 1439 (1971). . M Fisher, Phys. Rev. Lett. 401610M. Fisher, Phys. Rev. Lett. 40, 1610 (1978). . C Binek, W Kleemann, H Aruga Katori, J. Phys.: Condens. Matter. 13811C. Binek, W. Kleemann and H. Aruga Katori, J. Phys.: Condens. Matter 13, L811 (2001). N Moiseyev, Non-Hermitian quantum mechanicm. Cambridge University PressN. Moiseyev, Non-Hermitian quantum mechanicm, (Cambridge University Press, 2011). U Weiss, Quantum Dissipative Systems. SingaporeWorld Scientific3rd ed.U. Weiss, Quantum Dissipative Systems, 3rd ed. (World Scientific, Singapore, 2008). . S Diehl, A Micheli, A Kantian, B Kraus, H Büchler, P Zoller, Nature Phys. 4878S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Büchler, and P. Zoller, Nature Phys. 4, 878 (2008). . S Diehl, W Yi, A Daley, P Zoller, Phys. Rev. Lett. 105227001S. Diehl, W. Yi, A. Daley, and P. Zoller, Phys. Rev. Lett. 105, 227001 (2010). . C Bender, Rep. Prog. Phys. 70947C. Bender, Rep. Prog. Phys. 70, 947 (2007). . K Uzelac, P Pfeuty, R Jullien, Phys. Rev. Lett. 43805K. Uzelac, P. Pfeuty, and R. Jullien, Phys. Rev. Lett. 43, 805 (1979). S Sachdev, Quantum Phase Transitions. Cambridge University PressS. Sachdev, Quantum Phase Transitions(Cambridge Uni- versity Press, 1999). . G Von Gehlen, J. Phys. A: Math. Gen. 245371G. von Gehlen, J. Phys. A: Math. Gen. 24, 5371 (1991). . P Silvi, G Morigi, T Calarco, S Montangero, Phys. Rev. Lett. 116225701P. Silvi, G. Morigi, T. Calarco, and S. Montangero, Phys. Rev. Lett. 116, 225701 (2016). . B Wei, R Liu, Phys. Rev. Lett. 109185701B. Wei and R. Liu, Phys. Rev. Lett. 109, 185701 (2012). . X Peng, H Zhou, B Wei, J Cui, J Du, R Liu, Phys. Rev. Lett. 11410601X. Peng, H. Zhou, B. Wei, J. Cui, J. Du, and R. Liu, Phys. Rev. Lett. 114, 010601 (2015). . G Von Gehlen, J. Phys. A: Math. Gen. 245371G. von Gehlen, J. Phys. A: Math. Gen. 24, 5371 (1991). . K Uzelac, P Pfeuty, R Jullien, Phys. Rev. Lett. 43805K. Uzelac, P. Pfeuty, and R. Jullien, Phys. Rev. Lett. 43, 805 (1979).
[]
[ "Real-time X-ray Phase-contrast Imaging Using SPINNet -A Novel Speckle-based Phase-contrast Imaging Neural Network", "Real-time X-ray Phase-contrast Imaging Using SPINNet -A Novel Speckle-based Phase-contrast Imaging Neural Network" ]
[ "Zhi Qiao \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Xianbo Shi \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Yudong Yao \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Michael J Wojcik \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Luca Rebuffi \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "ANDMathew J Cherukara \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Lahsen Assoufid \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Zqiao@anl Gov \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n", "Xshi@anl Gov \nAdvanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA\n" ]
[ "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA", "Advanced Photon Source\nArgonne National Laboratory\n60439LemontIllinoisUSA" ]
[]
X-ray phase-contrast imaging has become indispensable for visualizing samples with low absorption contrast. In this regard, speckle-based techniques have shown significant advantages in spatial resolution, phase sensitivity, and implementation flexibility compared with traditional methods. However, their computational cost has hindered their wider adoption. By exploiting the power of deep learning, we developed a new speckle-based phase-contrast imaging neural network (SPINNet) that boosts the phase retrieval speed by at least two orders of magnitude compared to existing methods. To achieve this performance, we combined SPINNet with a novel coded-mask-based technique, an enhanced version of the speckle-based method. Using this scheme, we demonstrate a simultaneous reconstruction of absorption and phase images on the order of 100 ms, where a traditional correlation-based analysis would take several minutes even with a cluster. In addition to significant improvement in speed, our experimental results show that the imaging resolution and phase retrieval quality of SPINNet outperform existing single-shot speckle-based methods. Furthermore, we successfully demonstrate its application in 3D X-ray phase-contrast tomography. Our result shows that SPINNet could enable many applications requiring high-resolution and fast data acquisition and processing, such as in-situ and in-operando 2D and 3D phase-contrast imaging and real-time at-wavelength metrology and wavefront sensing.Recently, the speckle-based imaging (SBI) method has been applied to quantitative X-ray multi-contrast imaging, at-wavelength metrology, and wavefront sensing[13][14][15][16]. Compared with Talbot grating interferometry, the SBI method's primary advantages are spatial resolution, phase sensitivity, and experimental flexibility[14,15,17]. Most SBI schemes, including single-shot X-ray speckle tracking (XST), X-ray speckle-scanning tracking (XSS), and X-ray speckle-vector tracking (XSVT), mainly use digital image correlation (DIC) to track the speckle movement[16,18,19]. An optimization-based method called unified modulated pattern analysis (UMPA) was later proposed to deal with broader situations, including different scan positions, spatial resolution, and phase sensitivity for either random or periodic patterns[13]. Most recently, a coded-mask-based multi-contrast imaging (CMMI) technique has been developed [20], which uses a coded phase mask to generate a pre-known pattern with ultra-high-contrast. Combined with the advanced optimization method, CMMI has shown superior performance in spatial resolution and phase sensitivity compared with existing methods. However, the computational efficiency of current SBI methods, whether DIC-based or optimization-based, is a major obstacle for their application in in-situ and in-operando measurements, where real-time analysis is critical. Data processing using the current correlation-based analysis methods may take several minutes even with a cluster, even when high-performance computing resources are employed. The SBI method using multi-resolution analysis and wavelet-transform has significantly improved the data processing speed[21,22]. However, the improvement is still far from the requirement of many in-situ measurements. In addition, with the advent of next-generation synchrotron facilities, such as the Advanced Photon Source Upgrade project [23], the experimental data volume and data acquisition are expected to be many orders of magnitudes larger and faster compared with the current state. Therefore, high-speed image processing is essential and will open many new opportunities in real-time applications such as in-situ multi-contrast imaging [24] and real-time wavefront sensing and control[25].Machine learning has been actively investigated in computational imaging methods to achieve super-resolution, low-flux imaging, and high-speed imaging[26][27][28][29][30][31][32][33]. Inspired by the deeplearning-based optical flow methods[34][35][36][37][38][39]for object detection in the computer vision field, we propose a novel speckle-based phase-contrast imaging neural network (SPINNet) for single-shot imaging. It improves the data processing speed by more than two orders of magnitudes compared to existing X-ray speckle-based phase-contrast imaging methods. Using SPINNet, we demonstrate simultaneous phase and amplitude recovery within 100 ms. In addition, the image quality also outperforms the DIC-based XST method as quantified by the noise level. Our results show that SPINNet could enable in-situ and in-operando SBI measurements in 2D and 3D phase-contrast imaging.
10.1364/optica.453748
[ "https://arxiv.org/pdf/2201.07232v1.pdf" ]
246,035,209
2201.07232
7d52b2fe45e5e37a2cdd04812456b8fa41315da1
Real-time X-ray Phase-contrast Imaging Using SPINNet -A Novel Speckle-based Phase-contrast Imaging Neural Network Zhi Qiao Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Xianbo Shi Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Yudong Yao Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Michael J Wojcik Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Luca Rebuffi Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA ANDMathew J Cherukara Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Lahsen Assoufid Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Zqiao@anl Gov Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Xshi@anl Gov Advanced Photon Source Argonne National Laboratory 60439LemontIllinoisUSA Real-time X-ray Phase-contrast Imaging Using SPINNet -A Novel Speckle-based Phase-contrast Imaging Neural Network X-ray phase-contrast imaging has become indispensable for visualizing samples with low absorption contrast. In this regard, speckle-based techniques have shown significant advantages in spatial resolution, phase sensitivity, and implementation flexibility compared with traditional methods. However, their computational cost has hindered their wider adoption. By exploiting the power of deep learning, we developed a new speckle-based phase-contrast imaging neural network (SPINNet) that boosts the phase retrieval speed by at least two orders of magnitude compared to existing methods. To achieve this performance, we combined SPINNet with a novel coded-mask-based technique, an enhanced version of the speckle-based method. Using this scheme, we demonstrate a simultaneous reconstruction of absorption and phase images on the order of 100 ms, where a traditional correlation-based analysis would take several minutes even with a cluster. In addition to significant improvement in speed, our experimental results show that the imaging resolution and phase retrieval quality of SPINNet outperform existing single-shot speckle-based methods. Furthermore, we successfully demonstrate its application in 3D X-ray phase-contrast tomography. Our result shows that SPINNet could enable many applications requiring high-resolution and fast data acquisition and processing, such as in-situ and in-operando 2D and 3D phase-contrast imaging and real-time at-wavelength metrology and wavefront sensing.Recently, the speckle-based imaging (SBI) method has been applied to quantitative X-ray multi-contrast imaging, at-wavelength metrology, and wavefront sensing[13][14][15][16]. Compared with Talbot grating interferometry, the SBI method's primary advantages are spatial resolution, phase sensitivity, and experimental flexibility[14,15,17]. Most SBI schemes, including single-shot X-ray speckle tracking (XST), X-ray speckle-scanning tracking (XSS), and X-ray speckle-vector tracking (XSVT), mainly use digital image correlation (DIC) to track the speckle movement[16,18,19]. An optimization-based method called unified modulated pattern analysis (UMPA) was later proposed to deal with broader situations, including different scan positions, spatial resolution, and phase sensitivity for either random or periodic patterns[13]. Most recently, a coded-mask-based multi-contrast imaging (CMMI) technique has been developed [20], which uses a coded phase mask to generate a pre-known pattern with ultra-high-contrast. Combined with the advanced optimization method, CMMI has shown superior performance in spatial resolution and phase sensitivity compared with existing methods. However, the computational efficiency of current SBI methods, whether DIC-based or optimization-based, is a major obstacle for their application in in-situ and in-operando measurements, where real-time analysis is critical. Data processing using the current correlation-based analysis methods may take several minutes even with a cluster, even when high-performance computing resources are employed. The SBI method using multi-resolution analysis and wavelet-transform has significantly improved the data processing speed[21,22]. However, the improvement is still far from the requirement of many in-situ measurements. In addition, with the advent of next-generation synchrotron facilities, such as the Advanced Photon Source Upgrade project [23], the experimental data volume and data acquisition are expected to be many orders of magnitudes larger and faster compared with the current state. Therefore, high-speed image processing is essential and will open many new opportunities in real-time applications such as in-situ multi-contrast imaging [24] and real-time wavefront sensing and control[25].Machine learning has been actively investigated in computational imaging methods to achieve super-resolution, low-flux imaging, and high-speed imaging[26][27][28][29][30][31][32][33]. Inspired by the deeplearning-based optical flow methods[34][35][36][37][38][39]for object detection in the computer vision field, we propose a novel speckle-based phase-contrast imaging neural network (SPINNet) for single-shot imaging. It improves the data processing speed by more than two orders of magnitudes compared to existing X-ray speckle-based phase-contrast imaging methods. Using SPINNet, we demonstrate simultaneous phase and amplitude recovery within 100 ms. In addition, the image quality also outperforms the DIC-based XST method as quantified by the noise level. Our results show that SPINNet could enable in-situ and in-operando SBI measurements in 2D and 3D phase-contrast imaging. Introduction With the development of high-brightness synchrotron facilities and free electron lasers, advanced X-ray imaging techniques including near-field phase-contrast imaging and far-field diffraction imaging are well-developed and widely used in material science, environmental science, and biomedical imaging [1][2][3][4]. Compared with far-field diffraction-based methods, near-field phasecontrast imaging is a full-field imaging technique with lower coherence requirement, larger field of view, higher experimental flexibility, and can be performed with small-scale laboratory X-ray sources. In addition, phase-contrast X-ray imaging is more sensitive to sample density with low-Z materials than traditional transmission X-ray imaging, making it a preferred tool for studying low-absorption soft tissues such as muscles. During the past few decades, various phase-contrast imaging methods have been developed [5,6], including the propagation-based method and grating interferometry, applied in 2D and 3D imaging of biomedical samples. However, quantitative information is missing from the propagation-based method [7,8] where extra constraints such as the sample's size and properties are required for the phase retrieval and reconstruction. Talbot grating interferometry can provide quantitative multi-contrast imaging, including absorption, phase, and dark-field [9][10][11][12], but suffers from complicated setup, high source coherence requirement, phase warping, and relatively low spatial resolution. Fig. 1. Schematic of a typical speckle-based phase-contrast imaging setup with a collimated beam. Note that the coded mask can be replaced by any speckle generator. and are the mask-to-sample, mask-to-camera distances, respectively. The experiment was carried out at the 1-BM beamline of the Advanced Photon Source (APS). The X-ray photon energy was set to 14 keV using a Si(111) double-crystal monochromator. To obtain a high-contrast speckle pattern, a coded binary phase mask with a pitch size of 5 µm was used as the speckle generator [20]. The coded mask was fabricated by electroplating 2 µm thick Au into polymer molds patterned using electron beam lithography (see Supplement 1 for details on the coded-mask pattern). The detector camera system had an effective pixel size of 0.65 µm, which consisted of a 100 µm thick LuAG:Ce scintillator, a 10× objective, a 45°reflective mirror, and an Andor Neo sCMOS camera. The sample-to-camera distance was − = 628 mm. For the 2D imaging experiment, only one speckle image pair with one reference image and one sample image was recorded. For the 3D tomography measurement, the reference image was acquired only once before putting the sample in the beam, while the sample images were collected at each sample rotation angle. SPINNet design structure The design structure of SPINNet is based on the optical flow networks, which have been used for object detection in self-driving [34,36]. As shown in Fig. 2, SPINNet consists of three sub-networks: the feature extractor for reference and sample images, the estimator (PhaseNet and TNet for phase and transmission, respectively), and the refiner (PhaseRefiner and TRefiner for phase and transmission, respectively). The detailed structures for each sub-network can be found in Supplement 1. The inputs for SPINNet are one reference image ( ) and one sample image ( ), while the outputs are the transmission image ( ) and two displacement maps ( and ). The SPINNet structure also includes the multi-resolution analysis and 3D cost volume evaluation, essential features for speckle tracking to achieve the best efficiency and accuracy. The multi-resolution analysis has been demonstrated to improve the calculation efficiency of the conventional correlation-based algorithms by reducing the adequate searching window size in each of the multi-resolution levels [21]. It is particularly effective for samples with large phase gradients and coexistent fine features. Here we introduce the multi-resolution analysis into SPINNet in the feature extractor and estimator sub-networks. The effects of different resolution levels can be found in Fig. S3 in Supplement 1. The 3D cost volume is another key feature of SPINNet to account for the process of sub-window correlation, which is an extra but necessary physical constraint. The multi-resolution process starts with the feature extractor, which extracts feature information of different domain sizes from the input reference image ( ) and sample image ( ). Since the reference and sample images can have different features, two separate feature extractors are built for and , respectively. A max multi-resolution level of = 6 is used in the current SPINNet. For each resolution level, the image size will be down-sampled from a lower level by a factor of 2 in each image coordinate dimension ( and ) and, at the same time, the feature information will have an increasing channel number as the third dimension. Details of the extractor functions and the dimensions of can be found in Supplement 1. The estimator contains multi-resolution levels (named as Net-level , with ∈ [2, ]) as shown in Fig. 2. There are totally ( − 1) PhaseNets and ( − 1) TNets for image features , , and with different sampling resolution. Firstly, the initial displacement ( 0 and 0 ) and transmission ( 0 ) maps are generated randomly for the highest multi-resolution level (Net-level ). The reference image feature is propagated to sample image position using the displacement maps ( +1 and +1 ) from the higher level for each feature channel ( ∈ [1, ]) according to Eq. (1). This propagation process is called the prop-layer, denoted as ( , , ) = ( − +1 , − −1 , ).(2) In this step, we set the transmission = 1 to reduce the cross-talk between the phase shift and transmission in the neural network. Then the 3D cost volume is constructed as ( , , ℎ) = ( − , − , ) ( , , ) ,(3) where ℎ = × (2 + 1) + , and , ∈ (− , ) are distances of nearby pixels within the max searching range along and , respectively. The total channel dimension of the 3D cost volume in the ℎ coordinate is (2 + 1) 2 . Thanks to the multi-resolution approach, a search range of = 3 or 4 is adequate in each level, giving a channel dimension of to be 49 or 81, respectively. Once the 3D cost volume is built, the displacement and for the multi-resolution level can be obtained by passing through the PhaseNet. and also become the input for the next lower level estimator. Unlike the phase prediction, the multi-resolution reference image and sample image without going through feature extractors are used for the transmission prediction because the relative intensity information might be missing in the image feature and . and are obtained simply by image downsizing using bilinear interpolation, by a factor of 2 in and for each level.The inverse prop-layer, which is an inverse process of Eq. (2), is used to propagate the sample image ( ) to the reference image domain ( ). Then TNet will predict the transmission image based on and at each resolution level . The above procedure will be repeated to reach the last multi-resolution level (typically = 2 or 3). Finally, a PhaseRefiner and a TRefiner are added to remove the extra noise and improve the accuracy of the predicted displacement and transmission images. All subnets (PhaseNet, TNet, PhaseRefiner, and TRefiner) use similar conventional neural network structures described in detail in Fig. S2 in Supplement 1. Here a multi-resolution level of = 2 instead of = 1 is used for faster training and prediction speed by reducing the network size. From experience, we note that no obvious improvement can be observed by introducing the level of = 1. Considering that 1 and 1 are original reference image ( ) and sample image ( ), respectively, without passing through the feature extractors, which contains no feature information, it is reasonable to discard the multi-resolution level of = 1. Once the displacements and and the transmission are obtained by SPINNet, the following loss function is used to characterize the prediction error, = || − || 2 2 || || 2 2 + || − || 2 2 || || 2 2 + || − || 2 2 || || 2 2 ,(4) where || · || 2 represents 2 norm, , and are ground-truth images of the displacements and transmission, respectively. Training data generation The training data generation is critical for a neural network and can affect the accuracy and general applicability in analyzing actual experimental data. Below we discuss how to generate the reference and sample image pairs based on Eq. (1) to best approximate the experimental conditions. Reference image generation In the conventional speckle-tracking methods, sandpaper or filter membrane is usually used as the speckle generator, which is inexpensive and straightforward to implement. However, the random speckle pattern has various parameters such as speckle feature size, blurring effect, and contrast to add extra complexity to the neural network training. Thus, numerous training data points will be needed to cover different experimental conditions. On the other hand, the coded mask was proposed recently to generate speckle patterns with ultra-high contrast [20]. The pre-knowledge of coded-mask design parameters significantly reduces the data volume required for network training. Therefore, we choose to generate the simulation data based on the coded binary phase mask. Each reference image is generated by propagating a plane wave modulated by a randomly generated binary-phase-mask pattern. We consider all coded-mask patterns to have a total size of × pixels with a mask pitch size of pixels. Firstly, a random binary noise image with a size of / × / pixels is generated to have half pixels equal to 0 and the other half equal to 1. Then this binary noise image is upsampled by a factor of without interpolation to produce an image with the size of × pixels to represent the pattern of a coded mask. The reference image is obtained by propagating a plane wave with its amplitude and phase modulated by to a distance using the Fresnel diffraction formula, = ∬ ( 0 + 0 ) 0 ( − 0 ) 2 +( − 0 ) 2 2 0 0 2 ,(5) where = 2 / , 0 and 0 define the wavefront amplitude and the consequent image intensity and contrast, and 0 is the phase shift of the coded mask. At last, the reference image is Gaussian filtered with a kernel size of 3 pixels, which mimics the limited resolution of the detector system. We set the simulation parameters to = 0.06 nm, = 0.3 m, 0 = 0.7, 0 = 0.3, and 0 = . Note that these parameters do not need to be the same as the real condition, as long as the produced pattern images have similar feature sizes and contrast as the experimental data. Figure. 3(a) shows an example of a reference image. Sample image generation A sample image is generated using one of the above reference images and Eq. (1) with the displacement ( and ) and transmission ( ) maps created randomly as described below. Firstly, a Gaussian noise image ( × pixels) is generated with a mean value of 0.5, normalized to the range of [0, 1], and Gaussian filtered with a kernel size of 10 pixels to remove high-frequency noises. Note that the smallest sample feature size can be determined by choosing the Gaussian filter kernel size carefully. The Bézier curve is then used to create a sample contour image. The sample phase image is obtained by multiplying the noise image with the sample shape image and then scaled by a factor randomly generated within a range of [ , 20 ] for each image. Once the sample phase image is generated, the displacement maps ( and ) are calculated as the image gradients along and directions, respectively, following Eq. (1). The generation of sample transmission is similar to the phase generation process, except that the transmission image is scaled by a factor randomly selected in a range of [0.02, 0.2] for each image. Considering that the sample transmission image normally has the same shape and features as its phase image, we set 40% of the transmission images to have the same distribution as the phase image, and the rest 60% have independent distributions. Example images of sample transmission , horizontal displacement , and vertical displacement are shown in Figs. 3(b), (c) and (d), respectively. SPINNet training A dataset of 10,000 simulated reference-sample image pairs is generated with 80% for training and 20% for validation. SPINNet is trained within three stages from the low-resolution to the high-resolution level using the Adam optimizer with an initial learning rate of 1.0E-4. The first stage trains the Net-level 6 to Net-level 3 within 400 epochs until convergence. In the second stage, Net-level 2, initialized with the weight from the former Net-level 3, is added into the network and trained within 600 epochs. The third stage trains the whole network with the PhaseRefiner and TRefiner within 500 epochs. For a better convergence, the learning rate is reduced by half for every 100 epochs. The number of parameters for SPINNet is around 5 million. The whole training takes 8 hours using 16 NVIDIA A100 GPUs with a batch size of 96, 45, and 8 for the three stages. The training and validation loss based on Eq. (4) is shown in Fig. 4. The use of a refiner effectively reduces the training loss in stage 3. The validation loss and training loss are almost the same, indicating no obvious overfitting during training. Results and Analysis Simulation data reconstruction We first use 10,000 separately simulated data to evaluate the performance of SPINNet. Figures 5(a) and (b) show a simulated reference and sample images as an example. The histogram in Fig. 5(c) shows the statistics of the percentage error between the SPINNet prediction and the ground truth for all 10,000 data. We define a low error regime with a percentage error of less than 0.25%, a medium error regime with 0.25%~1.0%, and a high regime with a larger than 1% error. There are only 7% of data that have prediction errors above 1%, but all are still less than 4%. Examples of the predicted displacements and transmission and their ground truth from each error regime are also shown in Fig. 5(d). Even in the high error regime, the displacement prediction agrees well with the ground truth regarding low-frequency features and with only slight difficulty in predicting high-frequency components. Figure 5(d) also indicates that the displacement (or phase) prediction has less noise than the transmission prediction because PhaseNet is based on 3D cost volume with a search range of 3 pixels, so that an extra physical constraint is implemented. At the same time, TNet predicts the transmission in a pixel-wise fashion without any constraint. Although the prediction of transmission has more noise, the overall distribution and structure are all accurately reconstructed. Experimental data reconstruction In this section, we test the SPINNet performance with experimental data and compare the predicted results with the conventional DIC-based XST analysis. The SPINNet reconstruction was carried out using one NVIDIA A100 GPU, while the DIC-based XST analysis was performed on a cluster with 24-core Xeon E5-2670 CPU and 32 GB RAM. The DIC-based XST analysis used a template window size of 7×7 and a searching window size of 20×20 pixels. Quantitative phase reconstruction The reconstruction accuracy is essential for applications requiring quantitative information such as X-ray at-wavelength metrology and quantitative phase-contrast imaging. For X-ray at-wavelength metrology, state-of-the-art optics such as refractive lenses and total-reflection mirrors have figure errors that require a phase measurement accuracy in the /100 level [40]. Here we test the SPINNet reconstruction accuracy by measuring a Beryllium X-ray lens and comparing the results with conventional DIC-based XST analysis. A 2D parabolic beryllium lens with an apex radius of 200 µm was measured using the single-shot speckle-tracking setup in Fig. 1 with a sample-camera distance of 500 mm. SPINNet predicted the horizontal differential phase, vertical differential phase, transmission, and integrated phase profiles as shown in Figs. 6(a)-(d) using the measured reference and sample images. After subtracting the best fit parabola from the phase profile in Fig. 6(d), the residual phase error within a circular area of 800 µm diameter was obtained and shown in Fig. 6(e). For comparison, the lens phase was also reconstructed using the DIC-based XST analysis, and the resulting residual phase error is shown in Fig. 6(f). The relative rms error between the residual phase in Figs. 6(e) and (f) is about 0.15 rad (0.02 ), which is close to the theoretical phase sensitivity (0.01 ) of the experimental setup. Regrading the data processing speed, the DIC-based XST analysis took several minutes. In contrast, the SPINNet prediction took only 0.16 second with a batch size of 1, which can be further reduced by increasing the batch size: for example, 0.13 second computation time is obtained with a batch size of 3. The ultra-fast speed and high accuracy prove the feasibility of SPINNet in real-time X-ray wavefront sensing and beamline diagnostics at synchrotron light sources and free-electron lasers beamlines. Phase-contrast imaging comparison A flour bug was measured using the same setup in Fig. 1. The reconstructed results using SPINNet are shown in Fig. 7 and compared with results using DIC-based XST analysis. The differential phase images using SPINNet [ Fig. 7(a) and (b)] have much lower noise than those using the DIC-based XST analysis [ Fig. 7(e) and (g)]. As a result, the phase image using SPINNet [ Fig. 7(c)] has significantly better quality and higher spatial resolution than that of the DIC-based XST analysis [ Fig. 7(g)]. Also, the SPINNet predicted transmission image [ Fig. 7(d)] shows identical sample features as in the radiography image [ Fig. 7(h)], but with much higher contrast. Because of the optimized network structure, the SPINNet reconstruction outperforms the DIC-based XST analysis in image quality. Two main factors contribute to the superiority of SPINNet: (1) the network is based on convolution layers acting as low-pass filters to suppress image noise, and (2) SPINNet is not only based on the image pixel value, but also extracted features using feature extractors. In addition, the multi-resolution process takes into account the feature information with pyramid resolution levels, which may also help the image reconstruction quality. (a) (b) (c) (d) (e) (f) (g) (h) For the DIC-based XST analysis, the template window size directly limits the available spatial resolution. A small template window size gives rise to a better spatial resolution at the expense of a larger noise level. Instead, SPINNet predicts phase and transmission in a pixel-wise fashion based on the feature information and the 3D cost volume to obtain a higher spatial resolution. The high speed of SPINNet is also the result of the multi-resolution process and thus the reduced search window size for the 3D cost volume. The searching window size of the 3D cost volume is only 3×3 pixels in SPINNet, while that of the DIC-based XST analysis is 20×20 pixels. It should also be noted that even SPINNet was trained with simulated data using code mask patterns of 5 µm pitch size, the trained model also works well for experimental images with a different mask pitch size. Prediction results of the same flour bug with experimental data obtained using a coded mask of 2 µm pitch in Fig. S4 (see Supplement 1) show high image quality and resolution with only slightly higher noise. Therefore, SPINNet is not sensitive to the training data and can be a general tool for speckle-based phase-contrast imaging. Phase-contrast tomography For high-resolution tomography, 1800 projections of the flour bug were acquired with an angular resolution of 0.1°and an exposure time of 5 seconds. The exposure time is primarily restricted by the X-ray flux of the bending magnet beamline. Using one A100 GPU, the data processing time of each image pair with a size of 2112×2112 pixels was 0.16 seconds. Thanks to the fast data processing speed of SPINNet, the phase and transmission images of the full tomography dataset were obtained within only several minutes. Both the 3D transmission and phase volumes were reconstructed by filtered back projection (FBP) method with a ram-lak filter using the Astratoolbox [41,42] and shown in Fig. 8. Visualization 1 and Visualization 2 provide complementary information on the sample. Thanks to the fast speed of SPINNet, the 3D reconstruction can be carried out immediately after the data acquisition, which is beneficial for checking the data quality during the experiment time. The 3D resolution in Fig. 8 is mainly limited by the detector resolution and the angular resolution and stability of the tomography rotation stage. Discussion and Conclusion SPINNet has been developed for high-resolution real-time phase-contrast imaging and wavefront sensing. SPINNet has an optimized structure featured with multi-resolution analysis and 3D cost volume evaluation. The multi-resolution approach significantly accelerates the reconstruction speed and enables the simultaneous prediction of sample features with different length scales and phase variations. The 3D cost volume is added to the network to provide physical constraints for the network training, which is essential for the prediction accuracy of SPINNet. The 3D cost volume contains the correlation information in addition to the image intensity, which has a similar physical process as the conventional DIC-based analysis. Thanks to the optimized network structure, SPINNet can be trained with pure simulation data. This characteristic can significantly broaden applications of SPINNet where experimental training data are difficult to acquire. Although the network is trained with simulation data, SPINNet can be applied directly to the actual experimental dataset without any fine-tuning or transfer learning process. We successfully demonstrated applications in quantitative at-wavelength metrology, quantitative phase imaging, and phase-contrast tomography. SPINNet outperforms the traditional DIC-based XST analysis in terms of the imaging quality and, more significantly, the computation speed. Over two-order of magnitude improvement in speed has been demonstrated compared with the DIC-based analysis, enabling real-time and in-situ measurement for SBI methods. Furthermore, the reconstruction speed of SPINNet can be improved by slightly sacrificing spatial resolution and image quality. As shown in Fig. S3 in Supplement 1, a reconstruction speed in the 50 ms level can be achieved for each image pair by using a Net-level 3 model. In summary, to the best of our knowledge, this is the first time a deep learning method has been applied in the speckle-based imaging field. We anticipate SPINNet will solve the bottleneck of speckle-based imaging techniques in measurement speed. It will pave the way to real-time phase-contrast imaging measurements in a broad range of applications, including real-time X-ray wavefront sensing and in-situ phase-contrast imaging of biomedical samples and soft materials. Real-time X-ray Phase-contrast Imaging Using SPINNet -A Novel Speckle-based Phase-contrast Imaging Neural Network This document provides supplementary information to "Real-time X-ray Phase-contrast Imaging Using SPINNet -A Novel Speckle-based Phase-contrast Imaging Neural Network". We show details on the coded-mask pattern, the structure of the speckle-based phase-contrast imaging neural network (SPINNet), and the comparison of SPINNet predictions with different network levels. CODED PHASE MASK PATTERN The coded mask was fabricated by electroplating Au (2 µm thick) into polymer molds (poly(methyl methacrylate), PMMA) on a silicon nitride membrane. The bias-corrected coded-mask pattern has a binary pitch size of 5 µm generated using a Python program with a random number generator to create a script for a CNST Nanolithography Toolbox [1]. The detailed fabrication process can be found in Ref [2] with the primary difference being the pattern generation and development described herein. Figure S1 shows the SEM image of the used phase mask. DETAILED SPINNET STRUCTURE The detailed structures of SPINNet feature extractor and phase and transmission estimators and refiners are shown in Fig. S2. The blocks represent series of neural network operations. The feature extractor has a similar structure as the encoder of the well known encoder-decoder neural network structure. The green block in Fig. S2(a) represents a fundamental procedure consisting of a 2D convolution (Conv2d) layer with a stride of 1, a 2D batch normalization (Batchnorm2d) layer, and a leaky Relu nonlinear activation (LeakyRelu) layer. The yellow block includes a Conv2d layer with a stride of 2, a Batchnorm2d layer, and a LeakyRelu layer. Because of the different stride settings in the Conv2d layer, the yellow block can down-sample the input data, as shown in Fig. S2(a). The input simulation data for training has 512×512 pixels with a single channel, which will be down-sampled by a factor of 2×2 but with an increasing channel number from 1 to 128. The feature extractor networks for the reference and sample images have the same structure but are independent. Figures S2(b) and (c) are the detailed network structures of the phase and transmission estimators (PhaseNet and TNet), respectively. Both estimator networks consist of five consequent green blocks (Conv2d+Batchnorm2d+LeakyRelu) and a final Conv2d block with a kernel size of 3 [red blocks in Figs. S2(b) and (c)]. The only difference between the phase and transmission estimators is the final output layer with an output channel of 2 and 1, respectively. The phase and transmission refiners (PhaseRefiner and TRefiner) are shown in Fig. S2(d) and (e), respectively. Their structures are similar to the estimators, except the final Conv2d layer has a kernel size of 1 instead of 3. Test results showed that the kernel size of 1 can provide a higher spatial resolution because of a weaker low-pass filtering effect than a kernel size of 3. The difference between the phase and transmission refiners is again on the different output channel numbers. PREDICTION COMPARISON WITH DIFFERENT NETWORK LEVELS SPINNet was trained with three stages: stage 1, Net-level 6~3; stage 2, Net-level 6~2; and stage 3, Net-level 6~2 with refiners. Figure S3 shows the predicted differential phase, phase, and transmission images using the trained models of stage 1 [ Fig. S3(a)], stage 2 [ Fig. S3(b)], and stage 3 [ Fig. S3(c)]. The sample boundaries in the predicted phase and transmission images using the stage 1 model are less sharp and noisier than the prediction of stage 2 and stage 3 models. By adding the phase and transmission refiner, the stage 3 quality. The prediction time using the stage 1, stage 2, and stage 3 models for a full-field image pair is 72 ms, 125 ms, and 160 ms with a batch size of 1 (50 ms, 100 ms, and 130 ms with a batch size of 3), respectively. Considering the already good imaging quality of the stage 1 model as shown in Fig. S3(a), the SPINNet efficiency can be pushed to a 50 ms level with slightly sacrificing image quality. This can be potentially beneficial for applications requiring faster speed than better imaging quality, such as a real-time wavefront sensor in a fast feedback beamline control system. PREDICTION RESULTS FOR DIFFERENT CODED MASKS All simulated training data for SPINNet was generated based on different coded mask patterns with a pitch size of 5 µm, and it has been proved that the trained model worked on the experimental data using a coded mask with the same pitch size, as shown in Fig. S3. Here, we demonstrate that the same SPINNet model also works on data with different coded mask parameters. Figure S4 shows the SPINNet predicted differential phase, phase, and transmission images of the same flour bug but with experimental data using a coded mask of 2 µm pitch. Compared with the results in Fig. S3 (c1~c4), images in Fig. S4 have only slightly higher noise. Even though the trained SPINNet model has never seen a 2 µm speckle pattern, the predicted results still show high image quality and resolution. SPINNet predictions of (a) the horizontal differential phase, (b) the vertical differential phase, (c) the phase image, and (d) the transmission image of a flour bug. The SPINNet model was trained with simulated data using coded mask patterns with 5 µm pitch, while the experimental data was obtained using a coded mask with 2 µm pitch. Fig. 2 . 2Schematic of SPINNet structure. Fig. 3 . 3Examples of simulated training data for SPINNet: (a) reference image, (b) transmission image, (c) horizontal displacement, , and (d) vertical displacement, . Fig. 4 . 4Training and validation loss as a function of epoch within three stages. Stage 1 (epoch 1~400) is for Net-level 6~3. Stage 2 (epoch 401~1000) is for Net-level 6~2. Stage 3 (epoch 1001~1500) is for Net-level 6~2 with refiners. Fig. 5 . 5Simulated (a) reference image and (b) sample image. (c) Histogram of the prediction error distribution of a 10,000 test dataset using SPINNet. Example prediction results and their ground truth from each error regime. Fig. 6 . 6SPINNet reconstructed (a) horizontal differential phase, (b) vertical differential phase, (c) transmission image, (d) phase profile, and (e) residual phase error after removing the best fit parabola from (d) of a 2D parabolic beryllium lens. (f) Residual phase error result using DIC-based XST analysis for the same lens. Fig. 7 . 7Comparison of phase reconstruction results using SPINNet and DIC-based XST analysis of a flour bug. (a) Horizontal differential phase, (b) vertical differential phase, (c) phase, and (d) transmission image reconstructed by SPINNet. (e) Horizontal differential phase, (f) vertical differential phase, and (g) phase image reconstructed by the DIC-based XST analysis. (h) Raw radiography image. Fig. 8 . 8Tomography volume rendering of the 3D (a) phase and (b) transmission of a flour bug. Inserted figures are the line slices. Funding . U.S. Department of Energy (No. DE-AC02-06CH11357). Fig. S1 . S1SEM image of a coded binary phase mask with a pitch size of 5 µm. Fig. S2 . S2Detailed SPINNet structures for (a) the feature extractor, (b) the phase estimator (PhaseNet), (c) the transmission estimator (TNet), (d) the phase refiner (PhaseRefiner), and (e) the transmission refiner (TRefiner). Fig. S3 . S3SPINNet predictions of (1) the horizontal differential phase, (2) the vertical differential phase, (3) the phase image, and (4) the transmission image of a flour bug using (a) stage 1 model with Net-level 6~3, (b) stage 2 model with Net-level 6~2, and (c) stage 3 model with Net-level 6~2 and the refiners. Fig. S4. SPINNet predictions of (a) the horizontal differential phase, (b) the vertical differential phase, (c) the phase image, and (d) the transmission image of a flour bug. The SPINNet model was trained with simulated data using coded mask patterns with 5 µm pitch, while the experimental data was obtained using a coded mask with 2 µm pitch. model further improved the imaging2 (a1) (a2) (a3) (a4) (b1) (b2) (b3) (b4) (c1) (c2) (c3) (c4) Acknowledgement. This research used resources of the Advanced Photon Source, Argonne Leadership Quantitative phase imaging in biomedicine. Y Park, C Depeursinge, G Popescu, Nat. Photonics. 12Y. Park, C. Depeursinge, and G. Popescu, "Quantitative phase imaging in biomedicine," Nat. Photonics 12, 578-589 (2018). X-ray ptychography. F Pfeiffer, Nat. Photonics. 12F. Pfeiffer, "X-ray ptychography," Nat. Photonics 12, 9-17 (2018). Coherent x-ray diffraction imaging. J Miao, R L Sandberg, C Song, IEEE J. selected topics quantum electronics. 18J. Miao, R. L. Sandberg, and C. Song, "Coherent x-ray diffraction imaging," IEEE J. selected topics quantum electronics 18, 399-410 (2011). An overview of synchrotron radiation applications to low temperature geochemistry and environmental science. G E BrownJr, N C Sturchio, Rev. Mineral. Geochem. 49G. E. Brown Jr and N. C. Sturchio, "An overview of synchrotron radiation applications to low temperature geochemistry and environmental science," Rev. Mineral. Geochem. 49, 1-115 (2002). Principles of Different X-ray Phase-Contrast Imaging: A Review. S Tao, C He, X Hao, C Kuang, X Liu, Appl. Sci. 112971S. Tao, C. He, X. Hao, C. Kuang, and X. Liu, "Principles of Different X-ray Phase-Contrast Imaging: A Review," Appl. Sci. 11, 2971 (2021). X-ray phase-contrast imaging: from pre-clinical applications towards clinics. A Bravin, P Coan, P Suortti, Phys. Medicine Biol. 58A. Bravin, P. Coan, and P. Suortti, "X-ray phase-contrast imaging: from pre-clinical applications towards clinics," Phys. Medicine Biol. 58, R1-R35 (2012). Single-image geometric-flow x-ray speckle tracking. D M Paganin, H Labriet, E Brun, S Berujon, Phys. Rev. A. 9853813D. M. Paganin, H. Labriet, E. Brun, and S. Berujon, "Single-image geometric-flow x-ray speckle tracking," Phys. Rev. A 98, 053813 (2018). Single-shot x-ray speckle-based imaging of a single-material object. K M Pavlov, H T Li, D M Paganin, S Berujon, H Rougé-Labriet, E Brun, Phys. Rev. Appl. 1354023K. M. Pavlov, H. T. Li, D. M. Paganin, S. Berujon, H. Rougé-Labriet, and E. Brun, "Single-shot x-ray speckle-based imaging of a single-material object," Phys. Rev. Appl. 13, 054023 (2020). Systematic-error-free wavefront measurement using an X-ray singlegrating interferometer. T Inoue, S Matsuyama, S Kawai, H Yumoto, Y Inubushi, T Osaka, I Inoue, T Koyama, K Tono, H Ohashi, M Yabashi, T Ishikawa, K Yamauchi, Rev. Sci. Instruments. 8943106T. Inoue, S. Matsuyama, S. Kawai, H. Yumoto, Y. Inubushi, T. Osaka, I. Inoue, T. Koyama, K. Tono, H. Ohashi, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Systematic-error-free wavefront measurement using an X-ray single- grating interferometer," Rev. Sci. Instruments 89, 043106 (2018). High sensitivity X-ray phase contrast imaging by laboratory grating-based interferometry at high Talbot order geometry. J Vila-Comamala, L Romano, K Jefimovs, H Dejea, A Bonnin, A C Cook, I Planinc, M Cikes, Z Wang, M Stampanoni, Opt. Express. 292049J. Vila-Comamala, L. Romano, K. Jefimovs, H. Dejea, A. Bonnin, A. C. Cook, I. Planinc, M. Cikes, Z. Wang, and M. Stampanoni, "High sensitivity X-ray phase contrast imaging by laboratory grating-based interferometry at high Talbot order geometry," Opt. Express 29, 2049 (2021). X-ray phase microtomography with a single grating for high-throughput investigations of biological tissue. M.-C Zdora, J Vila-Comamala, G Schulz, A Khimchenko, A Hipp, A C Cook, D Dilg, C David, C Grünzweig, C Rau, P Thibault, I Zanette, Biomed. Opt. Express. 81257M.-C. Zdora, J. Vila-Comamala, G. Schulz, A. Khimchenko, A. Hipp, A. C. Cook, D. Dilg, C. David, C. Grünzweig, C. Rau, P. Thibault, and I. Zanette, "X-ray phase microtomography with a single grating for high-throughput investigations of biological tissue," Biomed. Opt. Express 8, 1257 (2017). Development and implementation of a portable grating interferometer system as a standard tool for testing optics at the Advanced Photon Source beamline 1-BM. L Assoufid, X Shi, S Marathe, E Benda, M J Wojcik, K Lang, R Xu, W Liu, A T Macrander, J Z Tischler, Rev. Sci. Instruments. 8752004L. Assoufid, X. Shi, S. Marathe, E. Benda, M. J. Wojcik, K. Lang, R. Xu, W. Liu, A. T. Macrander, and J. Z. Tischler, "Development and implementation of a portable grating interferometer system as a standard tool for testing optics at the Advanced Photon Source beamline 1-BM," Rev. Sci. Instruments 87, 052004 (2016). X-ray Phase-Contrast Imaging and Metrology through Unified Modulated Pattern Analysis. M.-C Zdora, P Thibault, T Zhou, F J Koch, J Romell, S Sala, A Last, C Rau, I Zanette, Phys. Rev. Lett. 118203903M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, "X-ray Phase-Contrast Imaging and Metrology through Unified Modulated Pattern Analysis," Phys. Rev. Lett. 118, 203903 (2017). X-ray phase tomography with near-field speckles for three-dimensional virtual histology. M.-C Zdora, P Thibault, W Kuo, V Fernandez, H Deyhle, J Vila-Comamala, M P Olbinado, A Rack, P M Lackie, O L Katsamenis, M J Lawson, V Kurtcuoglu, C Rau, F Pfeiffer, I Zanette, Optica. 71221M.-C. Zdora, P. Thibault, W. Kuo, V. Fernandez, H. Deyhle, J. Vila-Comamala, M. P. Olbinado, A. Rack, P. M. Lackie, O. L. Katsamenis, M. J. Lawson, V. Kurtcuoglu, C. Rau, F. Pfeiffer, and I. Zanette, "X-ray phase tomography with near-field speckles for three-dimensional virtual histology," Optica 7, 1221 (2020). State of the Art of X-ray Speckle-Based Phase-Contrast and Dark-Field Imaging. M.-C Zdora, J. Imaging. 460M.-C. Zdora, "State of the Art of X-ray Speckle-Based Phase-Contrast and Dark-Field Imaging," J. Imaging 4, 60 (2018). X-ray Multimodal Tomography Using Speckle-Vector Tracking. S Berujon, E Ziegler, Phys. Rev. Appl. 544014S. Berujon and E. Ziegler, "X-ray Multimodal Tomography Using Speckle-Vector Tracking," Phys. Rev. Appl. 5, 044014 (2016). X-ray microscopy using two phase contrast imaging techniques: two dimensional grating interferometry and speckle tracking. H Wang, S Berujon, I Pape, K Sawhney, J. Physics: Conf. Ser. 46312042H. Wang, S. Berujon, I. Pape, and K. Sawhney, "X-ray microscopy using two phase contrast imaging techniques: two dimensional grating interferometry and speckle tracking," J. Physics: Conf. Ser. 463, 012042 (2013). Near-field speckle-scanning-based x-ray tomography. S Berujon, E Ziegler, Phys. Rev. A. 9563822S. Berujon and E. Ziegler, "Near-field speckle-scanning-based x-ray tomography," Phys. Rev. A 95, 063822 (2017). Quantitative x-ray dark-field and phase tomography using single directional speckle scanning technique. H Wang, Y Kashyap, K Sawhney, Appl. Phys. Lett. 108124102H. Wang, Y. Kashyap, and K. Sawhney, "Quantitative x-ray dark-field and phase tomography using single directional speckle scanning technique," Appl. Phys. Lett. 108, 124102 (2016). Single-shot x-ray phase-contrast and dark-field imaging based on coded binary phase mask. Z Qiao, X Shi, M J Wojcik, L Rebuffi, L Assoufid, Appl. Phys. Lett. 11911105Z. Qiao, X. Shi, M. J. Wojcik, L. Rebuffi, and L. Assoufid, "Single-shot x-ray phase-contrast and dark-field imaging based on coded binary phase mask," Appl. Phys. Lett. 119, 011105 (2021). Single-shot speckle tracking method based on wavelet transform and multiresolution analysis. Z Qiao, X Shi, L Assoufid, Advances in Metrology for X-Ray and EUV Optics IX. International Society for Optics and Photonics11492114920Z. Qiao, X. Shi, and L. Assoufid, "Single-shot speckle tracking method based on wavelet transform and multi- resolution analysis," in Advances in Metrology for X-Ray and EUV Optics IX, vol. 11492 (International Society for Optics and Photonics, 2020), p. 114920O. Wavelet-transform-based speckle vector tracking method for x-ray phase imaging. Z Qiao, X Shi, R Celestre, L Assoufid, Opt. Express. 28Z. Qiao, X. Shi, R. Celestre, and L. Assoufid, "Wavelet-transform-based speckle vector tracking method for x-ray phase imaging," Opt. Express 28, 33053-33067 (2020). The upgrade of the advanced photon source. M Borland, BNLA Blednykh, BNLBrookhaven National Lab. Tech. rep.M. Borland and A. Blednykh, "The upgrade of the advanced photon source," Tech. rep., Brookhaven National Lab.(BNL), Upton, NY (United States) (2018). High-accuracy wavefront sensing for x-ray free electron lasers. Y Liu, M Seaberg, D Zhu, J Krzywinski, F Seiboth, C Hardin, D Cocco, A Aquila, B Nagler, H J Lee, Optica. 5Y. Liu, M. Seaberg, D. Zhu, J. Krzywinski, F. Seiboth, C. Hardin, D. Cocco, A. Aquila, B. Nagler, H. J. Lee et al., "High-accuracy wavefront sensing for x-ray free electron lasers," Optica 5, 967-975 (2018). In situ x-ray imaging of defect and molten pool dynamics in laser additive manufacturing. C L A Leung, S Marussi, R C Atwood, M Towrie, P J Withers, P D Lee, Nat. Commun. 9C. L. A. Leung, S. Marussi, R. C. Atwood, M. Towrie, P. J. Withers, and P. D. Lee, "In situ x-ray imaging of defect and molten pool dynamics in laser additive manufacturing," Nat. Commun. 9, 1-9 (2018). Deep learning for high-resolution and high-sensitivity interferometric phase contrast imaging. S Lee, O Oh, Y Kim, D Kim, D S Hussey, G Wang, S W Lee, Sci. Reports. 109891S. Lee, O. Oh, Y. Kim, D. Kim, D. S. Hussey, G. Wang, and S. W. Lee, "Deep learning for high-resolution and high-sensitivity interferometric phase contrast imaging," Sci. Reports 10, 9891 (2020). Multi-resolution convolutional neural networks for inverse problems. F Wang, A Eljarrat, J Müller, T R Henninen, R Erni, C T Koch, Sci. Reports. 105730F. Wang, A. Eljarrat, J. Müller, T. R. Henninen, R. Erni, and C. T. Koch, "Multi-resolution convolutional neural networks for inverse problems," Sci. Reports 10, 5730 (2020). Phase recovery and holographic image reconstruction using deep learning in neural networks. Y Rivenson, Y Zhang, H Günaydın, D Teng, A Ozcan, Light. Sci. & Appl. 717141Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, "Phase recovery and holographic image reconstruction using deep learning in neural networks," Light. Sci. & Appl. 7, 17141 (2017). Deep learning in holography and coherent imaging. Y Rivenson, Y Wu, A Ozcan, Light. Sci. & Appl. 885Y. Rivenson, Y. Wu, and A. Ozcan, "Deep learning in holography and coherent imaging," Light. Sci. & Appl. 8, 85 (2019). Deep speckle correlation: a deep learning approach toward scalable imaging through scattering media. Y Li, Y Xue, L Tian, Optica. 51181Y. Li, Y. Xue, and L. Tian, "Deep speckle correlation: a deep learning approach toward scalable imaging through scattering media," Optica 5, 1181 (2018). Low Photon Count Phase Retrieval Using Deep Learning. A Goy, K Arthur, S Li, G Barbastathis, 121243902A. Goy, K. Arthur, S. Li, and G. Barbastathis, "Low Photon Count Phase Retrieval Using Deep Learning," arXiv 121, 243902 (2018). Quantitative Phase Imaging and Artificial Intelligence: A Review. Y Jo, H Cho, S Y Lee, G Choi, G Kim, H Min, Y Park, IEEE J. Sel. Top. Quantum Electron. 25Y. Jo, H. Cho, S. Y. Lee, G. Choi, G. Kim, H.-s. Min, and Y. Park, "Quantitative Phase Imaging and Artificial Intelligence: A Review," IEEE J. Sel. Top. Quantum Electron. 25, 1-14 (2018). Lensless computational imaging through deep learning. A Sinha, J Lee, S Li, G Barbastathis, Optica. 41117A. Sinha, J. Lee, S. Li, and G. Barbastathis, "Lensless computational imaging through deep learning," Optica 4, 1117 (2017). D Sun, X Yang, M.-Y Liu, J Kautz, PWC-Net: CNNs for Optical Flow Using Pyramid, Warping, and Cost Volume. D. Sun, X. Yang, M.-Y. Liu, and J. Kautz, "PWC-Net: CNNs for Optical Flow Using Pyramid, Warping, and Cost Volume," arXiv (2017). P Fischer, A Dosovitskiy, E Ilg, P Häusser, C Hazırbaş, V Golkov, P Smagt, D Cremers, T Brox, FlowNet: Learning Optical Flow with Convolutional Networks. arXivP. Fischer, A. Dosovitskiy, E. Ilg, P. Häusser, C. Hazırbaş, V. Golkov, P. v. d. Smagt, D. Cremers, and T. Brox, "FlowNet: Learning Optical Flow with Convolutional Networks," arXiv (2015). T.-W Hui, X Tang, C C Loy, LiteFlowNet: A Lightweight Convolutional Neural Network for Optical Flow Estimation. arXivT.-W. Hui, X. Tang, and C. C. Loy, "LiteFlowNet: A Lightweight Convolutional Neural Network for Optical Flow Estimation," arXiv (2018). A Lightweight Optical Flow CNN -Revisiting Data Fidelity and Regularization. T.-W Hui, X Tang, C C Loy, arXivT.-W. Hui, X. Tang, and C. C. Loy, "A Lightweight Optical Flow CNN -Revisiting Data Fidelity and Regularization," arXiv (2019). LiteFlowNet3: Resolving Correspondence Ambiguity for More Accurate Optical Flow Estimation. T.-W Hui, C C Loy, arXivT.-W. Hui and C. C. Loy, "LiteFlowNet3: Resolving Correspondence Ambiguity for More Accurate Optical Flow Estimation," arXiv (2020). SelFlow: Self-Supervised Learning of Optical Flow. P Liu, M Lyu, I King, J Xu, arXivP. Liu, M. Lyu, I. King, and J. Xu, "SelFlow: Self-Supervised Learning of Optical Flow," arXiv (2019). How to specify super-smooth mirrors: simulation studies on nano-focusing and wavefront preserving x-ray mirrors for next-generation light sources. X Shi, L Assoufid, R Reininger, 8th Int. Symp. on Adv. Opt. Manuf. Test. Technol. Subnanometer Accuracy Meas. for Synchrotron Opt. X-Ray Opt. X. Shi, L. Assoufid, and R. Reininger, "How to specify super-smooth mirrors: simulation studies on nano-focusing and wavefront preserving x-ray mirrors for next-generation light sources," 8th Int. Symp. on Adv. Opt. Manuf. Test. Technol. Subnanometer Accuracy Meas. for Synchrotron Opt. X-Ray Opt. pp. 968703-968703-11 (2016). Fast and flexible x-ray tomography using the astra toolbox. W Van Aarle, W J Palenstijn, J Cant, E Janssens, F Bleichrodt, A Dabravolski, J De, K J Beenhouwer, J Batenburg, Sijbers, Opt. Express. 24W. Van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. De Beenhouwer, K. J. Batenburg, and J. Sijbers, "Fast and flexible x-ray tomography using the astra toolbox," Opt. Express 24, 25129-25147 (2016). The astra toolbox: A platform for advanced algorithm development in electron tomography. W Van Aarle, W J Palenstijn, J De, T Beenhouwer, S Altantzis, K J Bals, J Batenburg, Sijbers, Ultramicroscopy. 157W. Van Aarle, W. J. Palenstijn, J. De Beenhouwer, T. Altantzis, S. Bals, K. J. Batenburg, and J. Sijbers, "The astra toolbox: A platform for advanced algorithm development in electron tomography," Ultramicroscopy 157, 35-47 (2015). The nanolithography toolbox. K C Balram, D A Westly, M Davanco, K E Grutter, Q Li, T Michels, C H Ray, L Yu, R J Kasica, C B Wallin, J. Res. Natl. Inst. Standards Technol. K. C. Balram, D. A. Westly, M. Davanco, K. E. Grutter, Q. Li, T. Michels, C. H. Ray, L. Yu, R. J. Kasica, C. B. Wallin et al., "The nanolithography toolbox," J. Res. Natl. Inst. Standards Technol. pp. 464-476 (2016). Measurement of x-ray beam coherence along multiple directions using 2-d checkerboard phase grating. S Marathe, X Shi, M J Wojcik, A T Macrander, L Assoufid, J. Vis. Exp. JoVE. S. Marathe, X. Shi, M. J. Wojcik, A. T. Macrander, and L. Assoufid, "Measurement of x-ray beam coherence along multiple directions using 2-d checkerboard phase grating," J. Vis. Exp. JoVE (2016).
[]
[ "THE KOHNEN-ZAGIER FORMULA FOR MAASS FORMS FOR Γ 0 (4)", "THE KOHNEN-ZAGIER FORMULA FOR MAASS FORMS FOR Γ 0 (4)" ]
[ "Nickolas Andersen " ]
[]
[]
We extend a formula of Duke, Imamōglu, and Tóth (which itself is a generalization of the Katok-Sarnak formula) to prove the Kohnen-Zagier formula for Maass forms for Γ 0 (4).
null
[ "https://arxiv.org/pdf/2203.00704v1.pdf" ]
247,218,158
2203.00704
3d40b1d048038aace2c93c257b6d6d93cc1effee
THE KOHNEN-ZAGIER FORMULA FOR MAASS FORMS FOR Γ 0 (4) 1 Mar 2022 Nickolas Andersen THE KOHNEN-ZAGIER FORMULA FOR MAASS FORMS FOR Γ 0 (4) 1 Mar 2022arXiv:2203.00704v1 [math.NT] We extend a formula of Duke, Imamōglu, and Tóth (which itself is a generalization of the Katok-Sarnak formula) to prove the Kohnen-Zagier formula for Maass forms for Γ 0 (4). Introduction Let d be a fundamental discriminant. The Kohnen-Zagier formula [12] relates the |d|-th coefficient of a holomorphic Hecke eigenform g of half-integral weight on Γ 0 (4) to L( 1 2 , f × d · ), where f is the Shimura lift of g. The formula is an explicit version of the general relation of Waldspurger [16]. Here we show how the ideas of [6] and [1] can be combined to give a short proof of the Kohnen-Zagier formula for Maass cusp forms for Γ 0 (4). We adopt the notation of [6]; see the next section for details. such that for any fundamental discriminant d ≡ 0, 1 (mod 4) we have 12π|d||b(d)| 2 = ϕ, ϕ −1 Γ( 1 2 − sgn d 4 + ir 2 ) 2 L( 1 2 , ϕ × χ d ). (1.1) Here χ d = d · and L(s, ϕ × χ d ) denotes the analytic continuation of the L-function L(s, ϕ × χ d ) = ∞ n=1 a(n)χ d (n) n s . The d = 1 case of Theorem 1.1 is a corollary of the main result of Katok and Sarnak [11], which relates the product b(d)b (1) 12 √ π|b(1)| 2 = ϕ, ϕ −1 ∞ 0 ϕ(iy)y −1 dy. The latter integral evaluates to a multiple of L( 1 2 , ϕ). To prove their formula, Katok and Sarnak modify the theta lift of Shintani [15] and Niwa [13]. With some extra work, their method can probably produce a formula for |b(d)| 2 where d is any positive fundamental discriminant. Using the Kuznetsov trace formula and some ideas from the proof of the Selberg trace formula, Biró [3] extended the Katok-Sarnak formula to general level for a pair of positive discriminants d and d ′ such that d is fundamental. The main result of [3] is a relation between b(d)b(d ′ ) and the twisted sums Q∈Γ\Q dd ′ χ d (Q) C Q ϕ(z)y −1 |dz|, where χ d is a character of the finite group Γ\Q dd ′ . When d = d ′ the sum above evaluates to a multiple of L( 1 2 , ϕ × χ d ). It is not clear whether the methods of [11] or [3] can be extended to cover the case where d, d ′ are negative. In [6], Duke, Imamoḡlu, and Tóth generalized the formulas of Katok-Sarnak and Bíro in the case of Maass forms for Γ 0 (4) to allow for two discriminants d, d ′ of any sign, as long as d is fundamental. In the new case, when d and d ′ are both negative (and dd ′ is not a square), Theorem 4 of [6] gives a relation between b(d)b(d ′ ) and Q∈Γ\Q dd ′ χ d (Q) F Q ϕ(z) dxdy y 2 , where F Q is a finite area hyperbolic surface with boundary C Q . The case d = d ′ is not covered in that theorem because the proof relies on being able to compute the integral over C Q of a certain Poincaré series, and the corresponding integral when dd ′ is a square does not converge. Here we use the main idea of [1] to modify the Poincaré series in the case d = d ′ and give a short proof of Theorem 1.1. The generalization of Theorem 1.1 to Maass forms for Γ 0 (4N), with N odd and squarefree, was proved by Baruch and Mao in [2]. Their proof utilizes the powerful tools of automorphic representation theory. Background Throughout this paper we make use of several special functions, especially the Bessel functions I ν (x), J ν (x), and K ν (x), and the Whittaker functions M µ,ν (x) and W µ,ν (x). Definitions and properties of these functions can be found in Sections 10 and 13 of [4]. In the rest of this section, we give some background information on the objects in the introduction, including some standard facts we will need for the proof of the main theorem. We are mostly following the notation and setup of [6]. Other standard references are [7,9,14]. Maass cusp forms of weight 0. Let Γ = PSL 2 (Z) and let ∆ k = y 2 ∂ 2 x + ∂ 2 y − iky∂ x denote the weight k hyperbolic Laplacian. A function ϕ : H → C is a Maass form of weight 0 for Γ if it is Γ-invariant and is an eigenfunction of ∆ 0 with eigenvalue normalized by (∆ 0 + λ)ϕ = 0 and λ = 1 4 + r 2 with r ≥ 0. The quantity r is called the spectral parameter of ϕ. We say that ϕ is a Maass cusp form if the constant term in its Fourier expansion is zero, i.e. ϕ(z) = 2 √ y n =0 a ϕ (n)K ir (2π|n|y)e(nx) for some coefficients a ϕ (n) ∈ C. For each r ≥ 0 let U r denote the vector space of Maass cusp forms of weight 0 with spectral parameter r. For each prime p, the Hecke operator T p acts on U r via Fourier expansions as (T p ϕ)(z) = 2 √ y n =0 a ϕ (pn) + p −1 a ϕ (n/p) K ir (2π|n|y)e(nx). The Hecke operators commute with each other and with ∆ 0 , so we can find an orthogonal (with respect to the Petersson inner product ·, · ) basis B r of U r consisting of Hecke eigenforms. We will normalize the elements of B r , which are called Hecke-Maass cusp forms, so that a(1) = 1. We can also assume that each ϕ is even or odd, meaning that a(−n) = ±a(n) respectively. Maass cusp forms of weight 1 2 . A function ψ : H → C is a Maass form of weight 1/2 for Γ 0 (4) if it satisfies ψ(γz) = J(γ, z)ψ(z) for all γ ∈ Γ 0 (4), where J(γ, z) = θ * (γz) θ * (z) , θ * (z) = y 1/4 n∈Z e(n 2 z), and if (∆ k + λ)ψ = 0 for some λ. If ψ is not a constant multiple of θ * then λ ≥ 1 4 and we define the spectral parameter r as before. Such a ψ is a cusp form if the constant term in its Fourier expansion at each of the cusps of Γ 0 (4)\H is zero. In this case the Fourier expansion is written ψ(z) = n =0 b ψ (n)W1 4 sgn(n),ir (4π|n|y)e(nx). Let V r denote the vector space of Maass cusp forms of weight 1/2 on Γ 0 (4) with spectral parameter r/2. 1 The Kohnen plus space is the subspace V + r of V r comprising forms whose Fourier coefficients are supported on indices n ≡ 0, 1 (mod 4). For each prime p ≥ 3, the Hecke operator T p 2 acts on V + r via Fourier expansions as (T p 2 ψ)(z) = 0 =n≡0,1(4) b ψ (p 2 n) + n p p −1 b ψ (n) + p −1 b ψ (n/p 2 ) W1 4 sgn(n), ir 2 (4π|n|y)e(nx). The Shimura lift. In Theorem 1.2 of [2], Baruch and Mao show that for each ϕ ∈ U r there is a unique ψ ∈ V + r , spectrally normalized so that ψ, ψ = 1, such that for each prime p ≥ 3 we have T p 2 ψ = a ϕ (p)ψ. (2. 1) The form ϕ is called the Shimura lift of ψ. A computation involving (2.1) and T p ϕ = a ϕ (p)ϕ shows that a ϕ (m)b ψ (d) = m n|m n − 3 2 d n b ψ (m 2 d/n 2 ) for all fundamental discriminants d. Quadratic forms and cycles. For each positive discriminant D, let Q D denote the set of (indefinite) integral binary quadratic forms Q = [a, b, c] with b 2 − 4ac = D. The group Γ acts on Q D in the usual way, and the set Γ\Q D is finite. For Q = [a, b, c] ∈ Q D , the equation ax 2 + bxy + cy 2 = 0 has two solutions (x : y) in P 1 (R). When D is not a square, each x/y is a real quadratic irrationality, and when D is a square we have either (x : y) = (1 : 0), corresponding to the point at i∞, or x/y ∈ Q. Let S Q denote the geodesic in H connecting the two solutions, and let Γ Q ⊆ Γ denote the isotropy subgroup {γ ∈ Γ : γQ = Q}. We follow [6] in orienting S Q clockwise if a > 0, counterclockwise if a < 0, and downward if a = 0 (if a = 0 the geodesic is a vertical line). 2 When D is not a square, Γ Q is infinite cyclic, and when D is a square Γ Q is trivial. Let C Q = Γ Q \S Q be the cycle corresponding to Q; it has finite length when D is not a square, and infinite length otherwise. Let D = dd ′ be a factorization of D into a fundamental discriminant d and a discriminant d ′ . The generalized genus character χ d associated to the factorization D = dd ′ is In [8] it is shown that χ d (Q) is well-defined on equivalence classes Q ∈ Γ\Q D . It will be helpful to have an explicit description of Γ\Q D when D = d 2 and d is a fundamental discriminant. The following is a straightforward generalization of Lemma 3 of [1]. χ d (Q) = Proof of Theorem 1.1 We begin by borrowing a few intermediate results from [6]. For Re(s) > 1 let F m (z, s) denote the Poincaré series The function F 0 (z, s) is the usual real analytic Eisenstein series (see [10,Chapter 15]) and has Fourier expansion In Proposition 5 of [6] the authors show that the cycle integrals of F m (z, s) and ∂ z F m (z, s) over finite geodesics yield weighted sums of Kloosterman sums. The next proposition is a complementary result that evaluates the cycle integrals over infinite geodesics, provided that we make a small modification to the integrand as in [1]. Suppose that Q = [a, b, c] ∈ Q D with D a square and let a 1 , a 2 be the rational projective solutions to ax 2 + bxy + cy 2 = 0. For each j = 1, 2 there is a unique γ j ∈ Γ ∞ \Γ such that γ j a j = ∞, and we define Since F m,σQ (z, s) = F m,Q (σz, s) for all σ ∈ Γ, the integrals C Q F m,Q (z, s)y −1 |dz| and F 0 (z, s) = y s + Λ(2s − 1) Λ(2s) y 1−s + 2 √ y n =0 |n| s− 1 2 σ 1−2s (|n|) Λ(2s) K s− 1 2 (2π|n|y)e(nx),(3.C Q ∂ z F m,Q (z, s) dz (3.3) are well-defined, assuming they converge. To show convergence, using Lemma 2.1 we may assume that Q = [0, |d|, c] and 0 ≤ c < |d| for D = d 2 . Then we can take a 1 = ∞ and γ 1 = I. The Fourier expansions (3.1) and (3.2) show that for Re(s) > 1 the integrals in (3.3) converge at ∞, and the observation F m,Q (γ −1 2 z, s) = γ =I,γ −1 2 f m (γz, s) shows that the integrals converge at a 2 . For Re(s) > 1 and d a fundamental discriminant, define T m (d) = Q∈Γ\Q d 2 χ d (Q)          C Q F m,Q (z, s)y −1 |dz| if d > 0, C Q i∂ z F m,Q (z, s) dz if d < 0. Then we have the following analogue of Proposition 5 of [6]. T m (d) =            6π 1/2 |d| 3/2 m n|m n −3/2 d n Φ + d, m 2 n 2 d; 2s+1 4 if m > 0, Γ( s 2 + 1−sgn d 2 ) 2 |d| s L(s, χ d ) 2 Γ(s)ζ(2s) if m = 0, where, for p, q ≡ 0, 1 (mod 4) and pq > 0 we have Φ + (p, q, s) = Γ(s − sgn p 4 )Γ(s − sgn q 4 ) 3 √ π 2 2−2s Γ(2s − 1 2 ) (pq) − 1 2 4|c>0 K + (p, q, c) c J 2s−1 4π √ pq c . Here K + (p, q, c) is the half-integral weight Kloosterman sum K + (p, q, c) = (1 − i) d mod c c d ε d e pd + qd c × 1 if c/4 is even, 2 if c/4 is odd, with ε d = 1 if d ≡ 1 (mod 4) and ε d = i if d ≡ 3 (mod 4). Proof. When d, m > 0 this is (4.4) of [1] (see also Proposition 4 of that paper). For the case d > 0, m = 0, (4.4) of [1] reads T 0 (d) = Γ( s 2 ) 2 4Γ(s) d s L(s, χ d ) ∞ c=1 K + (d, 0; 4c) c s+1/2 . By Lemma 4 of [5] we have T 0 (d) = Γ( s 2 ) 2 d s L(s, χ d ) 2 Γ(s)ζ(2s) . Now assume that d < 0. We will closely follow the proof of Proposition 4 of [1]. From the proof of Lemma 5 of [6] (see (9.2) especially) we have 2i∂ z F m,Q (z, s) = γ∈Γ∞\Γ γ =γ 1 ,γ 2 f 2,m (γz, s) d(γz) dz , where f 2,m (z, s) = φ 2,m (y, s)e(mx) and φ 2,m (y, s) = sy s−1 if m = 0, sm −1/2 (2πy) −1 Γ(s) Γ(2s) M 1,s− 1 2 (4πmy) if m > 0. We choose representatives [c, |d|, 0] for Γ\Q d 2 as in Lemma 2.1 so that χ d (Q) = d c . Since T m (d) = 1 2 Q∈Γ∞\Q d 2 Q=[a,b,c],a =0 χ d (Q) C Q e(mx)φ 2,m (y, s) dz. Since χ d (−Q) = −χ d (Q) and the geodesic C −Q is the same set as C Q but with opposite orientation, we have T m (d) = Q∈Γ∞\Q d 2 Q=[a,b,c],a>0 χ d (Q) C Q e(mx)φ 2,m (y, s) dz. Each cycle C Q with Q = [a, b, c] and a > 0 can be parametrized by z = Re z Q − e −iθ Im z Q , 0 ≤ θ ≤ π, where z Q = − b 2a + i |d| 2a is the apex of the geodesic. Thus C Q e(mx)φ 2,m (y, s) dz = e −mb 2a H m |d| 2a , where H m (t) = it π 0 e(−mt cos θ)φ 2,m (t sin θ, s)e −iθ dθ. It follows that T m (d) = ∞ a=1 H m |d| 2a b(2a) b 2 ≡d 2 (4a) χ d a, b, b 2 −d 2 4a e −mb 2a . By Lemma 7 of [6] we have H m (t) = 2 √ πΓ( s+1 2 )t 1/2 Γ( s 2 ) J s− 1 2 (2π|m|t) when m = 0, while when m = 0 we have by [4, (5.12.2)] that H 0 (t) = 2 √ πt s Γ( s+1 2 ) Γ( s 2 ) . The result follows after using Lemma 8 of [6]. We would like to apply Proposition 3.1 to the integrals appearing in Proposition 3.2, but the integrals in (3.3) do not converge for Re(s) = 1 2 . However, the integrals C Q (F m,Q (z, s) − c(s)F 0,Q (z, s))y −1 |dz| and C Q ∂ z (F m,Q (z, s) − c(s)F 0,Q (z, s)) dz, where c(s) = 2|m| 1/2−s σ 2s−1 (|m|) (2s − 1)Λ(2s − 1) , do converge for Re(s) > 0, as long as s is not one of the poles of the integrands. This is because the coefficient of y 1−s equals zero in the Fourier expansion of F m,Q (z, s) − c(s)F 0,Q (z, s) at the cusps corresponding to the endpoints of C Q . Note that by Proposition 3.1 we have Res s= 1 2 +ir (2s − 1)(F m,Q (z, s) − c(s)F 0,Q (z, s)) = Res s= 1 2 +ir (2s − 1)F m (z, s) = ϕ∈Br ϕ, ϕ −1 2a ϕ (m)ϕ(z) because c(s)F 0,Q (z, s) is analytic at s = 1 2 + ir, r = 0, and f m (γ j z, s) is analytic for s ∈ C. So we get the next result by following the proof of Proposition 6 of [6] with only minor changes. Proposition 3.3. For any even Hecke-Maass cusp form ϕ ∈ U r there is a unique Hecke-Maass cusp form ψ ∈ V r such that ϕ is the Shimura lift of ψ and such that for any fundamental discriminant d we have 12π 1/2 |d| 3 2 |b ψ (d)| 2 = 1 ϕ, ϕ Q∈Γ\Q d 2 χ d (Q)          C Q ϕ(z)y −1 |dz| if d > 0, C Q i∂ z ϕ(z) dz if d < 0. for some function g(n, y) which satisfies g(−n, y) = g(n, y) and g(n, y) ≪ |n| 1/2 e −2π|n|y as |n|y → ∞ and g(n, y) ≪ n y −1/2 as y → 0. So if Re(s) > 1 we have Q∈Γ\Q D χ d (Q) C Q i∂ z ϕ(z)y s dz = −2πi n =0 na ϕ (n)G(n, d) ∞ 0 y s+ 1 2 K ir (2π|n|y) dy because a ϕ (−n)G(−n, d)g(−n, y) = −a ϕ (n)G(n, d)g(n, y). Again using (3.4) we find that Q∈Γ\Q D χ d (Q) C Q i∂ z ϕ(z)y s dz = π −s− 1 2 |d| Γ( s 2 + ir 2 + 3 4 )Γ( s 2 − ir 2 + 3 4 )L(s + 1 2 , ϕ × χ d ). The result follows as in the previous case. Theorem 1 . 1 . 11Let ϕ be an even Hecke-Maass cusp form of weight 0 for SL 2 (Z) with Fourier expansion ϕ(z) = 2 √ y n =0 a(n)K ir (2π|n|y)e(nx).Then there exists a unique Maass cusp form ψ of weight 1/2 for Γ 0 (4) (a, b, c, d) = 1 and Q represents n, 0 if gcd(a, b, c, d) > 1. Lemma 2 . 1 . 21If D = d 2 then the sets {Q = [c, |d|, 0] : 0 ≤ c < |d|} and {Q = [0, |d|, c] : 0 ≤ c < |d|} are both complete sets of representatives for Γ\Q D . In both cases we have χ d (Q) = d c . F m (z, s) = γ∈Γ∞\Γ f m (γz, s), where f 0 (z, s) = y s and for m = 0 f m (z, s) 1) where σ a (n) is the sum of the a-th powers of the divisors of n, and Λ(s) = π −s/2 Γ(s/2)ζ(s). The modified Eisenstein series Λ(2s)F 0 (z, s) is analytic in C \ {0, 1} and is invariant unders → 1 − s. For m = 0 the Fourier expansion of F m (z, s) is given in Theorem 3−1 (4π |mn| c −1 ) if mn < 0, J 2s−1 (4π |mn| c −1 ) if mn > 0and K(m, n, c) is the ordinary (weight 0) Kloosterman sum. The following result is Proposition 3 of[6].Proposition 3.1. For any m = 0, the function F m (z, s) has a meromorphic continuation to Re(s) > 0 with Res s= 1 2 +ir (2s − 1)F m (z, s) = ϕ∈Br ϕ, ϕ −1 2a ϕ (m)ϕ(z). F m,Q (z, s) = γ∈Γ∞\Γ γ =γ 1 ,γ 2 f m (γz, s). Proposition 3. 2 . 2Let m ≥ 0 and Re(s) > 1. Suppose that d is a fundamental discriminant. Then gcd(c, d) > 1 we can restrict the sum to those c which are coprime to d. For Q = [c, |d|, 0] with gcd(c, d) = 1 we have a 1 = (0 : 1), γ 1 = ( 0 −1 1 0 ), a 2 = (d : c), γ 2 = a b c |d| , for some a, b ∈ Z. Thus T m (d) = 1 2 c mod |d| gcd(c,d)=1 Q=[c,|d|,0] d c γ∈Γ∞\Γ γ =γ 1 ,γ 2 C γQ e(mx)φ 2,m (y, s) dz. to one of the quantities depending on whether d is negative or positive, respectively. Here Q d is a set of integral binary quadratic forms of discriminant d, Γ = PSL 2 (Z), z Q is the root of Q(z, 1) in the complex upper half-plane H, and C Q is a hyperbolic geodesic, finite if d > 1 and infinite if d = 1 (see Section 2 for details). In the case d = 1 there is one term in the sum, namely Q = [0, 1, 0], and their formula readsQ∈Γ\Q d ϕ(z Q ) or Q∈Γ\Q d C Q ϕ(z)y −1 |dz|, Having U r and V r correspond to spectral parameters r and r/2, respectively, follows[6] and is convenient when working with the Shimura lift, which sends an element of V r into U r . Note that the papers[5,1] give S Q and C Q the opposite orientation. Brigham Young University, Provo, UT 84602 Proof of Theorem 1.1. Suppose that d is a fundamental discriminant. Then by Lemma 2.1 the quadratic forms [0, |d|, c] with 0 ≤ c < |d| form a complete set of representatives for Γ\Q d 2 andSuppose first that d > 0. If Re(s) > 1 thenThus, using that a ϕ (n) = a ϕ (−n) we find that Periods of the j-function along infinite geodesics and mock modular forms. Nickolas Andersen, Bull. Lond. Math. Soc. 473Nickolas Andersen. Periods of the j-function along infinite geodesics and mock modular forms. Bull. Lond. Math. Soc., 47(3):407-417, 2015. A generalized Kohnen-Zagier formula for Maass forms. Ehud Moshe Baruch, Zhengyu Mao, J. Lond. Math. Soc. 822Ehud Moshe Baruch and Zhengyu Mao. A generalized Kohnen-Zagier formula for Maass forms. J. Lond. Math. Soc. (2), 82(1):1-16, 2010. Cycle integrals of Maass forms of weight 0 and Fourier coefficients of Maass forms of weight 1/2. A Biró, Acta Arith. 942A. Biró. Cycle integrals of Maass forms of weight 0 and Fourier coefficients of Maass forms of weight 1/2. Acta Arith., 94(2):103-152, 2000. . F W J Olver, A B Daalhuis, D W Lozier, B I Schneider, R F Boisvert, C W Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClainF. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. Cycle integrals of the j-function and mock modular forms. W Duke, Ö Imamoḡlu, Tóth, Ann. of Math. 1732W. Duke,Ö. Imamoḡlu, andÁ. Tóth. Cycle integrals of the j-function and mock modular forms. Ann. of Math. (2), 173(2):947-981, 2011. Geometric invariants for real quadratic fields. W Duke, Ö Imamoḡlu, Tóth, Ann. of Math. 1842W. Duke,Ö. Imamoḡlu, andÁ. Tóth. Geometric invariants for real quadratic fields. Ann. of Math. (2), 184(3):949-990, 2016. Fourier coefficients of the resolvent for a Fuchsian group. John D Fay, J. Reine Angew. Math. 293John D. Fay. Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math., 293(294):143-203, 1977. Heegner points and derivatives of L-series. B Gross, W Kohnen, D Zagier, II. Math. Ann. 2781-4B. Gross, W. Kohnen, and D. Zagier. Heegner points and derivatives of L-series. II. Math. Ann., 278(1- 4):497-562, 1987. The Selberg trace formula for PSL(2, R). Dennis A Hejhal, Lecture Notes in Mathematics. 2Springer-VerlagDennis A. Hejhal. The Selberg trace formula for PSL(2, R). Vol. 2, volume 1001 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1983. Analytic number theory. Henryk Iwaniec, Emmanuel Kowalski, American Mathematical Society53Providence, RIHenryk Iwaniec and Emmanuel Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004. Heegner points, cycles and Maass forms. Svetlana Katok, Peter Sarnak, Israel J. Math. 841-2Svetlana Katok and Peter Sarnak. Heegner points, cycles and Maass forms. Israel J. Math., 84(1-2):193- 227, 1993. Values of L-series of modular forms at the center of the critical strip. W Kohnen, D Zagier, Invent. Math. 642W. Kohnen and D. Zagier. Values of L-series of modular forms at the center of the critical strip. Invent. Math., 64(2):175-198, 1981. Modular forms of half integral weight and the integral of certain theta-functions. Shinji Niwa, Nagoya Math. J. 56Shinji Niwa. Modular forms of half integral weight and the integral of certain theta-functions. Nagoya Math. J., 56:147-161, 1975. Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II. Walter Roelcke, Math. Ann. 167ibid.Walter Roelcke. Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II. Math. Ann. 167 (1966), 292-337; ibid., 168:261-324, 1966. On construction of holomorphic cusp forms of half integral weight. Takuro Shintani, Nagoya Math. J. 58Takuro Shintani. On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J., 58:83-126, 1975. Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J.-L Waldspurger, J. Math. Pures Appl. 609Email address: [email protected]. Waldspurger. Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. (9), 60(4):375-484, 1981. Email address: [email protected]
[]
[ "A Thermodynamic Classification of Real Numbers", "A Thermodynamic Classification of Real Numbers" ]
[ "Thomas Garrity [email protected] \nDepartment of Mathematics and Statistics\nWilliams College Williamstown\n01267MA\n" ]
[ "Department of Mathematics and Statistics\nWilliams College Williamstown\n01267MA" ]
[]
A new classification scheme for real numbers is given, motivated by ideas from statistical mechanics in general and work of Knauf [17] and Fiala and Kleban [9] in particular. Critical for this classification of a real number will be the Diophantine properties of its continued fraction expansion.
10.1016/j.jnt.2010.01.016
[ "https://arxiv.org/pdf/0811.1369v3.pdf" ]
16,738,807
0811.1369
05558f2687ac40cf8973e7a5eaf08daea800d1b6
A Thermodynamic Classification of Real Numbers 15 Mar 2009 Thomas Garrity [email protected] Department of Mathematics and Statistics Williams College Williamstown 01267MA A Thermodynamic Classification of Real Numbers 15 Mar 2009 A new classification scheme for real numbers is given, motivated by ideas from statistical mechanics in general and work of Knauf [17] and Fiala and Kleban [9] in particular. Critical for this classification of a real number will be the Diophantine properties of its continued fraction expansion. Introduction Though this paper is about number theory in general and about a classification scheme for real numbers in particular, it has its roots in Thermodynamic Formalism, which was developed in the 1960s by Ruelle [32] [33] , Sinai [35] and others in an attempt to put statistical mechanics on a firm mathematical foundation. Once done, the underlying mathematical scheme can then, in principle, be applied to non-physical situations, using the original real-world interpretations to guide and influence what questions are to be asked and what structure is to be discovered. This process has been begun in number theory. In [17], Knauf developed a one-dimensional thermodynamic system based on the Farey fractions that ex-hibited phase transition. In [9], Fiala and Kleban generalized Knauf's work and showed that their generalization has the same free energy as Knauf's. We will put these earlier works into a common linear algebra framework, allowing us to make a seemingly minor, but actually significant, change in the original partition function. We will produce, for each positive real number, a thermodynamic system. Different real numbers will exhibit different free energies, giving us a new classification scheme for positive real numbers. (This classification scheme can easily be extended to also include negative reals.) In Section 2, we give a brief overview of the parts of the statistical mechanics formalism that we will be using. In particular, we will see the key importance of the partition function. In Section 3, we tie this formalism to number theory, in particular to the Farey matrices. In Section 3.3, we put Knauf's work into this language and do the same thing in Section 3.4 for Fiala and Kleban's work. In Section 3.5, we show how to alter the earlier partition functions that will put us into the world of Diophantine analysis. This is the section in which we are not just changing the notation from earlier work. In Section 4.1, we use our Diophantine partition function to give a new classification scheme for real numbers. In particular we develop the idea of a real number having a 1-free energy limit. In Section 4.2, we show how this is naturally linked to continued fractions. The rest of Section 4 deals with proving that there are real numbers with 1-free energy limits, that there are reals without a 1-free energy limit, that all algebraic numbers have a k-free energy limits with k > 1 and that all quadratic irrationals have 1-free energy limits. We also show that e has a √ N log N -free energy limit. We will conclude with open questions in Section 5. There has been a lot of other work linking statistical mechanics to number theory. There is other work of Knauf [19] [20] [21], of Guerra and Knauf [12], of Contucci and Knauf [6], of Fiala, Kleban andÖzlük [10], of Kleban andÖzlük [16], of Prellberg, Fiala and Kleban [28], of Feigenbaum, Procaccia and Tel [8] and others. There is also the transfer operator method, applied primarily to the Gauss map, which allows, in a natural way, tools from functional analysis to be used. We believe this was pioneered by Mayer (see his [25] for a survey) , and nontrivially extended by Prellberg [27], by Prellberg and Slawny [29], by Isola [14] and recently by Esposti, Isola and Knauf [7]. An introduction to this work is in chapter nine of Hensley [13]. We will not be following this approach here. The Partition Function and the Free Energy This is a rapid fire overview of basic terms in statistical mechanics. For each N ∈ N, we have a finite set S N , called the state space. Let E : S N → R + be a function that we call energy. The partition function is defined to be Z N (β) = σ∈SN e −βE(σ) . If we were modeling a physical system, the elements in the state space correspond to what can happen. The variable β corresponds to the inverse of the temperature. The underlying physical assumption is that the probability that a system is in a state σ ∈ S N will be Probability in state σ = e −βE(σ) Z N (β) . While far from a proof, this interpretation makes sense, in that at high temperatures (meaning for β close to zero), all states become increasingly likely, while at low temperatures the most likely state increasingly becomes the state with the lowest energy. There is a free energy if the following limit exists: f (β) = lim N →∞ log(Z N (β)) N , with the function f (β) being called, naturally enough, the free energy. It is believed that phase transitions occur at values of β for which f (β) fails to be analytic. For almost all of this paper, our state space will be S N = {σ = (σ 1 , . . . , σ N ) : σ i = 0 or 1}. Thus each of our S N will have order 2 N . We can think of our state space as having N site points, each having value 0 or 1. S N = {σ = (σ 1 , . . . , σ N ) : σ i = ±1}. The energy function for the Ising model is E(σ) = N i=1 σ i σ i+1 . Ising, in his 1925 thesis, showed that for this model there is no phase transition, meaning he showed that the free energy is an analytic function. For the twodimensional analog, it is one of the great discoveries (originally by Onsager in 1944) that phase transition does occur. Most texts on statistical mechanics, such as [37], describe the Ising model in detail. Note that in the Ising model, a site will only interact with those other sites that are immediately adjacent to it. This is an example of finite range interaction. Since there is no phase transition for the one-dimensional Ising model, it was long believed that there would be no phase transition for any onedimensional system. But in the 1960s, it was discovered that phase transition can occur if the interactions are not of finite range but over possibly arbitrarily long distances. A good introduction to this work is in Mayer's The Ruelle-Araki transfer operator in classical statistical mechanics [24]. Such interactions are called long range interactions. In the following number theoretic models, it is key that the interactions are long range. For matrices A = (a ij ) and B = (b ij ), we denote the Hilbert-Schmidt product (which is also called the Hadamard product) as A * B = T r(AB T ) = 1≤i,j≤n a ij b ij . For example, thinking of a 2 × 2 matrix as an element of R 4 , then A * B = T r(AB T ) is simply the dot product of the two vectors. Let M n denote the space of n × n matrices. For a function f : M n → R and for two n × n matrices M and A, define f (M )(β)|A = 1 |M * A| β f (M A T ), following notation as in [9]. For k n × n matrices A 1 , A 1 , . . . , A k , define f (M )(β)|(A 1 , . . . , A k ) = k i=1 f (M )(β)|A i . Consider the map Z : N × M n × R × M k n × Γ(M n , R) → R * , where N is the natural numbers, M n is the space of n × n matrices, Γ(M n , R) is the space of functions from n × n matrices to the real numbers and R * is the extended real numbers, defined by setting Z(N, M, β, (A 0 , . . . , A k−1 ), f ) = f (M )(β)|(A 0 , . . . , A k−1 ) N . Here the notation (A 0 , . . . , A k−1 ) N is referring to the above newly defined product of tuples of matrices and hence can be viewed as short-hand for all products A i1 · · · A iN , with 0 ≤ i l ≤ k − 1. We want to link this with partition functions. Fix an n × n matrix M and also k n × n matrices A 0 , A 1 , . . A σ1 · · · A σn . In order to get the above partition function, we set the energy of a state to be: E(σ) = log |M * (A σ1 · · · A σn )|. In our applications, it is more natural not to emphasize the energy function. Using this language, we have natural recursion relations linking the partition function Z n+1 with partition functions for various Z n , with different choices for the matrix M . More precisely Lemma 3.1. Z N +1 (M )(β) = Z N (M A T 1 )(β) + . . . + Z N (M A T k )(β). This is a simple calculation. It offers a more general and natural form for the key recursion relation (2) in [9]. Farey Matrices This section continues the building of needed machinery. (See, also, section 4.5 in [11]. Another source would be [23]) We develop the Farey partitioning of the extended real numbers. Start with the set F 0 = 1 0 , 0 1 and define the Farey sum of two fractions, in lowest terms, to be p q ⊕ r s = p + r q + s . We now extend a given nth Farey set F n to the (n + 1)st Farey set by adding to F n all of the terms obtained by applying the Farey sum. Thus we have, F 0 = 1 0 , 0 1 F 1 = 1 0 , 1 1 , 0 1 F 2 = 1 0 , 2 1 , 1 1 , 1 2 , 0 1 F 3 = 1 0 , 3 1 , 2 1 , 3 2 , 1 1 , 2 3 , 1 2 , 1 3 , 0 1 . By reversing the order of the terms in F n , we get a partitioning of R + ∪ ∞. Here we are thinking of 1 0 as the point at infinity, which is why we are working with the extended real numbers R * . We now describe this partitioning in terms of iterations of matrix multiplication. Let A 0 = 1 0 1 1 and A 1 = 1 1 0 1 . These two matrices are key to understanding the Farey decomposition of the unit interval and continued fraction expansions. Further, these two matrices will be key to the partition function of Knauf [17], of Fiala and Kleban [9], and to our eventual use of partition functions for Diophantine approximations. Note that p r q s A 0 = p + r r q + s s . and p r q s A 1 = p p + r q q + s . Then F 1 , save for the point at infinity 1 0 , can be obtained in the natural way by examining the right columns of first A 1 , then A 0 . Likewise, F 2 , save for the point at infinity 1 0 , can be obtained in a similar natural way by examining the the right columns of first A 2 1 , then A 1 A 0 , then A 0 A 1 and finally A 2 0 . In general the elements of F n , save of course for the point 1 0 , are given by the right columns of the 2 n products of matrices of A 1 and A 0 . These allow us to recover the continued fraction of a positive real number α. We know that any real number α can be written α = a 0 + 1 a 1 + 1 a2+ 1 a 3 +··· , which is usually denoted by α = [a 0 ; a 1 , a 2 , a 3 , . . .], where a 0 is an integer and the remaining a i are positive integers. The number α is rational if and only if its continued fraction expansion terminates. We say that the rational pm qm is the mth partial fraction for the number α if p m q m = [a 0 ; a 1 , . . . , a m ]. We want to use our Farey matrices to find the nth partial sum pn qn for a given number α. We return to our Farey numbers, but reverse the orders of the numbers: F 0 = 0 1 , 1 0 F 1 = 0 1 , 1 1 , 1 0 F 2 = 0 1 , 1 2 , 1 1 , 2 1 , 1 0 F 3 = 0 1 , 1 3 , 1 2 , 2 3 , 1 1 , 3 2 , 2 1 , 3 1 , 1 0 . We can thus view This pattern continues. Consider the matrix A a0 1 A a1 0 · · · A aN−1 1 A aN 0 with a 0 ≥ 0 and a i > 0. Then the left column of the matrix will correspond to [a 0 ; a 1 , . . . , a N ] while the right column will correspond to [a 0 ; a 1 , . . . , a N −1 ]. For matrices A a0 1 A a1 0 · · · A aN−1 0 A aN 1 with a 0 ≥ 0 and a i > 0, then the left column will correspond to [a 0 ; a 1 , . . . , a N −1 ] while now the right column will [a 0 ; a 1 , . . . , a N ]. Thus to determine the continued fraction expansion for a given positive real number α, we just have to keep track of which interval α is in for a given F N . Knauf's work This section will show how Knauf's number theoretic partition function [17] can easily be put into the language of this paper. Consider the sets F N ∩ [0, 1] F 1 ∩ [0, 1] = 0 1 , 1 1 F 2 ∩ [0, 1] = 0 1 , 1 2 , 1 1 F 3 ∩ [0, 1] = 0 1 , 1 3 , 1 2 , 2 3 , 1 1 F 4 ∩ [0, 1] = 0 1 , 1 4 , 1 3 , 2 5 , 1 2 , 3 5 , 2 3 , 3 4 , 1 1 . As is well known, as N → ∞, these sets will eventually contain all rational numbers in the unit interval [0, 1]. Then Knauf defines his partition function as Z K N (β) = p q ∈FN ∩(0,1) 1 q β . (Note that the K is not being used as an index but stands for 'Knauf'; the subscript N is an index.) In the limit we get Z K (β) = lim N →∞ Z K N (β) = ∞ n=1 φ(n) n β , where φ(n) is the Euler totient function. In turn, ∞ n=1 φ(n)/n β is well-known to equal to ζ(β − 1) ζ(β) , showing that there is critical point phenomena for this one-dimensional system. The ζ(β − 1)/ζ(β) will show up throughout this paper and is why many of the later theorems are only true for β > 2. We now show how this can be put into the language of Section 4.1. Let M K = 0 0 0 1 , and let A 0 and A 1 be the above Farey matrices. Then the Knauf partition function is Z K N (β) = 1(M K )|A 0 (A 0 , A 1 ) N . The initial A 0 is just to insure that we are in the unit interval. Also, for A 0 (A 0 , A 1 ) N , we are using the new product for matrices defined in section 3.1 and not traditional matrix multiplication. Fiala-Kleban Work In [9], Fiala and Kleban considered a different number theoretic partition function. In the language of this paper, let M F = 0 0 x 1 . (Again, the F is not an index but stands for "Fiala-Kleban".) Then the new partition function will be Z F N (x, β) = 1(M F )|A 0 (A 0 , A 1 ) N . As mentioned earlier, the recursion relation (2) in Section 2 of [9] is simply a special case of Lemma 3.1. Two partition functions are said to have the same thermodynamics if their free energies are equal. Thus, the Knauf partition function Z K N (β) and the Fiala-Kleban partition function Z F N (β) have the same thermodynamics if lim N →∞ log(Z K N (β)) N = lim N →∞ log(Z F N (β)) N . This equality is shown in Section 4 of [9], using as an intermediary tool a certain transfer operator and depending on the earlier work of Knauf [17]. In [10], Fiala, Kleban andÖzlük showed the thermodynamics of the Knauf partition function is thermodynamically equivalent to a number of other number theoretic partition functions. It is certainly the case, though, that thinking of the various matrices as vectors in R 4 will yield more straightforward proofs of these equivalences. A Diophantine approach We now make a seemingly minor change in our choice for the matrix M that will create a quite different thermodynamics, leading in the next section to a new classification of the real numbers. Set M = 0 −1 0 α for some real number α. Define the Diophantine partition function to be Z N (α; β) = 1(M )|(A 0 , A 1 ) N . Then we have Z N (α; β) = p q ∈FN 1 |p − αq| β . Note that the terms that dominate the above sum occur when |p − αq| is small. This places us firmly in the realm of Diophantine analysis. We will see that we can classify real numbers α by understanding the existence of free energy limits for the statistical systems associated to the partition function Z N (α; β). Again, while this partition function is cast in the same overall language as Knauf and Fiala and Kleban, its thermodynamic properties will be quite different. This is what separates the present work from [17] and [9]. For k = 1, this is a number-theoretic version of the free energy of the system. By an abuse of notation we will also say that α has a f (N )-free energy limit, for an increasing function f (N ) if lim N →∞ log(Z N (α; β)) f (N ) exists for all β > β c . To see that this is a meaningful classification scheme for real numbers, we will establish the following three theorems: Theorem 4.2. There exists a number that has a 1-free energy limit for β > 2. for all relatively prime integers p and q. Then α has a k-free energy limit for β > 2, for any k > 1, and in fact, the k-free energy limit is zero. An easy consequence of the above is that all algebraic numbers have k-free energy limits equal to zero, for any k > 2. While we do not know if algebraic numbers have 1-free energy limits, we will show Theorem 4.5. All quadratic irrationals have 1-free energy limits for β > 2. We will also show, in Section 4.7 that the number e has √ N log(N )-free energy limit. Links to continued fractions The goal of this paper is not just to give a new way to classify real numbers but also to show how such a classification scheme follows from the thermodynamical formalism, fitting into a more general framework. But if all we wanted was the classification scheme, then it is possible to reframe our definitions so that there is no need for the language of statistical mechanics. The goal of this section is to state the theorems that would allow us to avoid thermodynamics. They will also be key to proving the five theorems of the previous section. We now set some notation for the rest of the paper. Given the positive real number α, for each positive integer N there is associated a positive integer m and nonnegative integer k such that N = a 0 + a 1 + · · · + a m + k, with 0 ≤ k < a m+1 . We create a subsequence of N, denoted by N 0 , N 1 , N 2 , . . . by setting N m = a 0 + a 1 + · · · + a m . In this notation, given any N ∈ N, we have N m ≤ N < N m+1 , or, in other words, a 0 + a 1 + · · · + a m ≤ N = a 0 + a 1 + · · · + a m + k < a 0 + a 1 + · · · + a m + a m+1 . d N = 1 |p N − αq N | . We have the following chain of inequalities that will be key: d Nm−1 < d Nm+1 < d Nm+2 < · · · < d Nm+am+1−1 < d Nm , which are well-known (for motiviation, see the chapter on continued fractions in [36]). We will show Theorem 4.6. For any β > 2 and for any positive real number α, we have for all positive integers N that log(d β N ) N ≤ log(Z N (α; β)) N ≤ log( ζ(β−1) ζ(β) N d β Nm ) N . To show that there are numbers that do not have 1-free energy limits, we will construct an α so that a subsequence of log(d β N ) N approaches infinity. In turn, to show that that there are numbers that have 1-free energy limits, we will construct an α so that log( ζ(β−1) ζ(β) N d β Nm ) N approaches zero, forcing log(ZN (α;β)) N to approach zero (and thus guaranteeing that the limit exists). Preliminaries The reason why we look at the products of the matrices A 0 and A 1 is that the right columns will correspond to all of the integer lattice points p q with p and q relatively prime in the the first quadrant of the plane. If A i1 · · · A in has right column p q , then Let C N (v 1 , v 2 ) be the subset of C(v 1 , v 2 ) consisting of vectors that are the right columns of all possible A i1 · · · A iN . Then set M * (A i1 · · · A in ) = (−1, α) · p q = αq − p Let v 1 = p 1 q 1 , v 2 = p 2 q 2 be vectors that satisfy det(v 1 , v 2 ) = ±1. Let C(v 1 , v 2 ) denoteZ N (α; β, v 1 , v 2 ) = p q ∈CN (v1,v2) 1 |αq − p| β . In the same way, we set Z(α; β, v 1 , v 2 ) = p q ∈C(v1,v2) 1 |αq − p| β .Z N (α; β) ≤ m−1 i=1 Z N (α; β, v i , v i+1 ) + Z N (α; β, v m , w n ) + n−1 i=1 Z N (α; β, w i , w i+1 ). We need the above to be an inequality since there is "overcounting" on the right hand side, since, for example, the part of Z N (α; β, v i−1 , v i ) coming from the vector v i also appears as a term in Z N (α; β, v i , v i+1 ). The key, as we will see, is that the Z N (α; β, v m , w n ) term contributes the most to the partition function Z N (α; β). For the rest of this section, we want to control the sizes of the various Z N (α; β, v i , v i+1 ) and Z N (α; β, w i , w i+1 ). We first return to the more general case of two vectors For v 1 = p1 q1 , v 2 = p2 q2 with det(v 1 , v 2 ) = ±1,d 1 = 1 |p 1 − αq 1 | , d 2 = 1 |p 2 − αq 2 | , suppose that d 1 < d 2 , which means that the line through the origin and the point (p 2 , q 2 ) is closer to the line x = αy than the line through the origin and (p 1 , q 1 ). We want to show that Z(α; β, v 1 , v 2 ) < |d 2 | β ζ(β − 1) ζ(β) . We know that 1 d 1 = (−1, α) · p 1 q 1 , 1 d 2 = (−1, α) · p 2 q 2 . Let v be some integer lattice point in the cone C(v 1 , v 2 ). Then there are relatively prime positive integers a and b with v = av 1 + bv 2 . We have (a + b)|(−1, α) · v 2 | < |(−1, α) · (av 1 + bv 2 )| = |(−1, α) · v| < (a + b)|(−1, α) · v 1 |. Inverting and raising everything to the power of β, we have |d 1 | β (a + b)| β < 1 |(−1, α) · v| β < |d 2 | β (a + b)| β . Summing over every vector in C(v 1 , v 2 ), we get |d 1 | β 1 |a + b| β < Z(α; β, v 1 , v 2 ) < |d 2 | β 1 |a + b| β , where in the first and third summation we are summing over all relatively prime positive integers a and b. It is well-known, as mentioned in Section 4.3 of Knauf [17], that, 1 |a + b| β = ζ(β − 1) ζ(β) . It is here that the ζ(β−1) ζ(β) makes its critical appearance. We have our desired inequality. Proof of Theorem 4.6 First, the partition function Z N (α; β) is the sum of many positive terms, including d β N . Thus we have the lower bound log(d β N ) N ≤ log(Z N (α; β)) N . Now for the upper bound. We are at the Nth stage. Letting t ≤ N , expressing N as N = a 0 + · · · + a m + k, with 0 ≤ k < a m+1 and using the notation from section 4.2, we assume that the vectors pt−1 qt−1 and pt qt lie on the same side of the line (x = αy). This is equivalent to there being some s ≤ m with a 0 + · · · + a s ≤ t − 1 < t < a 0 + · · · + a s+1 . We know that det p t−1 p t q t−1 q t = ±1. We have Z N α; β; p t−1 q t−1 , p t q t < Z α; β; p t−1 q t−1 , p t q t < ζ(β − 1) ζ(β) d β t < ζ(β − 1) ζ(β) d β Nm , using that d t ≤ d Nm . Since there will be N such cones, we have log(Z N (α; β)) N ≤ log N ζ(β−1) ζ(β) d β Nm N , finishing the proof. Proof of Theorem 4.2 We know that the best rational approximations to a real number α are the p Nm q Nm = [a 0 ; a 1 , . . . , a m ]. It is well known that q Nm+1 = a m+1 q Nm + q Nm−1 . Further (as in Lemma 7.2 of [4]) a m+1 q Nm ≤ 1 |p Nm − αq Nm | = d Nm ≤ (a m+1 + 2)q Nm . Our goal in this section is to construct a real number α for which lim N →∞ log N ζ(β−1) ζ(β) (d Nm ) β N = 0, for β > 2, which by Theorem 4.5 will force α to have a 1-free energy limit. If lim N →∞ log N ζ(β−1) ζ(β) (d Nm ) β N exists, then it equals lim N →∞ log N N + lim N →∞ log ζ(β−1) ζ(β) N + lim N →∞ β log d Nm N . The first two terms in the above certainly go to zero. Thus we must construct a real α so that lim N →∞ β log dN m N = 0. Recalling our notation that N m = a 0 + · · · + a m ≤ N = N m + k, with 0 ≤ k < a m+1 , we have for each N log d Nm N ≤ log d Nm N m . Thus all we have to do is construct an α so that lim N →∞ β log dN m Nm = 0. For α = [a 0 ; a 1 , . . .], define the function f (m) by setting a m+1 = q f (m) Nm . We have β log(d Nm ) N ≤ β log(d Nm ) N m ≤ β log[(a m+1 + 2)q Nm ] N m ≤ β log(2a m+1 q Nm ) N m = β log(2) N m + β log q f (m) Nm q Nm N m = β log(2) N m + β[f (m) + 1] log(q Nm ) N m Since the first term in the last equation goes to zero as N → ∞, we have, if the limits exist, that lim N →∞ β log(d Nm ) N m ≤ lim N →∞ β[f (m) + 1] log(q Nm )] N m We now start with a q 0 and a q 1 and a function f (m) and use these to create our number α, Let q 0 = 1, q 1 = 2, and f (m) = m for m ≥ 1. Then define for m ≥ 2, a m+1 = q f (m) Nm . Now log(q Nm ) = log(a m q Nm−1 + q Nm−2 ) ≤ log(2a m q Nm−1 ) = log(2q f (m−1)+1 Nm−1 ) = log(2) + (f (m − 1) + 1) log(q Nm−1 ) Then we have β[f (m) + 1] log(q Nm ) N m ≤ β[f (m) + 1] log(2) N m + β[f (m) + 1](f (m − 1) + 1) log(q Nm−1 ) N m We will now use that N m = a 0 + . . . + a m > a m = q f (m−1) Nm−1 Then β[f (m) + 1] log(q Nm ) N m < β(m + 1) log(2) q m−1 Nm−1 + β(m + 1)m log(q Nm−1 ) q m Proof of Theorem 4.3 We use from Theorem 4.5 that log(d β N ) N ≤ log(ZN (α;β)) N . We will construct a real number α so that lim Nm→∞ log(d β Nm ) N m = ∞. This will mean that lim N →∞ log(ZN (β)) N will not exist, which is the goal of Theorem 4.3. Proceeding as in the proof of Theorem 4.2, we define α = [a 0 ; a 1 , a 2 , . . .] inductively on the a ′ m s by setting, as before, a 0 = 1 and a m+1 = q f (m) Nm but now defining f (m) as f (m) = a 0 + · · · + a m . We use that a m+1 q Nm ≤ d Nm and N m = a 0 + · · · + a m , we have log d β Nm N m ≥ log (a m q Nm ) β N m = β log q f (m) Nm q Nm N m = β(f (m) + 1) log q Nm N m = β(a 0 + · · · a m + 1) log q Nm a 0 + · · · + a m . Since the last term goes to infinity as N m → ∞, we are done. Proof of Theorem 4.4 We assume that α is a positive real number such that there is a positive constant C and a constant d ≥ 2 with C q d ≤ |p − αq| for all relatively prime integers p and q. In particular, we have C q d Nm ≤ |α − p Nm q Nm |. In our notation, this is d Nm ≤ cq d−1 Nm , where c is a constant depending on α but not on N m . In a similar argument as in Theorem 4.2, the number α will have a k-free energy limit if we can show We know that q Nm = a m q Nm−1 + q Nm−2 . Then we get Since it is always the case that a 0 + . . . + a m ≥ m + 1, the above limit must always be zero for any k > 1. q Nm = a m q Nm−1 + q Nm−2 ≤ 2a m q Nm−1 = 2a m (a m−1 q Nm−2 + q Nm−3 ) ≤ 2 2 a m a m−1 q Nm−2 . . . Quadratic Irrationals have 1-free energy limits The key will be that every quadratic irrational has an eventually periodic continued fraction expansion (see section 7.6 in [36]). We will show first, for a quadratic irrational, that lim m→∞ log q Nm /N m exists and then show that the existence of this limit is equivalent to the number having a 1-free energy limit. Let α be a quadratic irrational. We can write The period length is l. For notational convenience, set b = b 0 + · · · + b p c = c 1 + · · · + c l B = 0 1 1 b p · · · 0 1 1 b 0 C = 0 1 1 c l · · · 0 1 1 c 1 . such that m = p + kl + n. For this n, set d n = c 1 + · · · + c n D n = 0 1 1 c n · · · 0 1 1 c 1 . Note that there are only a finite number of possibilities for the d n and the D n . From section 7.6 of Stark, we have q Nm−1 p Nm−1 q Nm p Nm = D n C k B. Now N m = b + kc + d n grows linearly with respect to k. Since q Nm is the lower left term of the matrix D n A k B, we also have that log q Nm grows linearly with respect to k, with constant leading coefficient. Thus lim m→∞ log qN m Nm must exist. Now to show why this will imply that α has a 1-free energy limit. Since d Nm−1 ≤ d N ≤ d Nm and since N m ≤ N ≤ N M+1 , we have from Theorem 4.6 that log(d β Nm−1 ) N m+1 ≤ log(Z N (α; β)) N ≤ log( ζ(β−1) ζ(β) N d β Nm ) N m . From our earlier work, we know that α will have a For background on φ, see Chapter 1.7 in [30]. The key is that continued fraction expansion for φ is φ = [1 : 1, 1, 1, 1, . . .]. From the above theorem, we know that the 1-free energy limit, for β > 2 is lim m→∞ log q Nm N m . For φ, we know that N m = m + 1 and that q Nm = F m , where F m is the mth Fibonacci number. We know that F m = 1 √ 5   1 + √ 5 2 m − 1 − √ 5 2 k   . Then the 1-free energy limit will be lim m→∞ log q Nm N m = lim m→∞ log F m m + 1 = log φ, as desired. Of course, similar arguments can be used to find the 1-free energy limits for many quadratic irrationals. 4.9 The number e has √ N log(N)-free energy limit Thus for e − 1, we have a 3k = a 3k+1 = 1 and a 3k+2 = 2(k + 1). Let q Nm denote the denominators of the fraction [a 0 ; a 1 , . . . a m ] associated to e − 1, where N m = a 0 + · · · + a m . In an analogous fashion to the proof that quadratic irrationals have 1-free energy limits, we will first show for e − 1 that lim Nm→∞ log q Nm √ N m log(N m ) exists and then show that the existence of this limit will give us our result. We know that a m q Nm−1 < q Nm = a m q Nm−1 + q Nm−2 . Then we have a 0 a 1 · · · a m < q Nm ≤ 2 m+1 a m a m−1 . . . a 0 . Then, provided the limits exist, we have lim Nm→∞ log a 0 a 1 · · · a m √ N m log(N m ) ≤ lim Nm→∞ log q Nm √ N m log(N m ) ≤ lim Nm→∞ log 2 m+1 a m a m−1 . . . a 0 √ N m log(N m ) . We will show that lim Nm→∞ log a 0 a 1 · · · a m √ N m log(N m ) = lim Nm→∞ log 2 m+1 a m a m−1 . . . a 0 √ N m log(N m ) , which will force lim Nm→∞ log q Nm / √ N m log(N m ) to exist. We now have to look at the explicit values for the a m . There are three cases, depending on if m is 0, 1 or 2 mod 3. We will let m = 3k + 2 and show that the above limits exist and are equal. The other two cases are similiar. The above denominator is 1 + 1 + 2 + 1 + 1 + 4 + 1 + . . . + 2(k + 1) = 2(k + 1) + (k + 1)(k + 2), which has quadratic growth in k. The numerator on the left hand side of the above limit is log a 0 + · · · log a m = log 1 + log 1 + log 2 + log 1 + log 1 + log 4 + · · · log 2(k + 1) = k log 2 + log 1 + log 2 + · · · log(k + 1), which has k log k growth. Similarly for the right hand side of the above limit, we have log 2 m+1 a m a m−1 . . . a 0 = log(a m+1 +2)+(m+1) log(2)+log(a 0 )+. . .+log(a m ), which also has k log k growth, with the same leading coefficient. The above limits will exist. Now to show that lim N →∞ log(Z N (α; β)) √ N log N exists. In an argument similar to the proof of Theorem 4.6, we have log(d β Nm−1 ) N m+1 log N m+1 ≤ log(Z N (α; β)) √ N log N ≤ log( ζ(β−1) ζ(β) N d β Nm ) √ N m log N m . Following the proof in the last section, we see that Since a m+1 q Nm ≤ d Nm ≤ (a m+1 + 2)q Nm q Nm , if the limits exist, we have lim m→∞ log a m+1 q Nm √ N m log N m ≤ lim m→∞ log d Nm √ N m log N m ≤ lim m→∞ log(a m+1 + 2)q Nm √ N m log N m . Since the a m+1 grow at most linearly with respect to m and the N m grow quadratically, we have Conclusion There are a number of directions for future work. It would certainly be interesting to know if there are algebraic numbers besides the quadratics that have 1-free energy limits. Also, what type of free energies are there for various other real numbers. We have shown that some real numbers have 1-free energy limits while others do not. This too is just a beginning. What types of limits are possible? Are there numbers α for which the sequence log(ZN (α;β)) N has any possible limit behavior? For example, certainly we should be able to rig α so that any number can be the limit of the sequence. In fact, we should be able to find such sequences with two accumulation points, three accumulation points, etc. All of these should provide information about the initial real number α. Once we know that there is a free energy limit, then the most pressing question is to find for which values of β is the free-energy non-analytic. These points will be the analogs to critical point phenomena in physics. For the Knauf But certainly other M can be chosen and studied. Also, Z N (β) depends on the choice of the matrices A 0 and A 1 . Other choices for these matrices will lead to different thermodynamical systems, each with its own number-theoretic implications. Of course, why stick to two-by-two matrices. This leads to multi-dimensional continued fractions. There are many different multi-dimensional continued fraction algorithms. See Schweiger [34] for a sampling of some. Also, Major [23] has some preliminary work on this. Each of these will give rise to a thermodynamical system, again with meaning in number theory. Finally, there is the question of putting these results into the language of transfer operators. (See [31], [24], [1] for general references.) positive integer k. For each positive integer N , our state space will be S N = {(σ 1 , . . . , σ N ) : σ i = 0, 1, . . . , k − 1}. Thus S N contains k N elements. We define a new type of product of an N -tuple of n × n matrices with an M -tuple of such matrices to be the M N -tuple: (A 1 , . . . , A N )(B 1 , . . . B M ) = (A 1 B 1 , A 1 B 2 , . . . , A N B M ). . , A k−1 . Let the function f be the constant function 1, or in words, let f (B) = 1(B) = 1 for all matrices B ∈ M n . Then define the partition function to beZ N (M )(β) = Z(N, M, β, (A 0 , . . . , A k−1 ), 1) = 1(M )(β)|(A 0 , . . . , A k−1 ) N .The "physical" intuition is as follows. Think of a one-dimensional lattice with n sites. At each site, there are k possible states, each of which can either be indexed by a number σ i between 0 and k − 1 or by a matrix A σi . Then the states can be viewed as either all possible σ = (σ 1 , . . . , σ n ) ∈ Z/kZ or all matrices of the form can be thought of as the flipping of the columns of A 1 . Likewise, F 2 will split the positive reals into four intervals: [ 0 1 , 1 2 ] (which can be thought of as the flipping of the columns of A 0 A 0 ), [ 1 2 , 1 1 ] (which can be thought of as the flipping of the columns of A 0 A 1 ), [ 1 1 , 2 1 ] (which can be thought of as the flipping of the columns of A 1 A 0 ) and [ 2 1 , 1 0 ] (which can be thought of as the flipping of the columns of A 1 A 1 ). the notation above, we have for each positive real number α and each positive integer N the partition function Z N (α; β). Definition 4.1. A real number α has a k-free energy limit if there is a number β c such that lim N →∞ log(Z N (α; β)) N k exists for all β > β c Theorem 4. 3 . 3There exists a number that does not have a 1-free energy limit, for any value of β. Theorem 4. 4 . 4Let α be a positive real number such that there is a positive constant C and constant d ≥ 2 with C q d ≤ |p − αq| Let our positive real number α have continued fraction expansion [a 0 ; a 1 , a 2 , . . .]. We know that the best rational approximations to α are given by the rational numbers [a 0 ; a 1 , a 2 , . . . , a m ]. The fractions [a 0 ; a 1 , a 2 , . . . , a m , k], with 0 < k < a m+1 , are called the secondary convergents to α. For a given m, we know that all the vectors corresponding to the [a 0 ; a 1 , a 2 , . . . , a m−1 , k], with 0 ≤ k < a m , are on the same side of the line x = αy, while [a 0 ; a 1 , a 2 , . . . , a m ] jumps to the other side of the line. 0 ; a 1 , . . . , a m , k], with 0 ≤ k < a m+1 , with p N and q N having no nontrivial common factors. We know that the fractions pN m qN m are the best rational approximations to the initial real α. Denote the cone of integer lattice points defined by: C(v 1 , v 2 ) = {av 1 + bv 2 : a, b relatively prime nonnegative integers}. Suppose we have integer lattice vectors v 1 = 0 1 , v 2 , . . . , v m on one side of the line x = αy and integer lattice vectors w 1 = 1 0 , w 2 , . . . , w n on the other side of the line x = αy such that det(v i , v i+1 ) = −1, det(w i , w i+1 ) = 1 and det(v m , w n ) = −1. Then we have under the additional assumption that v 1 and v 2 lie on the same side of the line x = αy. limit zero as N → ∞. Thus with the choice of the function f (m) = m we have constructed a real number that has 1-free energy limit, finishing the proof of Theorem 4.2. ≤ 2 m+1 a m a m−1 . . . a 0 . Thus we want to examine lim Nm→∞ log(2 m+1 a m a m−1 . . . a 0 ) (N m ) k or lim Nm→∞ (m + 1) log(2) + log(a 0 ) + . . . + log(a m ) (a 0 + . . . + a m ) k . Corollary 4. 7 . 7All positive algebraic numbers of degree greater than or equal to two have k-free energy limits equal to zero, for any k > 1.Liouville's Theorem (see Theorem 1.1 in[5]) states that all irrational algebraic numbers α have the property there is a constant d ≥ 2 with C q d ≤ |p − αq| for all relatively prime integers p and q, allowing us to immediately use the above theorem. α = [b 0 :, b 1 , . . . , b p , c 1 , c 2 , . . . , c l , c 1 , c 2 , . . . , c l , . . .]. will actually show this for e − 1. The continued fraction expansion for . . .]. since N m+1 − N m−1 grows at most linearly with respect to m and since each N m grows quadratically with respect to m. The various partition functions Z N (β) depend on the choice of the two-by-two matrix M . Different choices of M give rise to different thermodynamics. Knauf's M K and Fiala's and Kleban's M F are of the As of August 2008, the web site http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm offers many other attempts over the years to find links between statistical mechanics and number theory.Finally, I would like to thank Edward Burger for many interesting conversations about this work and Steven Miller and L. Pedersen for comments on an earlier draft. Also, I would like to thank Peter Kleban, AliÖzlük and Thomas Prellberg for finding a significant error in an earlier draft. The most famous example is the one-dimensional Ising model. For convenience, we let each site have the value of 1 or −1. Thus for the Ising model, we have We know that a m+1 q Nm ≤ d Nm ≤ (a m+1 + 2)q Nm . Then, if the limits exist,Since the a m+1 are bounded and the N m → ∞, we have Corollary 4.8. The golden ratio φ = 1+ has 1-free energy limit equal to log φ, for β > 2.1-free energy limit, for β > 2, if we can show lim m→∞ log d Nm−1 N m+1 = lim m→∞ log d Nm N m . We will see that these limits will exist if and only if lim m→∞ log qN m Nm exists. m > p, a m and a m+1 can only be chosen from finite set {c 1 , . . . c l }. Thus lim m→∞ log d Nm−1 N m+1 = lim m→∞ log d Nm−1 N m−1 since the denominators, for all m, differ by a fixed amount (and of course since N m → ∞). we have lim m→∞ log a m+1 q Nm N m ≤ lim m→∞ log d Nm N m ≤ lim m→∞ log(a m+1 + 2)q Nm N m . lim m→∞ log a m+1 N m = lim m→∞ log(a m+1 + 2) N m = 0, giving us that lim m→∞ log d Nm N m = lim m→∞ log q Nm N m , concluding the proof. √ 5 2 approach, there have been a number of good papers (such as in [17] [18] [19] [20] [21] [2] [6] [7] [10] [12] [14] [15][22] ) exploring the nature of these critical points. For our Diophantine partition functions, these types of questions seem to be equally subtle. The Partition Function and the Free Energy Classifying real numbers via free energy Classifying real numbers via free energy Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics. V Baladi, World Scientific16V. Baladi, Positive Transfer Operators and Decay of Correlations (Ad- vanced Series in Nonlinear Dynamics, Volume 16), World Scientific, 2000. F Boca, Products of matrices. 11F. Boca, Products of matrices [ 1 1 . J. Reine Angew. Math. 606, J. Reine Angew. Math. 606 (2007), pp. 149-165. . W Bosma, Approximations by Mediants, Mathematics of Computation. 54W. Bosma, Approximations by Mediants, Mathematics of Computation, 54(1990), pp. 421-434. E Burger, Exploring the Number Jungle: A Journey into Diophantine Analysis. American Mathematical Society8E. Burger, Exploring the Number Jungle: A Journey into Diophantine Analysis, Student Mathematical Library Vol. 8, American Mathematical Society, 2000. Making Transcendence Transparent: An intuitive approach to classical transcendental number theory. E Burger, R Tubbs, SpringerE. Burger and R. Tubbs, Making Transcendence Transparent: An intuitive approach to classical transcendental number theory, Springer, 2004. The phase transition in statistical models defined on Farey fractions. P Contucci, A Knauf, Forem Math. 9P. Contucci and A. Knauf, The phase transition in statistical models defined on Farey fractions, Forem Math 9 (1997), pp.547-567. Generalized Farey Trees, Transfer Operators and Phase Transition. M Espoti, S Isola, A Knauf, Communications in Mathematical Physics. 275M. Espoti, S. Isola and A. Knauf, Generalized Farey Trees, Transfer Op- erators and Phase Transition, Communications in Mathematical Physics, 275 (2007), pp. 298-329. Scaling properties of multifractals as an eigenvalue problem. M Feigenbaum, I Procaccia, T Tel, Physical Review A. 39M. Feigenbaum, I. Procaccia and T. Tel, Scaling properties of multifractals as an eigenvalue problem, Physical Review A 39 (1989), pp. 5359-5372. Generalized Number Theoretic Spin-Chain Conditions to Dynamical Systems and Expectation Values. J Fiala, P Kleban, J. Stat. Phys. 121J. Fiala and P. Kleban, Generalized Number Theoretic Spin-Chain Condi- tions to Dynamical Systems and Expectation Values, J. Stat. Phys., 121 (2005), pp. 553-577. The Phase Transition in Statistical Models Defined on Farey Fractions. J Fiala, P Kleban, A Özlük, J. Stat. Phys. 110J. Fiala, P. Kleban and A.Özlük, The Phase Transition in Statistical Mod- els Defined on Farey Fractions, J. Stat. Phys., 110 (2003), pp. 73-86. R Graham, D Knuth, O Patashnik, Concrete Mathematics. Addison-WesleyR. Graham, D. Knuth and O. Patashnik, Concrete Mathematics, Addison- Wesley, 1989. Free energy and correlations of the numbertheoretical spin chains. F Guerra, A Knauf, J. Math. Physics. F. Guerra and A. Knauf, Free energy and correlations of the number- theoretical spin chains, J. Math. Physics, 39 (1998), pp. 3188-3202. . D Hensley, Continued Fractions. World Scientific. D. Hensley, Continued Fractions. World Scientific, 2006. On the spectrum of Farey and Gauss maps. S Isola, Nonlinearity. 15S. Isola, On the spectrum of Farey and Gauss maps, Nonlinearity 15 (2002), pp. 1521-1539. On asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics. J Kallies, A Özlük, M Peter, C Snyder, Comm. Math. Phys. 2221J. Kallies, A.Özlük, M. Peter, C. Snyder, On asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics. Comm. Math. Phys. 222 (2001), no. 1, pp. 9-43. A farey fraction spin chain. P Kleban, A Ozluk, Communications in Mathematical Physics. P. Kleban and A. Ozluk, A farey fraction spin chain, Communications in Mathematical Physics,203 (1999), pp. 635-647. On a Ferromagnetic Chain. A Knauf, Communications in Mathematical Physics. 153A. Knauf, On a Ferromagnetic Chain, Communications in Mathematical Physics, 153 (1993), pp. 77-115. Phases of the Number-Theoretical SPin Chain. A Knauf, J. Stat. Phys. 73A. Knauf Phases of the Number-Theoretical SPin Chain, J. Stat. Phys. 73, 423-431, 1993. On a ferromagnetic spin chain.part ii: thermodynamic limit. A Knauf, J. Math. Physics. 35A. Knauf, On a ferromagnetic spin chain.part ii: thermodynamic limit, J. Math. Physics, 35 (1994), pp. 228-236. The number-theoretic spin chain and the Riemann zeros. A Knauf, Communications in Mathematical Physics. 196A. Knauf, The number-theoretic spin chain and the Riemann zeros, Com- munications in Mathematical Physics, 196 (1998), pp.703-731. Number theory, dynamical systems and statistical mechanics. A Knauf, Rev Math. Phys. 11A. Knauf, Number theory, dynamical systems and statistical mechanics, Rev Math. Phys. 11 (1998), pp. 1027-1060. The limit distribution of a number theoretic function arising from a problem in statistical mechanics. P Manfred, J. Number Theory. 902P. Manfred, The limit distribution of a number theoretic function arising from a problem in statistical mechanics, J. Number Theory 90 (2001), no. 2, pp. 265-280. Phase Transitions of Multidimensional Generalizations of the Knauf Number-Theoretic Chain Modal, senior thesis. E Major, Williams CollegeE. Major, Phase Transitions of Multidimensional Generalizations of the Knauf Number-Theoretic Chain Modal, senior thesis, Williams College, 2003. The Ruelle-Araki transfer operator in classical statistical mechanics. D Mayer, SpringerD. Mayer, The Ruelle-Araki transfer operator in classical statistical me- chanics, Springer, 1980. Ergodic theory, symbolic dynamics, and hyperbolic spaces. D Mayer, Continued fractions and related transformations. TriesteOxford Univ. PressD. Mayer, Continued fractions and related transformations, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), 175-222, Oxford Univ. Press (1991), pp. 175-222. S Miller, R Takloo-Bighash, An Invitation to Modern Number Theory. Princeton University PressS. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, 2006. Towards a complete determination of the spectrum of the transfer operator associated with intermittency. T Prellberg, J. Phys. A. 36T. Prellberg, Towards a complete determination of the spectrum of the transfer operator associated with intermittency, J. Phys. A 36 (2003), pp. 2455-2461. Cluster Approximation for the Farey Fraction Spin Chain. T Prellberg, J Fiala, P Kleban, J. Stat. Phys. 123T. Prellberg, J. Fiala and P. Kleban, Cluster Approximation for the Farey Fraction Spin Chain J. Stat. Phys. 123 (2006), pp. 455-471. Maps of intervals with indiffferent fixed points: thermodynamic formalism and phase transition. T Prellberg, J Slawny, J. Stat. Phys. 66T. Prellberg and J. Slawny, Maps of intervals with indiffferent fixed points: thermodynamic formalism and phase transition, J. Stat. Phys. 66 (1992), pp. 503-514. A Rockett, P Szüsz, COntinued Fractions. World ScientificA. Rockett and P. Szüsz, COntinued Fractions, World Scientific, 1992. Dynamical zeta functions for maps of the interval. D Ruelle, Bull. Amer. Math. Soc.(NS). 30D. Ruelle, Dynamical zeta functions for maps of the interval, Bull. Amer. Math. Soc.(NS) 30 (1994), pp. 212-214. D Ruelle, Statistical Mechanics: Rigorous Results. World ScientificD. Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, 1999. D Ruelle, Thermodynamic Formalism, The Mathematical Structure of Equilibrium Statistical Mechanics. Cambridgesecond editionD. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics , second edition, Cambridge, 2004. F Schweiger, Multidimensional Continued Fractions. Oxford University PressF. Schweiger, Multidimensional Continued Fractions, Oxford University Press, 2000. Y Sinai, Theory of Phase Transitions. PergamonY. Sinai, Theory of Phase Transitions, Pergamon, 1983. H Stark, An Introduction to Number Theory. M.I.T. PressH. Stark, An Introduction to Number Theory, M.I.T. Press, 1994. C Thompson, Mathematical Statistical Mechanics. Princeton University PressC. Thompson, Mathematical Statistical Mechanics, Princeton University Press, 1972.
[]
[ "arXiv:1407.7533v1 [cond-mat.str-el] Fractionalized gapless quantum vortex liquids", "arXiv:1407.7533v1 [cond-mat.str-el] Fractionalized gapless quantum vortex liquids" ]
[ "Chong Wang \nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMAU. S. A\n", "T Senthil \nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMAU. S. A\n" ]
[ "Department of Physics\nMassachusetts Institute of Technology\n02139CambridgeMAU. S. A", "Department of Physics\nMassachusetts Institute of Technology\n02139CambridgeMAU. S. A" ]
[]
The standard theoretical approach to gapless spin liquid phases of two-dimensional frustrated quantum antiferromagnets invokes the concept of fermionic slave particles into which the spin fractionalizes. As an alternate we explore new kinds of gapless spin liquid phases in frustrated quantum magnets with XY anisotropy where the vortex of the spin fractionalizes into gapless itinerant fermions. The resulting gapless fractionalized vortex liquid phases are studied within a slave particle framework that is dual to the usual one. We demonstrate the stability of some such phases and describe their properties. We give an explicit construction in an XY -spin-1 system on triangular lattice, and interpret it as a critical phase in the vicinity of spin-nematic states.
10.1103/physrevb.91.195109
[ "https://arxiv.org/pdf/1407.7533v2.pdf" ]
8,043,323
1407.7533
889390efc3fd75ac4ed3e528448450f1ec05542e
arXiv:1407.7533v1 [cond-mat.str-el] Fractionalized gapless quantum vortex liquids 28 Jul 2014 Chong Wang Department of Physics Massachusetts Institute of Technology 02139CambridgeMAU. S. A T Senthil Department of Physics Massachusetts Institute of Technology 02139CambridgeMAU. S. A arXiv:1407.7533v1 [cond-mat.str-el] Fractionalized gapless quantum vortex liquids 28 Jul 2014(Dated: July 30, 2014) The standard theoretical approach to gapless spin liquid phases of two-dimensional frustrated quantum antiferromagnets invokes the concept of fermionic slave particles into which the spin fractionalizes. As an alternate we explore new kinds of gapless spin liquid phases in frustrated quantum magnets with XY anisotropy where the vortex of the spin fractionalizes into gapless itinerant fermions. The resulting gapless fractionalized vortex liquid phases are studied within a slave particle framework that is dual to the usual one. We demonstrate the stability of some such phases and describe their properties. We give an explicit construction in an XY -spin-1 system on triangular lattice, and interpret it as a critical phase in the vicinity of spin-nematic states. I. INTRODUCTION Quantum spin liquids are exotic phases of matter beyond Landau's paradigm of symmetry-breaking 1 . In contrast to other familiar ground states of quantum magnets (such as antiferromagnets or ferromagnets) the quantum spin liquid ground state has a non-local entanglement between its local degrees of freedom. Similar 'long range entanglement' also appears in the ground state of some other states of matter, for instance in the fractional quantum Hall states, and in Fermi/non-Fermi liquid metals. Since the original conception of the possibility of the quantum spin liquid, there has been tremendous progress in describing them theoretically. Many different kinds of quantum spin liquids are known to be theoretically possible. In the last decade a number of experimental candidates have also appeared. Interestingly all the existing experimental candidates seem to have gapless excitations which are not related to Goldstone modes of any broken symmetry. The theory of such gapless quantum spin liquids is however much less developed than the theory of gapped quantum spin liquid states. The currently known experimental candidate spin liquid materials may be conveniently grouped into two broad categories. The first -dubbed "weak Mott insulators" -are close to the Mott transition and have significant virtual charge fluctuations. Both the layered organics κ− (ET ) 2 Cu 2 (CN ) 3 and EtM e 3 Sb[P d(dmit) 2 ] 2 , and the three dimensional hyperkagome iridate N a 4 Ir 3 O 8 are all Mott insulating at ambient pressure but can be driven 2-4 through the Mott transition with application of moderate pressure. Indeed quantum spin liquid behavior may well be a common fate of weak Mott insulators. The second category -dubbed "strong Mott insulators"have large charge gaps that are well separated from their exchange scales. These two classes of spin liquids likely require different theoretical approaches. In weak Mott insulators gapless spin excitations are perhaps expected. At short length/time scales such insulators look roughly the same as a metal. As confirmed by various theoretical calculations, it is then reasonable that at longer length scales even though the charge localizes the spin continues to be carried by itinerant neu-tral fermions (the spinons). Remarkably gapless excitations are found even in candidate spin liquids which are strong Mott insulators. Striking examples are the Kagome systems 5,6 ZnCu 3 (OH) 6 Cl 2 (Herbertsmithite) and Cu 3 V 2 O 7 (OH) 2 .2H 2 O (Volborthite). Similarly the recently reported spin-1 spin liquid 7 Ba 3 N iSb 2 O 9 is also a strong Mott insulator. Recent progress in Density Matrix Renormalization Group calculations of the isotropic spin-1/2 Kagome magnet 8 reveal a large spin gap (0.14J) which is not seen in the experiments on Herbertsmithite 9 . The real model for this material is more complicated and must include Dzyaloshinski-Moriya as well as other anisotropies. Further there are significant impurity effects attributed to excess Cu spins sitting in between the Kagome planes. Other complications may exist in other materials. Nevertheless the surprisingly common occurance of gapless spin liquids in strongly Mott insulating materials leads to some fundamental questions in the theory of spin liquids. In what theoretical framework should we discuss these gapless spin liquids? Currently one framework that is known is to start with a slave particle description of the physical spin operator in terms of fermionic neutral spin-1/2 spinons. The resulting spinon Hamiltonian is then first treated at a mean field level. At this level of treatment the spinon spectrum may well be gapless (with Fermi points or even a Fermi surface). Going beyond mean field requires including fluctuations. The resulting theory typically includes a fluctuating gauge field. Thus in this approach gapless spin liquids are described by an effective theory that involves gapless fermionic spinons coupled to a fluctuating gauge field. If this theory is stable then this is a legitimate description of a possible gapless spin liquid phase. The slave particle approach described above is deservedly popular and it certainly enables description of a class of quantum spin liquids. However while this seems natural for weak Mott insulators (as is confirmed by many existing calculations) it is hardly obvious that this is the way forward in dealing with gapless spin liquids in, say, the Kagome magnets, or in the spin-1 magnet. As currently no other methods are known, fermionic spinon based approaches are the "knee-jerk" reaction of theo-rists to the announcement of any experimental candidate gapless spin liquid. A big open question in the field is whether there are other approaches that enables access to a different class of gapless spin liquids. More specifically do gapless spin liquids exist that are beyond the existing fermionic spinon (+ gauge field) paradigm? If so what is their phenomenology? Could they be more natural candidates for some of the spin liquids that are reported? In this paper we introduce a new theoretical route to gapless quantum spin liquids in spin systems with XY symmetry which appears to be distinct from the conventional fermionic spinon route. We utilize a dual description of such a spin system in terms of vortices in the XY spin. We show that quantum vortex liquid phases exist where there are gapless fermionic fields that carry the vorticity. We access these gapless vortex liquid phases through a 'dual' parton approach where we fractionalize the fundamental vortex field into fermionic half-vortices. These may then form a gapless state. We describe an example of this construction for spin-1 quantum XY models on a honeycomb lattice. The dual parton approach is complementary to the standard one which fractionalizes the physical spin itself. Indeed it is likely that the phases we access have no simple description within the standard approach. An interesting earlier attempt with a motivation similar to ours was made in Ref. 10-12, where the vortices were "fermionized" through a Chern-Simons fluxattachment. The fermions can then be put into a gapless band, and the resulting state becomes gapless and U (1)-symmetric. However, this construction has problems with implementing time-reversal symmetry. In the simplest context in which such a fermionized vortex duality was attempted, it was shown in Ref. 13 that such an approach would require an extra topological term into the original (un-dualized) description. Moreover, it was found recently 14,15 that such states realize time-reversal symmetry anomalously, and could appear only on the surface of certain bosonic topological insulators. Therefore such a state realized in two dimensions will break time-reversal symmetry, and hence is not suitable to describe symmetric quantum spin liquids. Closer to our approach is Ref. 16 which also employed a dual fermionic parton decomposition of the fundamental vortex field. The goal however was different from ours and that work did not attempt to find stable quantum spin liquid phases through the dual parton approach. II. DUALITY, VORTICES, AND FRACTIONALIZATION A spin system with XY symmetry can be fruitfully viewed as a system of interacting bosons (with S z playing the role of boson number and S + the role of the boson creation operator b † ). For a bosonic system with global U (1) symmetry, it is known that one can make a duality mapping and describe the system in terms of vortices 17 . Specifically, one can write the conserved U (1) current as the flux of a non-compact U (1) gauge field j µ = ǫ µνλ 2π ∂ ν a λ ,(1) The gauge field a µ couples to a formally bosonic field Φ that corresponds to vortices in the order parameter of the global U (1) symmetry. If the vortices are gapped, we get a superfluid/ordered magnet with the global U (1) symmetry broken, in which the gapless photons of the a µ gauge field corresponds to the Goldstone mode. But if the vortices are condensed instead, the whole system will be gapped due to the Higgs mechanism and we get a trivial Mott insulator/paramagnet. One can then ask the following question: is it possible for the vortices to be in a stable gapless phase, so that the whole system is gapless while the global U (1) symmetry is still preserved? The route we will take is to fractionalize the vortex into two fermions, schemetically we have Φ ∼ ψ 1 ψ 2 ,(2) where Φ represents the vortex field rather than the physical spin as in usual parton construction, and ψ 1,2 are fermions representing "fractionalized" vortices. Such a "dual" parton construction can easily be made timereversal invariant. As in the usual parton construction the dual parton representation introduces an SU (2) gauge redundancy. In this paper we will restrict ourselves to states where this SU (2) gauge structure is broken down to Z 2 . This will already be enough to produce a number of interesting states of the spin/boson system. Before describing the gapless states we are interested in let us briefly describe some conventional states that will help build intuition about these fractionalized vortices. Consider the simplest such fractionalized vortex state, in which the fermionic fractional vortices ψ is gapped, and couple to a µ with gauge charge 1/2. Then we may integrate them out to get a Maxwell action for the a µ . The gauge field fluctuations are thus gapless. Physically this is a superfluid phase of the original bosons. However the presence of the gapped fractional vortex means that it is a paired superfluid where boson pairs b 2 are condensed without condensation of individual bosons b. (In spin language this is a 'spin nematic' phase). The excitation spectrum of such a paired superfluid is well known. There is the usual gapless superfluid sound mode which in the dual description is identified with the propagating photon. The single boson survives as a gapped 'Bogoliubov' quasiparticle, and may be described as an Ising spin s. In addition there is a halfvortex excitation where the phase of b 2 winds by 2π. The Ising spin s in turn acquires a phase π upon encircling this vortex. Thus the Ising spin and the half-vortex are mutual semions. If we assign bose statistics to the halfvortex, its bound state with the Ising spin s yields an excitation that is a fermion and also carries half-vorticity. Clearly we identify this with the ψ particles in the dual parton description. Since we have assumed a state that has broken the dual SU (2) gauge structure to Z 2 , the ψ carry a Z 2 gauge charge (in addition to the U (1) gauge charge representing their vorticity). Correspondingly there is a Z 2 gauge vortex (the vision) which clearly must be identified with the s particle, i.e the unpaired boson in the paired superfluid. The original physical boson is the composite of a vison s and a 2π-flux of the U(1) gauge field. Condensing the original boson means condensing the vison s, which confines the half-vortices, in agreement with the usual description. One can also consider a different phase in which the ψ fermions are paired ψψ = 0. In such a phase a µ is gapped, and we get a fractionalized liquid with Z 2 topological order. The pair condensation quantizes the magnetic flux of a µ in units of 2π, which corresponds to an excitation b v with physical charge 1 and boson statistics. This b v is however not to be identified with the physical boson b. Indeed the unpaired ψ fermion survives as a Bogoliubov quasiparticle which is a mutual semion with the b v . This is in contrast with the physical boson b which is local with respect to all excitations. The state obtained this way has the topological order of a Z 2 quantum spin liquid but with symmetry realized in an unfractionalized manner. The most interesting situation -which we explore in this paper -is when we put the ψ fermions into a gapless band structure, such as a massless Dirac band. The gapless fermions will then couple to the gauge field a µ strongly, and form a gapless state which is not ordered. This is a gapless quantum spin liquid state which is potentially not accessible within the standard fermionic spinon-gauge field paradigm. III. CONSTRUCTION WITH FRUSTRATED QUANTUM XY MODEL We now illustrate the construction of an example of such a gapless fractionalized quantum vortex liquid. Consider a quantum XY antiferromagnet on a twodimensional triangular lattice. The Hamiltonian can be written as a rotor model (b ∼ e iφ ) in a background static gauge field A 0 : H = −J ij cos φ i − φ j + A 0 ij + U i n 2 i + ... (3) where A 0 gives a π flux on each triangular plaquette (corresponding to antiferromagnetic exchange). We can think of the π flux as requiring that there be an average vortex filling of 1/2 per site on the dual honeycomb lattice. Going then to the vortex picture, we get a theory of hard-core bosons (the vortices) at half-filling on the honeycomb lattice, coupled with a non-compact U(1) gauge field 10 : H = −2t ij e iaij Φ † i Φ j + h.c. + H Maxwell + ...(4) where one may also have short range vortex interaction terms in general. For spin-half antiferromagnets (i.e where the original rotor number is 1/2 per site on average), the vortices will themselves see a background π-flux on each plaquette. This system of hard-core bosonic vortices at half-filling could be fractionalized. To explore this possibility, we fractionalize the vortex operator Φ into two fermions using the slave-particle formulation: Φ i = 1 2 ǫ αβ ψ i,α ψ i,β , N i = 1 2 ψ † α ψ α ,(5) where N denotes the vortex density, and α, β = 1, 2 are the pseudo-spin indices, which transform under the internal SU (2) gauge symmetry as ψ α → U αβ ψ β . The lattice symmetries act on ψ i,α in the same ways as on Φ (up to an SU (2) gauge transform). For a spin model, time reversal acts on vortices as T : Φ i → Φ i , we have T : ψ i,α → ψ i,α (again up to a gauge rotation). The particle-hole symmetry (coming from π rotation of spins around x axis) transformation acting on the vortex is non-trivial: C : Φ i → Φ † i , which leads to C : ψ i,α → W i,αβ ψ † i,β where W is unitary with det(W ) = −1. Our goal is to explore phases in which the fermions ψ 1,2 are deconfined and gapless. The gaplessness of the fermions should be stable in the sense that it is protected by symmetries. It is instructive to reinterpret the "fermionized vortex" theory of Refs. 10-12 using this dual parton construction. It corresponds to putting ψ 1 in a Chern-insulator and ψ 2 in a gapless Dirac band. However, since time-reversal is broken in such a phase, the gaplessness is unprotected. Now consider a particular mean field ansatz that meets our need: H mean = − ij ψ † iα u αβ ij ψ jβ + h.c. ,(6) with the hopping matrices u ij given by u i,i+a1 = u i,i+a2 = u i,i+a3 = ητ 0 + λτ 3 ,(7)u i,i+a1+a2−a3 = u i,i+a1−a2+a3 = u i,i−a1+a2+a3 = ξτ 1 , where a i are the three nearest-neighbor vectors on the honeycomb lattice, η, λ, ξ are all real and τ l are Pauli matrices acting on the SU (2) gauge indices. It is easy to see that ψ † i τ µ ψ i = 0 on any site i due to the particle-hole and time-reversal symmetries preserved by the mean field band structure. Therefore the mean field ansatz satisfies the gauge constraints on average and no further chemical potential term is needed. To determine the remaining gauge structure in the phase described by Eq. (7), one needs to calculate the SU (2) gauge fluxes of the hopping matrices u ij on various loops, and all the fluxes must be invariant under the unbroken gauge group 1 . It is then straightforward to see that only the Z 2 gauge group ψ i → (−1) si ψ i survives. The ansatz given in Eq. (7) realizes all the lattice symmetries trivially, and is also manifestly time-reversal invariant. Hence ψ α transforms in exacly the same way as Φ. For charge conjugation C, by inspection one can see that we should choose C : ψ i,α → i(−1) i ψ † i,α , where (−1) i takes opposite values on different sublattices. The fermions ψ should also be coupled to the non-compact U (1) gauge field a µ , and from the structure of the ansatz it is clear that the only way to do this consistently is to assign charge-1/2 to both ψ 1,2 . The virtue of the ansatz Eq. (7) is that it supports a gapless band structure protected by symmetries. It is straightforward to show that the band structure is described by four Dirac cones (similar to Graphene) near ±Q, and the low energy 'mean field' Hamiltonian can be written as H ef f (k) = √ 3 2 ητ 0 + λτ 3 − 2ξτ 1 ⊗ k x σ 1 ⊗ v 3 − k y σ 2 ⊗ v 0 ,(8) where σ i acts on sub-lattice indices and v i on valley indices. The symmetry actions on the low energy fermions in the above basis can be worked out through standard procedures: we have the lattice translation T (1,0) : ψ → exp i 4π 3 σ 0 ⊗ v 3 ψ; π/3 rotation around the center of an honeycomb plaquette (a site of the original triangular lattice) R π/3 ψ = σ 2 ⊗ v 2 e −i π 6 σ 3 ⊗v 3 ψ; modified x-reflectioñ R x = R x C : ψ(k x , k y ) → τ 0 ⊗ σ 0 ⊗ v 1 ψ(−k x , k y ) (note that a simple reflection flips vorticity); charge conjugation C : ψ(k) → τ 0 ⊗ σ 3 ⊗ v 1 ψ † (−k); time reversal T (ψ(k x , k y ) → τ 0 ⊗ σ 0 ⊗ v 1 ψ(−k x , −k y ) and complex conjugation). We can now analyse generally what fermion-bilinear terms are allowed by symmetries in the low-energy theory. It is then straightforward to show that Eq. (8) is the most general form of symmetry-allowed low energy hamiltonian of the fermions. In particular, a mass term that opens up a fermion gap is not allowed by symmetries. Hence the gaplessness of the fermions are symmetry-protected, at least perturbatively. The above analysis can also be applied to a physical hard-core boson system on a honeycomb lattice at halffilling. The resulting state is a gapless Z 2 fractionalized liquid. The charge-1/2 fermions form four Dirac nodes, with a velocity anisotropy in the pseudo-spin space. As we will see below, when we view the theory instead as a vortex theory, the coupling to the U (1) gauge field a µ removes the velocity anisotropy at low energy. The low energy Lagrangian with the a µ field included can be written as L =ψ [−i(γ µ +γ µ )(∂ µ + iã µ )] ψ + 1 2e 2 f 2 µν .(9) We have chosen the normalizationã µ = a µ /2, η = 1 and ψ = iψ † γ 0 , where γ µ = (τ 0 ⊗σ 3 ⊗v 3 , τ 0 ⊗σ 2 ⊗v 0 , τ 0 ⊗σ 1 ⊗ v 3 ), andγ µ = (0, ( λτ 3 − 2ξτ 1 ) ⊗ σ 2 ⊗ v 0 , (λτ 3 − 2ξτ 1 ) ⊗ σ 1 ⊗ v 3 ) . This is not quite Dirac, but after including the fluctuation of the U(1) gauge field, it will renormalize to a Dirac theory with emergent Lorentz symmetry. For small λ amd ξ and large N f (here we have N f = 4), we have to first order 1 λ dλ dl = 1 ξ dξ dl = − 64 5π 2 N f .(10) Hence they are irrelevant to first order. The calculation is essentially identical to that in Ref. 18, where it was shown that the velocity anisotropy in real space was irrelevant (see Appendix A for details). Hence the low energy fixed point is simply the QED 3 with four flavors of Dirac fermions. It is believed that for flavor number N f not too small (greater than certain critical value N f,c ), the QED 3 fixed point is a CFT that is stable against spontaneous chiral symmetry breaking and fermion mass generation. The currently known 19 upper-bound for N f,c is N f,c < 6.6. If the actual value of N f,c is less than four, our theory would describe a stable critical phase, rather than just a fine-tuned critical point. One could also consider slightly modifying the system, by changing the flux on each plaquette in the rotor model Eq. (3) from π to (π + 2πδ). This changes the vortex filling to (1/2 + δ), which is also the filling fraction of the ψ fermions. The same mean field ansatz Eq. (6) would then describe small fermi surfaces coupled with the gauge field a µ . As discussed in Ref. 20, such a theory could describe a stable phase. However, we will not study this phase in detail since the modified system is harder to realize. Since our purpose is mainly to illustrate the new formalism, the QED 3 fixed point theory is enough to convey the message. The critical phase thus obtained has symmetries that are absent in the microscopic model, but emerge at low energy. These include the Lorentz invariance and the SU (4) flavor symmetry. The SU (4) group is generated by {τ 0 ⊗σ 0 ⊗v 3 , τ 0 ⊗σ 2 ⊗v 1 , τ 0 ⊗σ 2 ⊗v 2 , τ 1,2,3 ⊗σ 0 ⊗v 0 } and their tensor products, which gives 15 generators in total, denoted by T a , and by construction we have [T a , γ µ ] = 0. IV. PHYSICAL PROPERTIES Now we look at particular features of the specific gapless vortex liquid state constructed above by considering physical observables. As a critical theory, we expect many of the physical observables will have algebraic correlation functions, and the exponents can be calculated using the CFT description. A notable exception, however, is the in-plane spin-spin correlations. A spin-1 excitation S ± is represented as the composite of the vison s seen by the half-vortices ψ α and a half-monopole in a µ . Since the vison s is assumed to be gapped, we expect S ± to be also gapped, and the in-plane spin-spin correlations S + S − will thus be short-ranged. The out-of-plane spin-spin correlation functions S z S z , on the contrary, decays algebraically. In fact, since S z is conserved in the CFT with the corresponding current represented as j ∼ da, its scaling dimension must be h j = 2. We therefore have an interesting state with gapped S ± but critical S z . In fact, the rich symmetry structure of our theory gives many other conserved currents which all have scaling dimension h j = 2. These include the vorticity J µ = −iψγ µ ψ and the SU (4) flavor current J a µ = −iψγ µ T a ψ. The more interesting observables are nematic (spin-2) order parameters like (S + ) 2 . In the dual picture these nematic operators are represented as monopoles in QED 3 . There are four flavors of Dirac fermions and each of them gives a zero-mode in the presence of ±2π flux ofã µ . A gauge-invariant state created by a monopole event should have half of the zero-modes filled. Hence there are six possible monopoles, obtained by filling two of the four zero-modes. We show in Appendix B that the monopole operators indeed transform in the same way as (S ± ) 2 at the three low energy momenta (0, ±Q). The scaling dimension of the nematic operators is thus given by that of the monopole operators, which can be calculated in the large-N f limit 21,22 (here we have N f =4): h n ≈ 0.265N f − 0.038 ≈ 1.02. The relatively small scaling dimension reveals the proximity to nematically ordered phases. To actually go to a nematic phase, the fermions ψ α must acquire a mass gap. Since all the fermion mass terms break some global symmetries, the mass gap must be dynamically generated through spontaneous symmetry breaking, which agrees with the intuition that an ordered state on a frustrated lattice should break some symmetries other than the global U (1). Possible mass terms are the flavor SU (4) adjoint N a = −iψT a ψ and scalar M = −iψψ. It turns out 18 that M has a relatively large scaling dimension, so the primary instability comes from the N a terms. The scaling dimensions of all the N a operators (which must be the same due to the SU (4) symmetry) have been calculated 18 to leading order in 1/N f which gives h N ≈ 2 − 64/3π 2 N f ≈ 1.46. In particular, these include the coplanar order parameter (spin chirality) κ ∼ K z : τ µ =2 ⊗σ 3 ⊗v 0 , and the collinear order parameter (bond energy wave) K ± : τ µ =2 ⊗σ 1 ⊗(v 1 ±iv 2 ), which are expected to order in usual magnetic phases. The large number of operators with the same relatively small scaling dimensions gives a clear manifestation of the emergent SU (4) flavor symmetry. Physical observables that transform the same way with N a under microscopic symmetries will thus have the same scaling dimensions h N . It is straightforward to see that eight distinct physical operators are connected by the SU (4) flavor symmetry. We list all the physical operators in Appendix C. Finally we mention some of the thermodynamic properties of this state. Clearly the low-T heat capacity will be C ∝ T 2 , and the uniform spin susceptibility (for field coupling to S z ) will be χ z ∝ T . The proportionality constants will depend on the (non-universal) Dirac velocity v in a universal way such that the Wilson ratio T χ z C is a universal constant characteristic of the CFT (computable in the 1 N f expansion). There is another QED 3 fixed point for the theory in Eq. (9), by choosing the γ matrices differently. We discuss this fixed point in Appendix D. We show that physical observables behave differently in this new fixed point, so it is indeed a distinct phase from the one discussed so far. V. RELATIONSHIP TO OTHER STATES We now briefly consider how the gapless quantum vortex liquid state is related to other more familiar phases of the quantum XY magnet. We have already discussed in Sec. II and later that if the vortex fields ψ acquires a gap then the result is a phase with long range spin-nematic order (i.e where (S + ) 2 is ordered without ordering of S + ). As also discussed in Sec. II, if the ψ pair and condense, the result is a Z 2 quantum spin liquid but without fractionalization of the global U (1) quantum number. Although being conceptually close to a nematic phase, the gapless vortex liquid can also be found near other conventional states in principle, via a direct phase transition. To make a transition into a simple ordered state in which b = 0, simply condense the vison s seen by ψ, then the fermions ψ will be confined and the vortex Φ will be gapped, which is nothing but an ordinary superfluid. The trivial Mott insulator is also accessible through condensing the composite of the fermion half-vortex ψ and the vison s (which is a boson v ∼ ψs due to the mutual semion statistics), which will confine all the fractional particles and make the system gapped. VI. DISCUSSION We have described a concrete example of a gapless quantum spin liquid phase as a gapless fractionalized quantum vortex liquid. Can this phase be accessed directly within the usual spinon-gauge field paradigm based on some spinon mean field ansatz? We point out that since the vortices are fractionalized the short distance physics of our state is not that of fractionalization of the spin but rather that of spin nematic-like pairing. This suggests that the spinon mean field + fluctuation paradigm may not be able to reach the spin liquid state described here. Our construction thus presents a potential alternate class of gapless quantum spin liquids that may be beyond the spinon-gauge field paradigm. Clearly the dual parton approach developed here can be used to construct a variety of other gapless quantum vortex liquid states. An interesting example is a state where the fractionalized vortices form a gapless Fermi surface rather than Fermi points. The coupling of the vortices to the non-compact gauge field will lead to a low energy field theory similar to that of a spinon Fermi surface spin liquid 23,24 or the HLR state 25 of the half-filled Landau level. Of course as in the Dirac case discussed here the identification of physical operators in terms of the fields of the low energy theory will be different and will lead to different physical properties. The states described in this paper should open our eyes to other new possible routes to gapless spin liquid behavior and suggest alternate possibilities for building phenomenologies of existing experimental candidates. our calculation easier due to the cancellation of the sign ambiguities in Ref. 10. The filled negative Dirac sea is defined through |DS, q = e iqγ Π E<0 c † Eq |vac, q ,(B2) where the background flux is 2πq = ±2π, and |vac, q is the state with all the fermion levels unoccupied. One can choose the phases in the definition of |vac, q so that T |vac, q = |vac, −q , R x T |vac, q = |vac, q ,(B3) and choose the phase γ in Eq. (B2) so that C|DS, q = f † 1,R,−q f † 1,L,−q f † 2,R,−q f † 2,L,−q |DS, −q . (B4) The rest of the symmetry properties are determined by the filled Dirac sea, and are heavily constrained by the algebraic structure of the symmetry groups. The contributions from a filled Dirac sea with two flavors are calculated in Ref. 10, with some sign ambiguities that cannot be determined from the group structure. Fortunately we have two copies of the Dirac sea that transform identically under all the microscopic symmetries. Hence the sign ambiguities cancel, and the symmetry properties are uniquely determined from the group structure. One can then show that the symmetry properties of the filled negative Dirac sea are given by T δr |DS, q = |DS, q , R π/3 |DS, q = e iq2π/3 |DS, q , C|DS, q = f † 1,R,−q f † 1,L,−q f † 2,R,−q f † 2,L,−q |DS, −q , T |DS, q = |DS, −q , R x T |DS, q = R x CT |DS, q = |DS, q ,(B5) where f † fills a zero-mode, and q = ±1 is the monopole strength. The zero modes in the Coulomb gauge transform as T δr f R/L,q T −1 δr = e ±iQ·δr f R/L,q , R π/3 f R/L,q R −1 π/3 = ie −iqπ/6 f L/R,q , Cf R/L,q C −1 = f † L/R,−q , T f R/L,q T −1 = ±qf L/R,−q , (R x T )f R/L,q (R x T ) −1 = f R/L,q .(B6)ψ † γ 0 T a ψ Representative physical operators τ µ =2 ⊗ σ 3 ⊗ v 0 κp ∼ i i,j∈p (s + i s − j − s + j s − i ) τ µ =2 ⊗ σ 1 ⊗ (v 1 ± iv 2 ) B ± ij ∼ e ±iQ·(r i −r j ) (s + i s − j + s + j s − i ) τ 2 ⊗ σ 3 ⊗ v 0 κp i∈p s z i τ 2 ⊗ σ 1 ⊗ (v 1 ± iv 2 ) B ± ij s z i + s z j τ 1,3 ⊗ σ 3 ⊗ v 3 N v p i∈p s z i τ 2 ⊗ σ 3 ⊗ v 3 N v p i∈p s z i 2 TABLE I. Correspondence between slowly decaying fermion bilinears and microscopic operators in phase 1, where κp is the spin chirality defined on plaquette p, B ± ij is the bond-energy wave operator defined at the Brillouin zone coner ±Q, sz is the z-component of the physical spin, and N v p is the vorticity on plaquette p. The pseudo-spin SU(2) scalar N and M L/R transform as (S ± ) 2 at the three low energy momenta (0, ±Q), as expected. The emergence of the SU (2) vector L ±,0 as another set of spin-2 operators reveals the emergent flavor symmetry of the theory. Fermion bilinears ψ † γ 0 T a ψ Representative physical operators τ 0,1 ⊗ σ 3 ⊗ v 0 , the group structure of the total symmetry (microscopic, Lorentz and flavor) is now different from the previous theory, we expect these two theories to be physically distinct, seperated by a critical point at η = λ, where the velocity of one psuedospin component vanishes and the band structure changes drastically, although the microscopic symmetries are realized in exactly the same way. It is interesting to note that these phases are distinct solely by emergent symmetries. τ 2 ⊗ σ 0 ⊗ v 3 κp ∼ i i,j∈p (s + i s − j − s + j s − i ) τ 0,1 ⊗ σ 1 ⊗ (v 1 ± iv 2 ), τ 2 ⊗ σ 2 ⊗ (v 1 ± iv 2 ) B ± ij ∼ e ±iQ·(r i −r j ) (s + i s − j + s + j s − i ) τ 3 ⊗ σ 0 ⊗ v 3 κp i∈p s z i τ 3 ⊗ σ 2 ⊗ (v 1 ± iv 2 ) B ± ij s z i + s z j τ 1 ⊗ σ 3 ⊗ v 3 N v p i∈p s z i τ 3 ⊗ σ 0 ⊗ v 0 s + 1,i s − 2,i − s + 2,i s − 1,i /2i The operators connected by the emergent SU (4) flavor symmetry is listed in Table II, which is clearly distinct from the list given in Table I. Therefore the new fixed point is indeed qualitatively distinct from the phase discussed in the main text. One can then choose the phases in Eq.(B1) and define N = M 12,0 − M 21,0 , L + = M 11,0 , L − = M 22,0 and L 0 = Fermion bilinears M 12,0 + M 21,0 , such thatT δr : M L/R → e ±iQ·δr M L/R , N → N, L ±,0 → L ±,0 , R π/3 : M L/R → M R/L , N → N, L ±,0 → −L ∓,0 , R x T : M L/R → M L/R , N → N, L ±,0 → L ±,0 , C : M L/R → M † R/L , N → N † , L ±,0 → L † ∓,0 , T : M L/R → M † L/R , N → N † , L ± → L † ± , L 0 → −L † 0 . TABLE II . IICorrespondence between slowly decaying fermion bilinears and microscopic operators in the new fixed point. To find a simple correspondence of the last one, we can imagine having two species of spins on each site s1, s2 and then couple them in symmetric ways. . This work was partially supported by a Simons Investigator award from the Simons Foundation to Senthil Todadri. 1 X. G. Wen, Quantum Field Theory of Many-Body Systems, Oxford (2004). 2 Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, and G. Saito, Phys. Rev. Lett. 95, 177001 (2005). Acknowledgement:We thank F. Wang for helpful discussions. This work was supported by NSF DMR-Appendix A: RG of psuedo-spin velocity anisotropyWe can re-write the Lorentz-breaking perturbation asThe last term can be absorbed into the Dirac term by re-definingψ = (1 + 2λ/3τ 3 ) 1/2 ψ, simplifying the perturbation (to leading order in λ) toThe last two terms share the same structure with the velocity anisotropy term examined in Ref.18, from which the leading order RG flow follows directly:Appendix B: Quantum numbers of monopolesThe monopole operators are defined through their operations on the zero-flux ground state:where f † α,R/L,± occupies the zero-mode coming from psuedo-spin α and valley R/L in ±2π flux, and |DS, ± denotes the state with all the negative energy levels filled in ±2π flux. The symmetry properties of the zero-modes f α,R/L,± and the filled negative Dirac sea |DS, ± can be obtained. The calculation is almost identical to that in Ref.10The operators corresponding to the flavor SU (4) adjoint −iψT a ψ are listed inTable I.Appendix D: Another fixed pointThere is another QED 3 fixed point for the theory in Eq. (9), by choosing the γ matrices differently. For example one can take γ µ = (τ 0 ⊗ σ 3 ⊗ v 3 , τ 1 ⊗ σ 2 ⊗ v 0 , τ 1 ⊗ σ 1 ⊗ v 3 ), by the same argument we can show that perturbations likeγ µ = (0, ητ 0 ⊗ σ 2 ⊗ v 0 , ητ 0 ⊗ σ 1 ⊗ v 3 ) are irrelevant. This theory still has an SU(4) flavor symmetry, generated by {τ 0,1 ⊗ σ 0 ⊗ v 3 , τ 0,1 ⊗ σ 2 ⊗ v 1,2 , τ 2,3 ⊗ σ 1 ⊗ v 1,2 , τ 2,3 ⊗ σ 3 ⊗ v 3,0 , τ 1 ⊗ σ 0 ⊗ v 0 }. However, since . T Itou, A Oyamada, S Maegawa, M Tamura, R Kato, Phys. Rev. B. 77104413T. Itou, A. Oyamada, S. Maegawa, M. Tamura and R. Kato, Phys. Rev. B 77, 104413 (2008). . Yoshihiko Okamoto, Minoru Nohara, Hiroko Aruga-Katori, Hidenori Takagi, Phys. Rev. Lett. 99137207Yoshihiko Okamoto, Minoru Nohara, Hiroko Aruga- Katori, and Hidenori Takagi, Phys. Rev. Lett. 99, 137207 (2007). . J S Helton, K Matan, M P Shores, E A Nytko, B M Bartlett, Y Yoshida, Y Takano, Y Qiu, J.-H Chung, D G Nocera, Phys. Rev. Lett. 98107204J.S. Helton, K. Matan, M.P. Shores, E.A. Nytko, B.M. Bartlett, Y. Yoshida, Y. Takano, Y. Qiu, J.-H. Chung, D.G. Nocera, Phys. Rev. Lett. 98, 107204 (2007). . Satoshi Yamashita, Tomoya Moriura, Yasuhiro Nakazawa, Hiroyuki Yoshida, Yoshihiko Okamoto, Zenji Hiroi, J. Phys. Soc. Jpn. 7983710Satoshi Yamashita, Tomoya Moriura, Yasuhiro Nakazawa, Hiroyuki Yoshida, Yoshihiko Okamoto, Zenji Hiroi, J. Phys. Soc. Jpn., 79, 083710 (2010). . J G Cheng, G Li, L Balicas, J S Zhou, J B Goodenough, Cenke Xu, H D Zhou, Phys. Rev. Lett. 107197204J. G. Cheng, G. Li, L. Balicas, J. S. Zhou, J. B. Goode- nough, Cenke Xu, and H. D. Zhou Phys. Rev. Lett. 107, 197204 (2011). . Simeng Yan, David A Huse, Steven R White, Science. 3321173Simeng Yan, David A. Huse and Steven R. White, Science 332, 1173 (2011). . Tian-Heng Han, Joel S Helton, Shaoyan Chu, Daniel G Nocera, Jose A Rodriguez-Rivera, Collin Broholm, Young S Lee, Nature. 492Tian-Heng Han, Joel S. Helton, Shaoyan Chu, Daniel G. Nocera, Jose A. Rodriguez-Rivera, Collin Broholm and Young S. Lee, Nature 492, 406-410 (2012). . J Alicea, O I Motrunich, M Hermele, M P A Fisher, Phys. Rev. B. 7264407J. Alicea, O. I. Motrunich, M. Hermele, and M. P. A. Fisher, Phys. Rev. B 72, 064407 (2005). . J Alicea, O I Motrunich, M P A Fisher, Phys. Rev. Lett. 95247203J. Alicea, O. I. Motrunich, M. P. A. Fisher, Phys. Rev. Lett. 95, 247203 (2005). . J Alicea, O I Motrunich, M P A Fisher, Phys. Rev. B. 73174430J. Alicea, O. I. Motrunich, M. P. A. Fisher, Phys. Rev. B 73, 174430 (2006). . T Senthil, M P A Fisher, Phys. Rev. B. 7464405T. Senthil and M. P. A. Fisher, Phys. Rev. B 74, 064405 (2006). . A Vishwanath, T Senthil, Phys. Rev. X. 311016A. Vishwanath, T. Senthil, Phys. Rev. X 3, 011016 (2013). . C Wang, T Senthil, Phys. Rev. B. 87235122C. Wang and T. Senthil, Phys. Rev. B 87, 235122 (2013). . Michael Hermele, Phys. Rev. B. 79184429Michael Hermele, Phys. Rev. B 79, 184429 (2009). . C Dasgupta, B I Halperin, P A Matthew, D H Fisher, Lee, Phys. Rev. Lett. 472756Phys. Rev. BC. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981), Matthew P. A. Fisher and D. H. Lee, Phys. Rev. B 39, 2756 (1989). . M Hermele, T Senthil, M P A Fisher, Phys. Rev. B. 72104404M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev. B 72, 104404 (2005). . Tarun Grover, Phys. Rev. Lett. 112151601Tarun Grover, Phys. Rev. Lett. 112, 151601 (2014). . Max A Metlitski, David F Mross, Subir Sachdev, T Senthil, arXiv:1403.3694Max A. Metlitski, David F. Mross, Subir Sachdev and T. Senthil, arXiv:1403.3694. . V Borokhov, A Kapustin, X Wu, JHEP. 1244V. Borokhov, A. Kapustin, and X. Wu, JHEP. 12, 044 (2002). . S Silviu, Pufu, Phys. Rev. D. 8965016Silviu S. Pufu, Phys. Rev. D 89, 065016 (2014). . O I Motrunich, Phys. Rev. B. 7245105O.I. Motrunich, Phys. Rev. B 72, 045105 (2005). . S.-S Lee, P A Lee, Phys. Rev. Lett. 9536403S.-S. Lee and P. A. Lee, Phys. Rev. Lett. 95, 036403 (2005). . B I Halperin, Patrick A Lee, Nicholas Read, Phys. Rev. B. 477312B. I. Halperin, Patrick A. Lee and Nicholas Read, Phys. Rev. B 47, 7312 (1993).
[]
[ "Towards the optimal window for the 2MASS dipole", "Towards the optimal window for the 2MASS dipole" ]
[ "1⋆Michał J Chodorowski \nCopernicus Astronomical Center\nBartycka 1800-716WarsawPoland\n", "Jean-Baptiste Coiffard \nUniversité Paris XI\n91400OrsayFrance\n", "Maciej Bilicki \nCopernicus Astronomical Center\nBartycka 1800-716WarsawPoland\n", "Stéphane Colombi \nInstitut d'Astrophysique de Paris\nCNRS\n98 bis Boulevard Arago75014ParisFrance\n", "Paweł Ciecielag \nCopernicus Astronomical Center\nBartycka 1800-716WarsawPoland\n" ]
[ "Copernicus Astronomical Center\nBartycka 1800-716WarsawPoland", "Université Paris XI\n91400OrsayFrance", "Copernicus Astronomical Center\nBartycka 1800-716WarsawPoland", "Institut d'Astrophysique de Paris\nCNRS\n98 bis Boulevard Arago75014ParisFrance", "Copernicus Astronomical Center\nBartycka 1800-716WarsawPoland" ]
[ "Mon. Not. R. Astron. Soc" ]
A comparison of the 2MASS flux dipole to the CMB dipole can serve as a method to constrain a combination of the cosmological parameter Ω m and the luminosity bias of the 2MASS survey. For this constraint to be as tight as possible, it is necessary to maximize the correlation between the two dipoles. This can be achieved by optimizing the survey window through which the flux dipole is measured. Here we explicitly construct such a window for the 2MASS survey. The optimization in essence reduces to excluding from the calculation of the flux dipole galaxies brighter than some limiting magnitude K min of the near-infrared K s band. This exclusion mitigates nonlinear effects and shot noise from small scales, which decorrelate the 2MASS dipole from the CMB dipole. Under the assumption of negligible shot noise we find that the optimal value of K min is about five. Inclusion of shot noise shifts the optimal K min to larger values. We present an analytical formula for shot noise for the 2MASS flux dipole, to be used in follow-up work with 2MASS data.The misalignment angle between the two dipoles is a sensitive measure of their correlation: the higher the correlation, the smaller the expectation value of the angle. A minimum of the misalignment is thus a sign of the optimal gravity window. We model analytically the distribution function for the misalignment angle and show that the misalignment estimated by Maller et al. is consistent with the assumed underlying model (though it is greater than the expectation value). We predict with about 90% confidence that the misalignment will decrease if 2MASS galaxies brighter than K min = 5 mag are excluded from the calculation of the flux dipole. This prediction has been indirectly confirmed by the results of Erdogdu et al. The measured misalignment constitutes thus an alternative way of finding the optimal value of K min : the latter corresponds to a minimum of the former.
10.1111/j.1365-2966.2008.13432.x
[ "https://arxiv.org/pdf/0706.0619v2.pdf" ]
14,711,268
0706.0619
01081014c57181060e50452b6d79309285669953
Towards the optimal window for the 2MASS dipole 18 April 2008 1⋆Michał J Chodorowski Copernicus Astronomical Center Bartycka 1800-716WarsawPoland Jean-Baptiste Coiffard Université Paris XI 91400OrsayFrance Maciej Bilicki Copernicus Astronomical Center Bartycka 1800-716WarsawPoland Stéphane Colombi Institut d'Astrophysique de Paris CNRS 98 bis Boulevard Arago75014ParisFrance Paweł Ciecielag Copernicus Astronomical Center Bartycka 1800-716WarsawPoland Towards the optimal window for the 2MASS dipole Mon. Not. R. Astron. Soc 00018 April 2008arXiv:0706.0619v2 [astro-ph] (MN L A T E X style file v2.2)methods: analytical -cosmology: large-scale structure of Universe -cosmol- ogy: cosmic microwave background -galaxies: general -galaxies: infrared -galaxies: Local Group A comparison of the 2MASS flux dipole to the CMB dipole can serve as a method to constrain a combination of the cosmological parameter Ω m and the luminosity bias of the 2MASS survey. For this constraint to be as tight as possible, it is necessary to maximize the correlation between the two dipoles. This can be achieved by optimizing the survey window through which the flux dipole is measured. Here we explicitly construct such a window for the 2MASS survey. The optimization in essence reduces to excluding from the calculation of the flux dipole galaxies brighter than some limiting magnitude K min of the near-infrared K s band. This exclusion mitigates nonlinear effects and shot noise from small scales, which decorrelate the 2MASS dipole from the CMB dipole. Under the assumption of negligible shot noise we find that the optimal value of K min is about five. Inclusion of shot noise shifts the optimal K min to larger values. We present an analytical formula for shot noise for the 2MASS flux dipole, to be used in follow-up work with 2MASS data.The misalignment angle between the two dipoles is a sensitive measure of their correlation: the higher the correlation, the smaller the expectation value of the angle. A minimum of the misalignment is thus a sign of the optimal gravity window. We model analytically the distribution function for the misalignment angle and show that the misalignment estimated by Maller et al. is consistent with the assumed underlying model (though it is greater than the expectation value). We predict with about 90% confidence that the misalignment will decrease if 2MASS galaxies brighter than K min = 5 mag are excluded from the calculation of the flux dipole. This prediction has been indirectly confirmed by the results of Erdogdu et al. The measured misalignment constitutes thus an alternative way of finding the optimal value of K min : the latter corresponds to a minimum of the former. INTRODUCTION The dipole anisotropy of the cosmic microwave background (CMB) is interpreted as a direct measure, via the Doppler shift, of the motion of the Local Group (LG) relative to the CMB rest frame. The components of this motion of non-cosmological origin (the motion of the Sun in the Milky Way and the motion of the Milky Way in the LG) are known and can be subtracted (e.g., Courteau & van den Bergh 1999). When transformed to the barycenter of the LG, the motion is towards (l, b) = (273 • ± 3 • , 29 • ± 3 • ), and of amplitude vLG = 627 ± 22 km · s −1 , as inferred from the first-year WMAP data (Bennett et al. 2003). The kinematic interpretation of the CMB dipole is strongly ⋆ E-mail: [email protected] supported by its remarkable alignment with the dipole component of the large-scale galaxy distribution (often called the 'clustering dipole'), inferred from various all-sky surveys. In the gravitational instability scenario, this alignment is expected: peculiar velocities of galaxies are induced gravitationally and are thus strongly coupled to the large-scale matter distribution. Linear theory predicts the peculiar velocity of the LG, v, to be proportional to the LG peculiar acceleration, caused by the gravitational pull of surrounding matter inhomogeneities. Let us denote by δ the mass density contrast, δ ≡ ̺/̺ b − 1, where ̺ is the mass density of matter and ̺ b is its average value. The clustering dipole, g ≡ d 3 r 4π δ(r) r r 3 ,(1) is a quantity proportional to the peculiar gravitational acceleration c 0000 RAS (so we will call it interchangeably 'scaled gravity'), and can be estimated from a three-dimensional all-sky galaxy survey. In the linear regime, the relation between the velocity and the scaled gravity is v = H0f (Ωm)g . ( 2) Here, H0 is the Hubble constant, Ωm is the cosmic matter density parameter and f (Ωm) ≃ Ω 0.6 m (e.g., Peebles 1980). For a spherical survey d 3 r r/r 3 = 0, hence we can write g = d 3 r 4π 1 + δ(r) r r 3 = ̺ −1 b d 3 r 4π ̺(r)r r 2 .(3) In the following we will assume that dark matter (DM) in the Universe is entirely locked in DM halos of luminous galaxies. Modelling galaxies as point particles, the observed density field is ̺(r) = i miδD(r − ri), where δD is Dirac's delta; mi and ri are respectively the mass and the position of the i-th galaxy. Substituting this equation into Equation (3) (4) thus we see that the true gravitational acceleration equals to 4πG̺ b g. We will assume further that 'light traces mass', or that the mass-to-light ratio for galaxies is a universal constant, Υ. Then we can write g = ̺ −1 b i ΥLi 4πr i r 2 i = Υ ̺ b i Siri .(5) Here, Li is the luminosity of i-th galaxy and Si is its observed flux, Si = Li/4πr 2 i . In other words, since both the gravity and the flux fall off as distance squared, the gravitational acceleration of the LG is proportional to the dipole of the light distribution (i.e., the flux dipole) for a constant mass-to-light ratio. The sum in Equation (5) is in principle over all galaxies in the Universe, while in practice we have at our disposal only finite, usually flux-limited, catalogs of galaxies. In such catalogs, lower-mass dark matter halos will be underrepresented by the survey galaxies. To account for this, we write g = Υ ̺ b bL N i=1 Siri ,(6) where bL is the resulting luminosity bias and N is the total number of galaxies in a given survey. Combining Equation (2) with Equation (6) we obtain finally (Erdogdu et al. 2006;hereafter E06) v = H0Ω 0.6 m ̺LbL N i=1 Siri . In the above we have used the fact that the mass-to-light ratio Υ = ̺ b /̺L, where ̺L is the luminosity density of the Universe. Equation (7) shows that in the linear theory one can predict the LG peculiar velocity using solely an angular (two-dimensional) all-sky survey, bypassing the lack of radial information, i.e. distances. Specifically, a comparison between the CMB dipole and the flux dipole of a given survey can yield an estimate of the parameter β ≡ Ω 0.6 m /bL. Such a comparison was first performed by Yahil, Sandage & Tamman (1980) using the revised Shapley-Ames catalogue and by Davis & Huchra (1982) using the CfA catalogue, leading to the estimates of the flux dipoles that were within 30 • from the CMB dipole. The inclusion of redshift information, usage of progressively larger redshift surveys and theoretical improvements of the analyses led to smaller measured values of the misalignment. In particular, using the IRAS 1.2 Jy survey, Strauss et al. (1992, hereafter S92) found that the clustering dipole points around 25 • away from the CMB dipole. Using the further completed IRAS PSCz survey, Schmoldt et al. 1999 (hereafter S99) obtained the clustering dipole within 15 • of the CMB dipole. A similar analysis, based also on the IRAS PSCz survey, performed by Rowan-Robinson et al. (2000), determined the misalignment angle to be around 13 • . Two most recent analyses of the clustering dipole employed the Two Micron All Sky Survey (2MASS; Skrutskie et al. 1997). In particular, to compute the flux dipole, Maller et al. (2003;hereafter M03) used the angular 2MASS extended source catalogue, with a limiting magnitude of Ks = 13.57. (Approximately 740,000 galaxies covering 90% of the sky.) E06 used the Two Micron All Sky Redshift Survey (2MRS): approximately 23,200 2MASS galaxies with measured redshifts, selected from a total sample of about 24,800 galaxies with (extinction-corrected) magnitudes smaller than Ks = 11.25. 2MASS is the first near-infrared (JHKs passbands) all-sky survey. While most passbands tend to be sensitive to the instantaneous star formation rate, Ks passband is most sensitive to total stellar mass (Bell & de Jong 2001;Bell et al. 2003), making this band a better tracer of total mass. 2MASS has an effective image resolution of 1" and a hundred times greater sensitivity than the farinfrared IRAS survey. The photometric uniformity of the 2MASS survey is better than 4 per cent over the entire sky including the celestial poles (e.g., Jarrett et al. 2003). The median depth of the survey is 220 h −1 Mpc (Bell et al. 2003), a distance past where the clustering dipole has been shown to converge. 1 Given all these advantages of 2MASS over other all-sky galaxy surveys, it is perhaps surprising that the misalignment between the CMB dipole and the 2MASS flux dipole is not smaller than the corresponding one for IRAS galaxies. The value obtained by M03 is 16 • . For the 2MRS flux dipole, E06 obtained approximately 21 • . 2 In this paper we aim at answering the following questions. First: do we understand fully the origin of this misalignment? Second: can one do better with 2MASS, and if so, how? The answer to these questions is essential for optimal estimation of the parameter β by comparing the CMB dipole to the 2MASS dipole. The stronger is the correlation between the two dipoles, the smaller are statistical errors of such an estimate. Therefore, the observational window, through which the 2MASS dipole is measured, should be adapted to obtain the best correlation possible. The misalignment angle is a sensitive measure of this correlation: the higher the correlation, the smaller the angle. In other words, a minimum of the misalignment angle is a sign of the optimal 2MASS window. In this paper we will formally prove these statements. First, we will derive the 2MASS window. Next, we will optimize it under the assumption of negligible shot noise. Finally, we will demonstrate that a minimum of the expectation value of the angle corresponds to minimal variance of the resulting estimate of β. 1 The inclusion of galaxy redshifts in the dipole analyses allowed the estimation of the convergence depth, i.e. the distance at which most of the clustering dipole is generated. There is a controversy whether this convergence depth is about 50 h −1 Mpc, or rather 200 h −1 Mpc (for details see E06). In either case, 2MASS is deep enough to provide a reliable estimate of the clustering dipole. (But see Basilakos & Plionis 2006.) 2 E06 computed two kinds of the clustering dipole. The second one, the number dipole, was even more misaligned with the CMB dipole. Let us enumerate possible sources of the misalignment between the CMB dipole and an all-sky galaxy survey flux dipole. • M/L = const. The constant mass-to-light ratio is probably a good assumption for (almost) all galaxies when averaged over many galaxies of the same luminosity. For individual galaxies, however, M/L is expected to have some scatter. On the other hand, as stated earlier, 2MASS, unlike IRAS surveys, is mainly sensitive to total stellar mass. Consequently, the mass-to-light ratio of 2MASS galaxies is expected to have smaller scatter than that of IRAS galaxies. • Nonlinear bias. Writing Equation (6) we have implicitly assumed that the total flux dipole and the magnitude-limited flux dipole differ in the amplitude, but not in the direction. However, if large-scale distribution of low-mass DM halos is different from the distribution of high-mass halos, then the two dipoles will not be collinear. • Nonlinear dynamics. The peculiar velocity of the LG is equal to the temporal integral of the LG gravitational acceleration along the LG trajectory. Therefore, loosely speaking, while the response of the LG acceleration to growing nonlinearities is 'instantaneous", the response of the LG velocity is time-averaged and 'retarded'. As a result, the acceleration of the LG is more non-linear and higher in amplitude (in velocity units) than the velocity of the LG (Ciecielag et al. 2003). What is more relevant here, at orders higher than linear non-local character of gravity reveals itself and tends to misalign the velocity vector of the LG with the vector of its acceleration. However, the mean misalignment angle between the velocity and gravity of the LG-like regions simulated in numerical experiments is about 8 • (Davis et al. 1991, Ciecielag et al. 2001. • Observational effects: shot noise, finite volume of the survey, and the mask (due to the zone of avoidance, ZoA). Shot noise and finite volume are more an issue for 2MRS, which has a median depth of only 60 h −1 Mpc. Still, we devote Subsection 5.1 to a study of shot noise of the 2MASS dipole. M03 perform two standard treatments of the masked area: in one of them they clone the sky above and below the masked region; in another they fill the masked region with randomly chosen galaxies such that it has the same surface density as the unmasked area. A recent paper by Tully et al. (2008) puts these methods, at least partly, in question. They show that there lies a void in the ZoA, which they call the Local Void; the LG lies on its boundary. Therefore, in a part of the mask there is really nothing, and filling this region with faked galaxies leads to a systematic error of the estimate of the LG acceleration. However, the role of the Local Void is a recently raised issue and we will study it elsewhere. M03 and E06 notice that the misalignment is substantially reduced if they remove the brightest galaxies in the catalog. M03 remove all galaxies brighter than Ks = 8 mag (375 galaxies), while E06 remove just five the brightest. They suggest that these galaxies have M/L = const and/or non-linearly contribute to the acceleration. We will study these issues here. Specifically, the outline of this paper is as follows. In Section 2, we will present a formalism which will allow us to model semi-analytically the distribution function for the misalignment angle between the CMB dipole and the 2MASS flux dipole. In Section 3, we will model nonlinear effects which appear in such an analysis. The 2MASS gravity window will be derived and optimized in Section 4. In Section 5 we will account for observational errors. In Section 6 we will present a formal proof that, under the assumption of negligible shot noise, our window is indeed optimal. We will also demonstrate how to optimize the window in presence of shot noise. In Section 7 we will show the resulting distribution function for the misalignment angle. A summary and conclusions will be given in Section 8. ANALYTICAL DESCRIPTION OF THE MISALIGNMENT In this Section we will model theoretically the probability distribution function (PDF) for the misalignment angle between the CMB dipole and the 2MASS flux dipole. The CMB dipole estimates the peculiar velocity of the LG, v. The 2MASS flux dipole, Equation (6), estimates the gravitational acceleration -more specifically, the scaled gravity -of the LG, g, induced by large-scale matter inhomogeneities traced by 2MASS galaxies. Let p(g, v) denote the joint PDF for the LG scaled gravity and peculiar velocity. It is a standard practice to approximate it by a multivariate Gaussian (S92; S99). Numerical simulations (Kofman et al. 1994, Ciecielag et al. 2003 show that nongaussianity of fully nonlinear g and v is indeed small. This is not surprising since, e.g. gravity is an integral of density over a large volume (Eq. 1), so the central limit theorem can at least partly be applicable. Using statistical isotropy of g and v, their joint PDF can be simplified to the form (Juszkiewicz et al. 1990;Lahav, Kaiser & Hoffman 1990): p(g, v) = (1 − ρ 2 ) −3/2 (2π) 3 σ 3 g σ 3 v exp − x 2 + y 2 − 2ρµxy 2(1 − ρ 2 ) ,(8) where σg and σv are the r.m.s. values of a single Cartesian component of gravity and velocity, respectively. From isotropy, σ 2 g = g · g /3 and σ 2 v = v · v /3, where · denote the ensemble averaging. Next, (x, y) = (g/σg, v/σv), and µ = cos θ with θ being the misalignment angle between g and v. Finally, ρ is the crosscorrelation coefficient of gm with vm, where gm (vm) denotes an arbitrary Cartesian component of g (v). From isotropy, ρ = g · v g 2 1/2 v 2 1/2 .(9) Also from isotropy, xmyn = ρ δmn ,(10) where δmn denotes the Kronecker delta. In other words, there are no cross-correlations between different spatial components. For a given all-sky galaxy survey, the LG gravity is measured effectively through the window of the survey, Wg (cf. Eq. 1): g = d 3 r 4π δ(r)Wg(r) r r 3 .(11) In contrast, the LG velocity is not estimated from a velocity survey (i.e., from a catalog of peculiar velocities of galaxies), but measured directly from the dipole anisotropy of the CMB. Still, to relate it to theoretical quantities, we write: v = d 3 r 4π ϑ(r)Wv(r) r r 3 .(12) Here ϑ ≡ −∇ · v is the (minus) velocity divergence and we assume that the velocity field is irrotational. 3 Thus, similarly to g, v can be expressed as a Coulomb (Newton) integral over its source, i.e. the field of the velocity divergence. Here we do not assume that we know the latter from observations, but we know from theory its statistical relation to the density field (see this Section and Section 3). This is sufficient for our purposes in this work. Since v is directly measured from the CMB dipole, the effective velocity window, Wv, which we have introduced in Equation (12), is essentially unity. (Contributions from all perturbations are included.) 4 We modify slightly this form of the window to reflect the finite size of the LG. Following S92 and S99, we adopt Wv = 0, r < rLG , 1, otherwise ,(13) which has a small-scale cutoff, rLG = 1 h −1 Mpc. This window is markedly different from those appropriate for velocity surveys: the latter are not spherical, have complicated shapes and finite depth (Sarkar, Feldman & Watkins 2007). The gravity window, Wg, of the 2MASS survey is derived in Section 4. In Fourier space, relations (11) and (12) read: g k = ik k 2 δ k Wg(k),(14)v k = ik k 2 ϑ k Wv(k),(15) where the subscript k denotes the Fourier transform. The quantity W is related to the window W by the following equation (S92): W (k) ≡ k ∞ 0 W (r)j1(kr)dr .(16) Here and below j l represents the spherical Bessel function of first kind of order l. In particular, Wv(k) = j0(krLG) .(17) From equations (14) and (15) we have g · g = 1 2π 2 ∞ 0 W 2 g (k)P (k)dk ,(18) and v · v = 1 2π 2 ∞ 0 W 2 v (k)P ϑ (k)dk .(19) Here, P (k) and P ϑ (k) are respectively the power spectrum of the density and the power spectrum of the velocity divergence. Defining R(k) = P ϑ (k) P (k) ,(20) we have v · v = 1 2π 2 ∞ 0 W 2 v (k)R(k)P (k)dk .(21) Furthermore, g · v = 1 2π 2 ∞ 0 Wg(k) Wv(k)C(k)P 1/2 ϑ (k)P 1/2 (k)dk,(22) where C(k) is the so-called coherence function (CF), or the correlation coefficient of the Fourier components of the gravity and velocity fields (S92): C(k) ≡ g k · v * k |g k | 2 1/2 |v k | 2 1/2 = δ k ϑ * k |δ k | 2 1/2 |ϑ k | 2 1/2 .(23) Hence, we obtain finally ρ = ∞ 0 Wg(k) Wv(k)C(k)R 1/2 (k)P (k)dk ∞ 0 W 2 g (k)P (k)dk 1/2 ∞ 0 W 2 v (k)R(k)P (k)dk 1/2 .(24) Equations (18), (21) and (24) specify all the parameters (the variances and the correlation coefficient) that determine the joint PDF for g and v, Equation (8), in the absence of observational errors. The deviation of the correlation coefficient from unity is then due to different windows, through which the gravity and the velocity of the LG are measured, and due to nonlinear effects. The latter are described by two functions: the CF, and the ratio of the power spectra (Ciecielag & Chodorowski 2004; hereafter C04). The distribution for the misalignment angle can be derived from the joint distribution (8). Here we are interested in the distribution for the misalignment angle with the observed value of the LG velocity as a constraint. The conditional distribution function, p(g|v), readily results from (8): (Juszkiewicz et al. 1990;Lahav et al. 1990). The distribution for the amplitude of the LG acceleration and the cosine of the misalignment angle is p(g, µ|v) = 2πg 2 p(g|v). The distribution for µ is obtained by marginalizing over g, p(g|v) = (2π) −3/2 σ −3 g (1 − ρ 2 ) −3/2 exp − (x − ρy) 2 2(1 − ρ 2 )(25)p(µ|v) = 2π ∞ 0 dg g 2 p(g|v) ,(26) and the distribution for the angle itself is p(θ|v) = |dµ/dθ| p(µ|v) = sin(θ) p(µ|v). This yields (Juszkiewicz et al. 1990;Lahav et al. 1990) p(θ|v) = sin(θ) exp −q 2 qµ √ π + 1 2 + q 2 µ 2 exp q 2 µ 2 [1 + erf(qµ)] ,(27) where q = ρy 2(1 − ρ 2 ) .(28) We remind that y is the amplitude of the LG peculiar velocity in units of the 1D velocity dispersion, y = vLG/σv. For vLG = 627 km · s −1 (Bennett et al. 2003) and the values of the cosmological parameters adopted here (as described in Section 3), y = 2.64. This might suggest that the velocity of the LG is a rare event; however, this is on the contrary. First, one should compare the amplitude of the LG velocity to the 3D velocity dispersion, σv,3D = √ 3σv. Therefore, the relevant parameter here is y ′ ≡ vLG/σv,3D = 1.52. Second, the probability that a randomly chosen region will have velocity greater than vLG is equal to ∞ 1.52 dy ′ h(y ′ ), where h(y) = 2/π y 2 e −y 2 /2 is the Maxwellian distribution. For the lower limit of the integral equal to 2.64, the value of the integral is 0.07 = 7%. However, for 1.52, its value is 0.51 = 51%. The misalignment of only several degrees corresponds to a strong coupling between g and v. In the strong coupling limit erf(s) ≃ 1 − 1 √ πs e −s 2 for s ≫ 1 .(29) We then obtain a small-angle approximation of the distribution for the misalignment angle (Lahav et al. 1990): p(θ|v) ≃ θ θ 2 * exp − θ 2 2θ 2 * ,(30)with θ * = 1 − ρ 2 ρy .(31) Thus, in the strong coupling limit the misalignment angle, given the velocity constraint, is Rayleigh-distributed. The parameter θ * , much smaller than unity (in radians), is a characteristic measure of the misalignment. 5 The expectation value of the angle is θ|v = π 2 θ * .(32) Other quantities characterizing the distribution which are of interest here are quantiles. In our, slightly modified notation, the quantile θq of a distribution p(θ) is such a number, that θq θ min p(θ) dθ = q 100 .(33) For the Rayleigh distribution, θmin = 0. For our purposes, interesting quantiles are θ10 = 2 ln(10/9) θ * , and θ90 = √ 2 ln 10 θ * . NONLINEAR EFFECTS Using numerical simulations, C04 modelled the CF (Eq. 23) and the ratio of the power spectra (Eq. 20). The simulations were evolved from Gaussian initial conditions. As the initial power spectrum of matter fluctuations, a cold dark matter (CDM) spectrum was adopted (as in Eq. 7 of Efstathiou, Bond & White 1992), with the shape parameter Γ = 0.19. Both the CF and the ratio of the power spectra were modelled as functions of the wavevector, k, and the amplitude of the matter fluctuations, σ8. For the CF, C04 found the following fit: C(k) = 1 + (a0k − a2k 1.5 + a1k 2 ) 2.5 −0.2 ,(35) with the coefficients given by the following, power-law, scaling relations in σ8: a0 = 4.908 σ 0.750 8 , a1 = 2.663 σ 0.734 8 , (36) a2 = 5.889 σ 0.714 8 . The fit was calculated for k ∈ [0, 1] h Mpc −1 and σ8 ∈ [0.1, 1], with the imposed constraint C(k = 0) = 1. This constraint assures that for sufficiently large, linear scales, the relation between the gravity and the velocity is deterministic and linear (see Eq. 2). Formula (35) is a better fit to the CF than an earlier formula of Chodorowski & Ciecielag (2002), which was less accurate for low values of k. Chodorowski & Ciecielag (2002) investigated numerically also the dependence of the CF on Ωm and found it to be extremely weak. Defining the scaled velocity divergence,θ ≡ Ω −0.6 m ϑ = −Ω −0.6 m ∇ · v, C04 found the following fit for the ratio of the power spectra: R(k) = [1 + (7.071k) 4 ] −α ,(37) where α = −0.06574 + 0.29195σ8 for 0.3 < σ8 < 1 .(38) C04 argued that the ratio of the power spectra practically does not depend on the background cosmological model. This ratio is unity in the linear regime (k ≪ 1) but decreases in the nonlinear regime, because the velocity grows slower than it would be expected from the linear approximation. In this paper we use Equations (35) and (37) as the formulas respectively for the CF and the ratio of the power spectra. For σ8 we adopt the value obtained from a joint analysis of third-year WMAP and SDSS, σ8 = 0.77 (Spergel et al. 2007). In Equations (18), (21) and (24), as the power spectrum we use a CDM spectrum. For zero baryon content, the shape parameter of the spectrum, Γ, equals simply to Ωmh. Non-zero baryon content of the Universe modifies the shape parameter to (Sugiyama 1995 ) Γ eff = Ωmh exp −Ω b 1 + √ 2h/Ωm .(39) Here we adopt Γ eff = 0.15, the value obtained both from firstyear WMAP (Spergel et al. 2003) and a joint analysis of third-year WMAP and SDSS (Spergel et al. 2007). This value is in excellent agreement with the constraint on the shape of the power spectrum of 2MASS galaxies, Γ eff = 0.14 ± 0.02, obtained by Frith, Outram and Shanks (2005; assuming a flat ΛCDM cosmology, a primordial scale-invariant power spectrum and negligible neutrino mass). It is slightly higher than the corresponding result of Maller et al. (2005), Γ eff = 0.12 ± 0.01, obtained using a measure of the three-dimensional power spectrum via an inversion of the 2MASS angular correlation function. GRAVITY WINDOW OF 2MASS In this Section we derive the gravity window of the 2MASS survey. The 2MASS survey is dense, uniform and has an unprecedented sky coverage. Therefore, to a good accuracy it can be described by a spherical window. Since distances of 2MASS galaxies are unknown, the galaxies are weighted only by their fluxes and not by their distances (like, e.g., by the inverse of the selection function). This is the central assumption of the calculation below. The background light intensity due to uniform distribution of discrete sources is I = S dN,(40) where S = L/(4πr 2 ) is the observed flux from the sources with intrinsic luminosity L and dN is the number of sources per steradian. In the case of uniformly distributed sources with the luminosity function (LF) Φ(L), the contribution from a shell of thickness dr and radius r is dN = Φ(L) dL r 2 dr. For a flux-, or magnitudelimited survey, only galaxies with L > 4πr 2 Smin are observed, where Smin is the limiting (minimal) flux. Hence, I = ∞ 0 ∞ 0 Θ(r, L) L 4πr 2 Φ(L) dL r 2 dr,(41) where Θ(r, L) is the Heaviside step-function, ΘH (L−4πr 2 Smin). Writing Lmin ≡ 4πr 2 Smin, this yields I = L N0 4π ∞ 0 dr Wg(r).(42) Here , N0 = ∞ 0 Φ(L) dL, and L = ∞ 0 LΦ(L) dL ∞ 0 Φ(L) dL(43) is the average luminosity of the population. The flux window of the survey is Wg(r) = ∞ L min LΦ(L) dL ∞ 0 LΦ(L) dL .(44) Wg gives the percentage of the total light from distance r which is included in the survey. Loosely speaking, it suppresses contributions from distances larger than L /(4πSmin). Its detailed form is determined by the LF. The LF of 2MASS galaxies has been estimated by Bell et al. (2003), by matching a spectroscopic sample of Early Data Release SDSS galaxies with the 2MASS extended source catalog, to obtain redshifts for a subsample of 2MASS galaxies. Bell et al. fitted the 2MASS LF by the Schechter function: Φ(L) dL = Φ * L L * α exp − L L * dL L * ,(45) where Φ * is the LF normalization, L * is the characteristic luminosity at the 'knee' of the LF, where the form changes from exponential to power law, and α is the 'faint end slope'. For Ks-band, they found that α = −0.77 and the absolute magnitude M * , corresponding to the absolute luminosity L * , is M * = −23.29 + 5 log 10 h. We adopt this form of the 2MASS LF here. The flux window of the 2MASS survey can be compared with the selection function of the survey, defined as φ(r) = ∞ L min Φ(L) dL ∞ 0 Φ(L) dL .(46) The selection function gives the probability that a randomly selected galaxy at distance r will be included in the survey. E06 call the selection function the 'number-weighted selection function', and the flux window the 'flux-weighted (or luminosity-weighted) selection function'. They note that "the number-weighted selection function drops with the distance faster than the luminosityweighted selection function. At large distances, we observe only the most luminous galaxies, so the amount of 'missing' luminosity from a volume of space is not as big as the number of 'missing' galaxies". We fully agree. This implies in practice that when estimated from a galaxy survey, the flux dipole is a more robust quantity than the number dipole. If we want to exclude also the brightest sources, then Θ(r, L) in Equation (41) becomes the product of two Heaviside functions, Θ(r, L) = ΘH (L − 4πr 2 Smin) · ΘH(4πr 2 Smax − L). Here, Smax is the upper limiting (maximal) flux. It is simple to check that the survey window then becomes Wg(r) = Lmax L min LΦ(L) dL ∞ 0 LΦ(L) dL ,(47) where Lmax ≡ 4πr 2 Smax. This windows suppresses also contributions from distances smaller than about L /(4πSmax), or, for the Schechter LF, just about L * /(4πSmax). What would be the gravity window for the number dipole? The answer depends on the weighting scheme. In case of the number dipole all galaxies are weighted equally -in a sense that they are not weighted by their fluxes or masses -but they may, or may not, be weighted proportionally to the inverse of the selection function. If they are not, it is clear from the above analysis that then the gravity window is the selection function, φ. (Now, instead of missing some percentage of the total light from distance r, we miss some percentage of all galaxies located there.) However, if they are weighted as 1/φ(r), the gravity window is simply unity. This is so because the 1/φ(r) weighting corrects for, on average, missing signal from large distances. The price to pay for this correction is huge variance of such an estimator of the LG gravity. This variance is called shot noise (see Subsection 5.1) and has dominant contributions from large scales. Number-, rather than mass-, weighting of galaxies is another source of variance of this estimator, for simplicity also called shot noise. That shot noise comes predominantly from small scales. Using the IRAS 1.2 Jy survey, S92 measured the number dipole, weighting galaxies originally as 1/φ(r). To mitigate shot noise and nonlinear effects from small scales and shot noise from large scales, S92 decided to modify these weights. They did this introducing the so-called standard IRAS window, WI RAS = (r/rs) 3 , r < rs , 1, rs < r < Rmax , 0, Rmax < r .(48) This window is characterized by a small-scale smoothing, rs, and a sharp large-scale cutoff, Rmax. S99 adopted the values rs = 5 h −1 Mpc and Rmax = 150 h −1 Mpc, appropriate for the complete PSCz catalog (Saunders et al. 2000). Modified weights assigned to IRAS galaxies by S92 and S99 were weight(i) = WI RAS (ri) φ(ri) .(49) It is clear that in this case, the gravity window of the IRAS number dipole is Wg, I RAS = WI RAS . We will return to this point later. To specify completely the distribution for the misalignment angle (Eq. 27, or its small-angle approximation, Eq. 30), we need the value of the velocity variance, Equation (21), and of the correlation coefficient, Equation (24). In order to calculate the latter, we need to derive the Fourier form (Eq. 16) of the 2MASS window, given above. In Equation (16), we can use the fact that the spherical Bessel function, j1(x) = −(d/dx)j0(x), and integrate by parts. This yields Wg(k) = ∞ 0 j0(kr) W ′ g (r) dr.(50) Let's cast Equation (47) to the form Wg(r) = Lmax L min Ψ(L) dL ∞ 0 Ψ(L) dL ,(51) where Ψ(L) ≡ L Φ(L) .(52) We can then write W ′ g (r) = ∂Wg(r) ∂Lmax dLmax dr + ∂Wg(r) ∂Lmin dLmin dr = Ψ(Lmax) 8πrSmax − Ψ(Lmin) 8πrSmin ∞ 0 Ψ(L) dL .(53) Using Equations (50) and (53), and the fact that j0(x) = sin x/x, we obtain Wg(k) = 8πSmax k ∞ 0 Ψ(L) dL ∞ 0 sin(kr)Ψ(Lmax)dr − 8πSmin k ∞ 0 Ψ(L) dL ∞ 0 sin(kr)Ψ(Lmin)dr,(54) where Ψ(L) is defined by Equation (52). Note that the flux (or gravity) window does not appear in Equation (21) for the velocity variance, and in Equation (24) for the correlation coefficient it appears in such a way that its absolute normalization cancels out. In other words, the PDF for the misalignment angle is sensitive only to the shape of the gravity window. To relate the limiting fluxes to the limiting magnitudes, we remind that the observed minimal flux Smin is related to the apparent maximal magnitude Kmax in the following way: Smin = S0 10 −0.4Kmax ,(55) where S0 is the reference flux, which appears also in the relation between the absolute magnitude M * and absolute luminosity L * , M * = −2.5 log 10 L * 4π(10 pc) 2 S0 . In Equations (55)-(56) we can therefore eliminate S0, obtaining Smin = 1.803 × 10 −5 L * 4π(1h −1 Mpc) 2 .(57) Calculating the numerical coefficient in the above equation we have adopted M * = −23.29 + 5 log 10 h (Bell et al. 2003). Following M03, for Kmax we have adopted the value 13.57. The reason for this choice of Kmax is twofold. First, our aim here is to improve the 2MASS window used by M03 properly accounting for nonlinear effects, which affect only the choice of optimal Kmin (the minimal magnitude). Second, M03 chose Kmax = 13.57 because "the extended source catalog is 97.5% complete within the SDSS early data release for extinction-corrected Kron magnitudes of Ks 13.57 mag" (Bell et al. 2003, Jarrett 2004. The 2MASS window is the 2MASS flux-weighted selection function under the assumption that the survey is complete within the flux limits. If we wanted to go deeper, we should account for increasing incompleteness as a function of distance. However, Figure 1 of M03, showing the convergence of the 2MASS dipole as a function of the limiting magnitude, suggests that contributions from all galaxies (i.e. even from those not included in the survey) fainter than 13.57 mag are most likely negligible. Even for 2MASS galaxies brighter than 13.57 mag (where the catalog is complete), "the faintest 300,000 galaxies only change the dipole value by less than 5%" (M03). For Smax we have simply Smax = Smin10 0.4(13.57−K min ) .(58) If the brightest galaxies are not excluded, then either directly from Equation (44), or from Equation (54), performing the limit Smax → ∞, we obtain (54) and describe the cases of excluding all 2MASS galaxies brighter than K min = 8 mag and K min = 5 mag, respectively. Since all 2MASS windows have the same Kmax, they are similarly suppressed at large scales, i.e., at small k. For reference, we also plot the standard IRAS window (Eq. 60), with rs = 5 h −1 Mpc and Rmax = 150 h −1 Mpc (dot-long-dashed line). Wg(k) = 1 − 8πSmin k ∞ 0 Ψ(L) dL solid lines are plotted using Equation (54) and describe respectively the cases of excluding all 2MASS galaxies brighter than Kmin = 8 mag (as done by M03), and Kmin = 5 mag (our choice, as justified below). Let's try to understand the influence of the limiting magnitudes on the shape of the gravity window. Since all 2MASS windows have the same Kmax, they are similarly suppressed at large scales (small k). For a given Kmin (corresponding to maximal limiting flux), all objects brighter than L * are excluded from distances smaller than rmin = L * /(4πSmax), and at distances r < rmin, all sources brighter than L * (r/rmin) 2 are excluded. For the limiting magnitude Kmin = 5, rmin ≃ 4.5 h −1 Mpc, while for Kmin = 8, rmin ≃ 18.1 h −1 Mpc. Consequently, the window for no exclusion of the brightest galaxies (dotted line) does not drop down at all for large k (small scales). The window for Kmin = 5 (solid line) does drop down but is fairly wide, while the window for Kmin = 8 (dashed line) drops very rapidly. Since, as explained earlier, Wg, I RAS = WI RAS , using Equation (48) we have Wg, I RAS (k) = 3j1(krs) krs − j0(kRmax) .(60) For reference, we plot this standard IRAS window in Figure 1 (dotlong-dashed line). The small scale smoothing of the IRAS window is rs = 5 h −1 Mpc, while for the 2MASS window with Kmin = 5, the effective smoothing scale rmin is about 4.5 h −1 Mpc. It is not therefore surprising that at small scales the IRAS window, except for its oscillatory behaviour, decreases fairly similarly to this 2MASS window. Even neglecting shot noise, suppressing contributions to the flux dipole from small scales is necessary, since nonlinear effects should be mitigated. For large k the coherence function of velocity with gravity (Eq. 35) drops significantly below unity, decreasing the value of the cross-correlation coefficient (Eq. 24). Then suppressing the gravity window for large k has almost no effect on the cross-term (which is the numerator of Eq. 24), while it decreases the gravity variance, the square root of which appears in the denominator of this equation. This manipulation on the gravity window helps therefore to achieve the best possible correlation between the LG velocity and gravity. However, when one suppresses the gravity window for scales which are linear enough so that the CF is close to unity, one worsens the correlation again. This is so because even for linear fields (CF and the ratio of power spectra equal to unity) the correlation coefficient decreases for increasingly different windows of velocity and gravity. As a result, for some value of rmin, or Kmin, the correlation coefficient will have a maximum. We calculate the correlation coefficient, Equation (24) (using the appropriate formulas for the CF and the ratio of the power spectra, and the velocity window given by Eqs. 13 and 17), for the 2MASS gravity window, for a range of values of the limiting magnitude Kmin. Results are shown in Figure 2. We see that ρ has a maximum (1 − ρ has a minimum) for Kmin ≃ 4.5. Either not suppressing small scales at all (Kmin = −∞), or suppressing them excessively (Kmin greater than, say, 6) clearly decreases the correlation coefficient. This implies larger statistical errors of the estimated cosmological parameters when comparing the 2MASS dipole to the CMB dipole (see Sec. 6); choosing the optimal value for Kmin is therefore very important. Instead of Kmin = 4.5, as the optimal value we have adopted Kmin = 5. We have done this because the correlation coefficient changes in the range of Kmin from 4 to 5 hardly at all, while the number of excluded galaxies for Kmin < 5 would become very small, resulting in big Poisson noise. In our analysis so far we have not addressed the effect of shot noise. If shot noise is not negligible it also increases the optimal Kmin. This will be explained in Section 6. Shot noise for the 2MASS flux dipole will be discussed in detail in Subsection 5.1. OBSERVATIONAL ERRORS An estimate of the flux dipole from an all-sky galaxy survey is subject to observational errors. One of them is shot-noise, due to dilute sampling, by distant galaxies, of the underlying mass density field. Another one is the lack or deficit of galaxies in the Zone of Avoidance (at low Galactic latitudes). We will describe these errors correspondingly in Subsections 5.1 and 5.2. Shot noise The contribution to the LG gravity from a small volume element located at a distance r, estimated from the 2MASS survey is gE = Υ ̺ b i ′ Siri ≃ a i ′ Li r ,(61) where a = Υ/(4πr 2 ̺ b ), ′ i denotes the sum over galaxies included in the survey andr is the unit vector towards the volume element. We model this quantity theoretically introducing a window function, W : Figure 2. The correlation coefficient, ρ (Eq. 24), for the 2MASS gravity window (Eq. 54), for a range of values of the limiting magnitude K min . On the ordinate, the value of 1 − ρ is plotted. The coefficient has a maximum (respectively, 1 − ρ has a minimum) for K min ≃ 4.5. gM = Υ ̺ b i W (ri)νiSiri ≃ aW (r) i νiLi r .(62) In the above we have accounted for the fact that the mass to light ratio for an individual galaxy, Υi, may not be equal to its average value, Υ, but may have some scatter. The quantity νi ≡ Υi/Υ; hence νi = 1. In Equation (62) In the second step we have used the fact that the 2MASS gravity window we have constructed in Section 4, Wg, gives the percentage of the total light from distance r which is included in the survey. N (∆V ) is the number of all galaxies in the volume element (regardless whether or not included in the survey). The expectation value of the modelled gravity is gM = aW (r) i νi Li r = aW (r) L N (∆V )r .(64) In Equation (64) we have assumed that the scatter in the massto-light ratio is independent of luminosity. However, the above result for gM is also correct for fairly broad classes of luminositydependent scatter, e.g. for ν = 1 + αF(L), where the luminosityindependent random variable α has zero mean. [In order for the variable ν to be always positive, we have to impose an additional constraint |αmin| < F −1 max , where αmin (negative since α = 0) is the minimum value of α and Fmax is the maximum value of the function F.] Comparing Equation (63) to (64) we see that if we adopt W = Wg, then the estimated gravity is an unbiased estimator of the modelled gravity. (From a different perspective, the modelled gravity is an unbiased estimator of what we really observe.) Our 2MASS gravity window is constructed precisely in such a way to assure this. However, the quantity gE has scatter around gM . For flux (or number) dipoles estimated from flux-limited galaxy catalogs, the dilute sampling at large distances introduces significant scatter, called shot noise. The scatter in the values of gE around gM due to the scatter in the mass-to-light ratio is not, properly speaking, shot noise (S92). However, for simplicity, we will call both these effects 'shot noise'. (S92 also follow this convention.) A lucid derivation of shot noise for the number dipole of IRAS galaxies can be found in Appendix A of S92. Shot noise for the 2MASS dipole can be calculated in a similar way. However, except for the fact that the 2MASS dipole is a flux one, there is another important difference. While in the derivation of S92, galaxies are assigned weights essentially proportional to the inverse of the selection function, 2MASS galaxies are given equal weights (their distances are unknown). We will see below that this introduces a qualitative difference in the resulting formula for shot noise. To compute the variance of gE we take the difference between gE and gM for a full shell of thickness dr, we square it and calculate its expectation value. Finally, we sum up contributions to the total variance from all shells. The result is: 6 σ 2 SN = ̺ −2 L i ′ S 2 i F (ri) 1 − 2Wg(ri) + (1 + Q) W 2 g (ri) φ(ri) . (65) Here, Q ≡ ν 2 − 1 0 quantifies the amount of scatter in M/L. Were there no scatter, the value of Q would be zero. The function F is F (r) = L min L 2 Φ(L) dL · L min Φ(L) dL L min LΦ(L) dL 2 .(66) The upper limit in the above integrals is either Lmax = 4πr 2 Smax or infinity, depending on whether we exclude the brightest objects or not (in the latter case, Smax = ∞). Let us now investigate contributions to shot noise from small (r → 0) and large (r → ∞) scales. For r → 0, consider first the case of no exclusion of the brightest objects. Then, both φ(r) and Wg(r) tend to unity. From Equation (66) it is obvious that then F (0) is a constant. Therefore, ∆σ 2 SN, nearby ∝ Q nearby, i ′ S 2 i .(67) If Q is significantly greater than zero, then the RHS of the above proportionality blows up (since for ri → 0, Si → ∞). This is shot noise from small scales, mentioned already in Section 4. It similarly plagues the number dipole (see Eq. 35 of S92, where there are similar r −4 i divergences). As already mentioned, to mitigate shot noise from small scales S92 introduced a window for the IRAS dipole. With inclusion of the IRAS window, contributions to shot noise from small scales in Equation (35) of S92 are proportional to W 2 I RAS (ri)/r 4 i . For the standard IRAS window (Eq. 48), they scale as r 2 i → 0 for ri → 0, so shot noise from small scales is indeed strongly suppressed. As already stated, 2MASS galaxies are assigned equal weights. Still, shot noise from small scales can be mitigated. This is achieved by excluding from the calculation of the dipole contributions from the brightest objects, as described below. For finite Smax, the selection function is 6 In this paper we need a formula for shot noise only for illustrative purposes, therefore the derivation will be presented in follow-up work. φ(r) = Lmax L min Φ(L) dL ∞ 0 Φ(L) dL .(68) Therefore, for r → 0 and finite Smax, both Wg(r) and φ(r) tend to zero. However, although φ(r) tends to zero, it is straightforward to verify that the quantity W 2 g (r)/φ(r) also tends to zero (at least for the Schechter form of the luminosity function). Finally, F (r) tends to a constant (though different from that for the case Smax = ∞). Hence, ∆σ 2 SN, nearby ∝ nearby, i S i <Smax ′ S 2 i .(69) The above sum is limited to objects with Si < Smax, what prevents it to blow up. Therefore, excluding the brightest objects is a good way to mitigate shot noise from small scales having at one's disposal angular data only. Contributions to shot noise from large scales do not depend on the choice whether we exclude the brightest objects, or not. For r → ∞, both φ(r) and Wg(r) tend to zero; it is straightforward to check that then also W 2 g (r)/φ(r) tends to zero. The limit of F (r) for r → ∞ is unity. Hence, using Equation (65) we obtain ∆σ 2 SN, distant = ̺ −2 L distant, i ′ S 2 i .(70) We see that in the case of the flux dipole calculated with equal weights assigned to all galaxies, shot noise from large scales does not blow up; on the contrary, it decreases. As a result, in an analysis of the 2MASS dipole one does not have to exclude any data from large distances. The analysis presented here assumed Kmin = 13.57, but it is now clear that when calculating the 2MASS dipole one can include contributions from 2MASS galaxies fainter than this magnitude. (Although, as mentioned earlier, Fig. 1 of M03, showing the convergence of the dipole as a function of the limiting magnitude, suggests that their contribution will be negligible. See also Fig. 3 of Jarrett 2004.) Large-scale asymptotic behaviour of shot noise for the flux dipole (Eq. 70) with equal weighting is in contrast to the corresponding behaviour of shot noise for the number dipole calculated with galaxy weights proportional to the inverse of the selection function. In the latter case, contributions to shot noise from large scales diverge as φ −2 (ri) (see Eq. 35 of S92; at large distances φ(ri) ≪ 1). 7 To cure this problem, S92 introduced in their standard IRAS window a sharp large-scale cutoff, Rmax (see Eq. 48). With inclusion of the IRAS window, contributions from large scales are proportional to W 2 I RAS /φ 2 (ri). If WI RAS is truncated at some Rmax, then there are no contributions to shot noise from scales beyond Rmax. Still, this does not imply that '1/φ(r)' weighting of distant galaxies is a good one. S92 were aware of this fact and constructed the optimal window for the IRAS survey, i.e. such that it minimized variance of the estimator of the LG velocity (Eq. 45 of S92). At large scales this window behaves asymptotically as J3(r)φ(r), where J3(r) = r 0 d 3 r ′ ξ(r ′ ) and ξ is the mass twopoint correlation function. Therefore, the optimal weighting at large distances is proportional to J3(r) (see Eq. 49), so instead of increasing [as φ −1 (r) does] it decreases to zero, suppressing shot noise from large scales. (2MASS weighting is intermediate between these two extremes.) At small scales, in the absence of the scatter in the masses of galaxies, the window approaches unity. Therefore, the 1/φ(r) weighting is then indeed the optimal one. 8 In the presence of scatter the window filters out small scales, as desired. Surprisingly, S92 resigned from using this window in the analysis of the LG acceleration and employed instead the standard IRAS window. The reason was that in the derivation of the optimal window they also attempted to account for nonlinear effects, but the coherence function they used was wrong (Chodorowski & Ciecielag 2002). As a consequence, the resulting window filtered out small scales excessively. In the present paper, working with only angular data we have no choice: we have to assign equal weights to all galaxies. Though this is not the optimal weighting, this is still quite good: shot noise from large scales does not blow up. Moreover, excluding the brightest objects helps to mitigate shot noise from small scales. Let us recall: we denote the estimated gravity of the LG by gE (Eq. 61) and its modelled gravity by gM (Eq. 62). Although gE is an unbiased estimator of gM , it is still a biased estimator of the true gravity of the LG. Large depth of the 2MASS survey makes the estimated dipole to converge, but in order to mitigate shot noise and nonlinear effects from small scales we have to suppress contributions from small distances. (For angular data the only way to do this is to exclude the brightest galaxies, located preferentially nearby.) This reduction of the signal introduces bias in the estimate of the LG gravity. However, applying a Maximum Likelihood analysis enables one to correct for this bias and to obtain an unbiased estimate of the parameter β = Ω 0.6 m /bL. This will be discussed in Section 6. How big is actual shot noise for the 2MASS survey? To answer this question, M03 performed bootstrap resampling on the 2MASS galaxy catalog (100 times). They found that the standard deviation of the dipole direction was a fraction of a degree, and of the dipole magnitude a fraction of a percent. They concluded that 'the systematic uncertainties are much larger than the shot noise'. Shot noise is certainly less an issue for the 2MASS dipole than for the IRAS PSCz dipole (S99) and for the 2MRS dipole (E06). It is smaller for the 2MASS dipole partly due to much bigger number of galaxies in this survey compared to IRAS PSCz and 2MRS: there are about 13, 000 galaxies in the PSCz catalog and 23, 000 galaxies in 2MRS, while for the limiting magnitude Ks = 13.57, the 2MASS catalog contains about 740,000 galaxies (M03). The main reason, however, is non-weighting of galaxies when calculating the 2MASS dipole. (E06 weighted 2MRS galaxies inversely to the 'flux-weighted selection function', or, in our terminology, the gravity window, Wg). Still, M03 analysed shot noise including all (so also the brightest) 2MASS galaxies. Therefore, it is somewhat surprising that they did not found a trace of shot noise from small scales. A forthcoming paper of some of us (Bilicki & Chodorowski, in preparation) will be devoted to the optimal measurement of the 2MASS dipole. We are planning to reexamine carefully the issue of shot noise there. Specifically, we are going to repeat the bootstrap resampling analysis and to compare its results to our analytical formula for shot noise, Equation (65). At first sight, it may seem surprising that Equation (65) can be used in the case of only angular data, since radial functions φ(r), Wg(r) and F (r) appear in it. However, these functions are uniquely determined by specifying Kmax (corresponding to Smin), Kmin (corresponding to Smax) and the luminosity function of the 2MASS galaxies. As described before, this luminosity function has been estimated e.g. by Bell et al. (2003). The only data employed in Equation (65) are fluxes, Si. (One also needs an estimate of Q, quantifying the amount of scatter in M/L.) Since we do not have these data at our disposal yet, for the rest of this paper we will accept the claim of M03 that shot noise for the 2MASS flux dipole is negligible. The mask The source of the biggest systematic error in 2MASS remains the lack or deficit of galaxies in the Zone of Avoidance (at low Galactic latitudes). M03 masked the region of the ZoA, and repopulated it with 'synthetic galaxies'. In one method they cloned the sky above and below the masked region. In another method, they filled 'the masked region with randomly chosen galaxies such that it has the same surface density as the unmasked area'. The first method gave a dipole pointing towards l = 263 • , b = 40 • . The second method resulted in a dipole pointing towards l = 266 • , b = 47 • . M03 adopted the mean of these two measurements as the best-fit dipole. However, the error bars they attributed to the mask-filling uncertainty were somewhat underestimated. We will return to this point later. The misalignment can be fully represented as a twodimensional vector lying on the celestial sphere. In the absence of shot noise, the total misalignment is a vectorial sum of the cosmologically-originated misalignment θc, described in Section 2, and the misalignment due to mask, θm: θ = θc + θm.(71) We have θm = (∆l, ∆b), where l and b are respectively the Galactic longitude and latitude. Under the simplest assumption, the distribution function for θm is a bivariate Gaussian of two uncorrelated variables of the same variance: p(∆l, ∆b) = (2π) −1 σ −2 exp − ∆l 2 + ∆b 2 2σ 2 .(72) The distribution for the modulus θm = √ ∆l 2 + ∆b 2 results immediately from Equation (72). It is a Rayleigh distribution (cf. Eq. 30), p(θm) = θm σ 2 exp − θ 2 m 2σ 2 .(73) Let us now invert the above reasoning and apply it to the variable θc. Since the distribution for θc is (approximately) Rayleigh, the distribution for θc is (approximately) a bivariate Gaussian. The variable θ is therefore a sum of two independent bivariate Gaussians, which itself is a bivariate Gaussian (of uncorrelated variables). Hence, the variable θ is Rayleigh-distributed, with the parameter θ ′2 * = θ 2 * + σ 2 .(74) Here, θ 2 * = θ 2 c /2, and σ 2 = θ 2 m /2. The parameter θ * is defined by Equation (31) and determined by the LG velocity variance (Eq. 21) and the correlation coefficient (Eq. 24). Let us find an estimate for the mask variance σ 2 . We have σ 2 = 1 N − 1 N i=1 x 2 i ,(75) where xi = (li −l) 2 + (bi −b) 2 , and (l,b) are the means for the sample. As stated above, M03 study the effects of two different methods of 'repopulating' the masked regions with galaxies, so N = 2. Then x2 = x1, henceσ = √ 2x1; the factor √ 2 mustn't be neglected. This yields (in degrees) σ ≃ 5.4 •(76) (as opposed to 3.4 • , or 4.5 • , finally adopted by M03). LIKELIHOOD FOR β We mentioned in Section 1 that a comparison between the CMB dipole and the 2MASS flux dipole (the latter given by Equation 7) can serve as a method to measure the parameter β = Ω 0.6 m /bL. Of course, it cannot be done by naive equating of the two dipoles: such an estimate would be biased. Here we outline a likelihood estimation of β (for details see C04). In a Bayesian approach, one ascribes a priori equal probabilities to values of unknown parameters, which allows us to express their likelihood function, given v and g of the LG, via the probability distribution function for v and g: L(param.) = p(v, g | param.) .(77) As the parameters to be estimated here we adopt β and bL; p is given by Equation (8). Theoretical quantities in this distribution are σg, σv, and ρ. Since now we account for observational errors, the variance of a single spatial component of measured gravity, σ 2 g , is a sum of the 1D cosmological component, σ 2 g,c , and errors, ǫ 2 /3 (ǫ denoting 3D errors, including shot noise and the mask). Here, gravity is inferred from a galaxian, rather than mass, density field. Therefore, σ 2 g,c = b 2 L g 2 /3, where g 2 = g · g is given by Equation (18). To sum up, σ 2 g = b 2 L g 2 + ǫ 2 3 .(78) Errors in the measured velocity of the LG are negligible compared to those in the gravity. The relation between the physical velocity, v ph , and the scaled velocity used in this paper, v, is v ph = Ω 0.6 m v, hence 1D velocity variance is σ 2 v = Ω 1.2 m s 2 v 3 = β 2 b 2 L s 2 v 3 ,(79) where s 2 v ≡ v 2 = v · v is given by Equation (21). Finally, errors in the estimate of the LG gravity do not affect the crosscorrelation between the LG gravity and velocity, but increase the gravity variance. This has the effect of lowering the value of the cross-correlation coefficient. Specifically, ρ ′ = ρ 1 + ǫ 2 b 2 L g 2 −1/2 ,(80) where ρ is given by Equation (24). From Equation (8), the logarithmic likelihood for β and bL takes the form: ln L(β, bL) = −3 ln (2π) − 3 ln σgbLsv(1 − ρ ′2 ) 1/2 − 3 ln β − 1 2(1 − ρ ′2 ) g 2 m σ 2 g + 3v 2 m β 2 b 2 L s 2 v − 2 √ 3ρ ′ µmgmvm σgβbLsv .(81) In the above likelihood, the 'data' are the measured values of the LG gravity and velocity, gm and vm, respectively, as well as µm, i.e. cosine of the misalignment angle. The model parameters σg and ρ ′ depend solely on bL; sv depends neither on β nor on bL. We have written down the expression for the likelihood only for illustrative purposes. Therefore, for simplicity we will restrict our analysis to the case of given bL. Then, to find a maximum of the likelihood we calculate its partial derivative with respect to β and equate it to zero. This yields the following equation: 3(1 − ρ ′2 )β 2 + √ 3ρ ′ µmgmvm σgbLsv β − 3v 2 m b 2 L s 2 v = 0 .(82) The LG gravity, inferred from the 2MASS survey, is tightly coupled to its velocity: 1 − ρ ′ ≪ 1 and 1 − µm ≪ 1. (See Table 1; θ obs = 16 • corresponds to µm = 0.96). At first approximation we can therefore assume ρ ′ = µm = 1, hencê β ≃ √ 3 σg bLsv vm gm = b 2 L g 2 + ǫ 2 b 2 L v 2 1/2 vm gm .(83) Thus, the estimate of β is not just the ratio of the LG velocity to its gravity: it is modified by nonlinear effects (which affect v 2 through the function R), different observational windows (which affect differently g 2 and v 2 ), and observational errors. If all these factors are properly accounted for, then the estimate of β is unbiased. An optimal estimator is such that is not only unbiased but also has minimal variance. Expanding the logarithmic likelihood (Eq. 81) around its maximum up to second order in β enables one to find an estimator of the variance of β. In the strong-coupling regime (1 − ρ ′ ≪ 1, 1 − µm ≪ 1), it iŝ σ 2 β = v 2 m v 2 b 2 L g 2 + ǫ 2 g 2 m 2 1 − ρ ′2 = v 2 m b 2 L g 2 + ǫ 2 v 2 g 4 m b 2 L g 2 1 − ρ 2 + ǫ 2 .(84) If errors are constant, i.e. they do not depend on Kmin, then a minimum of the variance corresponds to a maximum of the crosscorrelation coefficient ρ. (The dependence of g 2 on Kmin is very weak.) Including higher-order corrections to the above formula does not change this fact. In the present paper, errors are indeed constant: the error due to the mask obviously does not depend on Kmin and shot noise is assumed to be negligible. The window function of the 2MASS survey we have constructed here maximizes ρ (see Fig. 2). This is why we call this window, under the assumption of negligible shot noise, optimal. It exactly corresponds to the minimal expectation value of the misalignment angle. As mentioned earlier, in follow-up work we will estimate shot noise ourselves. If we find that it is in fact not negligible, then it will influence the optimal value of Kmin. Shot noise as a function of Kmin monotonically decreases (see Eq. 69). The factor 1 − ρ 2 , starting from the value of Kmin which maximizes ρ, monotonically increases (see Fig. 2). The interplay between these two opposing effects in Equation (84) shifts the optimal Kmin (corresponding to a minimum of the variance of the estimator of β) to a larger value, compared to the case of negligible shot noise. Table 1. Parameters of the distribution for the misalignment angle between the 2MASS and CMB dipoles, for various forms of the 2MASS gravity window. First column shows the limiting magnitude of the excluded brightest galaxies, K min . Second column shows the correlation coefficient of the LG velocity and gravity, ρ, calculated according to Equation (24). Third column shows the characteristic value of the misalignment angle (in radians), θ * , calculated using Equation (31). Fourth column shows the corresponding value of the misalignment angle (in radians), including observational errors due to the mask, θ ′ * , calculated using Equation (74). Fifth column shows the expectation value of the misalignment angle (in degrees), θ . Sixth and seventh columns show, respectively, the quantiles θ 10 and θ 90 (in degrees), defining the confidence intervals of 10 and 90% (for details see text). The last column shows the observed values of the misalignment angle (in degrees), obtained with and without exclusion of the brightest galaxies. Figure 3 shows the resulting PDFs for the misalignment angle, for various forms of the 2MASS gravity window. Like previously, dotted line corresponds to the case where the brightest galaxies are not excluded from the calculation of the flux dipole. The vertical stripe shows the value of the misalignment between the CMB dipole and the 2MASS flux dipole as calculated by M03, including all galaxies brighter than Ks = 13.57. The 'observed' value (16 • ) is greater than the expectation value for the angle (11.1 • ), but smaller than θ90 = 19.0 • (see Table 1). K min ρ θ * θ ′ * θ [ • ] θ 10 [ • ] θ 90 [ • ] θ RESULTING DISTRIBUTIONS FOR THE MISALIGNMENT To decrease the misalignment, in the second step M03 excluded from the analysis all galaxies brighter than Kmin = 8 mag. A dashed line is plotted for the gravity window corresponding to this case. Consistently with Figure 2, the expectation value for the angle does not decrease; on the contrary, it increases to 14.8 • (Table 1). Consequently, one would then expect the misalignment rather to increase. However, M03 noticed a substantial decrease of the misalignment, to about 5.2 • . This value is smaller than the corresponding θ10 = 5.4 • . Therefore, there is less than 10% chance that the decrease might have been accidental. Rather, an error in the analysis is more likely. Indeed, E06 repeated the procedure of M03 for the 2MRS data and essentially did not observe the decrease of the misalignment. 2MRS survey misses faint galaxies (fainter than Ks = 11.25), but does not miss bright galaxies. Therefore, if the effect was real, one should observe it also when using the 2MRS data. Solid line in Figure 3 is plotted using the window excluding 2MASS galaxies brighter than Kmin = 5 mag. As described in Section 4, we expect this window to be close to optimal. Indeed, the resulting distribution is the narrowest among the three plotted; the expectation value of the misalignment drops to 9.7 • and θ90 = 16.6 • (Table 1). Therefore, with (almost) 90% confidence we can expect the angle to decrease when performing such a preselection on 2MASS galaxies. Of course, this assumes constant mass-to-light ratio for all remaining (i.e., included) galaxies. The window with Kmin = 8 mag is not optimal because it excessively mitigates nonlinear effects. This window excludes too Figure 3. Probability distribution functions for the misalignment angle between the 2MASS and CMB dipoles, for various forms of the 2MASS gravity window. Dotted line is for the case where the brightest galaxies are not excluded from the calculation of the flux dipole. Dashed line describes the case of excluding all 2MASS galaxies brighter than K min = 8 mag; solid line describes the case of excluding all 2MASS galaxies brighter than K min = 5 mag. Thick solid vertical line shows the measured value of the misalignment between the CMB dipole and the 2MASS flux dipole as calculated by M03, including all galaxies brighter than Ks = 13.57 (16 • ). many galaxies: while the window with Kmin = 5 mag excludes all L * and brighter galaxies closer to the LG than about 5 h −1 Mpc, the window with Kmin = 8 mag does the same for the distance of about 18 h −1 Mpc (see Sec. 4). The signal from scales 5-18 h −1 Mpc is sufficiently 'linear' to increase (if included) the correlation between the LG velocity and its measured gravity. M03 found 375 2MASS galaxies brighter than Kmin = 8 mag. Based on this number and the relation N excl ∝ S −3/2 max , where N excl is the number of excluded galaxies, we predict about six 2MASS galaxies to be brighter than Kmin = 5 mag. To reduce the misalignment calculated using their sample, E06 excluded five the brightest galaxies in 2MRS. They noticed a significant decrease of the misalignment, from about 21 • to 14 • . The five most luminous galaxies in 2MRS are also the five most luminous galaxies in 2MASS. Therefore, exclusion of these galaxies should work also for denser and deeper 2MASS survey. SUMMARY • An ultimate goal of comparing the CMB dipole to the 2MASS dipole is an estimation of the cosmological parameter β ≡ Ω 0.6 m /bL. • To obtain an unbiased estimate of β, a good and standard method is Maximum Likelihood. • An important ingredient of this Likelihood analysis is the observational window through which the 2MASS flux dipole is measured, called here the gravity window of 2MASS. This window should be properly modelled. • By definition, the optimal window minimizes variance in the estimate of β; optimizing the 2MASS window is therefore important. • In this paper, we have modelled the 2MASS gravity window and optimized it under the assumption of negligible shot noise. This optimization has been achieved by excluding contributions to the dipole from the brightest galaxies (Sec. 4). Such an exclusion mitigates nonlinear effects from small scales, which decorrelate the LG velocity from the estimated LG gravity. We have found that the optimal value of the minimal limiting magnitude, Kmin (corresponding to maximal limiting flux), is about 5. • We have also demonstrated how to optimize the window in presence of shot noise. We have shown that the optimal value of Kmin will increase. • The misalignment angle is a sensitive measure of the correlation between the two dipoles: the higher the correlation, the smaller the expectation value of the angle (Eqs. 31-32). We have shown that a minimum of the misalignment corresponds to minimal variance of the estimator of β. A minimum of the misalignment is thus a sign of the optimal gravity window. • We have modelled analytically the probability distribution function for the misalignment angle (Sec. 2, App. A). We have shown that the misalignment estimated by M03 is consistent with the assumed underlying model (though it is greater than the expectation value). We have predicted that the misalignment is likely to decrease if 2MASS galaxies brighter than Kmin = 5 mag are excluded from the calculation of the flux dipole. This prediction has been indirectly confirmed by the results of E06. • In a future work, we plan to perform the optimal measurement of the value of β by comparing the CMB dipole to the 2MASS dipole. We will thus have to fully specify the optimal window in presence of shot noise (though M03 claim that shot noise of 2MASS survey is negligible). An estimate of shot noise can be obtained using methods described in Subsection 5.1. However, the misalignment angle can be used as an alternative way of optimizing the window. As a function of Kmin, the measured value of the misalignment will -with some scatter -initially decrease, reach a minimum and then increase (see Fig. 2). It is now clear that the value of Kmin for which the measured misalignment has a minimum will be close to that optimizing the measurement of β. exact distribution starts to deviate from the Rayleigh form, it is still well approximated by distribution (A3)-(A5), up to θ ′ * ∼ 0.5 (except for the very tail). In particular, the values of the quantiles θ10 and θ90 and of the mean angle remain within 2% from the exact values. Figure 1 1shows the 2MASS gravity windows for Kmax = 13.57 mag and different values of Kmin. Dotted line corresponds to Equation (59), i.e. to the case where the brightest galaxies are not excluded from the calculation of the flux dipole. Dashed and Figure 1 . 1Gravity windows for the 2MASS all-sky survey, for Kmax = 13.57 mag and different values of K min . Dotted line corresponds to Equation (59), i.e. to the case where the brightest galaxies are not excluded from the calculation of the flux dipole. Dashed and solid lines are plotted using Equation = the summation is over all galaxies in the volume element, regardless whether or not they are in the 2MASS survey.The expectation value of the estimated gravity is aWg(r) L N (∆V )r . obs [ • ]- 0.951 0.123 0.155 11.1 4.1 19.0 16.0 8 0.901 0.183 0.206 14.8 5.4 25.3 5.2 5 0.969 0.097 0.135 9.7 3.6 16.6 - Kelvin's circulation theorem assures that the cosmic velocity field is vorticity-free as long as there is no shell crossing. N-body simulations(Bertschinger & Dekel 1989, Mancinelli et al. 1994, Pichon & Bernardeau 1999 have shown that the vorticity of velocity is small in comparison to its divergence even in the fully nonlinear regime. c 0000 RAS, MNRAS 000, 000-000 The velocity of the LG is fully nonlinear and as such cannot be approximated by low-order moments of the velocity field. In particular, vLG is different from the bulk velocity of a region around it. − ρ ≪ 1, so q ≫ 1. Also, µ is then close to unity. Therefore, in equation(27)we can use an asymptotic formula for the error function, c 0000 RAS, MNRAS 000, 000-000 Consequently, we could approximate θ * by 2(1 − ρ)/y. However, exact expression(31)is not more complex, while it remains valid also for higher-order corrections to the distribution (30). c 0000 RAS, MNRAS 000, 000-000 These divergences are due to the weighting scheme and not to the type of the dipole. c 0000 RAS, MNRAS 000, 000-000 In a related paper,Feldman, Kaiser & Peacock (1994) constructed the optimal estimator for the density power spectrum inferred from redshift surveys. They derived a formula for the optimal weighting of galaxies (Eq. 2.3.4 ofFeldman et al. 1994). For small r the optimal weight behaves like φ −1 (r), while for large r it approaches asymptotically unity. ACKNOWLEDGMENTSThis work was carried out within the framework of the PAN/CNRS European Associated Laboratory (LEA) 'Astrophysics Poland-France'. It was also partially supported by the Polish Ministry of Science and Higher Education under grant N N203 0253 33, allocated for the period 2007-2010.APPENDIX A: BEYOND THE SMALL-ANGLE LIMITHere we check the accuracy of the small-angle approximation, Equation (30), of the distribution function for the misalignment angle, given in general byEquation (27). First, integrating by parts one can show that for ν ≫ 1,This yields for the error function a higher-order expansion (than Eq. 29):Using this expansion in Equation(27)for µ > 0, we obtainwhere θ * is given by Equation (31) and θ < π/2. For θ * → 0, this distribution simplifies to the Rayleigh distribution (Eq. 30), as expected.For µ < 0, one can show in a similar way thatUsing the latter expansion in Equation(27)yieldsDensity distribution (A3)-(A5) has analytical cumulative distribution function:This allows for a straightforward estimation of the quantiles. Moreover, using the fact that p = dF/dθ and integrating by parts, the expectation value of the angle can be readily calculated:Thus, for θ 2 * ≪ 1, distribution (A3)-(A5) has (almost) identical mean to the Rayleigh distribution (Eq. 32).In practice, the total misalignment angle is a convolution of the cosmologically-originated misalignment, θc, and the misalignment due to mask, θm. This convolution is not as simple as when both θm and θc are bivariate Gaussians. However, non-Gaussianity of θc + θm is smaller than of the variable θ ′ c with θ ′ * = θ 2 * + σ 2 (because θm is Gaussian). We see inTable 1that θ ′ * is at most 0.2. For θ ′ * ∼ < 0.2, both distribution (A3)-(A5) and the Rayleigh distribution approximate the exact one (Eq. 27) very well. Specifically, since then θ ′2 * ≪ 1, we can expand distribution (A3)-(A5), obtainingwhereIt is straightforward to check that approximate distribution (A8) is properly normalized. A simple calculation yieldsHence, for θ ′ * = 0.2, the Rayleigh distribution approximates the second moment of distribution (A8) (which, in turn, is then an excellent approximation of the exact distribution) to 1.3% accuracy. Similarly simple calculations can be performed for other even moments.Summing up, the distribution function for the misalignment angle between the CMB and 2MASS dipoles can be very well approximated by its small-angle limit, Rayleigh distribution. It may be worth noting for other applications that, while for θ ′ * > 0.2 the . S Basilakos, M Plionis, MNRAS. 3731112Basilakos S., Plionis M., 2006, MNRAS, 373, 1112 . E F Bell, R S De Jong, ApJ. 550212Bell E.F., de Jong R.S., 2001, ApJ, 550, 212 . E F Bell, D H Mcintosh, N Katz, M D Weinberg, ApJ. 149289Bell E.F., McIntosh D.H., Katz N., Weinberg M.D., 2003, ApJ, 149, 289 . C L Bennett, ApJS. 1481Bennett C.L., et al., 2003, ApJS, 148, 1 . E Bertschinger, A Dekel, ApJ. 3365Bertschinger E., Dekel A., 1989, ApJ, 336, L5 . M J Chodorowski, P Ciecielag, MNRAS. 331133Chodorowski M.J., Ciecielag P., 2002, MNRAS, 331, 133 . P Ciecielag, M J Chodorowski, MNRAS. 349C04945Ciecielag P., Chodorowski M.J., 2004, MNRAS, 349, 945 (C04) . P Ciecielag, M J Chodorowski, A Kudlicki, Acta Astron. 51103Ciecielag P., Chodorowski M.J., Kudlicki A., 2001, Acta Astron., 51, 103 . P Ciecielag, M J Chodorowski, M Kiraga, M A Strauss, A Kudlicki, F R Bouchet, MNRAS. 339641Ciecielag P., Chodorowski M.J., Kiraga M., Strauss M.A., Kudlicki A., Bouchet F.R., 2003, MNRAS, 339, 641 . S Courteau, S Van Den Bergh, AJ. 118337Courteau S., van den Bergh S., 1999, AJ, 118, 337 . M Davis, J P Huchra, ApJ. 254437Davis M., Huchra J.P., 1982, ApJ, 254, 437 . M Davis, M A Strauss, A Yahil, ApJ. 372394Davis M., Strauss M.A., Yahil A., 1991, ApJ, 372, 394 . G Efstathiou, J R Bond, S D M White, MNRAS. 2581Efstathiou G., Bond J.R., White S.D.M., 1992, MNRAS, 258, 1 . P Erdogdu, MNRAS. 368E061515Erdogdu P., et al., 2006, MNRAS, 368, 1515 (E06) . H A Feldman, N Kaiser, J A Peacock, ApJ. 42623Feldman H.A., Kaiser N., Peacock J.A., 1994, ApJ, 426, 23 . W J Frith, P J Outram, T Shanks, MNRAS. 364593Frith W.J., Outram P.J., Shanks T., 2005, MNRAS, 364, 593 . T H Jarrett, PASA. 21396Jarrett T.H., 2004, PASA, 21, 396 . T H Jarrett, T Chester, R Cutri, S E Schneider, J P Huchra, AJ. 125525Jarrett T.H., Chester T., Cutri R., Schneider S.E., Huchra J.P., 2003, AJ, 125, 525 . R Juszkiewicz, N Vittorio, R F G Wyse, ApJ. 349408Juszkiewicz R., Vittorio N., Wyse R.F.G., 1990, ApJ, 349, 408 . L Kofman, E Bertschinger, J M Gelb, A Nusser, A Dekel, ApJ. 42044Kofman L., Bertschinger E, Gelb J.M., Nusser A., Dekel A., 1994, ApJ, 420, 44 . O Lahav, N Kaiser, Y Hoffman, ApJ. 352448Lahav O., Kaiser N., Hoffman Y., 1990, ApJ, 352, 448 . A H Maller, D H Mcintosh, N Katz, M D Weinberg, ApJ. 598M031Maller A.H., McIntosh D.H., Katz N., Weinberg M.D., 2003, ApJ, 598, L1 (M03) . A H Maller, D H Mcintosh, N Katz, M D Weinberg, ApJ. 619147Maller A.H., McIntosh D.H., Katz N., Weinberg M.D., 2005, ApJ, 619, 147 P J Mancinelli, A Yahil, G Ganon, A Dekel, Proceedings of the 9th IAP Astrophysics Meeting 'Cosmic velocity fields. F. R. Bouchet and M. Lachièze-Reythe 9th IAP Astrophysics Meeting 'Cosmic velocity fieldsGif-sur-YvetteEditions Frontières215Mancinelli P. J., Yahil A., Ganon G., Dekel A., 1994, in Proceed- ings of the 9th IAP Astrophysics Meeting 'Cosmic velocity fields', ed. F. R. Bouchet and M. Lachièze-Rey, Gif-sur-Yvette: Editions Frontières, 215 P J E Peebles, The Large-Scale Structure of the Universe. PrincetonPrinceton University Press Pichon C., Bernardeau F343663Peebles P.J.E., 1980, The Large-Scale Structure of the Universe, Princeton: Princeton University Press Pichon C., Bernardeau F., 1999, A&A, 343, 663 . M Rowan-Robinson, MNRAS. 314375Rowan-Robinson M., et al., 2000, MNRAS, 314, 375 . D Sarkar, H A Feldman, R Watkins, MNRAS. 375691Sarkar D., Feldman H.A., Watkins R., 2007, MNRAS, 375, 691 . W Saunders, MNRAS. 31755Saunders W., et al., 2000, MNRAS, 317, 55 . I Schmoldt, MNRAS. 314S99893Schmoldt I., et al., 1999, MNRAS, 314, 893 (S99) The Impact of Large Scale Near-IR Sky Surveys. M F Skrutskie, ApJS. 148175Kluwer Academic Publishing Company Spergel DSkrutskie M.F., et al., 1997, The Impact of Large Scale Near-IR Sky Surveys, eds. F. Garzon et al., p. 25. Dordrecht: Kluwer Aca- demic Publishing Company Spergel D.N., et al., 2003, ApJS, 148, 175 . D N Spergel, ApJS. 170377Spergel D.N., et al., 2007, ApJS, 170, 377 . M A Strauss, A Yahil, M Davis, J P Huchra, K Fisher, ApJ. 397S92395Strauss M.A., Yahil A., Davis M., Huchra J.P., Fisher K., 1992, ApJ, 397, 395 (S92) . N Sugiyama, ApJS. 100281Sugiyama N., 1995, ApJS, 100, 281 . R B Tully, E J Shaya, I D Karachentsev, H M Courtois, D D Kocevski, L Rizzi, A Peel, ApJ. 676184Tully R.B., Shaya E.J., Karachentsev I.D., Courtois H.M., Ko- cevski D.D., Rizzi L., Peel A., 2008, ApJ, 676, 184 . A Yahil, A Sandage, G A Tamman, ApJ. 242448Yahil A., Sandage A., Tamman G.A., 1980, ApJ, 242, 448
[]
[ "Diverse Video Generation from a Single Video", "Diverse Video Generation from a Single Video" ]
[ "Niv Haim \nWeizmann Institute of Science\nRehovotIsrael\n", "Ben Feinstein \nWeizmann Institute of Science\nRehovotIsrael\n", "Niv Granot \nWeizmann Institute of Science\nRehovotIsrael\n", "Assaf Shocher \nWeizmann Institute of Science\nRehovotIsrael\n", "Shai Bagon \nWeizmann Institute of Science\nRehovotIsrael\n", "Tali Dekel \nWeizmann Institute of Science\nRehovotIsrael\n", "Michal Irani \nWeizmann Institute of Science\nRehovotIsrael\n" ]
[ "Weizmann Institute of Science\nRehovotIsrael", "Weizmann Institute of Science\nRehovotIsrael", "Weizmann Institute of Science\nRehovotIsrael", "Weizmann Institute of Science\nRehovotIsrael", "Weizmann Institute of Science\nRehovotIsrael", "Weizmann Institute of Science\nRehovotIsrael", "Weizmann Institute of Science\nRehovotIsrael" ]
[]
Equal contribution scaled to Full-HD videos. We also use the same framework to demonstrate video analogies and spatio-temporal retargeting. These observations show that classical approaches significantly outperform heavy deep learning machinery for these tasks. This sets a new baseline for single-video generation and manipulation tasks, and no less important -makes diverse generation from a single video practically possible for the first time.
10.48550/arxiv.2205.05725
[ "https://arxiv.org/pdf/2205.05725v1.pdf" ]
248,721,799
2205.05725
a0274068d613e389a4f14277b49eb879b81c54ef
Diverse Video Generation from a Single Video Niv Haim Weizmann Institute of Science RehovotIsrael Ben Feinstein Weizmann Institute of Science RehovotIsrael Niv Granot Weizmann Institute of Science RehovotIsrael Assaf Shocher Weizmann Institute of Science RehovotIsrael Shai Bagon Weizmann Institute of Science RehovotIsrael Tali Dekel Weizmann Institute of Science RehovotIsrael Michal Irani Weizmann Institute of Science RehovotIsrael Diverse Video Generation from a Single Video Generated #2 Generated #1 Input Video Generated #2 Generated #1 Input Video Equal contribution scaled to Full-HD videos. We also use the same framework to demonstrate video analogies and spatio-temporal retargeting. These observations show that classical approaches significantly outperform heavy deep learning machinery for these tasks. This sets a new baseline for single-video generation and manipulation tasks, and no less important -makes diverse generation from a single video practically possible for the first time. Figure 1 . Given an input video (red), we generate similarly looking videos (black) capturing both appearance of objects as well as their dynamics. The diversity of the outputs is both spatially (e.g., number of dancers and their positions are different from the input video) and temporally (generated dancers are not synced). As we present video results, the reader is encouraged to check our supplementary material GANs are able to perform generation and manipulation tasks, trained on a single video. However, these single video GANs require unreasonable amount of time to train on a single video, rendering them almost impractical. In this paper we question the necessity of a GAN for generation from a single video, and introduce a non-parametric baseline for a variety of generation and manipulation tasks. We revive classical space-time patches-nearest-neighbors approaches and adapt them to a scalable unconditional generative model, without any learning. This simple baseline surprisingly outperforms single-video GANs in visual quality and realism (confirmed by quantitative and qualitative evaluations), and is disproportionately faster (runtime reduced from several days to seconds). Our approach is easily Introduction Generation and editing of natural videos remain challenging, mainly due to their large dimensionality and the enormous space of motion they span. Most modern frameworks train generative models on a large collection of videos, producing high quality results for only a limited class of videos. These include extensions of GANs [15] to video data [2,25,33,39,43,47] and video-to-video translation [7,11,29,[44][45][46]50], autoregressive sequence prediction [3,5,6,12,14,[40][41][42] and more. While externallytrained generative models produce impressive results, they are restricted to the types of video dynamics in their training set. On the other side of the spectrum are single-video GANs. These video generative models train on a single input video, learn its distribution of space-time patches, and are then able to generate a diversity of new videos with the same patch distribution [4,17]. However, these take very long time to train for each input video, making them applicable to only small spatial resolutions and to very short videos (typically, very few small frames). Furthermore, their output oftentimes shows poor visual quality and noticeable visual artifacts. These shortcomings render existing single-video GANs impractical and unscalable. Video synthesis and manipulation of a single video sequence based on its distribution of space-time patches dates back to classical pre-deep learning methods. These classical methods demonstrated impressive results for various applications, such as video retargeting [21,32,36,49], video completion [19,28,48], video texture synthesis [10,13,20,[22][23][24] and more. With the rise of deep-learning, these methods gradually, perhaps unjustifiably, became less popular. Recently, Granot et al. [16] revived classical patchbased approaches for image synthesis, and was shown to significantly outperform single-image GANs in both runtime and visual quality. In light of the above-mentioned deficiencies of singlevideo GANs, and inspired by [16], we propose a fast and practical method for video generation from a single video that we term VGPNN (Video Generative Patch Nearest Neighbors). In order to handle the huge amounts of spacetime patches in a single video sequence, we use the classical fast approximate nearest neighbor search method Patch-Match by Barnes et al. [8]. By adding stochastic noise to the process, our approach can generate a large diversity of random different video outputs from a single input video in an unconditional manner. Like single-video GANs, our approach enables the diverse and random generation of videos. However, in contrast to existing single-video GANs, we can generate high resolution videos, while reducing runtime by many orders of magnitude, thus making diverse unconditional video generation from a single video realistically possible for the first time. In addition to diverse generation from a single video, by employing robust optical-flow based descriptors we use our framework to transfer the dynamics and motions between two videos with different appearance (which we call "video analogies"). We also show the applicability of our framework to spatio-temporal video retargeting and to conditional video inpainting. To summarize, our contributions are as follows: • We show that our space-time patch nearest-neighbors approach, despite its simplicity, outperforms single-video GANs by a large margin, both in runtime and in quality. • Our approach is the first to generate diverse high resolution videos (spatial or temporal) from a single video. • We demonstrate the applicability of our framework to other applications: video analogies, sketch-to-video, spatio-temporal video retargeting and conditional video inpainting. Our code and data will be released. Method Our main task is to generate diverse video samples based on a single natural input video, such that the generated outputs have similar appearance and motions as the original input video, but are also visually different from one another. In order to capture both spatial and temporal information of a single video, we start by building a spatio-temporal pyramid and operate coarse-to-fine to capture the internal statistics of the input video at multiple scales. At each scale we employ a Video-Patch-Nearest-Neighbor module (VPNN); VGPNN is in fact a sequence of VPNN layers. The inputs to each layer depend on the application, where we first focus on our main application of diverse video generation. Given an input video x, we construct a spatio-temporal pyramid {x 0 . . . , x N }, where x 0 = x, and x n = x n−1 ↓ r is a bicubically downscaled version of x n−1 by factor r (r = (r H , r W , r T ), where r H = r W are the spatial factors and r T is the temporal factor, which can be different). Q n = y n+1 ↑ r y N Q 0 = y 1 ↑ r V n = x n y n V 0 = x 0 y 0 Input Video Pyramid Generated Video Pyramid Q N = x N + z N V N = x N z N + x N = x N-1 ↓ r x n = x n-1 ↓ r VPNN VPNN VPNN Figure 2. VGPNN Architecture: Given a single input video x0, a spatio-temporal pyramid is constructed and an output video y0 is generated coarse-to-fine. At each scale, VPNN module ( Fig. 3) is applied to transfer an initial guess Qn to the output yn which shares the same space-time patch distribution as the input xn. At the coarsest scale, noise is injected to induce randomness. Multi-scale approach (Fig. 2): At the coarsest level, the input to the first VPNN layer is an initial coarse guess of the output video. This is created by adding random Gaussian noise z N to x N . The noise z N promotes high diversity in the generated output samples from the single input. The global structure (e.g., a head is above the body) and global motion (e.g., humans walk forward), is prompted by x N , where such structure and motion can be captured by small space-time patches. Each space-time patch of the initial coarse guess (x N + z N ) is then replaced with its nearest neighbor patch from the corresponding coarse input x N . The coarsest-level output y N is generated by choosing at each space-time position the median of all suggestions from neighboring patches (known as "voting" or "folding"). At each subsequent level, the input to the VPNN layer is the upscaled output of the previous layer (y n+1 ↑ r ). Each space-time patch is replaced with its nearest neighbor patch from the corresponding input x n (using the same patch-size as before, now capturing finer details). This way, the output y n in each level is similar in structure and in motion to the initial guess, but contains the same space-time patch statistics of the corresponding input x n . The output y n is generated by median voting as described above. To further improve the quality and sharpness of the generated output at each pyramid level (y n ), we iterate several times through the current level, each time using the current output y n as input to the current VPNN layer (similar to the EM-like approach employed in many patch-based works [e.g., 8,16,36,48]). QKV scheme: In several cases it is necessary to compare patches in another search space than the original RGB input space. To this end we adopt a QKV scheme (query, key and value, respectively) as used by [16]. For example, when comparing the upscaled output of previous layer to the corresponding level from the pyramid of the original video, the patches of the latter are sharper than the former. This is mitigated by setting V = x n and K = x n+1 ↑ r which has a similar degree of blur as Q = y n+1 ↑ where. Each patch Q i with nearest neighbour K j is replaced with V j (i, j are spatio-temporal positions). The QKV scheme is especially important in our video analogies application where it is used to include additional temporal information in the queries and the keys. Finding Correspondences: We use PatchMatch (Barnes et al. [8]) to find the nearest neighbors between Q and K (Fig. 3b). The algorithm is implemented on GPU using Py-Torch [30], with time complexity O(n × d) and O(n) additional memory (where n is the video size and d is the patch size). This dramatically reduces both run time and memory footprint used for video generation, making it possible to generate high-resolution videos (see Fig 6). An overview of VPNN module is shown in Fig. 3. Temporal Diversity and Consistency: To enhance the temporal diversity of our samples we set the temporal dimension of the output to be slightly smaller than that of the input video. Thus, motions in different spatial positions in the generated output are taken from different temporal po- Input Video Ours (VGPNN) HPVAEGAN Ours (VGPNN) SinGAN-GIF Input Video Figure 4. Comparing Visual Quality between our generated frames and those of HP-VAE-GAN [17] and SinGAN-GIF [4] (please zoom in on the frames). Note that our generated frames are sharper and also exhibit more coherent and plausible arrangements of the scene. sitions in the input video, increasing the overall temporal diversity (see for example the generated dancers in Fig. 1 that are not synced). We also found that the temporal consistency is best preserved in the generated output when the initial noise z N is randomized for each spatial position, but is the same (replicated) in the temporal dimension. Experimental Results We compare our results to those of HP-VAE-GAN [17] and SinGAN-GIF [4], both are methods for diverse video generation from single video. Our results are both qualitatively (Fig. 4) and quantitatively (Table 1) superior while reducing the runtime by a factor of 3×10 4 (from 8 days training on one video to 18 seconds for new generated video). While [4,17] are limited to generated outputs of small resolution (144×256), the use of efficient PatchMatch algorithm for nearest neighbors search dramatically reduces both run time and memory footprint used for video generation, making it possible to generate outputs in the same resolution of the input video (full-HD 1280×1920) Video analogies Video to video translation methods typically train on large datasets and are either conditioned on human poses or keypoint detection [e.g. 29,[44][45][46], or require knowledge of a human/animal model [e.g. 1,11,26,31,34,35,50]. We show that when videos' dynamics are similar in both their motion and semantic context within their video, one can use our framework to transfer the motion and appearance between the two (see Fig. 5). We term this task "video analogies" (inspired by image analogies [9,18,27]). More formally, we generate a new video whose spatio-temporal layout is taken from a content video C, and overall appearance and dynamics from a style video S. We first extract the "dynamic structure" of both videos - . Each generated video (black) takes the spatio-temporal layout from the input video in its row, and the appearance and dynamics of the input video from its column. Right: an example of sketch-to-video -the generated video (bottom) takes its spatio-temporal layout from the sketch video of morphed MNIST digits (middle) and its appearance and dynamics from the input video of parading soldiers (top). Table 1. Quantitative Evaluation: SVFID [17] measures the patch statistics similarity between the input video and a generated video. It computes the Fréchet distance between the statistics of the input video and the generated video using pre-computed C3D [38] features (lower is better). Note that our generated samples bear more substantial similarity to the input videos. In a user study, each of 100 AMT participants were asked to judge which of two generated samples was better in terms of sharpness, natural look and coherence. We report the percentage of users who favored our samples over the other. the magnitude of the optical flow (extracted via RAFT [37]), quantized into few bins (using k-means). We compute the spatio-temporal pyramids of the (i) style video S (ii) dynamic structure of the content video (dyn(C)) (iii) the dynamic structure the style video (dyn(S)). The output video is generated by setting Q, K, V at each level as follows: Level Q K V N (coarsest) dyn(C) N dyn(S) N S N n (any other) dyn(C) n Q n+1 ↑ dyn(S) n S n S n where denotes concatenation along the channels axis, and n denote the current level in the pyramid. Note that in the coarsest level, the two videos are only compared by their dynamic structure. In finer levels, the dynamic structure of C (the content video) is used to "guide" the output to the desired spatio-temporal layout. In Fig. 5left we show snapshots of the analogies of all possible pairs between three videos. In Fig. 5right we show an example for "sketch-tovideo" transfer, where the dynamic structure is given by a sketch video instead of an actual video. More Applications In the supplementary material we include examples for spatio-temporal retargeting and conditional video inpainting, as well as further technical details and ablations. Figure 3 . 3VPNN module gets as input RGB videos query, key and value (QKV respectively). Q and K can be concatenated to additional auxiliary channels. It outputs an RGB video. (a) Inputs are unfolded to patches (each position now holds a concatenation of neighboring positions); (b) Each patch in Q finds its nearest neighbor patch in K. This is achieved by solving the NNF using PatchMatch[8]; (c) Each patch in Q is replaced with a patch from V, according to the correspondences found in stage (b); (d) Resulting patches are folded back to an RGB video output. Figure 5 . 5Video Analogies: Left: an example of video analogies between all pairs of three input videos (red) Figure 6 . 6Comparing Generation Runtime between our approach (VGPNN), a naïve extension of GPNN[16] from 2D to 3D and HP-VAE-GAN [17] on 13-frames videos with different spatial resolutions (X-axis, all have 16:9 aspect ratio). Unpaired motion style transfer from video to animation. K Aberman, Y Weng, D Lischinski, D Cohen-Or, B Chen, ACM Transactions on Graphics (TOG). 394K. Aberman, Y. Weng, D. Lischinski, D. Cohen-Or, and B. Chen. Unpaired motion style transfer from video to ani- mation. ACM Transactions on Graphics (TOG), 39(4):64-1, 2020. 3 Futuregan: Anticipating the future frames of video sequences using spatio-temporal 3d convolutions in progressively growing gans. S Aigner, M Körner, arXiv:1810.01325arXiv preprintS. Aigner and M. Körner. Futuregan: Anticipating the fu- ture frames of video sequences using spatio-temporal 3d convolutions in progressively growing gans. arXiv preprint arXiv:1810.01325, 2018. 1 E Aksan, O Hilliges, Stcn, arXiv:1902.06568Stochastic temporal convolutional networks. arXiv preprintE. Aksan and O. Hilliges. Stcn: Stochastic temporal convo- lutional networks. arXiv preprint arXiv:1902.06568, 2019. 1 Singan-gif: Learning a generative video model from a single gif. R Arora, Y J Lee, Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision. the IEEE/CVF Winter Conference on Applications of Computer Vision14R. Arora and Y. J. Lee. Singan-gif: Learning a genera- tive video model from a single gif. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1310-1319, 2021. 1, 3, 4 M Babaeizadeh, C Finn, D Erhan, R H Campbell, S Levine, arXiv:1710.11252Stochastic variational video prediction. arXiv preprintM. Babaeizadeh, C. Finn, D. Erhan, R. H. Campbell, and S. Levine. Stochastic variational video prediction. arXiv preprint arXiv:1710.11252, 2017. 1 Delving deeper into convolutional networks for learning video representations. N Ballas, L Yao, C Pal, A Courville, arXiv:1511.06432arXiv preprintN. Ballas, L. Yao, C. Pal, and A. Courville. Delving deeper into convolutional networks for learning video representa- tions. arXiv preprint arXiv:1511.06432, 2015. 1 Recycle-gan: Unsupervised video retargeting. A Bansal, S Ma, D Ramanan, Y Sheikh, Proceedings of the European conference on computer vision (ECCV). the European conference on computer vision (ECCV)A. Bansal, S. Ma, D. Ramanan, and Y. Sheikh. Recycle-gan: Unsupervised video retargeting. In Proceedings of the Eu- ropean conference on computer vision (ECCV), pages 119- 135, 2018. 1 Goldman. Patchmatch: A randomized correspondence algorithm for structural image editing. C Barnes, E Shechtman, A Finkelstein, D B , ACM Trans. Graph. 2833C. Barnes, E. Shechtman, A. Finkelstein, and D. B. Gold- man. Patchmatch: A randomized correspondence algorithm for structural image editing. ACM Trans. Graph., 28(3):24, 2009. 2, 3 Structural analogy from a single image pair. S Benaim, R Mokady, A Bermano, L Wolf, Computer Graphics Forum. Wiley Online Library40S. Benaim, R. Mokady, A. Bermano, and L. Wolf. Structural analogy from a single image pair. In Computer Graphics Forum, volume 40, pages 249-265. Wiley Online Library, 2021. 3 Flow-based video synthesis and editing. K S Bhat, S M Seitz, J K Hodgins, P K Khosla, ACM SIGGRAPH 2004 Papers. K. S. Bhat, S. M. Seitz, J. K. Hodgins, and P. K. Khosla. Flow-based video synthesis and editing. In ACM SIGGRAPH 2004 Papers, pages 360-363. 2004. 1 Everybody dance now. C Chan, S Ginosar, T Zhou, A A Efros, Proceedings of the IEEE/CVF International Conference on Computer Vision. the IEEE/CVF International Conference on Computer Vision13C. Chan, S. Ginosar, T. Zhou, and A. A. Efros. Everybody dance now. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 5933-5942, 2019. 1, 3 Stochastic video generation with a learned prior. E Denton, R Fergus, PMLRInternational Conference on Machine Learning. E. Denton and R. Fergus. Stochastic video generation with a learned prior. In International Conference on Machine Learning, pages 1174-1183. PMLR, 2018. 1 Stylit: illumination-guided examplebased stylization of 3d renderings. J Fišer, O Jamriška, M Lukáč, E Shechtman, P Asente, J Lu, D Sỳkora, ACM Transactions on Graphics (TOG). 354J. Fišer, O. Jamriška, M. Lukáč, E. Shechtman, P. Asente, J. Lu, and D. Sỳkora. Stylit: illumination-guided example- based stylization of 3d renderings. ACM Transactions on Graphics (TOG), 35(4):1-11, 2016. 1 Stochastic latent residual video prediction. J.-Y Franceschi, E Delasalles, M Chen, S Lamprier, P Gallinari, PMLR, 2020. 1ternational Conference on Machine Learning. J.-Y. Franceschi, E. Delasalles, M. Chen, S. Lamprier, and P. Gallinari. Stochastic latent residual video prediction. In In- ternational Conference on Machine Learning, pages 3233- 3246. PMLR, 2020. 1 I J Goodfellow, J Pouget-Abadie, M Mirza, B Xu, D Warde-Farley, S Ozair, A Courville, Y Bengio, arXiv:1406.2661Generative adversarial networks. arXiv preprintI. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Ben- gio. Generative adversarial networks. arXiv preprint arXiv:1406.2661, 2014. 1 N Granot, B Feinstein, A Shocher, S Bagon, M Irani, arXiv:2103.15545Drop the gan: In defense of patches nearest neighbors as single image generative models. arXiv preprintN. Granot, B. Feinstein, A. Shocher, S. Bagon, and M. Irani. Drop the gan: In defense of patches nearest neighbors as single image generative models. arXiv preprint arXiv:2103.15545, 2021. 1, 2, 3, 4 Hierarchical patch vaegan: Generating diverse videos from a single sample. S Gur, S Benaim, L Wolf, arXiv:2006.1222614arXiv preprintS. Gur, S. Benaim, and L. Wolf. Hierarchical patch vae- gan: Generating diverse videos from a single sample. arXiv preprint arXiv:2006.12226, 2020. 1, 3, 4 Image analogies. A Hertzmann, C E Jacobs, N Oliver, B Curless, D H Salesin, Proceedings of the 28th annual conference on Computer graphics and interactive techniques. the 28th annual conference on Computer graphics and interactive techniquesA. Hertzmann, C. E. Jacobs, N. Oliver, B. Curless, and D. H. Salesin. Image analogies. In Proceedings of the 28th an- nual conference on Computer graphics and interactive tech- niques, pages 327-340, 2001. 3 Temporally coherent completion of dynamic video. J.-B Huang, S B Kang, N Ahuja, J Kopf, ACM Transactions on Graphics (TOG). 356J.-B. Huang, S. B. Kang, N. Ahuja, and J. Kopf. Temporally coherent completion of dynamic video. ACM Transactions on Graphics (TOG), 35(6):1-11, 2016. 1 Lazyfluids: appearance transfer for fluid animations. O Jamriška, J Fišer, P Asente, J Lu, E Shechtman, D Sỳkora, ACM Transactions on Graphics (TOG). 344O. Jamriška, J. Fišer, P. Asente, J. Lu, E. Shechtman, and D. Sỳkora. Lazyfluids: appearance transfer for fluid anima- tions. ACM Transactions on Graphics (TOG), 34(4):1-10, 2015. 1 A system for retargeting of streaming video. P Krähenbühl, M Lang, A Hornung, M Gross, ACM SIGGRAPH Asia. P. Krähenbühl, M. Lang, A. Hornung, and M. Gross. A sys- tem for retargeting of streaming video. In ACM SIGGRAPH Asia 2009 papers, 2009. 1 Graphcut textures: Image and video synthesis using graph cuts. V Kwatra, A Schödl, I Essa, G Turk, A Bobick, Acm transactions on graphics (tog). 223V. Kwatra, A. Schödl, I. Essa, G. Turk, and A. Bobick. Graphcut textures: Image and video synthesis using graph cuts. Acm transactions on graphics (tog), 22(3):277-286, 2003. 1 Texture optimization for example-based synthesis. V Kwatra, I Essa, A Bobick, N Kwatra, ACM SIGGRAPH 2005 Papers. V. Kwatra, I. Essa, A. Bobick, and N. Kwatra. Texture opti- mization for example-based synthesis. In ACM SIGGRAPH 2005 Papers, pages 795-802. 2005. V Kwatra, D Adalsteinsson, T Kim, N Kwatra, M Carlson, M Lin, Texturing fluids. IEEE transactions on visualization and computer graphics. 13V. Kwatra, D. Adalsteinsson, T. Kim, N. Kwatra, M. Carl- son, and M. Lin. Texturing fluids. IEEE transactions on visualization and computer graphics, 13(5):939-952, 2007. 1 A X Lee, R Zhang, F Ebert, P Abbeel, C Finn, S Levine, arXiv:1804.01523Stochastic adversarial video prediction. arXiv preprintA. X. Lee, R. Zhang, F. Ebert, P. Abbeel, C. Finn, and S. Levine. Stochastic adversarial video prediction. arXiv preprint arXiv:1804.01523, 2018. 1 J Lee, D Ramanan, R Girdhar, arXiv:1910.04742Metapix: Few-shot video retargeting. arXiv preprintJ. Lee, D. Ramanan, and R. Girdhar. Metapix: Few-shot video retargeting. arXiv preprint arXiv:1910.04742, 2019. 3 J Liao, Y Yao, L Yuan, G Hua, S B Kang, arXiv:1705.01088Visual attribute transfer through deep image analogy. arXiv preprintJ. Liao, Y. Yao, L. Yuan, G. Hua, and S. B. Kang. Visual attribute transfer through deep image analogy. arXiv preprint arXiv:1705.01088, 2017. 3 Video completion via motion guided spatial-temporal global optimization. M Liu, S Chen, J Liu, X Tang, Proceedings of the 17th ACM international conference on Multimedia. the 17th ACM international conference on MultimediaM. Liu, S. Chen, J. Liu, and X. Tang. Video completion via motion guided spatial-temporal global optimization. In Proceedings of the 17th ACM international conference on Multimedia, pages 537-540, 2009. 1 World-consistent video-to-video synthesis. A Mallya, T.-C Wang, K Sapra, M.-Y Liu, arXiv:2007.0850913arXiv preprintA. Mallya, T.-C. Wang, K. Sapra, and M.-Y. Liu. World-consistent video-to-video synthesis. arXiv preprint arXiv:2007.08509, 2020. 1, 3 Pytorch: An imperative style, high-performance deep learning library. A Paszke, S Gross, F Massa, A Lerer, J Bradbury, G Chanan, T Killeen, Z Lin, N Gimelshein, L Antiga, A Desmaison, A Kopf, E Yang, Z Devito, M Raison, A Tejani, S Chilamkurthy, B Steiner, L Fang, J Bai, S Chintala, Advances in Neural Information Processing Systems. H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. GarnettCurran Associates, Inc32A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Rai- son, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala. Pytorch: An imperative style, high-performance deep learning library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Process- ing Systems 32, pages 8024-8035. Curran Associates, Inc., 2019. URL http://papers.neurips.cc/paper/ 9015-pytorch-an-imperative-style-high- performance-deep-learning-library.pdf. 3 Human motion transfer from poses in the wild. J Ren, M Chai, S Tulyakov, C Fang, X Shen, J Yang, European Conference on Computer Vision. SpringerJ. Ren, M. Chai, S. Tulyakov, C. Fang, X. Shen, and J. Yang. Human motion transfer from poses in the wild. In European Conference on Computer Vision, pages 262-279. Springer, 2020. 3 Improved seam carving for video retargeting. M Rubinstein, A Shamir, S Avidan, ACM transactions on graphics (TOG). 273M. Rubinstein, A. Shamir, and S. Avidan. Improved seam carving for video retargeting. ACM transactions on graphics (TOG), 27(3):1-9, 2008. 1 Temporal generative adversarial nets with singular value clipping. M Saito, E Matsumoto, S Saito, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionM. Saito, E. Matsumoto, and S. Saito. Temporal generative adversarial nets with singular value clipping. In Proceed- ings of the IEEE international conference on computer vi- sion, pages 2830-2839, 2017. 1 Animating arbitrary objects via deep motion transfer. A Siarohin, S Lathuilière, S Tulyakov, E Ricci, N Sebe, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionA. Siarohin, S. Lathuilière, S. Tulyakov, E. Ricci, and N. Sebe. Animating arbitrary objects via deep motion trans- fer. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition, pages 2377-2386, 2019. 3 First order motion model for image animation. A Siarohin, S Lathuilière, S Tulyakov, E Ricci, N Sebe, Advances in Neural Information Processing Systems. 32A. Siarohin, S. Lathuilière, S. Tulyakov, E. Ricci, and N. Sebe. First order motion model for image animation. Ad- vances in Neural Information Processing Systems, 32:7137- 7147, 2019. 3 Summarizing visual data using bidirectional similarity. D Simakov, Y Caspi, E Shechtman, M Irani, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEE1D. Simakov, Y. Caspi, E. Shechtman, and M. Irani. Sum- marizing visual data using bidirectional similarity. In 2008 IEEE Conference on Computer Vision and Pattern Recogni- tion, pages 1-8. IEEE, 2008. 1, 2 Raft: Recurrent all-pairs field transforms for optical flow. Z Teed, J Deng, European conference on computer vision. SpringerZ. Teed and J. Deng. Raft: Recurrent all-pairs field trans- forms for optical flow. In European conference on computer vision, pages 402-419. Springer, 2020. 4 Learning spatiotemporal features with 3d convolutional networks. D Tran, L Bourdev, R Fergus, L Torresani, M Paluri, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionD. Tran, L. Bourdev, R. Fergus, L. Torresani, and M. Paluri. Learning spatiotemporal features with 3d convolutional net- works. In Proceedings of the IEEE international conference on computer vision, pages 4489-4497, 2015. 4 Mocogan: Decomposing motion and content for video generation. S Tulyakov, M.-Y Liu, X Yang, J Kautz, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionS. Tulyakov, M.-Y. Liu, X. Yang, and J. Kautz. Mocogan: Decomposing motion and content for video generation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1526-1535, 2018. 1 Decomposing motion and content for natural video sequence prediction. R Villegas, J Yang, S Hong, X Lin, H Lee, arXiv:1706.08033arXiv preprintR. Villegas, J. Yang, S. Hong, X. Lin, and H. Lee. Decom- posing motion and content for natural video sequence pre- diction. arXiv preprint arXiv:1706.08033, 2017. 1 Hierarchical longterm video prediction without supervision. R Villegas, D Erhan, H Lee, International Conference on Machine Learning. PMLRR. Villegas, D. Erhan, H. Lee, et al. Hierarchical long- term video prediction without supervision. In International Conference on Machine Learning, pages 6038-6046. PMLR, 2018. R Villegas, A Pathak, H Kannan, D Erhan, Q V Le, H Lee, arXiv:1911.01655High fidelity video prediction with large stochastic recurrent neural networks. arXiv preprintR. Villegas, A. Pathak, H. Kannan, D. Erhan, Q. V. Le, and H. Lee. High fidelity video prediction with large stochastic recurrent neural networks. arXiv preprint arXiv:1911.01655, 2019. 1 C Vondrick, H Pirsiavash, A Torralba, arXiv:1609.02612Generating videos with scene dynamics. arXiv preprintC. Vondrick, H. Pirsiavash, and A. Torralba. Gen- erating videos with scene dynamics. arXiv preprint arXiv:1609.02612, 2016. 1 Video-to-video synthesis. T.-C Wang, M.-Y Liu, J.-Y Zhu, G Liu, A Tao, J Kautz, B Catanzaro, arXiv:1808.0660113arXiv preprintT.-C. Wang, M.-Y. Liu, J.-Y. Zhu, G. Liu, A. Tao, J. Kautz, and B. Catanzaro. Video-to-video synthesis. arXiv preprint arXiv:1808.06601, 2018. 1, 3 Few-shot video-to-video synthesis. T.-C Wang, M.-Y Liu, A Tao, G Liu, J Kautz, B Catanzaro, arXiv:1910.12713arXiv preprintT.-C. Wang, M.-Y. Liu, A. Tao, G. Liu, J. Kautz, and B. Catanzaro. Few-shot video-to-video synthesis. arXiv preprint arXiv:1910.12713, 2019. Imaginator: Conditional spatio-temporal gan for video generation. Y Wang, P Bilinski, F Bremond, A Dantcheva, Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision. the IEEE/CVF Winter Conference on Applications of Computer Vision13Y. Wang, P. Bilinski, F. Bremond, and A. Dantcheva. Imagi- nator: Conditional spatio-temporal gan for video generation. In Proceedings of the IEEE/CVF Winter Conference on Ap- plications of Computer Vision, pages 1160-1169, 2020. 1, 3 Inmodegan: Interpretable motion decomposition generative adversarial network for video generation. Y Wang, F Bremond, A Dantcheva, arXiv:2101.03049arXiv preprintY. Wang, F. Bremond, and A. Dantcheva. Inmodegan: In- terpretable motion decomposition generative adversarial net- work for video generation. arXiv preprint arXiv:2101.03049, 2021. 1 Space-time video completion. Y Wexler, E Shechtman, M Irani, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. I-I. IEEEthe 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition1Y. Wexler, E. Shechtman, and M. Irani. Space-time video completion. In Proceedings of the 2004 IEEE Computer So- ciety Conference on Computer Vision and Pattern Recogni- tion, 2004. CVPR 2004., volume 1, pages I-I. IEEE, 2004. 1, 2 Non-homogeneous content-driven video-retargeting. L Wolf, M Guttmann, D Cohen-Or, Proceedings of the Eleventh IEEE International Conference on Computer Vision (ICCV). the Eleventh IEEE International Conference on Computer Vision (ICCV)L. Wolf, M. Guttmann, and D. Cohen-Or. Non-homogeneous content-driven video-retargeting. In Proceedings of the Eleventh IEEE International Conference on Computer Vision (ICCV), 2007. 1 Transmomo: Invariance-driven unsupervised video motion retargeting. Z Yang, W Zhu, W Wu, C Qian, Q Zhou, B Zhou, C C Loy, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern Recognition13Z. Yang, W. Zhu, W. Wu, C. Qian, Q. Zhou, B. Zhou, and C. C. Loy. Transmomo: Invariance-driven unsupervised video motion retargeting. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5306-5315, 2020. 1, 3
[]
[ "Spectral function of one hole in several one-dimensional spin arrangements", "Spectral function of one hole in several one-dimensional spin arrangements" ]
[ "R Hayn \nInstitute for Theoretical Physics\nInstitute for Solid State and Materials Research (IFW)\nInstitute for Materials Science\nUniversity of Technology\nKrjijanovskogo 3D-01062, D-01171, 252180Dresden, Dresden, KievGermany, Ukraine\n", "R O Kuzian \nInstitute for Theoretical Physics\nInstitute for Solid State and Materials Research (IFW)\nInstitute for Materials Science\nUniversity of Technology\nKrjijanovskogo 3D-01062, D-01171, 252180Dresden, Dresden, KievGermany, Ukraine\n" ]
[ "Institute for Theoretical Physics\nInstitute for Solid State and Materials Research (IFW)\nInstitute for Materials Science\nUniversity of Technology\nKrjijanovskogo 3D-01062, D-01171, 252180Dresden, Dresden, KievGermany, Ukraine", "Institute for Theoretical Physics\nInstitute for Solid State and Materials Research (IFW)\nInstitute for Materials Science\nUniversity of Technology\nKrjijanovskogo 3D-01062, D-01171, 252180Dresden, Dresden, KievGermany, Ukraine" ]
[]
The spectral function of one hole in different magnetic states of the one-dimensional t-J model including three-site term and frustration J ′ is studied. In the strong coupling limit J → 0 (corresponding to U → ∞ of the Hubbard-model) a set of eigenoperators of the Liouvillian is found which allows to derive an exact expression for the one-particle Green's function that is also applicable at finite temperature and in an arbitrary magnetic state. The spinon dispersion of the pure t-J model with the ground-state of the Heisenberg model can be obtained by treating the corrections due to a small exchange term by means of the projection method. The spectral function for the special frustration J ′ = J/2 with the Majumdar-Ghosh wave function is discussed in detail. Besides the projection method, a variational ansatz with the set of eigenoperators of the t-term is used. We find a symmetric spinon dispersion around the momentum k = π/(2a) and a strong damping of the holon branch. Below the continuum a bound state is obtained with finite spectral weight and a very small separation from the continuum. Furthermore, the spectral function of the ideal paramagnetic case at a temperature kBT ≫ J is discussed.
10.1103/physrevb.62.12156
[ "https://arxiv.org/pdf/cond-mat/0005462v1.pdf" ]
119,408,276
cond-mat/0005462
315715c188696cd9a0cbe5da248cef9399e9390c
Spectral function of one hole in several one-dimensional spin arrangements 26 May 2000 (November 20, 2018) R Hayn Institute for Theoretical Physics Institute for Solid State and Materials Research (IFW) Institute for Materials Science University of Technology Krjijanovskogo 3D-01062, D-01171, 252180Dresden, Dresden, KievGermany, Ukraine R O Kuzian Institute for Theoretical Physics Institute for Solid State and Materials Research (IFW) Institute for Materials Science University of Technology Krjijanovskogo 3D-01062, D-01171, 252180Dresden, Dresden, KievGermany, Ukraine Spectral function of one hole in several one-dimensional spin arrangements 26 May 2000 (November 20, 2018)71.27.+a, 73.20.Dx, 79.60.Bm The spectral function of one hole in different magnetic states of the one-dimensional t-J model including three-site term and frustration J ′ is studied. In the strong coupling limit J → 0 (corresponding to U → ∞ of the Hubbard-model) a set of eigenoperators of the Liouvillian is found which allows to derive an exact expression for the one-particle Green's function that is also applicable at finite temperature and in an arbitrary magnetic state. The spinon dispersion of the pure t-J model with the ground-state of the Heisenberg model can be obtained by treating the corrections due to a small exchange term by means of the projection method. The spectral function for the special frustration J ′ = J/2 with the Majumdar-Ghosh wave function is discussed in detail. Besides the projection method, a variational ansatz with the set of eigenoperators of the t-term is used. We find a symmetric spinon dispersion around the momentum k = π/(2a) and a strong damping of the holon branch. Below the continuum a bound state is obtained with finite spectral weight and a very small separation from the continuum. Furthermore, the spectral function of the ideal paramagnetic case at a temperature kBT ≫ J is discussed. I. INTRODUCTION The understanding of spin-charge separation is a central point in the physics of low-dimensional electronic systems. That issue is most clearly seen in one dimension where the exact solution of the 1D Hubbard model 1 reveals that the low-energy physics is dominated by decoupled, collective charge and spin excitations (also called holon and spinon, respectively). The idea that the spin and charge degrees of freedom separate has also been proposed to explain the properties of 2D cuprate superconductors. 2 However, even in one dimension, the Bethe ansatz solution did not yet give a complete answer for the spectral function A(k, ω). Only for U → ∞ an exact expression for A(k, ω) was found 3 due to the factorization property of the wave function according to Ogata and Shiba. 4 The results for the insulating half-filled case 3 could also be generalized to other filling factors. 5 The spin-charge separation was observed in ARPES measurements of the one dimensional, dielectric cuprate SrCuO 2 . 6 The spinon and holon branch of the spectral function were seen, in contrast to the analogous experiment of one hole in the CuO 2 plane 7 where spin and charge are coupled and the spin polaron quasiparticle has a dispersion proportional to J as proposed theoretically in Refs. [8][9][10]. The ARPES spectra in SrCuO 2 were analyzed using the pure t-J model. On the other hand, many 1D compounds, like for instance CuGeO 3 , 11 are characterized by frustration in the magnetic subsystem which may lead to a gap in the spin excitation spectrum. For the special frustration J ′ /J = 0.5 and in the limit J → 0 an analytic expression for A(k, ω) was derived recently, 12 under the assumption that the wave function factorizes. Besides the frustration, also temperature effects are important as it was observed in ARPES measurements on Na 0.96 V 2 O 5 . 13 So, there is a clear need to study the spectral function systematically under the influence of frustration and temperature and to derive analytic expressions. The present work focuses mainly on the effect of frustration and temperature on the spectral function in the insulating case. For that, we rederive first the exact solution of Sorella and Parola 3 in a straightforward way using Green's function technique. That is possible due to our finding of a set of eigenoperators of the Liouvillian (Sec. III) in the strong coupling limit J → 0. As a consequence, our derivation is applicable for any magnetic state and any temperature in that limit. Especially, one can show that the result which was derived in Ref. 12 does not depend on the assumption that the wave function factorizes. We present analytic expressions for the spectral function of one hole in several magnetic states: (i) the ground-state of the antiferromagnetic Heisenberg model, (ii) the Majumdar-Ghosh wave function 14 at the special frustration J ′ /J = 0.5, and (iii) the ideal paramagnetic state at temperatures much larger than the exchange energy k B T ≫ J. For large finite coupling we compare two methods to account for corrections J ∝ (t 2 /U ), namely the projection method and a variational ansatz using the set of eigenoperators of the t-term. We show that the former method yields a reasonable description of the spinon dispersion in the pure t-J model (Sec. IV) and an approximate result for the spectral function of the Majumdar-Ghosh model. For J ′ = J/2 it misses the bound state below the continuum which is obtained by the more accurate variational method (Sec. VB). The bound state has a finite spectral weight but a very small separation from the continuum. Both methods show in the Majumdar-Ghosh case that the low energy region for momenta k between π/2 and π (lattice constant a = 1) will be filled with states, that the spinon dispersion (i.e. that collective excitation corresponding to the lower edge of the continuum) becomes symmetric around π/2, and they indicate an overdamped holon branch. The damping of the holon branch is extremely large for very high temperatures (Sec. VI). Before presenting our results let us shortly discuss the different understandings of the term "spin-charge separation" as it can be met in the literature. The naive picture means that the low energy effective Hamiltonian may be written asĤ =Ĥ h +Ĥ s , [Ĥ h ,Ĥ s ] = 0 ,(1) and the electron operator is the product c iσ = s iσ h † i ,(2) where spinon s and holon h can be basically regarded as free particles. Then the normalized (i.e. A(k, ω)dω = 1) spectral function is A(k, ω) = 1 L Q 2f (Q)δ [ω − ǫ h (k − Q) − ǫ s (Q)] ,(3) where f (Q) = θ( π 2 − |Q|) is the Fermi distribution function of spinons, θ(x) is the Heaviside step function, L is the number of sites, ǫ h , ǫ s being holon and spinon energies, respectively. However, the naive understanding is not that one which is realized in 1D electron systems. 15 There, it was found that the eigenstates factorize in the limit U → ∞ in the form ψ(x 1 , . . . , x N , y 1 . . . y M ) = ψ SF (x 1 , . . . , x N )φ H (y 1 . . . y M ),(4) where x 1 , . . . , x N are the spatial coordinates of the N electrons on a L-site ring, and the y 1 . . . y M 'coordinates' label the position of the spin-up electrons on the squeezed Heisenberg ring, i.e. on the N occupied sites. The ψ SF is a spinless fermion state, and φ H is an eigenstate of an N -site Heisenberg Hamiltonian with periodic boundary conditions. The product form of equation (4) should not be interpreted as a trivial decoupling between charge and spin. In fact, the momentum of the spin wave function imposes a twisted boundary condition on the spinless fermion wave function. As a result, the Fermi distribution function f (Q) in (3) will be replaced by a function Z(Q) that is the expectation value of a chain of spin operators that has to be determined from the pure spin system 3 (for the details see Sec. III). The singularity of Z(Q) produces additional peaks in A(k, ω). This correct answer for the spectral function may be understood as a manifestation of the phase string effect. 16 It means that spinon and holon interact with each other via a nonlocal phase-string. Instead of (2) we should write c iσ = s iσ h † i exp π 2 l>i h † l h l + π 2 l>i (s † lσ s lσ − 1) . It should be noted that the phenomenon of spin-charge separation is not restricted to the limit U → ∞ in the 1D Hubbard model. At any finite U the spin and charge fluctuations propagate with different velocities. 1 That means that after some time the spin and charge degrees of freedom will be separated in space. But there is no analytic solution for the spectral function of the Hubbard model at arbitrary values of U and also the present calculation treats terms of order (t 2 /U ) as a perturbation. In that sense we will understand here spin-charge separation as a manifestation of the factorization property (4) in the spectral density. Sharp maxima in the continuum correspond to collective excitations whereas possible bound states indicate special eigenfunctions with a strong coupling between spin and charge. Another possible effect of additional terms in the Hamiltonian is the broadening (i.e. the damping) of the collective excitations. II. MODEL AND SPECTRAL DENSITY To describe the low energy physics of compounds with a 1D electronic structure it is sufficient in most cases to take into account only that band which is closest to the Fermi energy (see for instance Ref. 17). Treating the on-site Coulomb interaction explicitly, one obtains the well known 1D Hubbard model. In the present calculation we restrict ourselves to the strong coupling limit U ≫ t where we may project out the subspace of doubly occupied sites, and for the lower Hubbard band we obtain the effective Hamiltonian H =t +Ĵ +t 3 ,(5)wheret = −t i,g,α X α0 i X 0α i+g ,(6)J = J 2 i,α,β X αβ i X βα i+1 ,(7)t 3 = t 3 i,g,α,β X α0 i X βα i+g X 0β i+2g ,(8) and α, β =↑↓; g are the nearest neighbors g = ±1. The Hamiltonian is valid near half filling (X ++ i + X −− i = 1). The parameters J and t 3 are connected with the original values of the Hubbard model by J = 4t 3 = 4t 2 /U ,(9) but the Hamiltonian (5) is more general, if we relax the condition (9). It may be derived directly from the more realistic three-band Hubbard model. Then, the t 3 -term often becomes negligible and one obtains the t-J model. The Hamiltonian (5) is written in terms of Hubbard projection operators that act in the subspace of on-site states X αβ i ≡ |α, i β, i| , α, β = 0, ↑, ↓, 2 .(10) They are related with bare fermionic and spin operators through X σ0 i = c † i,σ (1 − n i,−σ ) , X σ2 i = −σc i,−σ n i,σ ,(11)X +− i = S + i = c † i,↑ c i,↓ , X σσ i = 1 2 + σ 2 c † i,↑ c i,↑ − c † i,↓ c i,↓ = 1 2 + σS z i ,(12) with σ = ±1. Other relations are easy to obtain with the use of the main property of Hubbard operator algebra X αβ i X γλ i = δ βγ X αλ i ,(13) that follows immediately from the definition (10). The commutation relations for operators on different sites are fermionic for operators that change the number of particles by odd integers, like (11), and bosonic for others. In the presence of frustration in the magnetic system, which is discussed for instance for CuGeO 3 , the t-J Hamiltonian may be generalized by inclusion of the J ′ -termĴ ′ = J ′ 2 i,α,β X αβ i X βα i+2 .(14) Our aim is to calculate the one-particle two-time retarded Green's function G(k, ω) and the spectral density of one hole in the magnetic state A(k, ω) = − 1 π ImG(k, ω + i0 + ),(15) that is roughly proportional to the ARPES signal intensity. We define 2πδ(k − k ′ )G(k, ω) = X σ0 k |X 0σ k ′ ≡ −i ∞ t ′ dte iω(t−t ′ ) {X σ0 k (t), X 0σ k ′ (t ′ )} ,(16) where X σ0 k = √ 2 +∞ m=−∞ e −ikm X σ0 m , X σ0 k , X 0σ k ′ = 2πδ(k − k ′ ) , and where {. . . , . . .} means the anticommutator. The expectation value denotes the thermal average over a grand canonical ensemble: ... = Q −1 Sp [e −β(Ĥ−µN ) ...], Q = Sp e −β(Ĥ−µN ) .(17) Here Sp implies taking the trace of an operator,N is the particle number operator, β = (kT ) −1 is an inverse temperature, and µ represents the chemical potential. The time dependence of the operator B(t) is given by B(t) = e it(Ĥ−µN ) Be −it(Ĥ−µN ) . III. EIGENOPERATOR AND HOLON DISPERSION Let us consider first the limit U → ∞ in the Hubbard model or J, t 3 → 0 in (5). Then only the t-term t = −t i,g,α X α0 i X 0α i+g , is nonzero. Note that it is a true many-body Hamiltonian due to the constraint of no double occupancy, as we see from Eq. (11). We introduce the set of operators v m,r = α1,...,αr X σα1 m X α1α2 m+g . . . X αr−1αr m+r−g X αr0 m+r , g = sign (r) ,(18)for which v m,r ,t = t (v m,r−1 + v m,r+1 )(19) holds at half filling. Any operator (18) can be considered as a string operator of a certain length consisting of a hole and an attached string of spin flips. Such a string can be produced by creating a hole in the Néel state and applying several times the kinetic energy (6) which creates misaligned spins. Similar string operators were used to describe the spin polaron quasiparticle in the 2D case. 10, 18 We make double Fourier transform v k,q = √ 2 +∞ m,r=−∞ e −ikm−iqr v m,r ,(20) and see that v k,q ,t = 2tv k,q cos q . The interpretation in terms of the string operator (18) is quite easy. We see that only the right end of the operator (18) was influenced by the t-term. Therefore, we may identify the right end of v m,r with the holon excitation. In the next Section, it will become clear that the left end of v m,r may be connected with the spinon. The operators v k,q (20) are eigenoperators of the Liouvillian L of the problem, where LÂ ≡ [Ĥ,Â]. Note that it is one of the rarest, if not the unique case in many-body physics that the explicit form for a set of eigenoperators can be given. From (21) we see that the equation of motion for the corresponding string operator Green's function closes and it has a simple pole form v k,q |v † k ′ ,q ′ = v k,q , v † k ′ ,q ′ ω − 2t cos q , v k,q , v † k ′ ,q ′ = 8π 2 δ(k − k ′ )δ(q − q ′ )Z(k − q + π) ,(22) where the spectral weight is the expectation value Z(q + π) = 1 2 +∞ r=−∞ e −iqr Ω r ,(23) of a chain of X-operators Ω r = α1,...,αr,σ X σα1 m X α1α2 m+g . . . X αr−1αr m+r−g X αrσ m+r = (2S m S m+g + 1 2 )(2S m+g S m+2g + 1 2 ) . . . (2S m+r−g S m+r + 1 2 ) ,(24) to be calculated for the pure spin-system without any hole. The expectation value Ω r of (24) cannot depend on the starting point m due to the translational symmetry of the problem. The operator (24) was introduced in Ref. 4 and explicit values on a 26-site Heisenberg ring were given for T = 0, when . . . becomes the average over the ground-state. Asymptotically, the following behavior was found Ω l → 1 √ l Re Ae iπl/2(25) which leads to a square root singularity of Z(Q). Using additionally the exact values Ω 0 = 1 and Ω 1 = 1 − 2 ln 2 the following formula may be derived 3 Z(Q) = −0.393 + 0.835/ cos Q θ( π 2 − |Q|) .(26) For the hole Green's function (16) we have X σ0 k |X 0σ k ′ = +π −π dq 2π +π −π dq ′ 2π v k,q |v † k ′ ,q ′ = 2πδ(k − k ′ ) +π −π dq π Z(k − q + π) ω − 2t cos q ,(27) and the spectral density is obtained in the way A(k, ω) = +π −π dQ π Z(Q)δ [ω + 2t cos(k − Q)] .(28) That gives the exact answer in the strong coupling limit (J → 0) where only the holon dispersion ǫ h (q) = −2t cos q is important. But the ratio J ′ /J may be arbitrary and (28) is not only exact in the Heisenberg case with Z(Q) from (26). Instead, from our derivation follows its validity for arbitrary magnetic states and it is not restricted to zero temperature. Then, however, Z(Q) is different. In the following we will give exact results for i) the Majumdar-Ghosh wave function at the special frustration J ′ /J = 0.5, and ii) the ideal paramagnetic case at k B T ≫ J. Two cases are quite trivial, namely the saturated ferromagnetic case and the classical Néel state. The former one leads to Z(Q + π) ∝ δ(Q) and a spectral function like for free fermions, whereas the latter case leads to the Brinkmann-Rice continuum 19 (Z(Q) = 1/2). A magnetic state inbetween the classical Néel and the Heisenberg case could, in principle, also be considered, for which the one-hole spectral function was derived in Ref. 20 treating the spin-fluctuations as perturbation. IV. SPINON DISPERSION In real systems the ratio J/t is roughly 0.3. It means that they are in the regime of strong coupling and the above consideration correctly describe the largest energy scale ∝ t. Now, we want to estimate the corrections that arise from other terms of the Hamiltonian (5). First we note that v k,q are eigenoperators for the t 3 -term v k,q ,t 3 = −2t 3 v k,q cos 2q ,(29) which leads to the replacement ǫ h (q) → ǫ h (q) + 2t 3 cos 2q in the denominator of (22). The commutation withĴ gives (see Appendix A for the details) v k,q ,Ĵ = −J cos(k − q)v k,q + J 2 (v ′ k,q + v ′′ k,q ) ,(30) where v ′ k,q and v ′′ k,q are Fourier transforms of the operators v ′ m,r = γ,α1,...,αr X σγ m X γα1 m−g X α1α2 m+g . . . X αr−1αr m+r−1 X αr0 m+r , (31) v ′′ m,r = − γ,α1,...,αr X σα1 m X α1α2 m+g . . . X αr−1αr m+r−g X αrγ m+r+g X γ0 m+r .(32) It is impressive that terms, which come from the commutation of "inner" X n operators in v m,r with n between the points m and m + r cancel each other and only the terms coming from the ends remain. The term v ′′ m,r presents a distortion of the right end of v m,r by means of the exchange part and may be interpreted as the loss of magnetic energy due to the presence of a holon. On the other hand, the term v ′ m,r , with a distorted left end will be shown to give rise to the spinon dispersion. We really observe the "separate" motion of the holon that is represented by the right end of v m,r and of the spinon that is the left end of v m,r . The holon motion is governed by the t-term and the spinon motion by the J-term. We put the word "separate" in quotes because the motion remains correlated due to the set of "inner" X n operators, connecting the ends of v m,r . We need an approximate approach to account for v ′ k,q + v ′′ k,q . For this purpose we use the projection technique v ′ k,q + v ′′ k,q ≈ v ′ k,q + v ′′ k,q , v † k,q v k,q , v † k,q v k,q .(33) Now, the Green's function for the string operator has the form v k,q |v † k ′ ,q ′ = 8π 2 δ(k − k ′ )δ(q − q ′ )Z(k − q + π) ω − 2t cos q + 2t 3 cos 2q − ǫ s (k − q) ,(34) where ǫ s (k − q) is defined by the equation v k,q ,Ĵ , v † k ′ ,q ′ ≡ 8π 2 δ(k − k ′ )δ(q − q ′ )ǫ s (k − q)Z(k − q + π) .(35) The contribution of the v ′′ m,r term to the spinon dispersion is determined by an expression of the form v ′′ m,r , v † m ′ ,r ′ = 1 2 δ m+r,m ′ +r ′ Ω ′′ r,r ′ ,(36) where the precise order of the X-operators in Ω ′′ r,r ′ can be easily inferred from (32) and is given in the Appendix A. There, it is also shown that for slowly decaying spin correlation functions (as in the present case, see (25)) the correlation functions Ω ′′ r,r ′ can be approximated to be a function of r − r ′ only, in the way: Ω ′′ r,r ′ ≈ Ω ′′ r−r ′ ,0 ≈ Ω 1 Ω r−r ′ .(37) That leads to a constant shift of the energy ǫ s as the only effect of v ′′ m,r which will be neglected further on. The contribution of the v ′ m,r term can be written analogously to (36), defining the spin correlation functions Ω ′ r,r ′ = Ω ′ r−r ′ ,0 . The correlation functions Ω ′ l,0 differ from Ω l+1 only by the exchange of two X-operators. Therefore, for large l, we may expect that Ω ′ l,0 ≈ Ω l+1 .(38) That leads after Fourier transformation to the contribution of the v ′ m,r term to the spinon dispersion (Appendix A). Together with the contribution of v k,q we obtain for the spinon dispersion ǫ s (Q − π)Z(Q) = J 2 cos Q Z(Q) + 1 2 + 1 2 Ω ′ 0,0 − Ω 1 − 1 2 sin Q 2π 0 dκ 2π Z(κ) cot Q − κ 2 + cot Q + κ 2 ,(39) and the hole spectral function becomes A(k, ω) = +π −π dQ π Z(Q)δ [ω + 2t cos(k − Q) − 2t 3 cos 2(k − Q) − ǫ s (Q − π)] .(40) The curve that we obtained for ǫ s with the formula (39) is close to ǫ s (Q − π) ≈ αJ cos Q , α ≈ 2 .(41) as shown in Fig. 1. The functional form (41) is consistent with Bethe-ansatz 3 and field-theoretical considerations. 15 (Sorella and Parola 3 derived a contribution Jπ/2 cos Q ≈ 1.6J cos Q.) We tested it also by comparing the first two terms of Fourier expansion of the product cos QZ(Q) with Ω ′ 0,0 and Ω ′ 1,0 that give values for α in (41) of 2.1 or 1.8, respectively (for the pair correlation functions we took the data of Ref. 4). Therefore, we are using the simplified formula (41) instead of (39) in the following analysis of the spectral density. We have checked that the differences are negligible. The spectral density (for t 3 = 0) is shown in Fig. 2. One can clearly distinguish between the spinon and holon features at the lower edge of the spectral density dispersing at an energy scale ∝ J (from k = 0 to k = π/2) or ∝ t (from k = π/2 to k = π). At k = k * , which is determined by t cos k * = J, another holon branch splits off the lower edge of the spectrum 15 and disperses towards k = 0 at an energy scale of t (and a corresponding holon branch splits off the upper edge of the spectrum). For k values inbetween 0 and k * one has three peaks in the spectral function (one spinon and lower and upper holon branch). One can easily imagine the situation in the doped case. Then the spinon and holon branches start at the Fermi energy with two different velocities. 21 In contrast to the 2D case, 9 there is no separate bound state at the lower edge of the spectrum indicating that there are only collective spin and charge excitations. Most of those features were also observed in the ARPES experiment. 6 In the naive picture of spin-charge separation (3) the spectral density would have square root singularities only either at the lower or at the upper edge of the spectrum. In Fig. 2, however, there are additional holon branches due to the square root singularity in Z(Q) (see also Ref. 15). For J → 0 the spinon feature in the spectral density, i.e. the lower edge of the spectrum, becomes completely flat between 0 and π/2. The corresponding pictures were already given in Ref. 3. Fig. 2 agrees also qualitatively with the finite cluster results. 6 V. MAJUMDAR-GHOSH MODEL We have shown that our approach is applicable for any magnetic state for J → 0. Now, we are going to present the spectral function of one hole in the t-J-J ′ model with the special frustration J ′ = J/2 (called here Majumdar-Ghosh (MG) model for simplicity). In that case rigorous analytic results may be obtained since the ground-state wave function of the MG spin Hamiltonian is exactly known. 14 It is the combination of two simple dimer states Ψ MG = (Φ 1 + Φ 2 )/ √ 2 ,(42) where Φ 1 = +∞ n=−∞ [2n, 2n + 1] , Φ 2 = +∞ n=−∞ [2n − 1, 2n] , and the singlet bond is denoted as [l, m] ≡ 1 √ 2 σ σX σ0 l X −σ0 m |vac . We are considering the MG model as a representative example for the case that there is a gap in the spin excitation spectrum (and also in the charge channel). To give the result for the spectral density in the strong coupling limit J → 0 one has to find the modified quasiparticle residue Z(Q) in (28). It can be simply derived from the correlation functions (see also Ref. 12) Ω l = 1 2 [ Φ 1 |Ω l |Φ 1 + Φ 2 |Ω l |Φ 2 ] Ω 2n = − 1 2 n , Ω 2n+1 = 1 2 − 1 2 n+1 , n ≥ 0 ,(43) in the following way Z(Q) = 1 2 + ∞ n=1 1 2 − e −iQ 4 − e 2iQ 2 n + h.c. = 3 2 1 + cos Q 5 + 4 cos 2Q .(44) The corrections for small J ≪ t may only be derived approximatively and we present two methods, projection method and variational procedure having different accuracy. A. Projection method First we calculate the spinon dispersion ǫ s in the same way as it was done in the Heisenberg case in Sec. IV. But we should keep in mind that its applicability is less justified for the MG model than for the pure t-J model due to the much faster decay of spin correlation functions (compare (43) with (25)). As before, we approximate Ω ′′ r,r ′ ≈ Ω 1 Ω r−r ′ which results in a constant energy shift from the v ′′ k,q term. Therefore, the first contribution to ǫ s coming fromĴ is merely determined by Ω ′ r,r ′ (see Appendix B) resulting in ǫ sJ (Q − π) = 2J cos Q .(45) We have a second contribution to ǫ s fromĴ ′ ǫ sJ ′ (Q − π) = J ′ −4 cos Q + 5 4 + cos 2Q .(46) We see that for J ′ = J/2 the terms proportional cos Q cancel and we find ǫ s (Q − π) = J 5 8 + 1 2 cos 2Q ,(47) which is symmetric around π/2. The spectral density is presented in Fig. 3. We see that in contrast to the t-J model the structures coming from Z(Q) (the holon branches) are much less pronounced, whereas square root singularities exist at the lower and upper edges of the spectrum. Their intensities are proportional to Z(k) or Z(k−π) at the lower and upper edges, respectively. Therefore, the square root singularity vanishes for k = π at the lower edge. Furthermore, one can see that the low energy region for k between π/2 and π being empty in Fig. 2 is now filled with states. The spectrum becomes more symmetric around π/2 and the low-energy edge is given by the spinon dispersion ǫ s (Q). The strong damping of the holon branch is due to the suppression of the singularity at the spinon Fermi edge (at Q = π/2 in Z(Q)). It is a universal feature for any 1D magnetic state having a gap in the spin excitation spectrum. The suppression of holon weight was also found by Voit 22 for the Luther-Emery phase in the Luttinger liquid. The form of the spectral density in Fig. 3 resembles also roughly the exact diagonalization study in Ref. 12. But a single bound state with a finite spectral weight that was obtained there, is missing in Fig. 3. That deficiency is due to the special projection procedure (33) which can only result in a continuous spectral density. Therefore, one has to go beyond the projection method. B. Variational ansatz Here we will use the set of string operators (18) as a set defining a variational wave function for the whole Hamiltonian. Due to the knowledge of the exact ground-state (42) all necessary matrix elements can be calculated without any further approximation. More precisely, we will diagonalize the HamiltonianĤ =t +Ĵ +Ĵ ′ in the space spanned by the set of basis operators v k,r = 1 √ L +∞ −∞ e ik(m+r) v m,r ,(48) where L is the number of lattice sites and v m,r was defined in (18). For that purpose one has to calculate the overlap matrix resulting in S r,r ′ = {v k,r , v † k,r ′ } = 1 2 Ω r−r ′ .(49) The kinetic energy part of the Hamilton matrix is given by: t 2 E k r,r ′ = [v k,r ,t], v † k,r ′ = t 2 e ik Ω r−r ′ −1 + e −ik Ω r−r ′ +1 .(50) The calculation of the exchange part of the Hamilton matrix is quite lengthy but straightforward. It shall not be given here in detail. To present the results we define a matrix E x : J 4 E x r,r ′ = [v k,r ,Ĵ +Ĵ ′ ], v † k,r ′ ,(51) whose matrix elements are listed in Appendix C. The Hamilton matrix is then given by E = t 2 E k + J 4 E x ,(52) and the matrix GF G r,r ′ = v k,r |v † k,r ′(53) can be found by solving the equation (ω + iΓ + −ES −1 )G = S , Γ > 0 .(54) Finally, the GF (16) can be obtained by G(k, ω) = 2G 0,0 . The numerical results for J = 0 and J = 0.4 at three different momenta are presented in Fig. 4. The curves for J = 0 coincide with the analytic expression (28,44). A number of 400 basis functions and a broadening of Γ = 0.05 are sufficient to reach the thermodynamic limit in contrast to the exact diagonalization method yielding only a sequence of δ-peaks. For J = 0.4 we can confirm the features found by the projection method, i.e. the low energy intensity between π/2 and π, the symmetric spinon dispersion and the overdamped holon branch. In addition, the exchange terms produce two new features not present in Sec. VA: a resonance peak near zero energy and a bound state below the continuum. The resonance peak is visible near k = π and becomes an antiresonance near k = 0. Careful inspection of the exact diagonalization data 12 indicates also a very high peak at the resonance position for k = π and a small gap at k = 0, but a better understanding of the resonance/antiresonance feature is still required. The bound state is not visible in Fig. 4 due to the broadening Γ which is too large. Instead, we present in Fig. 5 the spectral weight of the lowest eigenstate w 1 for k = π/2 and J = 0.4 in dependence on the number of basis functions. It is clearly seen that the weight tends to a constant value (w 1 ≈ 0.1) in difference to the weight w 3 of the third eigenstate. 23 At the same time, the separation e 1 = E 3 − E 1 between the first and the third eigenvalues E 1/3 stays finite for N → ∞ but the separation is very small (e 1 ≈ 0.02 in units of t). For J = 0, both w 1 and e 1 tend to zero for N → ∞. That means that the bound state is connected with the presence of a gap in the spin excitation spectrum. VI. IDEAL PARAMAGNETIC STATE Such a state is realized for very high temperatures T , much larger than the exchange energy k B T ≫ J. In that case spins at neighboring sites are completely uncorrelated. But the temperature is assumed to be lower than the Hubbard U such that the constraint of no double occupancy is preserved. Then the correlation functions become simply Ω l = 1 2 l , which results in Z(Q) = 3 8 1 5 4 + cos Q .(55) The calculation of the spinon part (without frustration) gives ǫ s (Q − π) = J 2 2 cos Q + 1 2 .(56) To calculate it one has to note that (38) is no approximation in the present case. The effect of the Ω ′′ r,r ′ terms can only be treated approximatively (see (37)) but it was checked by the variational method that its influence on the spectral function can be neglected. The information on Z(Q) and ǫ s is sufficient to calculate the spectral function (Fig. 6). It is surprising that the strong singularities at the band edges survive despite the large temperature. The lower edge disperses according to the dispersion of the spinon (56) with a width proportional to J and has its minimum at k = π (in contrast to the frustrated case Fig. 3 with a minimum at k = π/2). But a peak connected with the holon dispersion proportional to t is not seen in Fig. 6. Such a peak appears in the finite temperature spectral function of the 2D t-J model 24 and it can be expected since the first moment of the spectral function disperses according to t cos k. Its absence in 1D is a nontrivial and unexpected result. It can be understood in the present context since the holon branch is strongly damped due to the suppression of the singularity in Z(Q) at Q = π/2. Apparently, that suppression is more strong in (55) than in the frustrated case (44) such that the holon branch is still visible in Fig. 3 but it disappears nearly in Fig. 6. One should note that the above result holds only in the region J ≪ k B T ≪ U . One may speculate that a further increase of the temperature such that the constraint of no double occupancy is lifted should lead to drastic changes in the spectral function. The strong singularities at the lower or upper band edges should disappear and a free dispersion should become visible. VII. CONCLUSION In conclusion we could derive analytic expressions for the spectral function of one hole in several magnetic states. The expressions are rigorous in the limit J → 0, but our approach allows also to calculate the small J corrections. We analyzed the frustration and temperature effects. Results were given for the special frustration J ′ = J/2 with a gap in the spin excitation spectrum and for the ideal paramagnetic case. Both effects, frustration and temperature, lead to low-energy excitations between π/2 and π, and to a strong damping of the holon branches in the spectral function caused by the suppression of the singularity at the Fermi edge of spinons. The exchange terms in the MG model were found to be responsible for the finite weight of the lowest eigenstate and its finite, but small, energy separation from the rest of the spectrum, i.e. the bound state. The proposed scenario of holon branch damping seems to be a universal feature of frustration and temperature. Therefore, our results are of direct importance for photoemission experiments on strongly frustrated 1D compounds like CuGeO 3 , for instance. However, edge-shared cuprate chains have a smaller energy scale and less ideal 1D behavior in comparison with corner-shared compounds, 17 which hinders direct comparison with experiment. But it cannot be excluded that a small frustration is also present in SrCuO 2 such that our study gives one possible reason, why no real, separate holon branch could be observed in the experimental spectra of SrCuO 2 between k = 0 and π/2. 6 In the spin gap case we found a very small energy separation of the bound state from the continuum such that it is nearly impossible to detect it in a photoemission experiment. X αrσ nr ≡ (n 1 |n 2 |n 3 | . . . |n r−1 |n r ) . which means especially that (18) may be rewritten as v m,r ≡ (m|m + g|m + 2g| . . . |m + r]. In such a notation we obtain for the commutation with the Heisenberg Hamiltonian X σ0 m ,Ĵ = − J 2 g,γ X σγ m+g X γ0 m = − J 2 g (m + g|m] , (A.1) X αβ m ,Ĵ = J 2 g,γ X αγ m X γβ m+g − X αγ m+g X γβ m = J 2 g {(m|m + g| − (m + g|m|} , (A.2) and then v m,r ,Ĵ = J 2 {(m|m − g|m + g|m + 2g| . . . |m + r] − (m − g|m|m + g|m + 2g| . . . |m + r] −(m + g|m + 2g| . . . |m + r] − (m| . . . |m + r − g|m + r + g|m + r]} , (A.3) where g = sign(r). In deriving (A.3) it is important that the commutation of the "inner" operators X αβ m+l with l < r do not give rise to additional terms since the corresponding sums cancel each other. That is a direct consequence of one-dimensionality. Now, we consider the holon contribution to ǫ s coming from v ′′ k,q v ′′ k,q , v † k ′ ,q ′ = 2πδ(k − k ′ ) r,r ′ e iq ′ r ′ −iqr+ik(r−r ′ ) Ω ′′ r,r ′ (A.4) with Ω ′′ r,r ′ = (m| . . . |m + r − g|m + r + g|m + r|m + r − g ′ | . . . |m + r − r ′ ) , (A.5) and g = sign(r), g ′ = sign(r ′ ). We see that in general Ω ′′ r,r ′ depends both on r − r ′ and on r. Eqn. (A.4) can also be written as v ′′ k,q , v † k ′ ,q ′ = 2πδ(k − k ′ ) l e i(k−q)l S l , (A.6) with S l = +∞ r=−∞ e −i(q−q ′ )r Ω ′′ r,r−l . Due to the slow decay of spin correlation functions in the 1D Heisenberg state, one can expect that the main contribution to S l comes from regions where |r| ≫ |l|. There holds g = g ′ and we may rewrite and approximate (A.5) by Ω ′′ r,r−l = (0| . . . |l)(r + g|r) ≈ Ω l Ω 1 . (A.7) Then, the explicit dependence on r drops out and we obtain v ′′ k,q , v † k ′ ,q ′ = 8π 2 δ(k − k ′ )δ(q − q ′ )Z(k − q + π) Ω 1 , (A.8) i.e. a simple constant shift of the energy ǫ s . The contribution of the v ′ k,q term to the spinon dispersion is determined by the sequence of spin operators Ω ′ r,r ′ = (m|m − g|m + g| . . . |m + r|m + r − g ′ | . . . |m + r − r ′ ) (A.9) instead of (A.5). The expectation value of that term has to be calculated for the magnetic system without holes. In difference to Ω ′′ r,r ′ , it depends only on r − r ′ without further approximation Ω ′ r+l,r = Ω ′ l,0 = (0| − 1|1| . . . |l) , (l > 0) , (A.10) and Ω ′ −l,0 = Ω ′ l,0 . For l = 0, 1 it can be expressed through pair correlation functions Ω ′ 0,0 = 1 2 + 2 S 0 S 2 , Ω ′ 1,0 = 1 4 + 2 S 0 S 1 + S 0 S 2 . For large l > 0 we may expect Ω ′ l,0 ≈ Ω l+1 . (A.11) Using this approximation we obtain the following contribution to the spinon dispersion ǫ s which stems from the v ′ k,q term J 2 v ′ k,q , v † k ′ ,q ′ = 8π 2 δ(q − q ′ )δ(k − k ′ )ǫ ′ s (k − q)Z(k − q + π) (A.12) with ǫ ′ s (k)Z(k + π) = J 4 +∞ l=−∞ e −ikl Ω ′ l,0 . (A.13) After some algebra we find 4ǫ ′ s (Q − π)Z(Q)/J = Ω ′ 0,0 − 2 Ω 1 − 2 cos Q Z(Q) − 1 2 − sin QY (Q) , (A.14) where Y (Q) = +∞ l=1 2(−1) l sin(Ql) Ω l = 1 2π 2π 0 dκZ(κ) cot Q − κ 2 + cot Q + κ 2 . (A.15) The complete expression for the spinon dispersion follows from ǫ s (Q − π) = J cos Q + ǫ ′ s (Q − π) + const , and is given in Eqn. (39) neglecting the constant energy shift. Again we see that all terms coming from commutations at "inner" operators cancel. But now the motion of spinons becomes more complicated. The same considerations that show the absence of dispersion from the v ′′ m,r term are applicable to v and we obtain (46). Fig. 1: Comparison of the spinon dispersion ǫ s (Q − π) as calculated from the projection method (broken line, J = 0.4) with 2J cos Q (full line). Figures the following contributions to ǫ s = ǫ sJ + ǫ sJ ′ (we drop dispersionless terms). FromĴ comesǫ sJ (Q − π)Z(Q) = J 2 [2 cos QZ(Q) + Z ′ (Q)] sJ (Q − π) = 2J cos Q (B.4)has the same form (41) that we have assumed for the t-J model. For the term that comes from the left distorted end of v m,r due toĴ ′ we have Fig. 2 : 2Spectral density of the t-J model for J = 0.4 and t = 1. Fig. 3: Spectral density of the frustrated t-J model (J = 0.4 and t = 1) at the special frustration J ′ = 0.5J (using the Majumdar-Ghosh wave function) within the projection method. Fig. 4: Spectral density of the Majumdar-Ghosh model A(k, ω) for three different momenta k/π and t = 1, J = 0.4 (full lines) or J = 0 (dashed lines) with a variational set of 400 basis functions and a broadening of Γ = 0.05. Fig. 5: Weight w 1 and energy separation e 1 = E 3 − E 1 of the lowest eigenvalue E 1 at k/π = 0.5, t = 1, J = 0.4 (full lines) as a function of the inverse number of basis functions 1/N . The dashed lines are the weights w 3 and the energy separation e 3 = E 5 − E 3 of the third eigenvalue E 3 . Fig. 6: Spectral density of the t-J model (J = 0.4 and t = 1) in the ideal paramagnetic state. Appendix A: Spinon dispersion of the Heisenberg case In this Appendix we outline the main steps to derive the spinon dispersion of the pure t-J model using the projection method. For long chains of X-operators it is convenient to introduce the notations 25 α1,...,αrX σα1 n1 X α1α2 n2 X α2α3 n3 . . . X αr−1αr nr−1 X αr0 nr ≡ (n 1 |n 2 |n 3 | . . . |n r−1 |n r ] and σ,α1,...,αr X σα1 n1 X α1α2 n2 X α2α3 n3 . . . X αr−1αr nr−1 Appendix B: Majumdar-Ghosh model with projection method The commutation with the frustration Hamiltonian (14) is very similar to (A.3). It gives = {(m|m − 2g|m + g| . . . |m + r] − (m − 2g|m|m + g| . . . |m + r] −(m|m + g)(m + 2g| . . . |m + r] + (m|m + g|m − g|m + 2g| . . . |m + r] − (m|m − g|m + g| . . . |m + r]} , v (4) m,r = {−(m| . . . |m + r − g|m + r + 2g|m + r] −(m| . . . |m + r − 2g|m + r + g|m + r − g|m + r] + (m| . . . |m + r − g|m + r + g|m + r]} . (B.2)v m,r ,Ĵ ′ = J ′ 2 v (3) m,r + v (4) m,r (B.1) where v (3) m,r Appendix C: Matrix elements of the variational basis setThe matrixelements of E x r,r ′ in the neighborhood of r, r ′ = 0 are given by: One has two different regions in the matrix. The first one is defined for r > 0, r ′ > 0 and r ≥ r ′ + 2 where we have the matrix elements: . E H Lieb, F Y Wu, Phys. Rev. Lett. 201445E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20, 1445 (1968). A Career in theoretical Physics. P W Anderson, 20th Century Physics. SingaporeWorld Scientific7P.W. Anderson, A Career in theoretical Physics, World Scientific Series in 20th Century Physics (World Scientific, Singapore, 1994), Vol. 7. . S Sorella, A Parola, J. Phys. Condens. Matter. 43589S. Sorella and A. Parola, J. Phys. Condens. Matter 4, 3589 (1992); . A Parola, S Sorella, Phys. Rev. B. 4513156A. Parola and S. Sorella, Phys. Rev. B 45, 13156 (1992). . M Ogata, H Shiba, Phys. Rev. B. 412326M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 (1990). . K Penc, F Mila, H Shiba, Phys. Rev. Lett. 75894K. Penc, F. Mila, and H. Shiba, Phys. Rev. Lett. 75, 894 (1995). . C Kim, A Y Matsuura, Z.-X Shen, N Motoyama, H Eisaki, S Uchida, T Tohyama, S Maekawa, Phys. Rev. Lett. 774054C. Kim, A.Y. Matsuura, Z.-X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa, Phys. Rev. Lett. 77, 4054 (1996). . B O Wells, Z.-X Shen, A Matsuura, D M King, M A Kastner, M Greven, R J Birgeneau, Phys. Rev. Lett. 74964B.O. Wells, Z.-X. Shen, A. Matsuura, D.M. King, M.A. Kastner, M. Greven, and R.J. Birgeneau, Phys. Rev. Lett. 74, 964 (1995). . C L Kane, P A Lee, N Read, Phys. Rev. 396880C.L. Kane, P.A. Lee, and N. Read, Phys. Rev. B39, 6880 (1989). . G Martinez, P Horsch, Phys. Rev. 44317G. Martinez and P. Horsch, Phys. Rev. B44, 317 (1991). . R Eder, K Becker, Z. Phys. B. 78219R. Eder and K. Becker, Z. Phys. B 78, 219 (1990). . K Fabricius, A Klümper, U Löw, B Büchner, T Lorenz, G Dhalenne, A Revcolevschi, Phys. Rev. B. 571102K. Fabricius, A. Klümper, U. Löw, B. Büchner, T. Lorenz, G. Dhalenne, and A. Revcolevschi, Phys. Rev. B 57, 1102 (1998). . T Tohyama, S Maekawa, J. Phys. Chem. Solids. 591864T. Tohyama and S. Maekawa, J. Phys. Chem. Solids 59, 1864 (1998). . K Kobayashi, T Mizokawa, A Fujimori, M Isobe, Y Ueda, T Tohyama, S Maekawa, Phys. Rev. Lett. 82803K. Kobayashi, T. Mizokawa, A. Fujimori, M. Isobe, Y. Ueda, T. Tohyama, and S. Maekawa, Phys. Rev. Lett. 82, 803 (1999). . C K Majumdar, D K K Ghosh ; C, Majumdar, J. Math. Phys. 10911J. Phys. CC.K. Majumdar and D.K. Ghosh, J. Math. Phys. 10, 1399 (1969), C.K. Majumdar, J. Phys. C 3, 911(1970). . H Suzuura, N Nagaosa, Phys. Rev. B. 563548H. Suzuura and N. Nagaosa, Phys. Rev. B 56, 3548 (1997). . Z Y Weng, D N Sheng, C S Ting, Phys. Rev. B. 52637Z.Y. Weng, D.N. Sheng, and C.S. Ting, Phys. Rev. B 52, 637 (1995). . H Rosner, H Eschrig, R Hayn, S.-L Drechsler, J Malek, Phys. Rev. B. 563402H. Rosner, H. Eschrig, R. Hayn, S.-L. Drechsler, and J. Malek, Phys. Rev. B 56, 3402 (1997). . R Hayn, J.-L Richard, V Yu, Yushankhai, Solid State Commun. 93127R. Hayn, J.-L. Richard, and V.Yu. Yushankhai, Solid State Commun. 93, 127 (1995). . W F Brinkman, T M Rice, Phys. Rev. B. 21324W.F. Brinkman and T.M. Rice, Phys. Rev. B 2, 1324 (1970). . W Brenig, K Becker, Z. Phys. B. 76473W. Brenig and K. Becker, Z. Phys. B 76, 473 (1989). . J Voit, Rep. Prog. Phys. 57977J. Voit, Rep. Prog. Phys. 57, 977 (1994). . J Voit, J. Phys. Condens. Matter. 8779J. Voit, J. Phys. Condens. Matter 8, L779 (1996). Due to symmetry reasons, the weight of the lowest eigenstates with even number vanishes. Due to symmetry reasons, the weight of the lowest eigenstates with even number vanishes. . M Vojta, K Becker, Phys. Rev. B. 573099M. Vojta and K. Becker, Phys. Rev. B 57, 3099 (1998). . A F Barabanov, private communicationA.F. Barabanov, private communication.
[]
[ "Information retrieval in folktales using natural language processing", "Information retrieval in folktales using natural language processing" ]
[ "Adrian Groza [email protected] \nDepartment of Computer Science\nIntelligent Systems Group\nTechnical University of Cluj-Napoca\nRomania\n", "Lidia Corde [email protected] \nDepartment of Computer Science\nIntelligent Systems Group\nTechnical University of Cluj-Napoca\nRomania\n" ]
[ "Department of Computer Science\nIntelligent Systems Group\nTechnical University of Cluj-Napoca\nRomania", "Department of Computer Science\nIntelligent Systems Group\nTechnical University of Cluj-Napoca\nRomania" ]
[]
Our aim is to extract information about literary characters in unstructured texts. We employ natural language processing and reasoning on domain ontologies. The first task is to identify the main characters and the parts of the story where these characters are described or act. We illustrate the system in a scenario in the folktale domain. The system relies on a folktale ontology that we have developed based on Propp's model for folktales morphology.
null
[ "https://arxiv.org/pdf/1511.03012v1.pdf" ]
9,894,863
1511.03012
5cee621bc00e88bc461ba7071273aab1f31e70a7
Information retrieval in folktales using natural language processing Adrian Groza [email protected] Department of Computer Science Intelligent Systems Group Technical University of Cluj-Napoca Romania Lidia Corde [email protected] Department of Computer Science Intelligent Systems Group Technical University of Cluj-Napoca Romania Information retrieval in folktales using natural language processing Index Terms-Natural language processingontologiesliterary characterfolktales Our aim is to extract information about literary characters in unstructured texts. We employ natural language processing and reasoning on domain ontologies. The first task is to identify the main characters and the parts of the story where these characters are described or act. We illustrate the system in a scenario in the folktale domain. The system relies on a folktale ontology that we have developed based on Propp's model for folktales morphology. I. INTRODUCTION Recognising literary characters in various narrative texts is challenging both from the literary and technical perspective. From the literary viewpoint, the meaning of the term "character" leaves space to various interpretations. From the technical perspective, literary texts contain a lot of data about emotions, social life or inner life of the characters, while they are very thin on technical, straight-forward messages. To infer the character type from literary texts might pose problems even to the human readers [4]. Interactions between literary characters contain rich social networks. Extracting these social networks from narrative text has gained much attention [13] in different domains such as literary fiction [6], screenplays [1], or novels [9], [2]. Our aim is to correctly determine the relationships of a character in a tale and to find its role upon the development of the story. In line with [16], the first task is to identify the parts of the story where that character is involved. Our approach relies on interleaving natural language processing and ontology-based reasoning. We enact our method in the folktale domain. Information extraction systems usually have three components responsible for: named entity recognition, co-reference resolution and relationship extraction. These modules are integrated in a pipeline, in a layered manner, given that each task will use information provided by the previous neighbor. Natural language processing has been applied in the domain of folktales [14], [8]. Formal models for folktales have been proposed in [12], [15]. Character identification in folktales have been approached in [17], [19]. The remaining of the paper is organized as follows: Section II presents the ontology that we developed for modeling Name Description Villain The opponent of the hero -often the representation of evil. Dispatcher The person that sends the hero into the journey, or the person that informs the hero about the villainy. (Magical) Helper The one that helps the hero into its journey. Princess or Prize It represents what the hero receives when it is victorious. Donor Prepares the hero for the battle. Hero The main character in a story -often the representation of good. False hero The one that tries to steal the prize from the hero, or tries to marry the princess. the domain of folktales. Section III depicts the architecture of our system. Section IV illustrates our method to extract knowledge about characters. Section V presents the experimental results on seven folktales. Section VI browses related work, while section VII concludes the paper. II. ENGINEERING THE FOLKTALE ONTOLOGY To support reasoning in the folktale domain, we developed an ontology used to extract knowledge regarding characters. We assume the reader is familiarised with the syntax of Description Logic (DL). For a detailed explanation about families of description logics, the reader is referred to [3]. To support character identification and reasoning on these characters we need structured domain knowledge. Hence, we developed an ontology for the folktale domain as shown in Fig. 3. Our folktale ontology formalizes knowledge from three sources: 1) the folktale morphology as described by the Propp model [15]; 2) various entities specific to folktales (i.e., animals, witch, dragons); and 3) common family relations (i.e., child, fiancee, groom). In the following, these three knowledge sources are detailed: a) Folktale morphology: Firstly, we rely on the Propp's model [15] of the folktale domain. In the Propp's model the story broke down into several sections. Propp demonstrated that the sequence of sections appears in the same chronological order in Russian folktales. Propp identified a set of character types that appear in most of the folktales (see Table I). The corresponding formalization in Description Logic appears in Fig. 1 (axiom 1). In axiom 2, a false hero is a hero who is also a villain. Axiom 3 divides the characters into negative and positive ones. Note that positive and negative characters are not disjoint, as for instance the concept Prisoner belongs to both sets. b) Folktale main entities: Secondly, the common entities appearing in folktales were formalized in Fig. 2. The axioms depict the animals (axiom 21), witches or enchantresses which are women with a single social status (axioms 22 and 23), and supernatural characters like Giant in axiom 24. Specific characters like Goldsmith or King, and various objects (i.e. oven) are also modeled. A prince is defined in axiom 28 as a son that have a parent either a king or a queen. Similarly, the princess is a daughter with at least on parent of type king or queen (axiom 30). c) Family relationships in folktale: Fig. 4 lists part of the family relationships adapted to reason in the folktale domain. A significant part of these relationships are correlated with the recurrent theme of the main character who is finding his bride or fiancee. To facilitate reasoning on the ontology, we allow several extensions of the ALC version of description logics [3]. Using role inheritance we can specify that the role hasFather is more specific than the role hasParent. Hence, if we find in the folktale that a character has a father, the system deduces based on role inheritance that the character has also a parent. Similarly, inverse roles like hasChild and hasParent are used to infer new knowledge based on the partial knowledge extracted by natural language processing. If we identify that two individuals are related by the role hasChild, the system deduces that those individuals are also related by the role hasParent. The domain restriction specifies that only persons can have brothers. The range restriction constraints the range of the role hasGender to the concept Gender. III. SYSTEM ARCHITECTURE Extracting knowledge about characters is obtained by interleaving natural language processing (NLP) and reasoning on ontologies. The NLP component is based on GATE text engineering tool [5], while reasoning in DL on the OWLAPI [10], as depicted by the architecture in Fig. 5. Firstly, the folktale ontology is processed using OWLAPI to generate classes of characters from the ontology into GATE. The folktale corpus is analysed aiming to populate the ontology and to annotate each folktale with the identified named entities. In parallel to the annotation process, the Stanford parser creates the coreference information files. The task is challenging, as even a human might have a problem in decoreferencing some of the sentences, as example 1 illustrates. Example 1. "The Smiths went to visit the Robertsons. After that, they stayed home, watching tv.", where "they" might be tied to the Smiths, or the Robertsons, or to both of the families. For de-coreferencing, the following pipeline was designed (left part of Fig. 5). The tokenizer groups all the letters into words. Next, the sentence splitter (Ssplit) groups the sequence of tokens obtained in the previous step into sentences. The part of speech (POS) annotation labels all the tokens from a sentence with their POS tags. Lemma annotation generates the word lemmas for all the tokens in the corpus. The next step is to apply named entity recognition (NER) so that the numerical and temporal entities are recognized. This is done using a conditional random fields (CRF) sequence taggers trained on various corpora. The parse function provides a full syntactic analysis for each sentence in the corpora. Finally, the coreference chain annotation (Dcoref) obtains both the pronominal and nominal coreference resolution. After coreference resolution, the stories are updated with the coreference information. The Reverb information extraction tool [7] is used to generate triplets containing the following structure: nominal phrase, verb phrase, nominal phrase . For the sentence "Good heavens, said the girl, no strawberries grow in winter", the output of Reverb is exemplified in Table III. In order to obtain the triplets, each sentence has to be POS-tagged and NPchunked. IV. INTERLEAVING NATURAL LANGUAGE PROCESSING WITH REASONING ON ONTOLOGIES This section details three algorithms used to identify knowledge about characters. Algorithm 1 identifies characters in the folktale. Algorithm 3 is used for anaphora resolution of the named entities recognized as characters. Algorithm 2 extracts knowledge about characters from the de-coreferences. The execution flow of this pipeline, is presented in Fig. 6. Natural language processing is enacted to populate the folktale ontology. The extraction Algorithm 1 is performed repetitively on a document, each time using the newly populated ontology file. In this way, the algorithm interleaves reasoning on ontology with natural language processing based on Japes rules [18]. The first step is to apply the Jape rules JN on the folktale corpus aiming to identify all the definite and indefinite nominal phrases. Given that the characters are nominal phrases, this first step returns all the information needed, plus some extra phrases that have to be filtered out. Next, the Jape rules JC are enacted to select candidate characters from the set of nominal phrases previously identified. For each character found, a set of rules JR is used to match the character against a concept in the ontology. B-NP I-NP O B-VP B-NP I-NP O B-NP I-NP The king's daughter began to cry , for daughter was afraid of the cold frog which daughter did not like to touch, and which was now to sleep in daughter pretty, clean little bed. After identifying a concept for which the character is an instance, the algorithm exploits reasoning on ontology to identify all atomic concepts to which the character belongs. For instance, a character identified as Daughter will be an instance of Girl, Child, M aiden, SingleP erson (recall Fig. 4). For each concept to which the character belongs, the algorithm looks again in the corpus to see if there are other mentions of the newly introduced character. If this is the case, the character is related with the new knowledge. The decoreferencing algorithm (Alg. 2) uses as input the processing pipeline and the folktale corpus. The basic processing steps needed are the following: tokenize, ssplit, pos, lemma, ner, parse, dcoref. The decoreferencing algorithm is run on all stories at once, but it generates different output file for each story represented by the filename. In the first step, the Stanford parser applies the execution pipeline on the corpora of folktales. For each resulted file, the algorithm searches for coreference groups. In order to be able to return the modified text, the original text has to be stored in the returning argument of the algorithm. For each coreference group found, firstly the referenced word has to be processed and kept into a variable and then, each coreferenced word found, belonging to the group, has to be replaced in the original text with the referenced variable. In the end, the decoreferenced text for each corpus file is obtained. Algorithm 3 takes as input the result of algorithms 1 and (alg 2. The set of characters is used as the input, while the decoreferenced texts are used as an environment from which the algorithm extracts the perspective. For each character in the set of characters resulted from the extraction algorithm (alg 1), each line that resulted from reverb execution is processed. From each line, the sentence is extracted based on the output RR = run(R, D); if V = true then foreach c ∈ C do foreach line ∈ RR do sentence ← getSentence(line); if c ∈ sentence then P ← P ∪ sentence; end end end else foreach c ∈ C do foreach line ∈ RR do triplet ← getTriplet(line); if c ∈ triplet then P ← P ∪ triplet; end end end end Algorithm 3: Finding character's perspective. format of the Reverb service presented in Table III. If the character, from the character set, is mentioned in the sentence, then the sentence is appended to the output variable. These columns are combined in a triplet, and it is checked to see whether the current character appears is present in this triplet. In this case, the triplet is appended to the output variable. This algorithms score is represented by a subunitary number that represents the confidence that the extraction was correct. V. EXPERIMENTAL RESULTS A. Running scenario The system was tested against seven stories (Table V). This section illustrates the results of this pipeline for the secondary character Henry from the story "The frog king". The fragment on which the algorithms were applied is listed in Example 2. Example 2. "Then they went to sleep, and next morning when the sun awoke them, a carriage came driving up with eight white horses, which had white ostrich feathers on their heads, and were harnessed with golden chains, and behind stood the young king's servant Faithful Henry. Faithful Henry had been so unhappy when his master was changed into a frog, that he had caused three iron bands to be laid round his heart, lest it should burst with grief and sadness. The carriage was to conduct the young king into his kingdom. Faithful Henry helped them both in, and placed himself behind again, and was full of joy because of this deliverance. And when they had driven a part of the way the king's son heard a cracking behind him as if something had broken. So he turned round Henry was full of joy 6 the bands were springing from the heart of faithful Henry and cried, "Henry, the carriage is breaking." "No, master, it is not the carriage. It is a band from my heart, which was put there in my great pain when you were a frog and imprisoned in the well." Again and once again while they were on their way something cracked, and each time the king's son thought the carriage was breaking, but it was only the bands which were springing from the heart of Faithful Henry because his master was set free and was happy." The method has two kind of results -one for the long version, and one for the short version. Firstly, the results for the short version are listed in Table IV. Note that the output text is the decoreferenced one -this is the reason why the character might talk about itself in third person. Because of the de-coreferenced version of the stories part of text might not be correct from the human reader perspective. But it is the easiest way to understand the context of a character. Otherwise, it would be hard to see that when the text says "his master", that "his" refers to Henry, as Example 3 bears out. Example 3. 1. Then companion went to sleep, and next morning when the sun awoke companion, a band came driving up with eight white horses, which had white ostrich feathers on companion heads, and were harnessed with golden chains, and behind stood the young king's servant faithful Henry. 2. Faithful Henry had been so unhappy when henry master was changed into a frog, that Henry had caused three iron bands to be laid round henry heart, lest heart should burst with grief and sadness. 3. Faithful Henry helped bands both in, and placed Henry behind again, and was full of joy because of this deliverance. 4. Again and once again while you were on you way something cracked, and each time the king's son thought the band was breaking , but it was only the bands which were springing from the heart of faithful Henry because Henry master was set free and was happy." There are some cases in which there will be no result for a character (Example 4). Given that the character was extracted from the original file, by using Algorithm 1, there is a certainty that the character exists in the story. Example 4. When trying to search for the perspective of character "waiting-maid" in the story "Faithful John", the application will not be able to find any solution. In the unmodified text, the son character is introduced in the following way: "She took him by the hand and led him upstairs, for she was the waiting-maid." This happens because, when the anaphoric decoreference is run (Algorithm 2), the file is changed in the following way: "Girl took oh by the hand and led oh upstairs , for girl was the girl .". The change happened because the decoreferencing tool interpreted "the waiting-maid" as being tied up to the word "she", and, which is tied to "the girl" from the following phrase "Then said the girl ' the princess must see these , girl has such great pleasure in golden things , that girl will buy all you have . '". In this way, this character's part will be attributed to the "girl", which is the main character of the story. This situation in which the story is talking about a general character, but only after the main events, the character is finally revealed, is called cataphora [11]. B. Accuracy of the method The accuracy of our method is influenced by: 1) accuracy of character identification; 2) accuracy of identifying coreferences; 3) accuracy of Reverb when extracting triplets (the confidence indicator). Each of this services has an accuracy error that will be propagated from one component to another. We performed various tests on the corpus used for character identification, and we obtained an average accuracy of 70% (Table V). When calculating the accuracy, 20 characters were taken into consideration, meaning that for each story, about 3 characters were chosen. These characters were manually selected from the set of characters output by the character extraction system presented in [17], [19]. The characters were selected by choosing 2 main characters and a secondary character for each story. The testing was performed on seven different stories, and for each story, a set of main characters was chosen. The obtained overall accuracy is 74%, having an overall precision of 90% and a recall of 60%. The results are presented in Fig. 7. Figure 8 depicts the distribution of precision, recall and accuracy over the stories. The values were calculated using the following formulas: set, nor in the test set, f p means false positive and represents the number of sentences that are in the test set and not in the manually annotated set, and f n means false negative and represents the number of sentences that are in the manually annotated set, but not in the test set. In the folktale context, the tp represents the number of sentences that belong to the character's perspective, all those sentences that involve the character in any way. The average F-score for the Stanford-CoreNLP of 59.5 influences greatly the performance of the algorithm, as the characters perspective cannot be extracted, given that the character is not seen as being part of the sentence. The accuracy can be improved if a better decoreferencing tool will be used. Other coreference tools are For the anaphoric decoreference, there are several other tools (BART, JAVARAP, GuiTar and ARKref), but, from all, the Stanford-CoreNLP has be highest accuracy percentage. There is ongoing research in the coreference resolution domain, When calculating the performance scores, the extraction of the correct sentence was considered, and not on the correctness of the extracted sentence. Even though the right sentence was extracted, the information in the sentence will be according to the coreference resolution result. Hence, an error might be observed when reviewing the structure of the sentences. The algorithms performance is also influenced by the scores obtained by the Reverb tool. Also, the named entity recognition has an average precision of 79% and a recall of 72%. These scores do not influence directly the algorithms performance, but they have an effect on the number of characters for which the algorithm will try to find the roles they have on the development of the story. Together, all these scores combined, give the performance scores of the characters perspective in texts. The current version does not extract information about the characters' roles. The information extracted consists of the character identification, that is presented in [17], [19], and the story involving the character. The story can be presented in a standardized version. VI. DISCUSSION We can enact our solution in other domains instead of folktales. We exemplify he following three domains: a) software requirements, b) marketing and c) medical domain. Consider the domain of software requirements, where these requirements are written in natural language. Our system will support the identification of various actors appearing in the requirements document. First, one needs to replace the folktale ontology with a requirement ontology that provides knowledge on use cases, actors, their roles, etc. The same pipeline will be used to: 1) identify main actors (admin, various users, etc) and 2) extract knowledge about various actions these actors are supposed to perform. Another domain that could benefit from the same pipeline of execution, would be the marketing domain. Consider a dataset of product reviews or accommodation places in the tourism domain [20]. The system would extract only the sentences that reference the mentioned item. By having access to all the sentences of interest, further analysis is facilitated without having to process the entire text. Similar extraction systems have been proposed for the medical domain to extract information from clinical narratives. In this line, the MedEx system [21] aims to extract the medication information from clinical narratives. Similarly, there is also the OpenClinical system for assisting health care providers. In our approach, the extraction algorithm part is separated from the perspective searching part. Therefore, any ontology and any document can be used in order to find the character's or object's perspective in the document. We tested our method only on seven stories. With a complexity of O(n 3 ) in sentence length of syntactic parsing, our syntactic based on Stanford parser might be too slow for large corpus as the one of 15099 narratives analysed in [4]. VII. CONCLUSIONS Our method is able to extract knowledge on various characters. Our current accuracy for information extraction in the folktale domain is 74%. The experimental results were obtained for seven stories in the folktale domain. The precision score is above 90%, With an overall recall of only 60%, there are high chances that not all the information regarding a product was extracted. The developed algorithms aggregate three different services: Firstly, the named entity recognition was implemented by using an ontology based on Propps formal model. Based of this ontology, and some implemented Jape rules, the characters are extracted from a given story. Secondly, a coreference resolution tool was implemented by enacting anaphoric resolution to eliminate co-referenced words and to replace them with their representative, Thirdly, finding relationships between characters was integrated in order to link two noun phrases with a verbal phrase. Fig. 2 . 2Common entities in the folktale domain. Fig. 3 . 3Folktale ontology. A Fig. 5 . 5The System Architecture NN POS NN VBD TO VB , IN NN VBD JJ IN DT JJ NN WDT NN VBD RB IN TO VB , CC WDT VBD RB TO VB RP NN RB , JJ JJ NN . B-NP I-NP I-NP I-NP B-VP I-VP I-VP O B-PP B-NP B-VP B-ADJP B-PP B-NP I-NP I-NP B-NP I-NP B-VP O O B-VP I-VP O O B-NP B-VP B-ADVP B-VP I-VP B-NP I-NP B-ADVP O B-NP I-NP I-NP O When everything was stowed on board a ship, faithful John put on the dress of a merchant, and the king was forced to do the same in order to make king quite NN VBD VBN IN NN DT NN , NN NNP VBD IN DT NN IN DT NN , CC DT NN VBD VBN TO VB DT JJ IN NN TO VB NN RB JJ . B-ADVP B-NP B-VP I-VP B-PP B-NP B-NP I-NP O B-NP B-NP B-VP B-PP B-NP I-NP I-NP I-NP I-NP O O B-NP I-NP B-VP I-VP I-VP I-VP B-NP I-NP B-SBAR O B-VP I-VP B-NP B-ADJP I-ADJP O Sons each kept watch in turn, and sat on the highest oak and looked towards the tower. DT VBD NN IN NN , CC VBD IN DT JJS NN CC VBD IN DT NN . O B-NP B-VP B-NP B-PP B-NP O O B-VP B-PP B-NP I-NP I-NP O B-VP B-PP B-NP I-NP O Rapunzel grew into the most beautiful child under the sun. VBD IN DT RBS JJ NN IN DT NN . B-NP B-VP B-PP B-NP I-NP I-NP I-NP B-PP B-NP I-NP O The king's son ascended, but instead of finding son dearest rapunzel, son found the enchantress, who gazed at son with wicked and venomous looks. NN POS NN VBD , CC RB IN VBG NN NN NN , NN VBD DT NN , WP VBD IN NN IN JJ CC JJ NNS . B-NP I-NP I-NP I-NP B-VP O O B-PP I-PP B-VP B-NP I-NP I-NP O B-NP B-VP B-NP I-NP O B-NP B-VP B-PP B-NP B-PP B-NP I-NP I-NP I-NP O Input : O f -Folktale ontology; S -Corpus of folktales; JN -Jape rules to identify definite and indefinite nominal phrases; JC -Jape rules to identify candidate characters; JR -Jape rules to identify character's relation to the ontology; Result: C: Set of annotated characters; C ← ∅; N P ← applyRules(JN , S); while applyRules(JC, S, N P ) = null do N C ← applyRules(JC, S, N P ); Rel ← applyRules(JR, S, N C); foreach r ∈ Rel do foreach concept from r do if checkCast(N C, concept) then cast(N C, concept); end end end while is referred(S, N C) do Ref = getReference(); link(N C, Ref ); end C ← C ∪ N C; end Algorithm 1: Character extraction algorithm. Input : S: Corpus of folktales; P : Pipeline configuration for decoreferencing; F N : List with filenames for each S; SC: Stanford-CoreNLP command; Result: D: Decoreferenced texts of files from F N ;F iles = run(SC, P , F N ); foreach f ile in F iles do D ← S;foreach coref group ∈ f ile do rep ← findRepresentative(coref group); foreach coref word ∈ coref group do replace(D, coref word, rep ); end end end Algorithm 2: Decoreference algorithm. Fig. 6 . 6Main execution phases. Set of characters resulted from algorithm 1; D: Decoreferenced text resulted from algorithm 2; Result: P : String containing character's perspective in S; where tp means true positive, and represents the number of sentences that are found both in the manually annotated set and the test set, tn means true negative and represents the number of sentences that are neither in the manually annotated Fig. 7 . 7Precision, recall, and accuracy for the seven folktales analyzed. Fig. 8 . 8Comparing precision, recall and accuracy for each story. TABLE I MAIN ICHARACTERS IN THE PROPP'S MODEL. , where the characters are divided in nine types Prince ≡ Son ∃hasParent.King ∃hasParent.Queen A 29 Prince SingleSocialStatus A 30 Princess ≡ Daughter ∃hasParent.King ∃hasParent.Queen978-1-4673-8200-7/15 $31.00 c 2015 IEEE A 1 Agent Donor FalseHero Hero Prisoner Villain Dis- patcher MagicalHelper Princess Character A 2 Hero Villain FalseHero A 3 PositiveCharacter NegativeCharacter Character A 4 Villain FalseHero Prisoner NegativeCharacter A 5 Hero MagicalHelper Agent Donor Prisoner Dispatcher PositiveCharacter Fig. 1. Formalising the Propp's model of folktales. A 21 Bear Bird Dog Duck Frog Horse Lion SingleAnimal A 22 Enchantress ≡ Witch A 23 Enchantress Woman SingleSocialStatus A 24 Giant Supernatural A 25 Goldsmith Helmsman SingleSocialStatus A 26 King SingleSocialStatus A 27 Oven Object A 28 TABLE II EXPLOITING IIROLE CONSTRAINTS TO REASON ON THE ONTOLOGY.Extensios of ALC Folktale examples Role inheritance hasBrother hasSibling, hasFather hasParent, hasHusband hasConsort Inverse roles hasHusband ≡ hasWife − , hasChild ≡ hasParent − Transitive roles hasSibling t Domain restriction ∃hasBrother. Person Range restriction ∀hasBrother.Person, ∀hasGender.Gender symmetric roles hasConsort ≡ hasConsort − cardinality constraints ≤1 hasGender.Thing TABLE III EXTRACTING IIITRIPLETS FROM FOLKTALES USING REVERB. Good heavens, said the girl, no strawberries grow in winter.Original Sentence Nominal Phrase (arg1) Verb Phrase (arg2) Nominal Phrase (arg3) Extraction Confi- dence POS tags Chunk tags 10 3 4 5 9 11 12 no straw- berries grow in winter 0.505 JJ NNS , VBD DT NN , DT NNS VB IN NN . TABLE IV ELICITING IVKNOWLEDGE ABOUT HENRY.Character: Henry 1 Henry master was changed into a frog 2 Henry had caused three iron bands 3 faithful Henry helped bands 4 bands placed Henry 5 TABLE V ACCURACY VOF THE ALGORITHMS.Story Accuracy The Magic Swan-Geese 75% The Frog King 62% The King's Son who Feared Nothing 76% Faithful John 63% The Twelve Brothers 65% Rapunzel 74% The Three Little Men in the Woods 73% Average 70% ACKNOWLEDGMENTSWe thank the reviewers for their valuable comments. Part of this work was supported by the Department of Computer Science of Technical University of Cluj-Napoca, Romania. Parsing screenplays for extracting social networks from movies. A Agarwal, S Balasubramanian, J Zheng, S Dash, A. Agarwal, S. Balasubramanian, J. Zheng, and S. Dash, "Parsing screenplays for extracting social networks from movies," EACL 2014, pp. 50-58, 2014. Social network analysis of Alice in Wonderland. A Agarwal, A Corvalan, J Jensen, O Rambow, Workshop on Computational Linguistics for Literature. A. Agarwal, A. Corvalan, J. Jensen, and O. Rambow, "Social network analysis of Alice in Wonderland," in Workshop on Computational Linguistics for Literature, 2012, pp. 88-96. The description logic handbook: theory, implementation, and applications. F Baader, Cambridge university pressF. Baader, The description logic handbook: theory, implementation, and applications. Cambridge university press, 2003. A bayesian mixed effects model of literary character. D Bamman, T Underwood, N A Smith, Proceedings of the 52st Annual Meeting of the Association for Computational Linguistics (ACL14). the 52st Annual Meeting of the Association for Computational Linguistics (ACL14)D. Bamman, T. Underwood, and N. A. Smith, "A bayesian mixed effects model of literary character," in Proceedings of the 52st Annual Meeting of the Association for Computational Linguistics (ACL14), 2014. Evolving GATE to meet new challenges in language engineering. K Bontcheva, V Tablan, D Maynard, H Cunningham, Natural Language Engineering. 103-4K. Bontcheva, V. Tablan, D. Maynard, and H. Cunningham, "Evolv- ing GATE to meet new challenges in language engineering," Natural Language Engineering, vol. 10, no. 3-4, pp. 349-373, 2004. Extracting social networks from literary fiction. D K Elson, N Dames, K R Mckeown, Proceedings of the 48th annual meeting of the association for computational linguistics. the 48th annual meeting of the association for computational linguisticsAssociation for Computational LinguisticsD. K. Elson, N. Dames, and K. R. McKeown, "Extracting social networks from literary fiction," in Proceedings of the 48th annual meeting of the association for computational linguistics. Association for Computational Linguistics, 2010, pp. 138-147. Identifying relations for open information extraction. A Fader, S Soderland, O Etzioni, Proceedings of the Conference on Empirical Methods in Natural Language Processing. the Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsA. Fader, S. Soderland, and O. Etzioni, "Identifying relations for open information extraction," in Proceedings of the Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, 2011, pp. 1535-1545. Annotating with Propp's morphology of the folktale: reproducibility and trainability. B Fisseni, A Kurji, B Löwe, Literary and Linguistic Computing. 294B. Fisseni, A. Kurji, and B. Löwe, "Annotating with Propp's morphology of the folktale: reproducibility and trainability," Literary and Linguistic Computing, vol. 29, no. 4, pp. 488-510, 2014. Identification of speakers in novels. H He, D Barbosa, G Kondrak, ACL. H. He, D. Barbosa, and G. Kondrak, "Identification of speakers in novels." in ACL (1), 2013, pp. 1312-1320. The OWL API: A Java API for OWL ontologies. M Horridge, S Bechhofer, Semantic Web. 2M. Horridge and S. Bechhofer, "The OWL API: A Java API for OWL ontologies." Semantic Web, vol. 2, no. 1, pp. 11-21, 2011. Differential effects of constraints in the processing of Russian cataphora. N Kazanina, C Phillips, The Quarterly Journal of Experimental Psychology. 632N. Kazanina and C. Phillips, "Differential effects of constraints in the processing of Russian cataphora," The Quarterly Journal of Experimen- tal Psychology, vol. 63, no. 2, pp. 371-400, 2010. A declarative model for simple narratives. R Lang, Proceedings of the AAAI fall symposium on narrative intelligence. the AAAI fall symposium on narrative intelligenceR. Lang, "A declarative model for simple narratives," in Proceedings of the AAAI fall symposium on narrative intelligence, 1999, pp. 134-141. Structural analysis on social network constructed from characters in literature texts. G.-M Park, S.-H Kim, H.-G Cho, Journal of Computers. 89G.-M. Park, S.-H. Kim, and H.-G. Cho, "Structural analysis on social network constructed from characters in literature texts," Journal of Computers, vol. 8, no. 9, pp. 2442-2447, 2013. A description logic ontology for fairy tale generation. F Peinado, P Gervás, B Díaz-Agudo, Procs. of the Workshop on Language Resources for Linguistic Creativity, LREC. s. of the Workshop on Language Resources for Linguistic Creativity, LREC4F. Peinado, P. Gervás, and B. Díaz-Agudo, "A description logic ontology for fairy tale generation," in Procs. of the Workshop on Language Resources for Linguistic Creativity, LREC, vol. 4, 2004, pp. 56-61. Morphology of the Folktale. V I Propp, American Folklore Society9V. I. Propp, Morphology of the Folktale. American Folklore Society, 1958, vol. 9. An NLP-based cross-document approach to narrative structure discovery. N Reiter, A Frank, O Hellwig, Literary and Linguistic Computing. 29N. Reiter, A. Frank, and O. Hellwig, "An NLP-based cross-document approach to narrative structure discovery," Literary and Linguistic Com- puting, vol. 29, no. 4, pp. 583-605, 2014. Interleaving ontology-based reasoning and natural language processing for character identification in folktales. D Suciu, A Groza, IEEE 10th International Conference on Intelligent Computer Communication and Processing (ICCP2014). Cluj-Napoca, RomaniaD. Suciu and A. Groza, "Interleaving ontology-based reasoning and natural language processing for character identification in folktales," in IEEE 10th International Conference on Intelligent Computer Communi- cation and Processing (ICCP2014), Cluj-Napoca, Romania, 2014, pp. 67-74. Gate Jape grammar tutorial. D Thakker, T Osman, P Lakin, 1UK, Phil Lakin, UK, VersionNottingham Trent UniversityD. Thakker, T. Osman, and P. Lakin, "Gate Jape grammar tutorial," Nottingham Trent University, UK, Phil Lakin, UK, Version, vol. 1, 2009. Named entity recognition and resolution for literary studies. K Van Dalen-Oskam, J De Does, M Marx, I Sijaranamual, K Depuydt, B Verheij, V Geirnaert, K. van Dalen-Oskam, J. de Does, M. Marx, I. Sijaranamual, K. Depuydt, B. Verheij, and V. Geirnaert, "Named entity recognition and resolution for literary studies." Integrating DBpedia and SentiWordNet for a tourism recommender system. B Varga, A Groza, Intelligent Computer Communication and Processing. IEEE2011 IEEE International Conference onB. Varga and A. Groza, "Integrating DBpedia and SentiWordNet for a tourism recommender system," in Intelligent Computer Communication and Processing (ICCP), 2011 IEEE International Conference on. IEEE, 2011, pp. 133-136. Medex: a medication information extraction system for clinical narratives. H Xu, S P Stenner, S Doan, K B Johnson, L R Waitman, J C Denny, Journal of the American Medical Informatics Association. 171H. Xu, S. P. Stenner, S. Doan, K. B. Johnson, L. R. Waitman, and J. C. Denny, "Medex: a medication information extraction system for clinical narratives," Journal of the American Medical Informatics Association, vol. 17, no. 1, pp. 19-24, 2010.
[]
[ "Identification of Finite Dimensional Lévy Systems in Financial Mathematics", "Identification of Finite Dimensional Lévy Systems in Financial Mathematics" ]
[ "L Gerencsér \nMTA SZTAKI\n13-17 Kende StreetBudapestHungary\n", "· M Mánfay \nMTA SZTAKI\n13-17 Kende StreetBudapestHungary\n", "L Gerencsér \nMTA SZTAKI\n13-17 Kende StreetBudapestHungary; Central\n", "M Mánfay \nEuropean University\n9 Nádor StreetBudapestHungary\n" ]
[ "MTA SZTAKI\n13-17 Kende StreetBudapestHungary", "MTA SZTAKI\n13-17 Kende StreetBudapestHungary", "MTA SZTAKI\n13-17 Kende StreetBudapestHungary; Central", "European University\n9 Nádor StreetBudapestHungary" ]
[]
Lévy processes are widely used in financial mathematics to model return data. Price processes are then defined as a corresponding geometric Lévy process, implying the fact that returns are independent. In this paper we propose an alternative class of models allowing to describe dependence between return data. Technically such an alternative model class is obtained by considering finite dimensional linear stochastic SISO systems driven by a Lévy process. In this paper we consider a discrete-time version of this model, focusing on the problem of identifying the dynamics and the noise characteristics of such a so-called Lévy system. The special feature of this problem is that the characteristic function (c.f.) of the driving noise is explicitly known, possibly up to a few unknown parameters. We develop and analyze a variety of novel identification methods by adapting the so-called empirical characteristic function method (ECF) originally devised for estimating parameters of c.f.-s from i.i.d. samples. Precise characterization of the errors of these estimators will be given, and their asymptotic covariance matrices will be obtained. Their potential to outperform the prediction error method in estimating the system parameters will also be demonstrated.
null
[ "https://arxiv.org/pdf/1302.5221v1.pdf" ]
88,519,128
1302.5221
cbf4d908dc596459b5838f976a1a68fad833a64c
Identification of Finite Dimensional Lévy Systems in Financial Mathematics 21 Feb 2013 L Gerencsér MTA SZTAKI 13-17 Kende StreetBudapestHungary · M Mánfay MTA SZTAKI 13-17 Kende StreetBudapestHungary L Gerencsér MTA SZTAKI 13-17 Kende StreetBudapestHungary; Central M Mánfay European University 9 Nádor StreetBudapestHungary Identification of Finite Dimensional Lévy Systems in Financial Mathematics 21 Feb 2013Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor) 2 L. Gerencsér, M. Mánfaylinear stochastic systemsLévy processessystem identificationfinancial modelling Lévy processes are widely used in financial mathematics to model return data. Price processes are then defined as a corresponding geometric Lévy process, implying the fact that returns are independent. In this paper we propose an alternative class of models allowing to describe dependence between return data. Technically such an alternative model class is obtained by considering finite dimensional linear stochastic SISO systems driven by a Lévy process. In this paper we consider a discrete-time version of this model, focusing on the problem of identifying the dynamics and the noise characteristics of such a so-called Lévy system. The special feature of this problem is that the characteristic function (c.f.) of the driving noise is explicitly known, possibly up to a few unknown parameters. We develop and analyze a variety of novel identification methods by adapting the so-called empirical characteristic function method (ECF) originally devised for estimating parameters of c.f.-s from i.i.d. samples. Precise characterization of the errors of these estimators will be given, and their asymptotic covariance matrices will be obtained. Their potential to outperform the prediction error method in estimating the system parameters will also be demonstrated. Introduction The classical model for modelling market dynamics, namely geometric Brownian motion, was proposed by Louis Bacehelier [1]. This model is still the accepted core model despite the fact that empirical studies revealed that its assumptions are not realistic. For example, since price movements are induced by transactions which can be unevenly distributed in real time, it would be more natural to use a time changed Brownian motion to model price dynamics. If the time change is defined by a gamma process, we obtain the so-called VG (shorthand for Variance Gamma) process. VG processes reproduce a number of stylized facts of real price processes, such as fat tails and large kurtosis. It can be shown that the above time changed Brownian process itself is a Lévy process. Extending the above construction novel price dynamics have been proposed by a variety of authors, called the geometric Lévy processes obtained by exponentiating a Lévy process. A Lévy process (Z t ) is much like a Wiener process: a process with stationary an independent increments, but discontinuities or jumps are allowed. A good survey paper on Lévy processes used in financial modelling is the paper by Miyahara and Novikov, [18]. [12] studies several problems arising in the field of exponential Lévy processes. For an excellent introduction to the theory of Lévy processes see [3]. A key building block in the theory of Lévy processes is the compound Poisson process. A more general class Lévy process is formally obtained via Z t = t 0 R 1 xN (ds, dx),(1) where N (dt, dx) is a time-homogeneous, space-time Poisson point process, counting the number of jumps of size x at time t. In this case Z t is a pure jump process, which paradoxically means that the Lévy-Ito decomposition of Z t does not have a Brownian motion component (but it may have a drift term). The intensity of N (dt, dx) is defined by E[N(dt, dx)], which is due to time homogeneity can be written as E[N(dt, dx)] = dt · ν(dx), where ν(dx) is the Lévy-measure. The above representation given in (1) is mathematically rigorous if R 1 min(|x|, 1)ν(dx) < ∞. Under this condition the sample paths of Z t are of finite variation, a property supported by empirical evidence for most indices as emphasized in [6]. The characteristic function of a Lévy process can be written in the form E e iuZt = e tψ(u) ,(3) where ψ(u) is the characteristic exponent. The standard model of a price process within this framework is then S t = S 0 exp Z t ,(4) and (S t ) is called a geometric Lévy process. A variety of choices for (Z t ) has been proposed in the literature: it can be a stable process, a variance Gamma (VG) process, a tempered stable process, a special case of which is the (CGMY) process, a hypergeometric process or a Normal-inverse Gaussian (NIG) process. The motivation behind these models is the assumption that the returns of the stock process, say (S t+h − S t )/S t are independent and stationary. While this is an attractive assumption, its consequences are less attractive. In particular it follows that the variance of the price process tends to infinity, which is certainly unnatural for, say, prices of agricultural products. A closer look at data in fact reveals that there is a weak correlation between daily returns (S t+1 − S t )/S t . For example, considering data on IBM Coca Cola stock prices in a period of 20 years from Nov 1990 to Nov 2010 we found for the correlation coefficients of daily log-returns X t that corr(X t , X t−1 ) = −0.135. This small, but non-negligible, negative correlation calls for a refinement of the exponential Lévy model, allowing memory in the daily return process. An intuitive empirical argument can also be given in favor of the need for memory: namely an overreaction of the market is generally followed by a correction, resulting in a correlation between daily returns. The recently much studied popular Geometric fractional Brownian motion model gives return process with non-independent increments, for more details on fractional Brownian motion see papers of T.E. Duncan, for example [16]. We propose to introduce a new class of models, using the methodology of linear system theory, to capture the presence of decaying memory. The infinitesimal increments of the logarithm of the price process will be defined as a process dY t which is the output of a finite dimensional stable linear SISO (shorthand for singleinput-single-output) system, driven by a Lévy process: dY t = AdZ t , where A represents the linear mapping from input to output, and Z is a Lévy process. For the sake of convenience we let −∞ < t < +∞. In the case of a finite dimensional stable linear SISO system the mapping A can be described by a set of state-space equations, a well known example of such systems is defined by: dX t = HX t dt + dZ t (5) dY t = LX t dt + dZ t(6) From the above equations we get dY t = L t −∞ e H(t−s) dZs dt + dZ t .(7) The inverse system is formally obtained as dX t = (H − KL) X t dt + KdY t (8) dZ t = dY t − LdX t .(9) It is assumed that both systems A and A −1 are exponentially stable, equivalently, we assume that both H and (H − KL) are stable matrices. Such a system will be called a Lévy system. The inverse filter has the following form: dX t = (H − KL)X t dt + KdY t (10) dε t = dY t − LdX t .(11) Having defined the infinitesimal increments of the logarithm of the price process we define the price process according to (4): S t = S 0 exp Y t . In the statistical analysis of such systems, both the system dynamics and the fine characteristics of (Z t ) are to be identified. The first difficulty of applying a maximum-likelihood (ML) method lies in the fact that there is no natural reference measure in the space of sample paths. In addition, the computation of the Radon-Nikodym derivative is practically not feasible since t −∞ e H(t−s) dZs is not even a Lévy process. To avoid this problem we consider an alternative discrete-time model class, where the daily log-returns ∆yn are defined via a discrete time finite dimensional system ∆yn = A ∆Zn, where A represents the linear mapping from input to output, and ∆Zn is the increment of a Lévy process Z over an interval [(n − 1)h, nh), with some fixed h > 0. For the sake of convenience we let −∞ < n < +∞. A state space equation for this model is given by ∆X n+1 = H∆Xn + ∆Zn (13) ∆Yn = L∆Xn + ∆Zn.(14) We will call this model a discrete time finite dimensional Lévy system. Assume that A = A(θ * ) where θ * is an unknown parameter-vector, and similarly, let ν(dx) = ν(dx, η * ), where η * denotes an unknown parameter-vector. The ranges of of θ * and η * are assumed to be known. The fundamental problem to be discussed in this paper is to identify this system and to establish sharp results for the error of the estimator. If we knew the probability density function of the noise ∆Zn then we could apply an ML (Maximum Likelihood) estimation method, and establish sharp results for the estimation error, see [9]. The challenge of the present problem is that it is the characteristic function of the noise that is explicitly given. A natural approach to solve this problem is to combine techniques of system identification with the empirical characteristic function (ECF) method widely used in finance to analyze i.i.d. data. Before going into further details we present a few examples of Lévy processes used in finance. Lévy processes in finance To model the increments of the logarithm of a price process a wide range of geometric Lévy processes has been proposed by a variety of authors. Mandelbrot suggested to use α-stable process to model the price dynamics of wool, see [15]. An α-stable with 0 < α < 2 is defined via the Lévy measure ν(dx) = C − |x| −1−α 1 x<0 dx + C + |x| −1−α 1 x>0 dx. A recently widely studied class of Lévy processes is the CGMY process due to Carr, Geman, Madan and Yor [10]. It is obtained by setting C − = C + , and then, separately for x > 0 and x < 0, multiplying the Lévy-density of the original symmetric stable process with a decreasing exponential. The corresponding Lévymeasure, using standard parametrization, is of the form: ν(dx) = Ce −G|x| |x| 1+Y 1 x<0 dx + Ce −Mx |x| 1+Y 1 x>0 dx, where C, G, M > 0, and 0 < Y < 2. Intuitively, C controls the level of activity, G and M together control skewness. Typically G > M reflecting the fact that prices tend to increase rather than decrease. Y controls the density of small jumps, i.e. the fine structure. For Y < 1 the integrability condition (2) is satisfied, thus corresponding Lévy process is of finite variation. The characteristic exponent of the CGMY process is given by ψ(u) = CΓ (−Y ) (M − iu) Y − M Y + (G + iu) Y − G Y ,(15) where Γ denotes the gamma-function. Allowing C and Y to take on different values for x > 0 and x < 0 we get a more general class of processes called tempered stable process. see cite. Formally setting Y = 0 we get the Lévy density of the so-called Variance Gamma process (VG for short) that has been proposed by Madan, Carr and Chang [14]. The VG process is a time changed Brownian motion when the time change is a gamma process, which itself is a Lévy process, obtained by properly extending the definition of the inverse of a Poisson process from natural numbers to positive reals. Thus we can write V G(t) = W θ,σ (γµ,ν(t)), where W θ,σ (t) = θt + σW (t), with W being the standard Wiener process, and γ is a gamma process with mean rate µ, and variance rate ν, see [14]. Its characteristic function is given by ϕ V G(t) (u) = 1 − iuθν + u 2 σ 2 ν/2 −t/ν . This can be obtained by a formal limiting procedure taking into account the characteristic exponent given by (2.1) and taking Y → 0. The knowledge of the explicit form of the characteristic function is a common feature of distributions in finance. This is the case for tempered stable and related processes, see [5]. We will focus on the CGMY process. Discrete time Lévy systems A discrete time finite dimensional Lévy system is defined as ∆yn = A(θ * )∆Zn,(16) where ∆Zn is the increment of a Lévy process Z over an interval [(n−1)h, nh) with E [∆Zn] = 0, a property to be removed later and h > 0 is a fix sampling interval. The Lévy-measure of Z will be denoted by ν(dx) = ν(dx, η * ), where η * denotes an unknown parameter-vector, for example for a CGMY process η * = (C, G, M, Y ). The range of η * is assumed to be known. Condition 1 We assume that |x|≥1 |x| q ν(dx) < +∞ (17) for all 1 ≤ q ≤ Q with some constant Q. Note that Condition 1 holds with Q = ∞ in our benchmark examples. Let Dρ and D * ρ be compact domains such that ρ * ∈ D * ρ ⊂ int Dρ and Dρ ⊂ Gρ. Condition 2 A(θ) is assumed to be exponentially stable and exponentially inverse stable for θ ∈ G θ ⊂ R p , where G θ is a known open set. A system is exponentially stable if all the eigenvalues of A have strictly negative real parts. The application of the ML method would solve the full identification problem along standard lines, assuming that the density function of ∆Zn is known, see [21], which is unfortunately not the case. The objective of this paper is to present a combination of advanced techniques in systems identification with a specific statistical technique, widely used in the context in finance, called the ECF (shorthand for empirical characteristic function) method. The ECF method was originally designed for i.i.d. samples and A. Feuerverger and P. McDunnogh [13] showed that it can be interpreted as the Fourier transform of an ML method. Three identification problems In this section we formulate three identification problems related to discrete-time, finite dimensional Lévy systems, and sketch a possible path to their solution. The first, simplest problem is seemingly of mere technical interest: Known system parameters, unknown noise parameters. In this case define and compute εn(θ * ) = A −1 (θ * )∆yn = A −1 (θ * )A(θ * )∆Zn = ∆Zn, assuming, for the sake of simplicity, that ∆Zn = εn(θ * ) = 0 for n ≤ 0. After that we can apply the ECF method for i.i.d. samples to obtain the estimation of η * . This simple solution will be the base of the identification method presented in Section 7. Known noise parameters, unknown system parameters. This is the simplest, technically interesting and non-trivial problem. If we knew the probability density function of the noise, say f , we could obtain the maximum likelihood estimate of θ * via solving N n=1 f θ εn(θ), η * = 0,(18) where εn(θ) = A −1 (θ)∆yn(19) is the estimated innovation process of a SISO system, see [21]. Under certain conditions the asymptotic covariance matrix of the ML estimatê θ N is Σ ML = µ R * −1 , where µ = E f ′ (∆Zn, η * ) f (∆Zn, η * ) 2 , with f ′ being the derivative of f w.r.t the first variable and R * = lim n→∞ E ε nθ (θ * )ε T nθ (θ * ) . In our case, the p.d.f. of the noise distribution is not known. One might apply the prediction error method to estimate the system dynamics, i.e. θ * . However, we will show, in the case of CGMY noise, that we may estimate θ * in a more efficient way using an appropriate adaptation of the ECF method. In fact, this result is a special case of a more general result obtained for the general problem to be described in the next subsection. Both the system parameters and the noise parameters are unknown. The first method that we propose is quite straightforward: we estimate the system parameters using a PE method, then, using a certainty equivalence argument, we estimate the innovation process by inverting the system using the estimated parameters. Then, we estimate the noise parameters using ECF method for i.i.d. sequences. This method will be studied in Section 7. The second method, which is the main subject of this paper, estimates both the system parameters and noise parameters using an ECF method. First, an parameter-dependent, estimated innovation process εn(θ) is defined, then the characteristic function of the noise is fitted to empirical data defined in terms of εn(θ). Thus we get a score function that depends on both θ and η. The third method applies an extension of the ECF method using the blocks of the time-series of unprocessed data {∆yn} ∞ n=0 . More details can be found in the Discussion. Single term ECF method The ECF method has been widely used in finance as an alternative to the ML Method, assuming i.i.d. returns [7], [8], [17]. We adapt this technique to the problem of identifying the discrete-time Lévy system described in (12). Fix a realization of A in its innovation form, i.e. assume that A and its inverse are exponentially stable. The estimated innovation process (εn(θ)) is defined via the inverse filter: dX t = (H − KL)X t dt + KdY t (20) dε t = dY t − LdX t ,(21) for continuous time models. For discrete time Lévy systems we define the innovation process by εn(θ) = A −1 (θ)∆yn,(22) with zero initial conditions and θ ∈ Dρ. Let ε * n (θ) denote the stationary solution of (22) when −∞ < n < ∞. In general, the notation (.) * will be used throughout this paper if the corresponding stochastic process is obtained by passing through a stationary process through an exponentially stable linear filter starting at −∞, as opposed to initializing the filter at time 0 with some arbitrary initial condition, which is typically zero. Then we have for n ≥ 0 ε * n (θ) = εn(θ) + rn,(23)where rn = O Q M (α n ) with some 0 < α < 1, meaning that for all 1 ≤ q < Q sup n α −n E 1/q |rn| q < ∞. We will use this notation in a more general way: Definition 1 For a stochastic process Xn, and a function f : Z → R + we say that Xn = O Q M (f(n)) if for all 1 ≤ q ≤ Q sup n E 1/q |Xn| q f (n) < ∞ holds. The score functions to be used following the basic idea of the ECF method are defined as hn(u; θ, η) = e iuεn(θ) − ϕ(u, η) (24) h * n (u; θ, η) = e iuε * n (θ) − ϕ(u, η)(25) with u ∈ R. These are indeed appropriate score functions, since we obviously have E h * n (u; θ * , η * ) = 0, and hn(u; θ * , η * ) = h * n (u; θ * , η * ) + O Q M (αn) . While hn is the function that can be computed in practice, h * n is easier to handle, because its stationarity. Following the philosophy of the ECF method take a fix set u i -s, and define the k-dimensional vector hn(θ, η) = (hn(u 1 ; θ, η), . . . , hn(u k ; θ, η)) T . Let K > 0 be a fixed symmetric, positve definite k × k weighting matrix. Since the system of equations hn(θ, η) = 0 n = 1, . . . , N is overdetermined we seek a least-square solution. Therefore we define the cost functions as V N = V N (θ, η) = N n=1 |K −1/2 hn(θ, η)| 2 , V * N = V * N (θ, η) = N n=1 |K −1/2 h * n (θ, η)| 2 , and by solving V N θ (θ, η) = 0 (26) V N η (θ, η) = 0 (27) we obtain the estimationθ N andη N of θ * and η * , respectively. Analysis Differentiating V * N w.r.t θ and η we get the equations V * N θ (θ, η) = N n=1 h T * nθ (θ, η)K −1h * n(θ, η) + h T * n (θ, η)K −1h * nθ (θ, η) = 0, (28) V * N η (θ, η) = N n=1 h T * nη (θ, η)K −1h * n(θ, η) + h T * n (θ, η)K −1h * nη (θ, η) = 0, (29) whereh is the conjugate of h. Note that, setting θ = θ * , the second equation is just the optimality condition of the ECF method for i.i.d. samples [8]. As for the first equation, the derivative of the score function h with respect to θ is h nθ (u, θ, η) = e iuεn(θ) iuε nθ (θ).(30) Hence in the first equation h nθ (θ, η) and hn(θ, η) are not independent. However, the next lemma shows that their stationary approximation, h * nθ (θ, η) and h * n (θ, η), are uncorrelated. Lemma 1 For any η we have E [V * N θ (θ * , η)] = 0, and in addition E V * N η (θ * , η * ) = 0. Proof Consider the n th term in (28). We have E h * T nθ (θ * , η)K −1h * n (θ * , η) = = k l,m=1 K −1 l,m E e iu l ε * n (θ * ) iu l ε * nθ (θ * ) e −iumε * n (θ * ) − ϕ(−um, η) .(31) Compute the (l, m) th term using the tower law: E e iu l ε * n (θ * ) iu l ε * nθ (θ * ) e −iumε * n (θ * ) − ϕ(−um, η) = = E E e iu l ε * n (θ * ) iu l ε * nθ (θ * ) e −iumε * n (θ * ) − ϕ(−um, η) |F ∆Z n−1 ,(32)where F ∆Z n−1 = σ {∆Z k : k ≤ n − 1} . Here we used that ϕ(u, η) = ϕ(−u, η). Due to the fact that ε * nθ (θ * ) is F ∆Z n−1 measurable, (32) can be written as E iu l ε * nθ (θ * )E e i(u l −um)ε * n (θ * ) − e iu l ε * n (θ * ) ϕ(−um, η)|F ∆Z n−1 = E iu l ε * nθ (θ * ) ϕ(u l − um, η * ) − ϕ(u l , η * )ϕ(−um, η) = ϕ(u l − um, η * ) − ϕ(u l , η * )ϕ(−um, η) E iu l ε * nθ (θ * ) = 0. To reduce the last equation we used that E [∆Zn] = 0. Similarly for the n th term of (29) we have h * nη (u, θ, η) = −ϕη(u, η), which is non-random implying that E h * nη (u, θ * , η * )K −1h * n (u, θ * , η * ) = 0. ⊓ ⊔ The previous lemma also shows that the gradient of V N (θ, η) serves as an alternative score function. The following corollary is implied by the fact that E h T nθ (θ * , η)K −1h n(θ * , η) = E h * T nθ (θ * , η)K −1h * n (θ * , η) + O Q M (α n ).(33)Corollary 1 For any η we have E [V N θ (θ * , η)] = O Q M (α N ), and in addition E V N η (θ * , η * ) = O Q M (α N ). Define ρ = (θ, η), and define the asymptotic cost function by W (θ, η) = W (ρ) = E K −1/2 h * n (ρ) 2 . Condition 3 The equation Wρ(ρ) = 0 has a unique solution in D * ρ . A crucial object is the Hessian of W at ρ = ρ * : R * = Wρρ(ρ * ). It is easy to see that R * = W θθ (θ * ) 0 0 Wηη(η * ) is block diagonal matrix. The following result provides a precise characterization of the estimation error: Theorem 1 Under Conditions 1,2 and 3 we havê ρ N − ρ * = −(R * ) −1 1 N V N ρ (ρ * ) + O Q/(2(p+q)) M (N −1 ) First, we prove some lemmas that will be used in the proof of Theorem 1. For the definition of L-mixing processes and for other corresponding definitions and theorems see the Appendix. Lemma 2 Under Conditions 1,2,3 processes εn(θ), ε nθ (θ) and ε nθθ (θ) are L-mixing uniformly of order Q. Proof First, note that since ∆yn = q i=0 a i (θ * )∆Z i , holds, ∆yn is a linear combination of L-mixing processes of order Q. Using the fact that an uniformly exponentially stable filter with L-mixing input produces an uniformly L-mixing output [20] we get that ∆yn is L-mixing processes of order Q for each n. The innovation process and its derivatives with respect to θ can be written as εn(θ) = A −1 (θ)∆y i ε nθ (θ) = A −1 θ (θ)∆y i ε nθθ (θ) = A −1 θθ (θ)∆y i . Again, since A −1 (θ) and its derivative with respect to θ are uniformly exponentially stable we conclude the lemma. Proof First, note that since εn, ε nθ and ε nθθ are L-mixing processes uniformly of order Q, the processes hn, hnρ and hnρρ are L-mixing uniformly of order Q/2 as well. It follows that the process un(ρ) = ∂ ∂ρ h T n (ρ)K −1/2h n(ρ) (ρ) − Wρ(ρ)(34) and its derivative with respect to ρ are L-mixing uniformly of order Q/2. E [u * n (ρ)] = 0 implies E [un(ρ)] = O Q/2 M (α n ) uniformly in ρ and hence following Theorem 9 we have for δV N ρ = sup ρ∈Dρ 1 N V N ρ (ρ) − Wρ(ρ) ,(35)δV N ρρ = sup ρ∈Dρ 1 N V N ρρ (ρ) − Wρρ(ρ) ,(36)d ′ = inf |Wρ(ρ)| : ρ ∈ Dρ, ρ − ρ * ≥ d > 0, since Wρ is continuous and Dρ is compact. It follows that δV N ρ > d ′ , and we have seen that this event has probability O(N −s ). So for Ω N = δV N ρ > d, δV N ρρ > d ′ we have P (Ω N ) > 1 − O(N −s ) with any 0 < s ≤ Q/2. The equation Wρ(ρ) has a unique solution in Dρ. Hence by using the implicit function theorem, see Theorem 10, one can easily conclude that V N ρ (ρ) = 0 has a unique solution if d ′ and d ′′ are sufficiently small. ⊓ ⊔ Lemma 4 Under Conditions 1,2,3 we haveρ N − ρ * = O Q/2 M (N −1/2 ). Proof We have 0 = V N ρ (ρ N ) = V N ρ ρ * + V N ρρ ρ − ρ * ,(37) where V N ρρ = 1 0 V N ρρ (1 − λ) ρ * + λρ N dλ. Since εn(θ * ) = ∆Z k + O Q M (α n ) for some |α| < 1, and with un = ∂ ∂ρ hn(ρ)K −1/2 2 using the inequality in Theorem 7 with fn = 1 and q ≤ Q from E 1/q N n=1 un q ≤ CqN 1/2 M 1/2 q (u)Γ 1/2 q (u) (38) we conclude V N ρ (ρ * ) = O Q/2 M (N 1/2 ). Let W N ρρ = 1 0 Wρρ (1 − λ) ρ * + λρ N dλ. W is a smooth function, hence Wρρ ρ * + λ ρ N − ρ * − Wρρ(ρ * ) < c ρ N − ρ * < cd.(39) Clearly Wρρ(ρ * ) is positive definite, hence W N ρρ > cI, with some c > 0. Since on Ω N 1 N V N ρρ − W N ρρ < d ′ holds, choosing d ′ sufficiently small yields λ min 1 N V N ρρ > c(40) on Ω N , where λ min (M) denotes the smallest eigenvalue of M. Thus V −1 N ρρ < cN −1 on Ω N . Then using (37) we get that χ ΩN ρ N − ρ * = O Q/2 M (N −1/2 ). Furthermore, since P (Ω C N ) = O(N −s ) for any 0 < s ≤ Q/2, the lemma follows. ⊓ ⊔ Now we are ready to prove Theorem 1. Proof Using the previous lemma one can improve (39): Wρρ ρ * + λ ρ N − ρ * − Wρρ(ρ * ) < c ρ N − ρ * = O Q/2 M (N −1/2 ), and after integration with respect to λ we get W N ρρ − Wρρ(ρ * ) = O Q/2 M (N −1/2 ).(41)Since δV N ρρ = O Q/(2(p+q)) M (N −1/2 ), it implies 1 N V N ρρ − W ρρ = O Q/(2(p+q)) M (N −1/2 ).(42) Hence by triangle inequality from (41) and (42) 1 N V N ρρ − Wρρ(ρ * ) = O Q/(2(p+q)) M (N −1/2 )(43) follows. From (40) and (43) we get χ ΩN V −1 N ρρ − 1 N W −1 ρρ (ρ * ) = O Q/(2(p+q)) M (N −3/2 ).(44) Finally, χ ΩN ρ N − ρ * = −χ ΩN V −1 N ρρ V N ρ (ρ * ) = −χ ΩN 1 N W −1 ρρ (ρ * ) + O Q/(2(p+q)) M (N −3/2 ) V N ρ (ρ * ) = −χ ΩN 1 N W −1 ρρ (ρ * )V N ρ (ρ * ) + O Q/(2(p+q)) M (N −1 ) Since χ ΩN = 1 − O Q M (N −s ) for any 0 < s ≤ Q/(4p + 4q), from the last expression reads as − R * −1 1 N V N ρ (ρ * ) + O Q/(2(p+q)) M (N −1 ) ⊓ ⊔ The following theorem provides an explicit expression for the Hessian of W : Theorem 2 Under Conditions 1,2,3 we have R * = W θθ (θ * ) 0 0 Wηη(η * ) , i.e. R * is block diagonal, and here W θθ (θ * ) = w E ε * nθ (θ * )ε T * nθ (θ * ) , with w = k l,m=1 K −1 l,m (u 2 l + u 2 m )ϕ(u l , η * )ϕ(−um, η * ) − (u l − um) 2 ϕ(u l − um, η * ) , and (Wηη) j,j ′ (η * ) = k l,m=1 K −1 l,m ϕη j (u l , η * )ϕ η ′ j (−um, η * ) + ϕ η ′ j (u l , η * )ϕη j (−um, η * ) . Proof First let j, j ′ ≤ dim θ = p, then an entry (R * ) j,j ′ of R * is E   ∂ 2 ∂θ j ∂θ j ′ k l,m=1 K −1 l,m e iu l ε * n (θ) − ϕ(u l , η) e −iumε * n (θ) − ϕ(−um, η) θ=θ * η=η * Carrying out differentiation yields E k l,m=1 K −1 l,m e iu l ε * n (θ * ) (iu l ) 2 ε * nθj (θ * ) ε * nθ j ′ (θ * ) + e iu l ε * n (θ * ) (iu l ) ε * nθjθ j ′ (θ * ) × e −iumε * n (θ * ) − ϕ(−um, η * ) + +E k l,m=1 K −1 l,m e iu l ε * n (θ * ) (iu l ) ε * nθj (θ * ) e −iumε * n (θ * ) (−ium) ε * nθ j ′ (θ * ) + + k l,m=1 K −1 l,m e iu l ε * n (θ * ) (iu l ) ε * nθ j ′ (θ * ) e −iumε * n (θ * ) (−ium) ε * nθj (θ * ) + +E k l,m=1 K −1 l,m e iu l ε * n (θ * ) − ϕ(u l , η * ) × e −iumε * n (θ * ) (−ium) 2 ε * nθj (θ * ) ε * nθ j ′ (θ * ) + e −iumε * n (θ * ) (−ium) ε * nθjθ j ′ (θ * ) Now we use, like in the proof of Lemma 1, the tower rule and that ε * nθ (θ * ) is F ∆Z n−1 measurable and that E ε * nθj (θ * ) = E ε * nθ j ′ (θ * ) = E ε * nθjθ j ′ (θ * ) = 0. The previous formula reads as E ε * nθj (θ * )ε * nθ j ′ (θ * ) k l,m=1 K −1 l,m (ϕ(u l − um, η * ) − ϕ(u l , η * )ϕ(−um, η * )) (−u 2 l ) + 2E ε * nθj (θ * )ε * nθ j ′ (θ * ) k l,m=1 K −1 l,m ϕ(u l − um, η * )(u l um) + E ε * nθj (θ * )ε * nθ j ′ (θ * ) k l,m=1 K −1 l,m (ϕ(u l − um, η * ) − ϕ(u l , η * )ϕ(−um, η * )) (−u 2 m ) = E ε * nθj (θ * )ε * nθ j ′ (θ * ) × k l,m=1 K −1 l,m (u 2 l + u 2 m )ϕ(u l , η * )ϕ(−um, η * ) − (u l − um) 2 ϕ(u l − um, η * ) To double check the result note that the last formula gives real matrix since conjugation doest not modify the value of the double sum. If j ≤ p < j ′ ≤ p + q, then (R * ) j,j ′ equals to E ∂ 2 ∂θj ∂η j ′ k l,m=1 K −1 l,m e iu l ε * n (θ) − ϕ(u l , η) e −iumε * n (θ) − ϕ(−um, η) θ=θ * η=η * = 0, because the differentiation with respect to η j ′ yields a non-random constant of the form ϕη j (u, η * ) and the differentiation with respect to θ j yields the term E e iuε * n (θ * ) iuε * nθj (θ * ) = 0. Finally, if p < j, j ′ then (R * ) j,j ′ equals to E ∂ 2 ∂ηj ∂η j ′ k l,m=1 K −1 l,m e iu l ε * n (θ) − ϕ(u l , η) e −iumε * n (θ) − ϕ(−um, η) θ=θ * η=η * = k l,m=1 K −1 l,m ϕη j (u l , η * )ϕ η ′ j (−um, η * ) + ϕ η ′ j (u l , η * )ϕη j (−um, η * ) . To sum it up, R * = W θθ (θ * ) 0 0 Wηη(η * ) is block diagonal matrix, where W θθ (θ * ) = E ε * nθ (θ * )ε T * nθ (θ * ) × k l,m=1 K −1 l,m (u 2 l + u 2 m )ϕ(u l , η * )ϕ(−um, η * ) − (u l − um) 2 ϕ(u l − um, η * ) , and (Wηη) j,j ′ (η * ) = k l,m=1 K −1 l,m ϕη j (u l , η * )ϕη j ′ (−um, η * ) + ϕη j ′ (u l , η * )ϕη j (−um, η * ) . ⊓ ⊔ Remark 1 : Note that the expression for (Wηη) j,j ′ (η * ) is identical to what we would obtained for i.i.d. samples following [7]. Remark 2: Since we have w ≥ 0, the expression for w yields a non-trivial inequality for characteristic functions. The next step in calculating the asymptotic covariance matrix ofθ N is the computation of S * = Cov (V * N θ (ρ * ), V * N θ (ρ * ) ). For this we need to introduce the following auxiliary function: F (a, b, c, d, η) = ab ϕ(a + b + c + d, η) − ϕ(a + b + c, η)ϕ(d, η)− ϕ(a + b + d, η)ϕ(c, η) + ϕ(a + b, η)ϕ(c, η)ϕ(d, η) . Theorem 3 Under Conditions 1,2,3 we have S * = Cov V * N θ (ρ * ), V * N θ (ρ * ) = Cov (V * N θ (ρ * ), V * N θ (ρ * )) 0 0 Cov V * N η (ρ * ), V * N η (ρ * ) , where Cov (V * N θ (ρ * ), V * N θ (ρ * )) = s E ε * nθ (θ * )ε T * nθ (θ * ) , with s = N l,m,s,t=1 K −1 l,m K −1 s,t × F (u l , us, −um, −u t , η * ) + F (u l , −u t , −um, us, η * )+ F (−um, us, u l , −u t , η * ) + F (−um, −u t , u l , us, η * ) , and Cov V * N η (ρ * ), V * N η (ρ * ) j,j ′ = N l,m,s,t=1 K −1 l,m K −1 s,t × ϕη j (u l , η * )ϕη j ′ (us, η * )ϕ(−um − u t , η * ) + ϕη j (u l , η * )ϕη j ′ (−u t , η * )ϕ(−um + us, η * )+ ϕη j (−um, η * )ϕη j ′ (us, η * )ϕ(u l − u t , η * ) + ϕη j (−um, η * )ϕη j ′ (−u t , η * )ϕ(u l + us, η * ) The proof of the last theorem is a simple calculation like the previous one and the proof uses that E ε * nθi (θ * )ε * mθi (θ * ) = E E ε * nθi (θ * )ε * mθi (θ * )|F ∆Z n−1 = E ε * nθi (θ * ) E ε * mθi (θ * )|F ∆Z n−1 = 0 for m > n. The proof follows the line of arguments for Lemma 1. We note that calculations are considerably simplified if we take K = I. Note that both R * and S * are of the form c Σ P , where Σ P is the asymptotic covariance matrix for the prediction error method, see below (46), and c is a constant. The last two theorems and Theorem 1 together gives an exact formula for the asymptotic covariance matrix of the estimator. Theorem 4 Under Conditions 1,2 and 3 the asymptotic covariance matrix of the ECF estimator for θ * can be written as Σ E = (R * ) −1 S * (R * ) −1 = s w 2 Σ P ,(45) where the s and w are given in Theorems 3 and 2. Combining PE and ECF estimators In this section we estimate the dynamics in a natural way and then we estimate the noise parameters using the ECF method. We identify θ * using only the orthogonality of ∆Z by applying a prediction error method. This way we get an estimationθ N of θ * , without using the characteristic function of ∆Z. Then we apply an ECF method with the score function hn(u, η) = e iuεn(θN ) − ϕ(u, η) to estimate η * . First, we define the estimated innovation process as in the previous sections. The prediction error method is obtained by minimizing the cost function V P,N (θ) = 1 2 N n=1 ε 2 n (θ). In practice the estimatedθ N is defined as the solution of V P,N θ (θ) = N n=1 εn(θ)ε nθ (θ) = 0. The asymptotic cost function associated with the PE method is defined as W P (θ) = 1 2 lim n→∞ Eε 2 n (θ) = 1 2 Eε * 2 n (θ), recall that ε * n (θ) is the innovation process that is calculated with stationary initial values. We have W P,θ (θ * ) = 0 and R * P : = W P,θθ (θ * ) = E ε * nθ (θ * )ε T * nθ (θ * ) . The asymptotic covariance matrix of the PE estimate of θ * is given by Σ P = E ε * nθ (θ * )ε T * nθ (θ * ) −1 .(46) An ideal score function for the ECF method to estimate η * would be defined by h opt,n (u, η) = e iuε * n (θ * ) − ϕ(u, η).(47) Since we are not given θ * we define an alternative, θ-dependent score function via hn(u, θ, η) = e iuεn(θ) − ϕ(u, η). These are appropriate score functions since E [h * n (u, θ * , η * )] = 0. Fix a set of real numbers u 1 , · · · , u k , with N k ≥ dim η and define hn(θ, η) = (hn(u 1 , θ, η), · · · , hn(u k , θ, η)) T . Then we obtain the estimateη N of η * by finding a least squares solution to the over-determined system of equations hn(θ N , η) = 0 n = 1, . . . , N More precisely, define the θ-dependent cost function V E,N (θ, η) = N n=1 K −1 hn(θ, η) 2 , where K is a symmetric, positive definite weighting matrix. Then we obtain the estimateη N of η * by minimizing V E,N (θ N , η). Define the (θ-dependent) asymptotic cost function as W E (θ, η) = E K −1/2 h * n (θ, η) 2 . Let its Hessian w.r.t. η at η = η * be denoted by R * E = W E,ηη (θ * , η * ). To formulate our result we need some technical conditions. Conditions 1 and 2 have been already presented in Section 3. Let ρ be the joint parameter i.e. ρ = (θ, η). Let Dρ and D * ρ be compact domains such that ρ * ∈ D * ρ ⊂ int Dρ and Dρ ⊂ Gρ. Condition 3' The equations W P,θ (θ) = 0, and W E,η (θ * , η) = 0 have a unique solution in D * ρ . The following lemma, with minor variation, can be found in [19]. Lemma 5 Under Conditions 1,2,3' we haveθ N − θ * = O Q/2 M (N −1/2 ). Our next result characterizes the estimation error of the ECF method for the noise parameter η * . Theorem 5 Under Conditions 1,2 and 3' we havê η N − η * = −(R * E ) −1 1 N V E,N η (η * ) + O Q/(2(p+q)) M (N −1 ). The proof is obtained by the very same methods as Theorem 1 combined with the fact that In view of the efficiency of the ECF method for i.i.d. samples the question arises what can be achieved by the proposed adaptation of the ECF method when identifying the dynamics of a linear stochastic system. We do not have an answer to this general question, but we will show that the commonly used PE method can be outperformed by an appropriately calibrated ECF method when the noise is CGMY. Without loss of generality we may assume that Var (∆Zn) = 1. Wηη(θ * , η * ) − Wηη(θ N , η * ) = O Q/2 M (N −1/2 ),(48)which is implied byθ N − θ * = O Q/2 M (N −1/2 Surprisingly, we will see that the ECF method may outperform the PE method by using a single u sufficiently close to 0. Letting u tend to 0 the asymptotic covariance of the ECF estimate tends to the asymptotic covariance of the PE estimate. On the other hand, numerical investigations show that increasing the number of u-s used in the ECF method may not improve the efficiency significantly. For k = 1 the asymptotic covariance ofθ N obtained by the ECF method is lim N →∞ N Cov(θ N − θ * ), , which reads as, using Theorems 3 and 2, E ε * nθ (θ * )ε * T nθ (θ * ) −1 − 1 4u 2 ϕ(2u) ϕ 2 (u) + ϕ(−2u) ϕ 2 (−u) − 2 ϕ(u)ϕ(−u) . Recall that the asymptotic covariance ofθ N obtained by the PE method is Σ P = E ε nθ (θ * )ε T nθ (θ * ) −1 . Thus the ECF estimator outperforms the PE estimator if s w 2 = − 1 4u 2 ϕ(2u) ϕ 2 (u) + ϕ(−2u) ϕ 2 (−u) − 2 ϕ(u)ϕ(−u) < 1. Theorem 6 For all u = 0, sufficiently close to 0 we have s w 2 < 1, and thus the corresponding single-term ECF estimator of the system parameter θ * , with k = 1, outperforms the PE estimator. Proof First note that for g(u) = − ϕ(2u) ϕ 2 (u) + ϕ(−2u) ϕ 2 (−u) − 2 ϕ(u)ϕ(−u)(49) g(u) = g(u) holds, so g is a real-valued function. Let us compute the Taylor expansion of g around 0. The first three derivatives of ϕ(u) for a CGMY process with zero expectation are given by ϕ(0) = 1, ϕu(0) = iE [∆Zn] = 0, ϕuu(0) = −E (∆Zn) 2 = = −CΓ (2 − Y ) M Y −2 + G Y −2 = −1, ϕuuu(0) = −iE (∆Zn) 3 = 0. After a lengthy computation, that we omit, we get that g(u) = −4u 2 + 4 3 G −2 (Y − 2)(Y − 3)u 4 + O(u 6 ).(50)Thus s w 2 = − 1 4u 2 ϕ(2u) ϕ 2 (u) + ϕ(−2u) ϕ 2 (−u) − 2 ϕ(u)ϕ(−u) = 1 − 1 3 G −2 (Y − 2)(Y − 3)u 2 + O(u 4 ). Since G < 0 and 0 < Y < 2, the coefficient of u 2 is negative. Hence, by choosing u sufficiently small s w 2 < 1 can be achieved. Numerical investigations show that for a CGMY process with parameters C = 0.564, G = M = 1, Y = 0.5 the minimal value of g is approximately 0.73. We experienced that increasing the number of u-s that are used does not reduce s/w 2 significantly. For example, choosing (u 1 , . . . , u k ) = (0.1, 0.2, . . . , 0.1k) and K = I we get s/w 2 = 0.688. Discussion In the previous section we assumed that E [∆Zn] = 0. This is a standard assumption in system identification, but certainly not realistic for financial data. Thus e.g. in the case of a CGMY process this assumption would imply G = M , excluding possible skewness in the distribution. While the case E [∆Zn] = m * = 0 would pose no problem for the case of i.i.d. data, surprisingly the single term ECF method may break down. The reason for this is that V * N θ (θ, η) is no more a score function, since we cannot guarantee that E V * N θ (θ * , η) = 0 holds, see Lemma 1. Namely in the proof of Lemma 1 we make use of the equality E ε * nθ (θ * ) = 0,(51) which may not be valid. Note, however, that hn(u; θ, η) = e iuεn(θ) − ϕ(u, η) does have the property required for a score function, namely E e iuε * n (θ * ) − ϕ(u, η * ) = 0.(52) Thus, using an instrumental variable approach, we may choose an appropriate linear combination of these score functions, say N n=1 M hn(θ, η), where M is a (p + r) × k matrix, and consider the equation: Assuming that p + r < k we may rightly expect that taking mathematical expectation the resulting equation has (θ * , η * ) as an isolated solution, and we may proceed as in Section 5. The elaboration of the details is the subject of ongoing research. An alternative approach is to adapt our method of combining the PE method with the ECF method. For this we first need to extend the PE method to deal with the case m * = 0, which is a standard exercise. Write ∆Zn = ∆en + m * , where E [∆en] = 0. Then equation (12) reads as ∆yn = A(θ * ) ∆en + m * . Define the estimated innovation process by εn(θ) = A −1 (θ)A(θ * )(∆en + m * ). Clearly E [εn(θ * )] = m * , thus we define the cost function via V N (θ, m) = 1 2 N n=1 (εn(θ) − m) 2 . The estimate (θ N ,m N ) of (θ * , m * ) is obtained by solving ∂ ∂(θ, m) V N (θ, m) = 0, which can be written as 0 = V N θ (θ, m) = N n=1 (εn(θ) − m) ε nθ (θ) 0 = V N m (θ, m) = − N n=1 (εn(θ) − m) . Having estimated the system dynamics with this extended PE method one may estimate the noise parameters with the ECF method, as in Section 7. The shortcoming of the above approach is that it does not exploit fully the potentials of the ECF method in estimating the system dynamics. Therefore we suggest a second pass for estimating θ * via a single term ECF method, withη N considered as the true parameter, applied to the system ∆yn − A(θ N )m N = A(θ * )(∆Zn −m N ),(53) wherem N ,θ N ,η N are the first estimates. Define ∆ỹn = ∆yn − A(θ N )m N and ∆Zn the previous equation reads as ∆ỹn = A(θ * )∆Zn,(54) with E ∆Zn = 0. Thus we may proceed according to Section 5 to obtain the corrected estimate of θ * . What we have obtained is an extension of the single term ECF method, which is computationally simpler. Ongoing investigations suggest that the efficiency of this generalized single term ECF method is as good as the original single term ECF when m * = 0. Finally we mention one more very different approach to deal with the problem of non-zero expectation, having interest on its own. The idea is to use an ECF method directly for blocks of unprocessed data, i.e. for blocks of the time series (yn). For this purpose let us imbed our data into the class of time series ∆yn(θ, η) = A(θ)∆Zn(η). Note that for (θ, η) = (θ * , η * ) we recover (in a statistical sense) our observed data. Fix a block length, say r, and define the r-dimensional blocks ∆Y r n (θ, η) = (∆yn(θ, η), . . . , ∆y n+r−1 (θ, η)). Letting U be an arbitrary r-vector the characteristic function of ∆Y r n (θ, η) is given by ϕn(U, θ, η) = E e iU T ∆Y r n (θ,η) , and the corresponding score function will be defined as hn(U, θ, η) = e iU T ∆Y r n − ϕn(U, θ, η). The point is that the characteristic function can be explicitly computed, at least in theory, as ϕn(U, θ, η) = E exp{iU T ∆Y r n (θ, η)} = E   exp    i r j=1 U j ∞ l=0 h l (θ)∆Z n+j−1−l (η)      = ∞ j=0 ϕ ∆Z(η) (v j (θ)),(55) with some θ-dependent constants v j . Here ϕ ∆Z denotes the characteristic function of ∆Z 1 (η). The weakness of this approach is that the characteristic function ϕn(U, θ, η) is given in terms of an infinite product, therefore it is not clear how to use it in actual computations. Appendix Let θ be a d-dimensional parameter vector. Definition 2 We say that xn(θ) is M -bounded of order Q if for all 1 ≤ q ≤ Q, M Q q (x) = sup n>0,θ∈D E 1/q |xn(θ)| q < ∞ Define Fn = σ {e i : i ≤ n} and F + n = σ {e i : i > n} where e i -s are i.i.d. random variables. Definition 3 We say that a stochastic process (xn(θ)) is L-mixing of order Q with respect to Fn, F + where Cm = 2(2m − 1) 1/2 . Define ∆x/∆ α θ = |xn(θ + h) − xn(θ)| / |h| α for n ≥ 0, θ = θ + h ∈ D with 0 < α ≤ 1. Definition 4 We say that xn(θ) is M -Hölder continuous of order Q in θ with exponent α if the process ∆x/∆ α θ is M -bounded of order Q. Now let us suppose that (xn(θ)) is measurable, separable, M -bounded of order Q and M -Hölder of order Q in θ with exponent α for θ ∈ D. The realizations of (xn(θ)) are continuous in θ almost surely hence x * n = max θ∈D0 |xn(θ)| is well defined for almost all ω, where D 0 ⊂ int D is a compact domain. Since the realizations of (xn(θ)) are continuous, x * n is measurable with respect to F . where C depends only on p, q, s, α and D 0 , D. Choosing fn = 1 and α = 1 and using Theorem 7 and 8 we obtain Theorem 9 Let (un(θ)) be an L-mixing of order Q uniformly in θ ∈ D such that Eun(θ) = 0 for all n ≥ 0, θ ∈ D, and assume that ∆u/∆θ is also L-mixing of order Q, uniformly in θ, θ + h ∈ D. Then Theorem 10 Let D 0 and D be as above. Let W θ (θ), δW θ (θ), θ ∈ D ⊂ R p be R pvalued continuously differentiable functions, let for some θ * ∈ D 0 , W θ (θ * ) = 0, and let W θθ (θ * ) be nonsingular. Then for any d > 0 there exists positive numbers d ′ , d ′′ such that |δW θ (θ)| < d ′ and δW θθ (θ) < d ′′ for all θ ∈ D 0 implies that the equation W θ (θ) + δW θ (θ) = 0 has exactly one solution in a neighborhood of radius d of θ * . Suppose that Conditions 1,2,3 hold. Then for any given d > 0 the equation V N ρ (ρ) = 0 has a unique solution in Dρ and it is in the sphere S = {ρ : |ρ − ρ * | < d} with probability at least 1 − O(N −s ) for any 0 < s ≤ Q/2. Furthermore the constant C in O(N −s ) = CN −s depends only on d and s. P (δV N ρ > d) = O(N −s ) with any 0 < s ≤ Q/(4p +4q) by Markov's inequality. Applying the same argument yields P (δV N ρρ > d ′′ ) = O(N −s ), for any d ′′ > 0, and any 0 < s ≤ Q/(4p + 4q). Suppose now that equation V N ρ (ρ) = 0 has a solution outside S. Define n uniformly in θ if it is Fn progressively measurable, M-bounded of order Q with any positive r and γq(r, x) = γq(r) Let (un), n ≥ 0 be an L-mixing process of order Q with Eun = 0 for all n, and let (fn) be a deterministic sequence. Then we have for all 1 ≤ m ≤ Q Theorem 8 8Assume that (xn(θ)) is measurable, separable, M -bounded of order Q and M -Hölder of order Q in θ with exponent α for θ ∈ D. Then we have for all positive q ≤ Qα/s and p/α < s ≤ Q/q,Mq(x * ) ≤ C (Mqs(x) + Mqs(∆x/∆ α θ)) Théorie de la spéculation. L Bachelier, Annales Scientifiques de lcole Normale Suprieure. 3L. Bachelier Théorie de la spéculation, in Annales Scientifiques de lcole Normale Su- prieure, 3 (17) (1900), pp. 21 86. Scholes The pricing of options and corporate liabilities. F Black, M , Journal of Political Economy. F. Black, M. Scholes The pricing of options and corporate liabilities, in Journal of Po- litical Economy, 81 (1973), pp. 345-637. J Jacod, A N Shiryaev, Limit theorems for stochastic processes. SpringerJ. Jacod, A.N. Shiryaev, Limit theorems for stochastic processes (2. ed.), Springer, (2002) K Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University PressK. Sato Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, (1999) Woyczynski Rosinski measures for tempered stable and related Ornstein-Uhlenbeck processes. Gy, W A Terdik, Probability and Mathematical Statistics. 26Gy. Terdik, W.A. Woyczynski Rosinski measures for tempered stable and related Ornstein-Uhlenbeck processes, in Probability and Mathematical Statistics, 26 (2) (2006), pp. 213-243. The fine structure of asset returns: an empirical investigation. P Carr, H Geman, D Madan, M Yor, Journal of Business. 752P. Carr, H. Geman, D. Madan and M. Yor, The fine structure of asset returns: an empirical investigation, in Journal of Business, 75 (2) (2002), pp. 305-332. Generalization of GMM to a continuum of moment conditions. M Carrasco, J.-P Florens, Econometric Theory. 16M. Carrasco and J.-P. Florens, Generalization of GMM to a continuum of moment conditions, in Econometric Theory, 16 (06) (2000), pp. 797-834. Efficient GMM estimation using the empirical characteristic function. M Carrasco, J P Florens, Idei working papers. 140M. Carrasco and J. P. Florens, Efficient GMM estimation using the empirical charac- teristic function, in Idei working papers, 140 (2002) Reppa A two-step maxumum-likelihood identification of non-gaussian systems. L Gerencser, Gy, Z Michaletzky, Proceedings of the 15th IFAC World Congress. the 15th IFAC World CongressL. Gerencser, Gy. Michaletzky, Z. Reppa A two-step maxumum-likelihood identification of non-gaussian systems, in Proceedings of the 15th IFAC World Congress, (2002) P Carr, H Geman, D B Madan, M Yor, The Fine Structure of Asset Returns: An Empirical Investigation. 75P. Carr, H. Geman, D.B. Madan and M. Yor, The Fine Structure of Asset Returns: An Empirical Investigation, in The Journal of Business, 75 (2000), pp. 305-322. R Cont, P Tankov, Financial Modelling with Jump Processes. 101R. Cont, P. Tankov, Financial Modelling with Jump Processes, in Journal of the Amer- ican Statistical Association, 101 (2006), pp. 1315-1316. S Raible, Lvy Processes in Finance: Theory, Numerics, and Empirical Facts. PhD dissertationS. Raible, Lvy Processes in Finance: Theory, Numerics, and Empirical Facts, PhD disser- tation (2000), http://www.freidok.uni-freiburg.de/volltexte/51/pdf/511.pdf, Accessed on 19 November 2012 On the efficiency of empirical characteristic function procedures. A Feuerverger, P Mcdunnogh, J.R. Stat. Soc. B. 431A. Feuerverger and P. McDunnogh, On the efficiency of empirical characteristic func- tion procedures, in J.R. Stat. Soc. B, 43 (1) (1981), pp. 20-47. The Variance Gamma Process and Option Pricing. B Madan, P Carr, C Chang, European Finance Review. 2B. Madan, P. Carr and C. Chang, The Variance Gamma Process and Option Pricing, in European Finance Review, 2 (1998), pp. 79-105. The Variation of Certain Speculative Prices. B Mandelbrot, Journal of Business. 35B. Mandelbrot, The Variation of Certain Speculative Prices, in Journal of Business, 35, (1963) Some processes associated with a fractional Brownian motion. T E Duncan, Mathematics of Finance. 351T. E. Duncan, Some processes associated with a fractional Brownian motion, in Mathe- matics of Finance, (eds. G. Yin and Q. Zhang) Contemp. Math. 351 (2004), pp. 93-102. Empirical characteristic function estimation and its applications. J Yu, Econometric Reviews. 232J. Yu, Empirical characteristic function estimation and its applications, in Econometric Reviews, 23 (2) (2004), pp. 93-123. . L Gerencsér, M Mánfay, L. Gerencsér, M. Mánfay Geometric Lévy Process Pricing Model. Y Miyahara, A Novikov, Research Paper Series. 66Quantitative Finance Research Centre, University of TechnologyY. Miyahara and A. Novikov, Geometric Lévy Process Pricing Model, in Research Paper Series 66, Quantitative Finance Research Centre, University of Technology, Sydney, (2001) On the martingale approximation of the estimation error of ARMA parameters. L Gerencsér, System & Control Letters. L. Gerencsér, On the martingale approximation of the estimation error of ARMA pa- rameters, in System & Control Letters, 15 (1990), pp. 417-423. On a class of mixing processes. L Gerencsér, Stochastics. 26L. Gerencsér, On a class of mixing processes, in Stochastics, 26 (1989), pp. 165-191. Reppa A two-step maximum-likelihood identification of non-Gaussian systems. L Gerencsér, Gy, Z Michaletzky, Proceedings of the 15th IFAC World Congress. the 15th IFAC World Congress15L. Gerencsér, Gy. Michaletzky, Z. Reppa A two-step maximum-likelihood identifica- tion of non-Gaussian systems, in Proceedings of the 15th IFAC World Congress 15 (2002)
[]
[ "Reversibility Checking for Markov Chains", "Reversibility Checking for Markov Chains" ]
[ "Q Jiang \nDepartment of Mathematics and Statistics\n\n", "M Hlynka \nDepartment of Mathematics and Statistics\n\n", "P H Brill \nDepartment of Mathematics and Statistics\n\n\nDepartment of Management Science\nUniversity of Windsor Windsor\nN9B 3P4OntarioCanada\n", "C H Cheung \nDepartment of Mathematics and Statistics\n\n" ]
[ "Department of Mathematics and Statistics\n", "Department of Mathematics and Statistics\n", "Department of Mathematics and Statistics\n", "Department of Management Science\nUniversity of Windsor Windsor\nN9B 3P4OntarioCanada", "Department of Mathematics and Statistics\n" ]
[]
In this paper, we present reversibility preserving operations on Markov chain transition matrices. Simple row and column operations allow us to create new reversible transition matrices and yield an easy method for checking a Markov chain for reversibility.
10.31390/cosa.12.2.02
[ "https://arxiv.org/pdf/1806.10154v1.pdf" ]
119,645,592
1806.10154
7cd5a2126d0c5d7dfa3b1fd3d4e8b6db71844758
Reversibility Checking for Markov Chains 26 Jun 2018 June 1, 2018 Q Jiang Department of Mathematics and Statistics M Hlynka Department of Mathematics and Statistics P H Brill Department of Mathematics and Statistics Department of Management Science University of Windsor Windsor N9B 3P4OntarioCanada C H Cheung Department of Mathematics and Statistics Reversibility Checking for Markov Chains 26 Jun 2018 June 1, 2018Reversible Markov chaindetailed balance equationsKolmogorov criterion Mathematics Subject Classification: 60J1060J22 In this paper, we present reversibility preserving operations on Markov chain transition matrices. Simple row and column operations allow us to create new reversible transition matrices and yield an easy method for checking a Markov chain for reversibility. Introduction Reversible Markov chains show up in many diverse areas. For example, they occur in MCMC (Markov Chain Monte Carlo) analyses (see [1] Aldous and Fill, 2001). They have geological applications as in [7] Richman and Sharp, 1991. They have applications in genetics models and queueing networks (see [4] Kelly, 1978). McCullagh [5] (1982) and Sharp and Markham [8] (2000) look at quasi symmetry and reversibility. More recent work has been done by Pistone and Rogantin [6] (2013). Notation We use standard Markov chain notation, as in [2] Durrett, 2012. Let P = [p ij ] be the probability transition matrix for an ergodic Markov chain X(t), t = 0, 1, 2, . . . with states {1, 2, . . . , n} where p ij = P (X(t) = j|X(t − 1) = i) for t = 1, 2 . . . . Let P (k) = [p (k) ij ] be the k step transition matrix. For an ergodic Markov chain, the limiting probability and stationary row vector π = (π 1 , π 2 , . . . ) with π j = lim k→∞ p (k) ij exists and is independent of i. The limiting vector π can be found by solving the balance equations π = πP,(1) subject to n j=1 π j = 1. Reversible Process From Kelly [4], (1978), an ergodic Markov chain X(t) on state space S is reversible if (X(0), X(1), . . . , X(t)) has the same distribution as (X(t), X(t − 1), . . . , X(0)) for all t. Given P and π, the chain X(t) is reversible iff for all i, j, it satisfies the detailed balance equations π i p ij = π j p ji ,(2) Kolmogorov's Check for Reversibility It may be desirable to verify reversibility before solving for the stationary vector π, since reversibility allows a simple method for find-ing π. We can also be interested in reversibility for other reasons. We refer to a transition matrix as being reversible if the corresponding Markov chain is reversible. One method is to use Kolmogorov's loop criterion (see [4] Kelly, 1978, Chapter 1). An ergodic Markov chain is reversible if and only if p j 0 j 1 p j 1 j 2 . . . p j k−1 j k p j k j 0 = p j 0 j k p j k j k−1 . . . p j 2 j 1 p j 1 ,j 0 ,(3) for every finite sequence of distinct states j 0 , j 1 , j 2 , . . . , j k Some matrices can be easily checked for reversibility by Kolmogorov's loop criterion. For a two-state Markov chain X(t), Kolmogorov's criterion is always satisfied since p 12 p 21 = p 12 p 21 . Also, if the transition matrix P is symmetric, then p ij = p ji for all i, j, so Kolmogorov's criterion is satisfied and the chain is reversible. Usually loop checking involves much computational work. In words, Kolmogorov's loop criterion says that a Markov transition matrix is reversible iff for every loop of distinct states, the forward loop probability product equals the backward loop probability product. One difficulty with Kolmogorov's method is that the number of loops that need to be checked grows very quickly with n where n is the number of states. We first present a result about the number of equations that must be checked in order to apply Kolmogorov's criterion. Kelly [4] notes (Exercise 1.5.2) that if there is a state which can be accessed from every other state in exactly 1 step (i.e. a column of the transition matrix with no zero entries), then it is sufficient to check loops of only three states. However, it is possible that no such state exists. Another technique to check for reversibility is presented in Richman and Sharp, 1991([7]). In their paper, they basically suggest premultiplying the probability transition matrix P by a diagonal matrix D formed by ratios of the entries in a particular nonzero row and its corresponding column. One difficulty with their method is that there may not exist such a nonzero row, further their result is stated in terms of tally matrices rather than probability transition matrices. In this paper, we present a new method (that allows zero entries to appear). This new method is convenient and uses only traditional matrix row and column operations. Counting Kolmogorov Loops The work in this section appears in Jiang [3] (2011) Property 2.1. For an n state Markov chain, with n ≥ 3, the number of equations that must be checked for reversibility by Kolmogorov's method is n i=3 n i (i − 1)! 2 .(4) Proof: For a three-state Markov chain, we note that only one equation Reversibility preserving matrix operations We next present a result to transform transition matrices in such a way as to preserve their reversibility status (either reversible or non-reversible). These transformations will be useful in creating new reversible Markov chains from existing ones, and for checking reversibility of Markov chains. We introduce a row multiplication operation on row i of a Markov transition matrix as the multiplication of row i by a positive constant that leaves the sum of the non diagonal elements at most 1, followed by an adjustment to p ii to make the row sum exactly to 1. We introduce a column multiplication operation on column j of a Markov transition matrix as the multiplication of column j by a positive constant of allowable size (so no row sums exceed 1) followed by adjustments to all diagonal entires to make every row sum exactly 1. Theorem 3.1. A Markov chain matrix P maintains its reversibility status after a row multiplication operation or a column multiplication operation. Proof. Let the ith row of P correspond to state i. The Kolmogorov criterion states that P is reversible iff for all loops of distinct states, the forward loop probability product equals the backward loop probability product. If a loop does include state i, then a multiplication row operation on row i has no effect on the forward and backward loop products. Otherwise, note that state i appears in the first subscript of a forward loop probability iff it appears in the first subscript of a backward loop probability. So the row operation will have an identical effect on both sides of the loop product. A similar conclusion holds for column product operations. Note: If we let P * be the matrix resulting from a row (or column) multiplication operation on P , then the limiting probabilities for P * in the above theorem are generally different than the limiting probabilities for P . Proof. This follows by noting that the detailed balance equations fail for the nonsymmetric zero states, since the limiting probabilities all all entries of an ergodic chain are nonzero. ALGORITHM: If the matrix P is an n × n probability transition matrix, then a sequence of at most n − 1 row or column multiplication operations will be sufficient to determine whether or not P is reversible or not. (1) Pick two nonzero symmetric positions in P, say p i1,i2 and p i2,i1 . Let S 1 = {i1, i2}. If p i1,i2 = p i2,i1 , move to the next step. Otherwise, assume p i1,i2 < p i2,i1 . Multiply row i2 by p i1,i2 /p i2,i1 and adjust p i2,i2 to make the row sum to 1. If p i1,i2 > p i2,i1 , multiply column i2 by p i2,i1 /p i1,i2 and adjust all diagonal entries so that the rows sum to 1. The new matrix P * will now have p * i1,i2 = p * i2,i1 . (2) Choose another state i3 which has nonzero transition probabilities with a state in S 1 . Make the appropriate row or column multiplication operation on row or column i3. Set S 2 = {i1, i2, i3}. (3) Repeat step 2 with a new state until there are no states left to add. After n − 1 steps we have S n = {i 1 , . . . , i n }. Let P * be the final matrix. Theorem 3.3. Let P be an n × n transition matrix to which the Algorithm is applied, resulting in P * . Then P is reversible iff P * is symmetric. Proof. If P * is symmetric, then by earlier comments, P * is reversible. By Theorem 3.1, P is reversible. Next assume that P is reversible. Then by Theorem 3.1, P * is reversible. Let τ = (τ 1 , . . . , τ n ) be the stationary vector for P * . Note that P * was formed so that p * ij = p * ji for particular subcollection of (i, j) which will include each of the states 1, ..., n somewhere among the (i, j) pairs So let (i, j) be part of the subcollection. Since P * is reversible, we must have detailed balance so τ i p * ij = τ j p * ji , for that particular i, j. But p * ij = p * ji so τ i = τ j for the particular pair (i, j). But all the states 1, 2, . . . , n appear somewhere in the subcollection so we conclude τ 1 = τ 2 = · · · = τ n . Now we take an arbitrary pair (i, j). Since P * is reversible, we have detailed balance for all i, j. Hence τ i p * ij = τ j p * ji for all i, j and since τ i = τ j , it follows that p * ij = p * ji for all i, j so P * is symmetric. Notes (1) We observe that if we conclude that a Markov transition matrix is reversible (using the Algorithm), then detailed balance will greatly simplify the computation of stationary vector π. (2) The Algorithm choose the smaller of two matrix entries in order modify the matrix P . Also corrections were made to the diagonal elements to ensure that the rows sum to 1. In fact, this is not really necessary and the matrix p * no longer needs to be a transition matrix. The important issue is whether P * is symmetric or not. (3) We may be able to conclude that P is not a reversible matrix by looking for symmetry only in the upper left corner of P * while it is being formed. That can save considerable computation. (4) Although our results are stated for transition matrices for a finite state space, the same Algorithm could be used to check infinite state spaces if there were a patterned matrix (as in quasi birth and death processes). To check for reversibility, we transform P by column or row operations. Note that the zeros of P are symmetric (i.e. 0 = p 12 = p 21 and 0 = p 34 = p 43 ). We also note that p 14 = p 41 so that symmetry for states S = {1, 4} already exists. We next try make the (1,3) entry match the (3,1) entry. We do not want to lose our (1,4) and (4,1) symmetry, so we could either multiply column 3 by .300/.075 or we could multiply row 3 by .075/.300. We make the latter choice. We get a new matrix with row 3 equal to (.075, .0625, .1125, .000). This new matrix is not a transition matrix because row 3 no longer sums to 1. We simply change the diagonal entry (3,3) to 1−.075−.0625 = .8625. This yields the matrix Then P is reversible iff P (1) is reversible. We now have S = {1, 4, 3}. We next need to include state 2 in our computations. We need to preserve our values in entries (1,4) and (4,1), (1,3) and (3,1), so we can only change row 2 or column 2. We choose to multiply column 2 by 4 to make entries (4,2) and (2,4) equal. Our result is Now P is reversible iff P (3) is reversible. But P (3) is symmetric so we know that it is automatically reversible. Thus, by Theorem 3.3, P is reversible. Since P is reversible, using detailed balance, we have π 1 p 14 = π 4 p 41 and π 1 p 13 = π 3 p 31 and π 3 p 32 = π 2 p 23 so π 1 (.5) = π 4 (.5) and π 1 (.075) = π 3 (.3) and π 3 (.25) = π 2 (.25). Hence π 2 = π 3 = 3π 1 = 3π 4 . Since the sum of the probabilities is 1, we have π 1 = π 4 = 3/8 and π 2 = π 3 = 1/8. Example P (1) =     .P (2) =     . Conclusion The procedure we present in this paper is based on Kolmogorov's loop criterion and the detailed balance equations. Our procedure to check for reversibility differs from the traditional approach in that we are using a transformed matrix P * obtained the initial transition matrix P . The advantage of the new approach is it only requires at most k − 1 elementary row or column operations to obtain the matrix P * , and hence determine the reversibility status. If we begin with an n × n transition matrix with each entry equal to 1/n (and hence clearly symmetric), we can use the row and column operations to create a large variety of nonsymmetric transition matrices which are reversible. p 12 p 23 p 31 = p 13 p 32 p 21 is needed since no length 2 loops need to be checked and any other length 3 loop with the same states results in the same equation. For n = 4, we must check each loop of 3 states and each loop of 4 states. For three state loops paths, we choose any 3 out of 4 states and there is only one equation for each. For the four-state loops, we fix the starting state. Then there are 3! orders for the other states. However, since the other side of equation is just the reversed path, there are only 3! 2 paths involving four states, with the first state fixed. In total, we need Property 3. 2 . 2If the zeros in an ergodic Markov transition matrix are not symmetric, then the matrix is not reversible. Table 1 : 1Number of equations to be checked for n state system Thus the number of equations that most be checked via Kolmogorov's method grows rapidly with n (the number of states) and makes Kolmorogov's criterion computationally difficult for even moderate values of n. (See Online Encyclopedia of Integer Sequences, oeis A002807) 425 .000 .075 .500 .000 .550 .250 .200 .075 .0625 .8625 .000 .500 .050 .000 .450    425 .000 .075 .500 .000 2.2 .250 .200 .075 .250 .8625 .000 .500 .200 .000 .450Again, the rows do not sum to 1 so we change the diagonal entries of rows 2,3,4 to fix this. The result is    P (3) =     .425 .000 .075 .500 .000 .550 .250 .200 .075 .250 .675 .000 .500 .050 .000 .300     AcknowledgmentsThis research was funded through a grant from NSERC (Natural Sciences and Engineering Research Council of Canada). Reversible Markov Chains and Random Walks on Graphs. D Aldous, J A Fill, Aldous, D. and Fill, J.A. 2002. Reversible Markov Chains and Random Walks on Graphs. http://www.stat.berkeley.edu/users/aldous/RWG/book.html R Durrett, Essentials of Stochastic Processes. Springer2nd ed.Durrett, R. 2012. Essentials of Stochastic Processes (2nd ed.) Springer. Construction of transition matrices of reversible Markov chains. M.Sc. Major Paper. Q Jiang, Department of Mathematics and Statistics, University of WindsorJiang, Q. 2009. Construction of transition matrices of reversible Markov chains. M.Sc. Major Paper, Department of Mathemat- ics and Statistics, University of Windsor. Reversibility and Stochastic Networks. F Kelly, Cambridge University PressCambridgeKelly, F. 1978. Reversibility and Stochastic Networks. Cam- bridge University Press, Cambridge. Some Applications of Quasisymmetry. P Mccullagh, Biometrika. 69McCullagh, P. 1982. Some Applications of Quasisymmetry. Biometrika, 69: 303-308. The algebra of reversible Markov chains. G Pistone, M Rogantin, Annals of the Institute for Statistical Mathematics. 65Pistone, G. and Rogantin, M. 2013. The algebra of reversible Markov chains. Annals of the Institute for Statistical Mathe- matics 65: 269-293 A Method for Determining the Reversibility of a Markov Sequence. D Richman, W E Sharp, Mathematical Geology. 22Richman, D. and Sharp, W.E. 1991. A Method for Determining the Reversibility of a Markov Sequence. Mathematical Geology, 22: 749-761 Quasi-Symmetry and Reversible Markov Sequences. W E Sharp, T Markham, Mathematical Geology. 32Sharp, W.E. and Markham, T. 2000. Quasi-Symmetry and Re- versible Markov Sequences. Mathematical Geology, 32: 561- 579.
[]
[ "Generative Adversarial Networks for Recovering Missing Spectral Information", "Generative Adversarial Networks for Recovering Missing Spectral Information" ]
[ "Dung N Tran [email protected] \nDepartment of Electrical and Computer Engineering\nU.S. Army Research Laboratory\nJohns Hopkins University Baltimore\n2800 Powder Mill Rd Adelphi21218, 20783MD, MD\n", "Trac D Tran [email protected] \nDepartment of Electrical and Computer Engineering\nU.S. Army Research Laboratory\nJohns Hopkins University Baltimore\n2800 Powder Mill Rd Adelphi21218, 20783MD, MD\n", "Lam Nguyen \nDepartment of Electrical and Computer Engineering\nU.S. Army Research Laboratory\nJohns Hopkins University Baltimore\n2800 Powder Mill Rd Adelphi21218, 20783MD, MD\n" ]
[ "Department of Electrical and Computer Engineering\nU.S. Army Research Laboratory\nJohns Hopkins University Baltimore\n2800 Powder Mill Rd Adelphi21218, 20783MD, MD", "Department of Electrical and Computer Engineering\nU.S. Army Research Laboratory\nJohns Hopkins University Baltimore\n2800 Powder Mill Rd Adelphi21218, 20783MD, MD", "Department of Electrical and Computer Engineering\nU.S. Army Research Laboratory\nJohns Hopkins University Baltimore\n2800 Powder Mill Rd Adelphi21218, 20783MD, MD" ]
[]
Ultra-wideband (UWB) radar systems nowadays typical operate in the low-frequency spectrum to achieve penetration capability. However, this spectrum is also shared by many others communication systems, which causes missing information in the frequency bands. To recover this missing spectral information, we propose a generative adversarial network, called SARGAN, that learns the relationship between original and missing band signals by observing these training pairs in a clever way. Initial results shows that this approach is promising in tackling this challenging missing band problem.
10.1109/radar.2018.8378737
[ "https://arxiv.org/pdf/1812.04744v2.pdf" ]
49,191,905
1812.04744
b61645e4fcec36003bdeff3732f904939762a728
Generative Adversarial Networks for Recovering Missing Spectral Information Dung N Tran [email protected] Department of Electrical and Computer Engineering U.S. Army Research Laboratory Johns Hopkins University Baltimore 2800 Powder Mill Rd Adelphi21218, 20783MD, MD Trac D Tran [email protected] Department of Electrical and Computer Engineering U.S. Army Research Laboratory Johns Hopkins University Baltimore 2800 Powder Mill Rd Adelphi21218, 20783MD, MD Lam Nguyen Department of Electrical and Computer Engineering U.S. Army Research Laboratory Johns Hopkins University Baltimore 2800 Powder Mill Rd Adelphi21218, 20783MD, MD Generative Adversarial Networks for Recovering Missing Spectral Information Ultra-wideband (UWB) radar systems nowadays typical operate in the low-frequency spectrum to achieve penetration capability. However, this spectrum is also shared by many others communication systems, which causes missing information in the frequency bands. To recover this missing spectral information, we propose a generative adversarial network, called SARGAN, that learns the relationship between original and missing band signals by observing these training pairs in a clever way. Initial results shows that this approach is promising in tackling this challenging missing band problem. I. INTRODUCTION Over the past few decades, ultra-wideband (UWB) radar systems have been widely employed in various practical applications due to their penetration capability. For example, the U.S. Army has been developing UWB radar systems for detection of difficult targets in various applications such as foliage penetration [2], ground penetration [3], and sensingthrough-the-wall [4]. To achieve penetration capability, these systems must operate in the low-frequency spectrum that spans from under 100 MHz to several GHz. In addition to the lowfrequency requirement for penetration, they must employ widebandwidth signals to achieve the desired resolution. However, the signal occupies a wide spectrum that is also shared by radio, TV, cellular phones, and other systems. The frequency allocation and use problem thus becomes a major challenge and only worsens over time as additional radar and communication systems that need the penetration feature must operate in this low-frequency spectral region. There are two key challenges for any UWB system: 1) the system must operate in the presence of other systems and 2) the system must avoid transmitting energy in certain frequency bands that are specified by frequency management agencies. As a result, the receive data have a spectral content that includes multiple bands that are either corrupted (due to the presence of interference sources) or nonexistent (because of no transmission in the prohibited frequency bands). In this paper, we tackle the latter problem in which a large portion of the spectrum is notched due to the frequency allocation issue. Conventional techniques usually detect the corrupted frequency bands by searching for spikes in the spectral domain. The fast Fourier transform (FFT) bins that correspond to the contaminated frequency bands are zeroed out. This technique results in severe sidelobes in the time or spatial domains of the output data and imagery due to the sharp transitions (frequency samples with no information) in the frequency domain. To overcome these limitations, Do et. al. [5] proposed a technique to recover missing spectral information using sparse representation. It is based on the assumption that the full spectrum data and corrupted versions are similarly sparsely represented by a full spectrum dictionary and a missing band dictionary, respectively. Its limitation is a lack in the ability to distinguish near-by targets at fine resolution. Furthermore, the missing frequency bands are required a priori. Recently, a class of generative model in neural network literature, namely, Generative Adversarial Network (GAN) [6], has produced remarkable results in various applications in computer vision, speech processing, and other fields. A standard GAN takes a random noise vector as an input and generates samples that resemble real data. There are also many works that feed GAN with conditions, such that the generated image samples are not only realistic but also match the constraints imposed by the conditions. Some works conditioned GAN on discrete class labels [7], [8], while many other works synthesized images by conditioning GAN on images for the tasks such as domain transfer [9], [10], image super-resolution [11], [12], image synthesis from surface normal maps [14], and style transfer [13]. In this paper, we propose a GAN framework to recover missing spectral information in multiple frequency bands of UWB synthetic aperture radar (SAR) data that are either corrupted or nonexistent. Specifically, we propose a generator loss function that encourages the network to seek solutions on the SAR image manifold that are consistent with data in the frequency domain. Our proposed method can be seen as a variant of a conditional GAN framework, but conditioned on the spectral domain. The network is trained by observing various spectral missing patterns. The advantage of this technique is twofold. First, all computational complexity is at the training phase. The testing phase only consists of some simple matrix multiplication. Second, to recover a SAR image from its frequency corrupted version, the trained network requires zero information of the missing band locations. This is an advantage of our proposed method over traditional spectral recovery techniques in which missing frequencies are required a priori. To our knowledge, this is the first GAN-based framework for recovering missing spectral information in UWB radar systems. II. METHOD We aim to reconstruct a SAR image X from its missing band version Z. In our framework, we adopt a GAN structure. We train the network by minimizing a standard discriminator loss and a generator loss specifically designed for this missing spectral problem. The training data include a set of image pairs, each consisting of an uncorrupted image and its frequencycorrupted counterpart. Each corrupted image is obtained by notching out certain frequency bands of the original image. Original images are not available in the testing phase. Our goal is to train a generator G θ G , parameterized by θ G that reconstructs a SAR image from its frequency-corrupted version. Given a set of training data {(X j , Z j )} n j=1 , we train the generator by solving minimize θ G n j=1 L(G θ G (Z j ), X j ).(1) Then a SAR image can be recovered from its missing band counterpart Z asẐ = Gθ G (Z).(2) We describe our generator loss in detail in Section II-B. It conditions on the frequency domain of the generated sample and forces the generator to favor solutions on the SAR image manifold. A. Generative Adversarial Networks GANs are neural networks for training generative models in an adversarial manner. A GAN consists of two networks, a generator G and a discriminator D. The generative network G learns a mapping from a low-dimensional representation space to a high-dimensional space. The purpose of G is to generate samples that resemble the training data. The discriminator D maps an input to a likelihood. Its role is to distinguish between the sample generated by G and the sample from the data distribution. Directly applying standard GANs to the missing spectral recovery problem fails to reconstruct original images, as they produce samples that are inconsistent with the input data in the frequency domain. We therefore formulate our generator loss to favor solutions that contain available frequencies in the corrupted images. This guarantees consistency between the generated sample and the original image. Moreover, the input in our generator is a corrupted image instead of a lowdimensional encoding as in traditional GANs. This allows our network to learn a mapping from a corrupted input to a desired solution. B. Generator Loss We encourage the generator to seek for solutions on the SAR image manifold that are consistent with the input. To do so, we formulate the generator loss as a weighted sum of a content loss component and an adversarial loss component. A SAR image and its missing band counterpart is related by: Fig. 1. SARGAN architecture. The generator produces an estimate of a full spectrum image from its corrupted version to full the discriminator. The discriminator tries to distinguish this estimate with the original image. In a successfully trained SARGAN, the generator produces estimates that are close to the full spectrum image, thus successfully fool the generator. F Z = M • F X.(3) Here, F is the Fourier matrix, and M is a binary masking matrix defined as: M i,j = 1 if (i, j) is in an available band, 0 if (i, j) is in an notched band.(4) In other words, the masking matrix notches out missing frequency bands and preserves the available frequencies in the original image. As missing local information in the frequency domain results in a global deviation in the time domain, imposing data consistency in the time domain fails to recover notched spectral information. We therefore define a content loss that requires generated samples to preserve available frequencies in the input images: content (G θ G (Z), X) = M • F G θ G (Z) − M • F X 1 , (5) where the L 1 loss is defined as A 1 = i,j |A i,j |, for a given matrix A. Note that the L 1 loss can be replaced by other losses such as L 2 . In our experiments, we find that the L 1 loss results in a faster convergent rate and more robust reconstruction than the L 2 loss. To further improve the reconstruction quality, we impose an adversarial loss to the generator. This encourages the generator to fool the discriminator by seeking solutions on the SAR image manifold: adversarial (G θ G (Z)) = − log D θ D (G θ G (Z))(6) The generator loss is defined as a weighted sum of these two losses: L(G θ G (Z), X) = content (G θ G (Z), X)+λ adversarial (G θ G (Z)),(7) where λ > 0 is a positive constant controlling the tradeoff between the two terms. C. Discriminator Loss We adopt a standard discriminator network D θ D which we train to solve the following optimization problem: max θ D E X∼pdata(X) [log D θ D (X)] + E Z∼p G (Z) [1 − D θ D (G θ G (Z))] . This allows one to train a generator to produce realistic SAR images from corrupted inputs to fool a discriminator, which is trained to differentiate reconstructed SAR images from original ones. Our generator is thus encouraged to favor solutions on the SAR image manifold. III. RESULTS In this section, we demonstrate SARGAN for the spectral recovery problem using SAR data from the U.S. Army Research Laboratory (ARL) UWB SAR system. This SAR database consists of targets (metal and plastic mines, 155-mm unexploded ordinance [UXO], etc.) and clutter objects (a soda can, rocks, etc.) buried under rough ground surfaces. The electromagnetic (EM) radar data are simulated based on the full-wave computational EM method known as finite-difference, time-domain (FDTD) software [26], which was developed by ARL. The software was validated for a wide variety of radar signature calculation scenarios [27], [28]. Our volumetric rough ground surface grid with the embedded buried targets was generated by using the surface root-meansquare (rms) height and the correlation length parameters. The targets are flush buried at a 2-3 cm depth. Fig. 2 (left) shows original SAR raw data (using VV polarization) of some targets that are buried under a perfectly smooth ground surface. Each target is imaged at a random viewing aspect angle and an integration angle of 60 • . In our experiment, the SAR radar is configured in sidelooking mode. It travels in the horizontal direction, transmits impulses to the imaging area, and receives backscattered radar signals from the targets. In this scene, there might be many point targets that have different amplitudes and are located randomly throughout the scene. For demonstrating purposes, we use the raw data in a case where there is a random point target on the scene. The left image in Fig. 2 shows the full spectrum raw data for this simulation scenario. The data bandwidth is from 380 MHz to 2.08 GHz, which contains 90% of the signal energy. It serves as the baseline image for performance comparison purposes. Next, we consider the spectral notches case due to the frequency allocation restriction. In our experiments, we randomly zero out frequency sub-bands of the spectrum, each equivalent to 10 times the frequency resolution which is equal to 9.15 MHz. These random frequency bands can be overlapped and sum up to 90% of the data spectrum. Fig. 3 demonstrates the aforementioned randomly notching procedure in the frequency domain of the data. The middle image in Fig. 2 shows the raw data with 90% of the spectrum being notched. Fig. 4 presents the downrange profiles of the data. The large amount of missing frequencies results in severe sidelobes in the data. Recovering the original data is therefore challenging in this situation. We use SARGAN to recover missing spectrum information under this setup. We use a four-layer fully connected neural network for the generator, and a three-layer fully connected neural network for the discriminator. The first and last layers of the generator have the same dimensions as the input data. The two hidden layers are of length 128. The dimension of the first layer of the discriminator is equal to that of the input data. Its hidden layer has 128 nodes and its output has one node, which guesses whether the input is a true image or one produced by the generator. In our experiments, we use a stable alternative to GANs, called Wasserstein GAN (WGAN). The training data are obtained as follows. From a full spectrum raw data, we produce several randomly spectrally notched version of that data. Each such notched data matrix together with the full spectrum data constitute a training pair. We then train our network using these training pairs. In the testing phase, the full spectrum raw data are unavailable. Our goal is to recover it from a corrupted version that is not included in the training data. The locations of the notched band are unknown to the network. This is significantly different than other traditional spectral recovery techniques in which missing frequencies are required a priori. test reconstructed data using SARGAN after 100 training epoches are shown in Fig. 4. The recovered data closely follow the original data whereas the corrupted version show severe sidelobes. Fig. 6 visualizes the generator loss during the training phase. It can be seen that the network converges after around 50 epochs, which matches the above down-range profile visualization. To further qualitatively evaluate SARGAN, we compute the Signal-to-Noise Ratio (SNR) in dB scale between the original data X and the data recovered using SARGANẐ using the formula: SNR(X,Ẑ) = 20 log 10 RMS(X) RMS(Ẑ − X) ,(8) where RMS(X) = 1 √ # of elements in X X 2 .(9) We also compare that to the SNR between the original data and the corrupted data. Table III shows the SNR in these two cases, and the performance gain in dB scale obtained by SARGAN. It can be seen that our proposed method reduced the sidelobe level by more than 15 dB. This matches the normalized downrange profiles shown in Fig. 4. Remarkable, SARGAN obtains this performance gain without any information on the missing band locations. Popular methods such as FFT and sparse recovery fail in this case. IV. CONCLUSION We proposed a Generative Adversarial Network framework, called SARGAN, to tackle the missing spectral information recovery problem. A well-trained SARGAN is expected to produce a good estimate of the full spectrum SAR data from its spectrally notched counterpart without any spectral information. In the training phase, the network is encouraged to learn the relationship between a set of full and corrupted spectrum data pairs. This relationship is captured in our proposed generator loss function, which forces SARGAN to favor solutions on the SAR data manifold which are consistent with the input data in the frequency domain. Using the real UWB SAR database from database, we show that the proposed framework can successfully recover the information from the missing frequency bands. Remarkably, it obtains more than 15 dB gain without knowing the missing frequency locations. To our knowledge, it is the first method obtaining such performance gain in this situation. Fig. 2 . 2Raw data in time domain of target versus aspect angle. Fig. 3 . 3Spectrum of the raw data with 90% missing in the bandwidth. Fig. 4 . 4Normalized down-range profiles in dB scale of the raw data. The spectrally notched data show severe sidelobes, whereas the data reconstructed using SARGAN follow the ground-truth very well. The test reconstructed result was obtained after 100 training epochs. Fig. 5 5shows the normalized down-range profiles in dB of the recovered data, produced by the generator of SARGAN, after each 10 epochs. It can be seen that after 40 epochs, the generated samples already well approximate the original full spectrum data. The normalized downrange profile of the Fig. 5 . 5Normalized down-range profiles in dB of the reconstructed data during the first 60 epochs. Each figure shows the testing result after each 10 epochs. Fig. 6 . 6Generator loss values during the training phase. The network converges after roughly 60 iterations. TABLE I . IRECOVERY PERFORMANCE OF SARGAN ON SPECTRALLYNOTCHED DATA (SNR)Corrupted Data SNR (dB) Recovery SNR (dB) Recovery Gain (dB) 8.15 23.99 15.84 . H Kopka, P W Daly, Guide To L A T E X, Addison-WesleyHarlow, England3rd ed.H. Kopka and P. W. Daly, A Guide to L A T E X, 3rd ed. Harlow, England: Addison-Wesley, 1999. Detection algorithms for ultrawideband foliage-penetration radar. L H Nguyen, R Kapoor, J Sichina, Proceedings of SPIE. SPIE3066Nguyen, L. H., Kapoor, R., Sichina, J., ?Detection algorithms for ultrawideband foliage-penetration radar,? Proceedings of SPIE Vol. 3066, pp. 165-176 (1997). Mine field detection algorithm utilizing data from an ultrawideband wide-area surveillance radar. L Nguyen, K Kappra, D Wong, R Kapoor, J Sichina, Proc. SPIE Int. Soc. SPIE Int. Soc3392627L. Nguyen, K. Kappra, D. Wong, R. Kapoor, and J. Sichina, ?Mine field detection algorithm utilizing data from an ultrawideband wide-area surveillance radar,? Proc. SPIE Int. Soc. Opt. Eng. 3392, 627 (1998). Sensing through the wall imaging using the Army Research Lab ultra-wideband synchronous impulse reconstruction (UWB SIRE) radar. L Nguyen, M Ressler, J Sichina, Proceedings of SPIE. 694769470Nguyen, L., Ressler, M., Sichina, J., "Sensing through the wall imaging using the Army Research Lab ultra-wideband synchronous impulse reconstruction (UWB SIRE) radar," Proceedings of SPIE Vol. 6947, 69470B (2008). Recovery of missing spectral information in ultra-wideband synthetic aperture radar (SAR) data " Radar Conference (RADAR) 2012 IEEE. L H Nguyen, T Do, L. H. Nguyen T. Do "Recovery of missing spectral information in ultra-wideband synthetic aperture radar (SAR) data " Radar Conference (RADAR) 2012 IEEE pp. 0253 0256 May. 2012. Generative adversarial nets. I Goodfellow, J Pouget-Abadie, M Mirza, B Xu, D Warde-Farley, S Ozair, A Courville, Y Bengio, Advances in Neural Information Processing Systems (NIPS). I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems (NIPS), pages 2672?2680, 2014. InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversar-ial Nets. X Chen, Y Duan, R Houthooft, J Schulman, I Sutskever, P Abbeel, NIPS. X. Chen, Y. Duan, R. Houthooft, J. Schulman, I. Sutskever, and P. Abbeel. InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversar-ial Nets. In NIPS, 2016. M Mirza, S Osindero, arXiv:1411.1784Conditional Generative Adver-sarial Nets. arXiv preprintM. Mirza and S. Osindero. Conditional Generative Adver-sarial Nets. arXiv preprint arXiv:1411.1784, 2014. Pixel-Level Domain Transfer. D Yoo, N Kim, S Park, A S Paek, . S Kweon, ECCV. D.Yoo,N.Kim,S.Park,A.S.Paek,andI.S.Kweon.Pixel-Level Domain Transfer. In ECCV, 2016. Image-to-Image Translation with Conditional Adversarial Networks. P Isola, J.-Y Zhu, T Zhou, A A Efros, CVPR. P. Isola, J.-Y. Zhu, T. Zhou, and A. A. Efros. Image-to-Image Translation with Conditional Adversarial Networks. In CVPR, 2017. Perceptual Losses for Real-Time Style Transfer and Super-Resolution. J Johnson, A Alahi, L Fei-Fei, ECCV. J. Johnson, A. Alahi, and L. Fei-Fei. Perceptual Losses for Real-Time Style Transfer and Super-Resolution. In ECCV, 2016. Photo-Realistic Single Image Super-Resolution Us-ing a Generative Adversarial Network. C Ledig, L Theis, F Huszar, J Caballero, A Cunning-Ham, A Acosta, A Aitken, A Tejani, J Totz, Z Wang, W Shi, CVPR. C. Ledig, L. Theis, F. Huszar, J. Caballero, A. Cunning-ham, A. Acosta, A. Aitken, A. Tejani, J. Totz, Z. Wang, and W. Shi. Photo-Realistic Single Image Super-Resolution Us-ing a Generative Adversarial Network. In CVPR, 2017. Precomputed Real-Time Texture Syn-thesis with Markovian Generative Adversarial Networks. C Li, M Wand, ECCV. C. Li and M. Wand. Precomputed Real-Time Texture Syn-thesis with Markovian Generative Adversarial Networks. In ECCV, 2016. Generative Image Modeling Using Style and Structure Adversarial Networks. X Wang, A Gupta, ECCV. X. Wang and A. Gupta. Generative Image Modeling Using Style and Structure Adversarial Networks. In ECCV, 2016.
[]
[ "Experimental Study on the Aerodynamic Sealing of Air Curtains", "Experimental Study on the Aerodynamic Sealing of Air Curtains" ]
[ "João Carlos Viegas \nNational Laboratory of Civil Engineering\nAv. do Brasil 1011700-066LisbonPortugal\n", "Fernando Oliveira [email protected]. \nFaculty of Sciences and Technology\nUniversidade Nova de Lisboa\n2829-516CaparicaPortugal\n", "Daniel Aelenei [email protected].*correspondence:[email protected] \nFaculty of Sciences and Technology\nUniversidade Nova de Lisboa\n2829-516CaparicaPortugal\n" ]
[ "National Laboratory of Civil Engineering\nAv. do Brasil 1011700-066LisbonPortugal", "Faculty of Sciences and Technology\nUniversidade Nova de Lisboa\n2829-516CaparicaPortugal", "Faculty of Sciences and Technology\nUniversidade Nova de Lisboa\n2829-516CaparicaPortugal" ]
[]
Controlling the air quality is of the utmost importance in today's buildings. Vertical air curtains are often used to separate two different climatic zones with a view to reduce heat transfer. In fact, this research work proposes an air curtain aimed to ensure a proper separation between two zones, a clean one and a contaminated one. The methodology of this research includes: (i) small-scale tests on water models to ensure that the contamination does not pass through the air curtain, and (ii) an analytical development integrating the main physical characteristics of plane jets. In the solution developed, the airflow is extracted from the contaminated compartment to reduce the curtain airflow rejected to the exterior of the compartment. In this research work, it was possible to determine the minimum exhaust flow necessary to ensure the aerodynamic sealing of the air curtain. This article addresses the methodology used to perform the small-scale water tests and the corresponding results.
10.3390/fluids3030049
[ "https://arxiv.org/pdf/2002.11155v1.pdf" ]
125,491,375
2002.11155
b94ecf28583d367f65eaf47b3c5a6ee14969b5b1
Experimental Study on the Aerodynamic Sealing of Air Curtains 2018 João Carlos Viegas National Laboratory of Civil Engineering Av. do Brasil 1011700-066LisbonPortugal Fernando Oliveira [email protected]. Faculty of Sciences and Technology Universidade Nova de Lisboa 2829-516CaparicaPortugal Daniel Aelenei [email protected].*correspondence:[email protected] Faculty of Sciences and Technology Universidade Nova de Lisboa 2829-516CaparicaPortugal Experimental Study on the Aerodynamic Sealing of Air Curtains 349201810.3390/fluids3030049Received: 9 June 2018; Accepted: 12 July 2018; Published: dateArticleair curtainaerodynamic sealingindoor air qualitycontamination Controlling the air quality is of the utmost importance in today's buildings. Vertical air curtains are often used to separate two different climatic zones with a view to reduce heat transfer. In fact, this research work proposes an air curtain aimed to ensure a proper separation between two zones, a clean one and a contaminated one. The methodology of this research includes: (i) small-scale tests on water models to ensure that the contamination does not pass through the air curtain, and (ii) an analytical development integrating the main physical characteristics of plane jets. In the solution developed, the airflow is extracted from the contaminated compartment to reduce the curtain airflow rejected to the exterior of the compartment. In this research work, it was possible to determine the minimum exhaust flow necessary to ensure the aerodynamic sealing of the air curtain. This article addresses the methodology used to perform the small-scale water tests and the corresponding results. Introduction Air curtains consist of plane jets that are frequently used to separate different environmental zones. Air curtains are basically designed to reduce or control both heat and mass transfers, as well as to reduce the spreading of airborne contaminants between two zones. Their application is particularly useful when the physical barriers are not viable, for several reasons. In this framework, air curtains have been employed for several purposes, such as: HVAC (Heating, Ventilation and Air Conditioning) [1][2][3], smoke control in passageways [4,5], airborne pollutant, and biological control [6][7][8]. The plane jet induces a shear stress between the airflow and the stagnant ambient air, which promotes air entrainment. In the jets with significant flow rates, the shear stress induces the development of turbulent structures with several length scales. The turbulent structures promote air mass entrainment and mixing from both sides into the jet ('contaminated' and 'non-contaminated' compartments). Furthermore, the rejection of the jet flow to the clean compartment corresponds to the loss of air curtain tightness (loss of pollutant containment). The usual application of this technology to protect the non-contaminated compartment from airborne contaminants (microorganisms, bacteria, fungi, and particles), in an almost isothermal condition, uses low jet velocities for obtaining an approximately laminar flow. The extraction at the contaminated side or at the tip of the jet is necessary in order to avoid the dispersion of the jet flow into the non-contaminated compartment (Figure 1a). Reversely, this objective can also be attained if clean air is supplied to the non-contaminated compartment (Figure 1b). Treating the 'clean' air before supplying it to the non-contaminated compartment, or treating the 'dirty' air removed from the contaminated compartment before releasing it into the environment are expensive procedures. Therefore, it becomes essential to find a technical solution for creating a proper zone separation that will be able to minimize flow rate requirements. In the context of the Nanoguard2ar project, the basic requirement is to reduce, as far as possible, the exhaust flow rate from the contaminated compartment (or the supply flow rate to the non-contaminated compartment), in order to reduce the air treatment costs. Although, for an undisturbed environment (e.g., isothermal flow), the best solution is just to provide a very small uniform horizontal air stream through the opening, in practice, there are disturbances (e.g., temperature differences, wind, people crossing the opening) that require the use of an air curtain to improve the aerodynamic sealing of the opening. For this purpose, the most effective air curtain is the one that is able to reduce the airborne contaminants, minimizing the exhaust (or supply) flow rate, when exposed to disturbances. Hayes and Stoecker [9][10][11] studied the aerodynamic sealing performance of vertical air curtains under isothermal and non-isothermal conditions. Hayes [8] developed the deflection modulus (Dm), which is indicative of the deflection of the air curtain jet and expresses the ratio of the outlet momentum to the transverse forces on the jet [12]. They defined [10] the operation condition where the air curtain jet reaches the floor as the 'optimum condition' and other operation conditions where the air curtain jet does not reach the floor as 'break-through condition', which can be further divided into the 'inflow breakthrough condition', where the air curtain flow is curved inward and does not reach the floor, and the 'outflow breakthrough condition', where the air curtain flow is curved outwards and does not reach the floor [13]. In our research, the ideal condition corresponds to the limit between the optimum condition and the inflow breakthrough condition, when having Figure 1 as reference (Figure 1a), because in this condition, the contaminant mixed up in the jet flow is not released to the outside of the compartment. In the air curtain applications at door height openings, the flow is characterized, in practice, by a transition regimen. The overall jet flow rate is reduced if the jet velocity is decreased; and hence, the use of lower jet velocities is also a strategy that can be adopted to reduce the jet flow rate rejected to the 'non-contaminated' compartment. Nevertheless, the vertical downward air curtains must be properly designed so as to avoid splitting the jet flow at the floor impingement zone, and consequently losing the air curtain aerodynamic sealing. However, with low air curtain speeds, when people walk through the door, the air curtain reconstitution time is longer, which could increment pollutant leakage through the air curtain. Using air curtains in a transition regimen (rather than in a laminar regimen) can minimize this problem. (a) (b) This study is developed within the framework of the Nanoguard2ar project (European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement N 690968). Its main goals are to develop, design, test, validate, and demonstrate an innovative nanomaterial-based 'microbial free' engineering solution, to ensure indoor air quality in buildings. To achieve this, an advanced oxidation process will be employed to clean the air extracted from the contaminated compartment (or to clean the external air before supplying it to the non-contaminated compartment), in conjunction with the use of air curtains, ensuring a proper separation between the spaces that are to be kept without cross contamination. Therefore, the purpose of this study is to define the plane jet characteristics that enable the best aerodynamic sealing of the air curtain to be achieved, by ensuring low exhaust flow rates from the contaminated compartment (or supply flow rates to the non-contaminated compartment), in accordance with the air treatment procedures adopted. This research encompasses three phases, namely: (i) small-scale experiments using water as the working fluid; (ii) computational fluid dynamics (CFD) simulations in order to verify if the small-scale test results are applicable to full size air curtains; and (iii) full size air curtain experiments. This paper describes the research carried out in the first phase. There are several applications of this concept that aim to avoid contaminant spreading, such as: an operating room [7,[14][15][16][17][18][19], a tobacco smoke control [20,21], the protection of art works in museums and of cultural heritage [22,23], and open refrigerated display cabinets [24,25]. Several studies about air curtain efficiency have been conducted on these applications. However, there is no systematic approach to the contaminant aerodynamic sealing in the transition between the optimum condition and the 'inflow breakthrough condition'. The studies on the application of air curtains have been based on experiments [13,[26][27][28][29], on CFD models [23,29,30], and on semi-analytical models [12,31]. Rydock et al. [27] presented an experimental study about the efficiency of air curtains in a dedicated-smoking restaurant area. The air curtain was installed at a 0.8 m height and the slot widths corresponded to 10 mm, 5 mm, and 3 mm, respectively, with varying mounting angles (+15, 0, and −15), and varying supply and extraction airflow rates, as well as several non-smoking sections with different air supply configurations. The maximum air curtain efficiency was achieved using the 100 m³/h per meter of air curtain and for a slot width of 5 mm. The results seem to indicate the feasibility of the use of air curtains to achieve improved air quality. However, a smoke free environment cannot be completely attained in a section of a single room. In this study, the air curtain flow was fed by the contaminant air from the smoking zone. The authors of this work concluded that to achieve an aerodynamic sealing for the tobacco smoke, the air curtain flow must be fed by clean air. Shih et al. [17] presented a numerical study on ethanol spreading in clean rooms. For the purpose of the study, the authors analysed the influence of air curtain parameters, such as jet velocity, angle, and installation height. The minimum velocity studied was 3 m/s. The maximum efficiency was achieved for a jet speed of 5 m/s and an angle of 15°. In this study, it was observed that the loss of air curtain sealing occurred essentially close to the bottom, where a flow separation takes place. Santoli, Cumo, and Mariotti [23] presented a numerical and experimental study on a cultural heritage building, using an air jet with speeds between 4 and 5 m/s, to create a physical barrier to the flow. They concluded that the ventilation system has efficiencies of 70-75%. However, the parameters that allow for maximizing the efficiency of the air curtain were not analysed in this study. In operating rooms, the critical operating area is often isolated from the external contaminants by the installation of laminar diffusers, in conjunction with the use of air curtains [16]. Cook and Int-Hout [18] pointed out that another way to guarantee the asepsis level in the operating room is to ensure a hierarchical pressure in this space, between 10 to 15% of the volume of the room. According to the authors, the difficulty in guaranteeing a high asepsis level is due to the difficulty in achieving a laminar flow in the panels. This difficulty is associated with the undesirable speeds attained by the air entrainment and with the pressure gradient resulting from temperature differences, which eventually increase the turbulence intensity in the room. However, these studies are not conclusive with regards to the speed that should be used to achieve a certain level of asepsis. They only refer to the fact that low speeds are more efficient [16]. Zhai and Osborne [14] presented a numerical and experimental study, in which they concluded that there is no correlation between the laminar flow diffuser, the air curtain flow rate, and the concentration of contaminant. It was observed that it was preferable to use one-way diffusers rather than air curtains with high jet velocities. However, this did not demonstrate the existing relation between the inflow and outflow rates. Goubran et al. [13] present an experimental study to verify and further investigate the flow characteristics of the building entrances equipped with air curtains. In this study, the working fluid is the air and the nozzle thickness was 0.0635 m. Two different supply velocities were selected, namely: (1) the maximum supply speed of the unit is 13.75 m/s; and (2) a lower speed 9.1 m/s. A small-scale chamber (1/3 scale of the real case) of 2.44 m × 2.44 m × 1.3 m (L × W × H) was used for the measurements of the infiltration/exfiltration and pressure differentials, which were then used for developing the empirical model across the operating air curtain. The flow and pressure measurements confirmed that, for the tested pressure difference range, air curtains can significantly reduce the infiltration. Additionally, the experimental results indicated that the higher the air curtain supply speed, the better the air curtain performs (i.e., it is able to continue to operate in the optimum condition at higher pressure differences). Following this work, Qi et al. [29] presented a parametric study of air curtain performance based on reduced-scale experiments and full-scale numerical simulations. Using the same experimental setup of previous research [13], they found that increasing the air curtain supply angle improves the air curtain performance when it is operated under the optimum condition and inflow breakthrough condition, but creates excessive exfiltration under the 'outflow breakthrough condition'. Increasing the supply speed of the air curtain generally improves the air curtain performance, whereas this improvement deteriorates with the increase of the supply angle under the outflow breakthrough condition. These works clearly stress the relevance of the pressure difference between inside and outside the compartment, which is due to several effects, such as the buoyancy effect, the head loss of the flow through the opening, and many other factors including the wind effect and indoor systems operation; however, in the absence of the buoyancy effect, this pressure difference through the opening protected by the air curtain is related with the head loss of the flow through the opening protected by the air curtain. As the head loss (that is expressed in these works by a discharge coefficient) is dependent on the characteristics of the opening [32], in our research, we considered that the average velocity of the flow across the door (u ) is the variable of reference. In the mentioned works, no clear correlation was established between the inlet flow rate (flow entering the door opening) and the velocity at the jet nozzle, which assures both the maximum air curtain effectiveness, and the level of asepsis achieved under isothermal conditions or low temperature differences. As there is no concern in reducing the exhaust flow from the contaminated compartment (or the supply flow to the non-contaminated compartment), in order to increase the effectiveness of the curtain and to reduce the air cleaning costs, both the air velocities and the corresponding Reynolds numbers in these works are quite high when compared with those reported in this research work. Therefore, to the best knowledge of the authors of this work, this study constitutes an added value to research in the field. For the purposes of this work, it was considered as relevant to evaluate the balance between the average speed at the door opening and the jet characteristics, with a view to increase the air curtain effectiveness and thus enable its integration into the Nanoguard2ar system. Minimal requirements of average speed at the door are obtained for very low jet velocity. Therefore, very low jet speeds will also be used up to a maximum of 1.70 m/s. In previous research [4], the extreme condition of the curtain exposed to a high buoyancy action was studied; in our research presented here, the opposed extreme condition of non-buoyant flow is studied. A simplified analytical model is presented in the next sections, with a view to discuss the variables that can influence the performance of the jet planes under a turbulent transition regimen. Two different conditions are analysed, namely the simplified balance of the jet momentum (which demonstrates a strong influence of the jet angle and is relevant for smaller Reynolds Number values-146, 367 and 687) and the comparison between the flow rate entrained by the jet from the non-contaminated zone, and the flow rate across the opening required to avoid the loss of aerodynamic sealing. In this derivation, the following assumptions are applicable: Methods Analytical Model • the momentum is conserved in the non-disturbed jet; • the speed of the jet was assumed as uniform at the jet cross section; and • the momentum balance is expressed by the product of the flow rate by the average speed (actually, this is not true and is only considered here to analyze the variables influencing the process; empirical correcting values will be afterwards deduced). The momentum balance requires the definition of the quantities of Equations (2) to (5), as follows: M M M (1) M b L u (2) J b L u (3) J M u (4) J M u(5) The average jet velocity at the floor impingement zone was also considered to be equal to the average velocity of the flow rate rejected to the non-contaminated side and to the average velocity of the flow rate rejected to the contaminate side, as presented in Equation (6): u u u (6) The momentum balance of the jet at the floor impingement zone is expressed by Equation (7), as follows: J J J sin α ⇔ M u M u M u sin α(7) From Equation (7), it will be possible to estimate the split of the jet flow rate, in accordance with Equations (8) to (10), as follows: Contaminated side Non-contaminated side M M 1 sin α 1 − sin α (8) M M 1 sin α 2 (9) M M 1 − sin α 2(10) The momentum of the flow rate rejected to the non-contaminated side can be expressed by Equation (11), as follows: J J 1 − sin α 2 ⇔ J b L u 1 − sin α 2(11) It can be assumed that the momentum of the flow through the door J (corresponding to the exhaust mass flow rate of the 'contaminated' compartment, Equation [12]) will be, at least, higher or equal to the momentum of the flow rate rejected to the 'non-contaminated' side. The limit condition is presented in Equation (13). J w h u (12) J J ⇔ w h u b L u 1 − sin α 2(13) Assuming that w = L, the relation of Equation (14) can be deduced from Equation (13), as follows: u u b h 1 − sin α 2 ,(14) This equation will be used as reference in the analysis of the experimental results. It was also assumed as relevant to compare another threshold condition with the flow rate through the door. In addition to the nozzle jet flow rate supplied from the non-contaminated compartment, there is also a part of the jet flow rate that is entrained from the non-contaminated compartment. Consequently, it can be assumed that the flow rate crossing the door will be, at least, equal to the flow rate entrained by the jet from the non-contaminated compartment. The jet flow rate is given by Equation (15) [33]: Q !" 0.44 & 2 x b ( Q(15) Considering the values obtained from Equations (16) and (17), the threshold condition previously referred to is given by Equation (18), as follows: Q b L u (16) x h cos+α ,(17)u -0.22 & 2 h b cos α ( 0.5/ u b h(18) Although Equations (14) and (18) do not present explicitly the pressure difference between the interior of the compartment and the exterior, such correlation can be obtained from these equations if the discharge coefficient corresponding to the opening protected by the plane jet is introduced. Therefore, the influence of the pressure difference is indirectly considered in this equation through u . Although in a previous work [4] the deflection modulus (defined by Hayes [9]) was included in the physical analysis, in this research, it is not necessary to consider this non-dimensional variable, because no buoyancy is acting in these experiments. Experimental Water Modelling In this research, tests were carried out on a small-scale model (1/20) installed in a water tank, with a view to simulate the full-size prototype (air curtain). Water was adopted as the testing fluid in order to maintain, in the model, the same range of Reynolds number as in the prototype. The definition of the Reynolds number currently used for plane jets, Re +u b , ν ⁄ , was considered [33]. Although axisymmetric jets remain laminar up to Re = 1000 and became fully turbulent for Re > 3000, the plane jets from a long and narrow slot may present appreciable turbulent instabilities beyond Re as low as 30 [33]. Therefore, these small-scale tests, with Re ranging from 147 to 2125, will be carried out in the transition regimen. The results may not be easily extrapolated for full size air curtains unless complementary work will be done. .00 mm, respectively. The jet was connected to a pump with a flowmeter that was used to measure the flow in the jet nozzle. In the compartment, on the opposite end of the door, there were two openings, one for the contaminant supply (fed by gravity and consisting of water with coloring), with a diameter of 0.01 m, and another for the exhaust flow (with a pump), with a diameter of 0.05 m. The contaminant level was kept constant so as to ensure that its flow rate also remained constant ( Figure 3). In the photo of Figure 3, the red bottles were used to avoid the floating of the model. The model is upside down because, when used for studying non-isothermal flows, it is less expensive to simulate the hot fluid with saltwater, which is denser [4]. This position does not influence the isothermal results. As previously mentioned, the jet complied with a set of geometric parameters, among which only the jet thickness and the jet angle (four angles used) were changed from test to test. In the tests from 89 to 91, the same 15° angle was invariably used and different nozzle thicknesses were adopted, while tests 55 and 56 were performed with an inactive jet. Table 1 indicates the geometric characteristics of the nozzle and the opening in the model, which is protected by the jet. As this test is representative of the isothermal conditions, before initiating it, care was taken to ensure a constant water supply temperature, which was equal to the temperature of both the tank and the contaminant. Before every test, the jet angle and thickness were adjusted and the jet nozzle flow rate was set to the desired value, and was measured. During the test, the contaminant was released into the compartment and the exhaust flow rate was increased until no contaminant transport through the opening was visible. In the end of every test, the extraction flowrate from the compartment, corresponding to the contaminant sealing, was measured and was used to calculate the average velocity through the opening protected by the plane jet, considering the mass flow balance. No pressure difference measurements were carried out because its effect, the average velocity through the opening, was used in this assessment. The standard uncertainty [34] estimated for the nozzle area ranged from 3% to 13%. The standard uncertainty estimated for the nozzle flow rate measurement was less than 0.5%. The standard uncertainty estimated for the nozzle velocity calculation ranged from 4% to 13%. The standard uncertainty estimated for the exhaust flow rate measurement corresponded to 2.5%. In the test plan adopted, each selected variable was changed successively. Therefore, the tests started with a higher Reynolds number (with the values Re = 2125, Re = 1710, and Re = 1224) and a jet thickness of 1.25 mm, after which the jet velocity was reduced to obtain lower Reynolds number values of Re = 687, Re = 367, and Re = 147, respectively. After these tests, the variation in the jet thickness was studied for the following values: 2.50 mm, 3.75 mm, and 5.00 mm, respectively. In the study of the jet thickness, the Reynolds number Re = 367 was maintained, which is why the jet speed was reduced. Another additional test was carried out with the inactive jet under isothermal conditions and using the same method as the one previously described. During the tests, the contaminant transport through the door was assessed by naked eye observation. Reference must be made of the fact that the human eye is not an optimal instrument, and, as such, it was necessary to establish a criterion to determine whether the exhaust flow was in a minimum limit state, so that no contaminant was leaking through the curtain. Therefore, the following cases were considered (Figures 4-6 present a detail of the test setup where is possible to see just the cross section of the plane jet and of the opening; these pictures were rotated in order so that the jet is presented in a downward position; the inside water is coloured by the turbulent mixture with the jet coloured water): oversized exhaust flow rate ( Figure 4), limit state condition ( Figure 5), and insufficient exhaust flow rate ( Figure 6). In the first case, a triangle of discoloration in the water was observed in the compartment, which showed that the inflow at the door was too high; in the third case, some eddies were observed, which were leaking through the door (meaning that the hydrodynamic sealing of the curtain was lost); in the second case, the previously mentioned structures were not observed, which corresponded to the optimized hydrodynamic sealing conditions. Furthermore, it was necessary to demonstrate that the water colorant intensity was strong enough to evidence any contaminant release through the curtain. Several tests were carried out in the same conditions, but by changing the water colorant concentration. In every test, the measurements were performed on the exhaust flow rates obtained for the optimum hydrodynamic sealing of the curtain. From test to test, the colorant concentration was increased until two different water colorant concentrations led to obtaining the same exhaust flow rates. From these two tests, the lower water colorant concentration was adopted in the subsequent research works. Results and Discussion Test Results The experimental results are presented in Tables 2 and 3. Every result corresponds to the average obtained from the two tests carried out in the same conditions. As the walls of the test compartment are impermeable and the water colorant supply flow rate is known, it is possible to assess the average velocity through the door u , which is also presented in Table 2, measuring the exhaust flow rate V !5 67" and computing the flow rate balance. The average velocity through the door u is put in evidence, because the authors believe that it is the relevant variable that influences the hydrodynamic sealing performance of the curtain. Analysis of the Influence of the Jet Nozzle Thickness on the Average Speed at the Door Opening The test results presented in Table 3 made it possible to study the influence of the jet nozzle thickness on the average velocity of the flow across the opening. Equation (19) was obtained by the linear regression, presented in Figure 7. It can be seen that the variation in the jet nozzle thickness, by keeping the Reynolds number constant (Re = 367), has no effect on the average velocity of the flow across the opening. It denotes that the average velocity of the flow across the opening depends linearly on the jet velocity at the nozzle only. Hence, it is not necessary to consider any influence associated with the jet nozzle thickness. Analysis of the Influence of the Jet Angle The second parameter analyzed was the influence of the jet angle. For the purpose of this study, the ratio between the average velocity at the opening and the nozzle velocity was compared with sin α. The lines connecting the experimental results for the same Reynolds number are almost linear, if +u u ⁄ , . is used as a coordinate. Figure 8 shows that there is no influence of the jet angle for the Reynolds numbers close to Re = 1710 or above this value. Influence of E . +1 − HJK L, on the Average Speed at the Door Opening When coordinates u and u . +1 − sin α, are used to express the test results (Figure 9), it is possible to see that the test results almost collapse into a straight line. It should be noted that it was found that the angle does not influence the average velocity at the opening u for the Reynolds numbers of Re = 1710 and above, and therefore, it was deemed irrelevant to include the term sin α in its expression (Equation (20)). Figure 10 shows the best fit for the test results when the coordinates referred to above are considered. The correlation coefficient obtained corresponded to 0.9883. Figure 9 also includes the test results obtained in the study about the effect of the variation on the jet thickness (see Table 3), designated in the figure key as 15*; it is clear that these results are fully aligned with further test results obtained for the same jet angle. Equation (20), below, was deduced on the basis of the test results. Mention must be made of the fact that for Re M 1710, the influence of sin α was not introduced, as no influence of the angle has been included for that condition (as concluded before). Research was carried out on the best function that can be adjusted to the test results, having these coordinates as the basis. Therefore, the best fit using the least squares method was obtained for Equation (21). Variables A, B, C, f, and j were considered as the degrees of freedom. Table 4 presents the values for the variables corresponding to the best fit. Figure 11 shows, a good adjustment of the experimental values has been achieved. This has led to obtaining a correlation coefficient of 0.9886 (Figure 12), which is quite close to the previous relation. These results are considered in Equation (22). It was observed that some variables were very close to the unit or to 0.5. It was assumed that some deviation in relation to these numbers could be due to experimental errors and uncertainty. Therefore, the variables were rounded to B = 1.00, f = 0.50, and j = 1.00. Variable C was supposed to assume the value C = 0.0026 m/s, which is the velocity measured at the opening when the velocity of the jet is zero (see Table 2). Table 5 presents the values for the variables corresponding to the best fit, with A being the only degree of freedom. Figures 13 and 14 show, the test results are still approaching linearity, with a correlation coefficient of 0.987, when expressed by the appropriate variables. As a result, the most appropriate equations to express the experimental results were obtained (Equations (23) and (24) u A +B − sin α, V u C (21) The graph in Figure 15 depicts the experimental results (u ; u ), which were compared with the corresponding values obtained by Equation (23). It is possible to observe that Equation (23) Analytical Method We stress that the empirical Function (23) does not show the same dependence on jet velocity and on jet angle as the theoretical simplified Function (14). Figure 16 presents Equation (14), together with the experimental results (the horizontal coordinate in this figure is different from the one adopted in the previous figures). It is clear that Equation (14) is close to the experimental results just for the jet angle of 15° and for Re < 1224. For the jet angles lower than 15°, the corresponding average jet nozzle velocity is higher than the predicted by Equation (14). For higher Reynolds numbers, Equation (14) predicts an average velocity of the flow across the door higher than the one measured in the experiments. This shows that some physical processes, which are relevant for the transport of the contaminant, are not expressed in the theoretically derived equation. We consider that such processes are related to the turbulent behaviour of the jet at the floor impingement zone. Moreover, the experiments showed the following: • The velocity at the opening is still necessary to avoid the spread of the contamination, when the jet is not active (u +\ ]^] , ); • The jet angle exerts no influence if Re M 1710; and • No influence of the nozzle thickness was observed in the experiments. In a previous research project, using the same small-scale model, saltwater tests were carried out, with a view to test the use of a plane jet to avoid the dispersion of contaminant buoyancy driven through the plane jet [4]. In these tests, the nozzle velocity ranged from 0.142 m/s to 1.000 m/s (177< Re <3422), the nozzle thickness from 0.00125 to 0.00750 m, and the jet angle from 25° to 35°. Moreover, the parameters of the nozzle velocity, jet thickness, and jet angle were set before the beginning of the test, and the exhaust flow rate was adjusted during the test to achieve the hydrodynamic sealing of the curtain (the same methodology as the one followed in the tests reported in the previous section). In addition, the curtain velocity was reduced as much as possible, so as to find the lowest exhaust flow rate for every buoyant condition. In many tests, this optimal condition was not reached (the jet velocity was too high), and therefore, the turbulent mixing strongly influenced the hydrodynamic sealing of the curtain (as in the test results reported in the previous section). Although the saltwater test results are still strongly influenced by buoyancy, it is relevant to compare them with the experimental results reported in the previous section. On basis of the saltwater tests, the average velocity of the flow across the door u , which was necessary to avoid the contaminant transport through the plane jet, was co-related with the plane jet characteristics by Equation (25). Figure 17 compares the experimental values of u with the predictions of Equation (25). It shows that Equation (25) is able to predict fairly well the experimental results, but the dispersion of the test results is quite high because of the buoyancy effect. Equation (25) is similar to Equation (18), but multiplied by an empirical constant with a value of 1.178. As a result, Equation (25) indicates that the flow rate across the door must be approximately 17.8% above the flow rate driven by the jet from the non-contaminated compartment. Finally, Figure 18 compares the experimental results (left side) with Equation (23) (right side) and Equation (25). It is clear that for the Reynolds number Re > 1224 or for the jet angle of 15°, the experimental results show a similar development (but the dependence of u on u is not the same), even though the experimental results require a higher flow rate across the door to reach the hydrodynamic sealing of the curtain than one the predicted by Equation (25). It is relevant to notice that even in such different test conditions (with or without buoyancy), it is visible a similar trend. Conclusions In this research, small-scale water modelling was used to assess the hydrodynamic sealing of a curtain formed by a plane jet. This small-scale modelling is intended to reproduce the performance of a full-scale air curtain, with the Reynolds number being in the range between Re = 147 and Re = 2125. The authors believe that the empirical results obtained are innovative and representative of the optimal conditions to set the aerodynamic sealing of full-scale air curtains for the same Reynolds numbers in an isothermal environment. The small-scale water modelling led to obtaining the empirical law presented in Equation (26), which corresponds to the limit state of the hydrodynamic sealing at the curtain. Equation (26) shows that a simple model based on the flow momentum balance at the jet (Equation (14)) is unable to properly express the test results for higher Reynolds Numbers Re > 1224, but Equation (14) is close to the experimental results just for the jet angle of 15° and for low Reynolds Numbers (Re < 1224). On the basis of previous small-scale saltwater test results [4], the average velocity of the flow across the door u , which was necessary to avoid the contaminant transport through the plane jet, was co-related with the plane jet characteristics by Equation (27). For Reynolds number Re > 1224, the experimental results show a similar development (but the dependence of u on u is not the same), even though the experimental results require a higher flow rate across the door to reach the hydrodynamic sealing of the curtain than the one predicted by Equation (27). w-door width X-jet length, longitudinal coordinate of the jet α -slope of the jet ν-kinematic viscosity Figure 1 . 1Strategies for the containment of a contaminant within a compartment (a) and for the protection of a clean room (b). Figure 2 2presents a cross section of the plane jet system. The flow rate balance is presented in Equation(1), where M , M , and M are the jet flow rate at the floor impingement zone, the flow rate rejected to the contaminated side, and the flow rate rejected to the non-contaminated side, respectively. Figure 2 . 2Cross section of the plane jet system. For this reason, this research project encompasses further phases of full-scale CFD simulation of air curtains and final full-scale testing. The model was placed in a tank measuring 1.20 m × 0.49 m × 0.50 m, and the model compartment had the following dimensions 0.40 m × 0.25 m × 0.26 m, and comprised an opening that simulated the door with the dimensions of 0.125 m × 0.125 m. The jet was located at the soffit of the opening and had a width of 0.1456 m and thicknesses of 1.25 mm, 2.50 mm, 3.75 mm, and 5 Figure 3 . 3Test model-in the drawing, red corresponds to the contaminant circuit, dark blue to the exhaust circuit, green to the jet circuit, and purple to the water intake (dimensions in cm). Figure 4 . 4Oversized exhaust flow rate. Figure 5 . 5Insufficient exhaust flow rate. Figure 6 . 6Limit state of the exhaust flow rate. Figure 7 . 7Analysis of the variation in the average velocity of the flow across the opening with jet velocity. Figure 8 . 8Jet angle influence. Figure 9 .Figure 10 . 910Relation between the average velocity at the opening and u . +1 − sin α, for different jet angles (15* values correspond to tests with variation in jet thickness). Linear regression between average velocity at the opening and u . +1 − sin α, . Figure 11 .Figure 12 . 1112Relation between the average velocity at the opening and u .YZ +1.00 − sin α, . Z (15* values correspond to tests with variations in jet thickness). YZ +1.00 − sin α, . Z Linear regression between average velocity at the opening and u .YZ +1.00 − sin α, . Z . Figure 13 . 13Relation between the average velocity at the opening and u . +1.00 − sin α, (15* values correspond to tests with variations in jet thickness). Figure 14 . 14Linear regression average between the velocity at the opening and u . +1.00 − sin α, . Figure 15 . 15is a fair approximation of the experimental results. It is also clear that the results for the Reynolds number Re = 1224 (corresponding to U0 = 0.98 m/s) are still in a transition between the low Reynolds number results, in which the jet angle is relevant, and the higher Reynolds number results, in which the jet angle is not relevant. Comparison between experimental results and Equation(23). Figure 16 . 16Comparison between experimental results and Equation(14). .00 − sin α, . Figure 17 . 17Comparison of the test results of u with the prediction of Equation(25). Figure 18 . 18Comparison of Equation (25) (prediction Ua) with the test results of Section 3.1 and Equation (23). Author Contributions: Investigation, F.O.; supervision, J.C.V. and D.A. Funding: This project has received funding from the European Union's Horizon 2020 research and innovation programme, under the Marie Sklodowska-Curie grant agreement N 690968. Table 1 . 1Geometric characteristics of the jet.Test Thickness of the Jet b0 (m) Width of the Jet L (m) Section of the Jet (m²) Height of the Door h (m) Width of the Door w (m) 57-88; 92-107 0.00125 0.1456 0.000182 0.125 0.125 89 0.00250 0.000364 90 0.00375 0.000546 91 0.00500 0.000728 Table 2 . 2Test results obtained with both active (thickness of 0.00125 m) and inactive jets.Jet Thickness (8 9 = 0.00125 m) : 9 (m/s) ; 9 (°) < =>?:@A (L/s) : ? (m/s) Re 1.70 0 1.20 0.077 2125 5 1.20 0.077 10 1.20 0.077 15 1.20 0.077 1.37 0 1.07 0.068 1710 5 1.07 0.068 10 1.07 0.069 15 1.07 0.069 0.98 0 0.84 0.054 1224 5 0.81 0.052 10 0.79 0.051 15 0.69 0.044 0.55 0 0.73 0.046 687 5 0.69 0.044 10 0.64 0.041 15 0.58 0.037 Table 4 . 4Results of the parameters adjusted by the least squares method.Variable Value A 0.0595 B 1.00 C 0.00 f 1.05 j 0.470 As Table 5 . 5Rounded parameters and parameter A adjusted by the least squares method.Variable Value A 0.0564 B 1.00 C 0.0026 f 1.00 j 0.50 As Acknowledgments: The small scale jet device was developed by Mr. Paulo Morais from Scientific Instrumentation Centre (CIC) of National Laboratory of Civil Engineering (LNEC). This device was built at CIC/LNEC.Conflicts of Interest:The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; and in the decision to publish the results. Numerical Study and Experimental Optimization of air Curtains. H G Garcia, Barcelona, SpainUniversity Politècnica CatalunyaPh.D. ThesisGarcia, H.G. Numerical Study and Experimental Optimization of air Curtains. Ph.D. Thesis, University Politècnica Catalunya, Barcelona, Spain, 2015. CFD modelling of aerodynamic sealing by vertical and horizontal air curtains. J C Gonçalves, J J Costa, A R Figueiredo, A M Lopes, 52Gonçalves, J.C.; Costa, J.J.; Figueiredo, A.R.; Lopes, A.M.G. CFD modelling of aerodynamic sealing by vertical and horizontal air curtains. Energy Build. 2012, 52, 153-160. Energy savings by aerodynamic sealing with a downward-blowing plane air curtain-A numerical approach. J J Costa, L A Oliveira, M C G Silva, Energy Build. 38Costa, J.J.; Oliveira, L.A.; Silva, M.C.G. Energy savings by aerodynamic sealing with a downward-blowing plane air curtain-A numerical approach. Energy Build. 2006, 38, 1182-1193. Saltwater experiments with air curtains for smoke control in the event of fire. J C Viegas, J. Build. Eng. 8Viegas, J.C. Saltwater experiments with air curtains for smoke control in the event of fire. J. Build. Eng. 2016, 8, 243-248. Efficiency of air curtains used for separating smoke free. G Krajewski, Proceedings of the BS2013: 13th Conference of International Building Performance Simulation Association. the BS2013: 13th Conference of International Building Performance Simulation AssociationChambéry, FranceKrajewski, G. Efficiency of air curtains used for separating smoke free. In Proceedings of the BS2013: 13th Conference of International Building Performance Simulation Association, Chambéry, France, 26-28 August 2013. Methodology for minimizing risk from airborne organisms in hospital isolation rooms. F Memarzadeh, J Jiang, ASHRAE Trans. 106Memarzadeh, F.; Jiang, J. Methodology for minimizing risk from airborne organisms in hospital isolation rooms. ASHRAE Trans. 2000, 106. Comparison of Operating Room Ventilation Systems in the Protection of the Surgical Site. F Memarzadeh, A Manning, ASHRAE J. 108Memarzadeh, F.; Manning, A. Comparison of Operating Room Ventilation Systems in the Protection of the Surgical Site. ASHRAE J. 2002, 108, 3-15. Predictions and measurements of the stack effect on indoor airborne virus transmission in a high-rise hospital building. T Lim, J Cho, B S Kim, Build. Environ. 46Lim, T.; Cho, J.; Kim, B.S. Predictions and measurements of the stack effect on indoor airborne virus transmission in a high-rise hospital building. Build. Environ. 2011, 46, 2413-2424. Heat Transfer Characteristics of the Air Curtain: A Plane Jet Subjected to Transverse Pressure and Temperature Gradients. F C Hayes, Champaign, IL, USAUniversity of IllinoisPh.D. ThesisHayes, F.C. Heat Transfer Characteristics of the Air Curtain: A Plane Jet Subjected to Transverse Pressure and Temperature Gradients. Ph.D. Thesis, University of Illinois, Champaign, IL, USA, 1968. Design data for air curtains. F C Hayes, W F Stoecker, ASHRAE Trans. 75Hayes, F.C.; Stoecker, W.F. Design data for air curtains. ASHRAE Trans. 1969, 75, 168-180. Heat transfer characteristics of the air curtain. F C Hayes, W F Stoecker, ASHRAE Trans. 75Hayes, F.C.; Stoecker, W.F. Heat transfer characteristics of the air curtain. ASHRAE Trans. 1969, 75, 153- 167. Assessing dynamic efficiency of air curtain in reducing whole building annual energy usage. S Goubran, D Qi, L Wang, Build. Simul. 10Goubran, S.; Qi, D.; Wang, L. Assessing dynamic efficiency of air curtain in reducing whole building annual energy usage. Build. Simul. 2017, 10, 497-507. Experimental study on the flow characteristics of air curtains at building entrances. S Goubran, D Qi, W Saleh, L Wang, R Zmeureanu, Build. Environ. 105Goubran, S.; Qi, D.; Saleh, W.; Wang, L.; Zmeureanu, R. Experimental study on the flow characteristics of air curtains at building entrances. Build. Environ. 2016, 105, 225-235. Simulation-based feasibility study of improved air conditioning systems for hospital operating room. Z J Zhai, A L Osborne, Front. Archit. Res. 2Zhai, Z.J.; Osborne, A.L. Simulation-based feasibility study of improved air conditioning systems for hospital operating room. Front. Archit. Res. 2013, 2, 468-475. Performance of ventilation system in a non-standard operating room. T T Chow, X Y Yang, Build. Environ. 38Chow, T.T.; Yang, X.Y. Performance of ventilation system in a non-standard operating room. Build. Environ. 2003, 38, 1401-1411. Int-Hout, D. Air motion control in the Hospital Operating Room. G Cook, ASHRAE J. 31Cook, G.; Int-Hout, D. Air motion control in the Hospital Operating Room. ASHRAE J. 3(1):30-36 2009. Using air curtain to control pollutant spreading for emergency management in a cleanroom. Y C Shih, A S Yang, C W Lu, Build. Environ. 46Shih, Y.C.; Yang, A.S.; Lu, C.W. Using air curtain to control pollutant spreading for emergency management in a cleanroom. Build. Environ. 2011, 46, 1104-1114. A new idea that is 40 years old-Air curtain hospital operating room systems. G Cook, D Int-Hout, ASHRAE Trans. 113Cook, G.; Int-Hout, D. A new idea that is 40 years old-Air curtain hospital operating room systems. ASHRAE Trans. 2007, 113, 349-357. Air Distribution Strategy Impact on Operating Room Infection Control. J Swift, E Avis, B Millard, T M Lawrence, Proceedings of the Clima. the ClimaHelsinkiSwift, J.; Avis, E.; Millard, B.; Lawrence, T.M. Air Distribution Strategy Impact on Operating Room Infection Control. In Proceedings of the Clima 2007 WellBeing Indoors, Helsinki, June 2007. Reducing bartenders exposure to ETS by a local ventilation-Field evaluation of the solution. M Hyvärinen, K Hagström, I Grönvall, P Hynynen, 9th International Conference on Indoor Air Quality and Climate. Montery, CA, USAProceedings of the Indoor AirHyvärinen, M.; Hagström, K.; Grönvall, I.; Hynynen, P. Reducing bartenders exposure to ETS by a local ventilation-Field evaluation of the solution. In Proceedings of the Indoor Air 2002, 9th International Conference on Indoor Air Quality and Climate, Montery, CA, USA, 30 June-5 July 2002; pp. 133-137. Efficient Ventilation: Displacement Ventilation and Air Curtain Zoning. H Skistad, NorwaySkistad, H. Efficient Ventilation: Displacement Ventilation and Air Curtain Zoning. Norway, 2011 pp. 1-17. Efficacy of an air curtain system for local pit environmental control for relic preservation in archaeology museums. X Luo, Z Gu, C Yu, T Ma, K Kase, Indoor Built Environ. 25Luo, X.; Gu, Z.; Yu, C.; Ma, T.; Kase, K. Efficacy of an air curtain system for local pit environmental control for relic preservation in archaeology museums. Indoor Built Environ. 2016, 25, 29-40. Air curtain as a barrier against pollutants in cultural heritage: A case study. L De Santoli, F Cumo, M Mariotti, Air Pollut. XIV. 1De Santoli, L.; Cumo, F.; Mariotti, M. Air curtain as a barrier against pollutants in cultural heritage: A case study. Air Pollut. XIV 2006, 1, 385-392. Air curtains of open refrigerated display cases revisited: A new technique for infiltration rate measurements. M Amin, H K Navaz, D Dabiri, R Faramarzi, WIT Trans. Eng. Sci. 61Amin, M.; Navaz, H.K.; Dabiri, D.; Faramarzi, R. Air curtains of open refrigerated display cases revisited: A new technique for infiltration rate measurements. WIT Trans. Eng. Sci. 2008, 61, 179-190. Effects of jet inclination angle and geometrical parameters on air curtain performance. S R Traboulsi, A Hammoud, M F Khalil, ASHRAE Trans. 115Traboulsi, S.R.; Hammoud, A.; Khalil, M.F. Effects of jet inclination angle and geometrical parameters on air curtain performance. ASHRAE Trans. 2009, 115, 617-629. Educing coherent eddy structures in air curtain systems. K Loubière, M Pavageau, Lett. Appl. Microbiol. Loubière, K.; Pavageau, M. Educing coherent eddy structures in air curtain systems. Lett. Appl. Microbiol. 2005, 2705, 4-6. An Isothermal Air Curtain for Isolation of Smoking Areas in Restaurants. J P Rydock, T Hestad, H Haugen, J E Skaret, Proceedings of ROOMVENT 2000 -Air Distribution in Rooms. ROOMVENT 2000 -Air Distribution in RoomsOxfordRydock, J.P.; Hestad, T.; Haugen, H.; Skaret, J.E. An Isothermal Air Curtain for Isolation of Smoking Areas in Restaurants; Proceedings of ROOMVENT 2000 -Air Distribution in Rooms, Oxford, June 2000. Visualizing the Flow Induced by an Air Curtain over a Mannequin Using Stereo Particle Image Velocimetry. F K Lu, J E Fernandes, Proceedings of the ISFV13-13th international Symposium on Flow Visualization. the ISFV13-13th international Symposium on Flow VisualizationNice, FranceLu, F.K.; Fernandes, J.E. Visualizing the Flow Induced by an Air Curtain over a Mannequin Using Stereo Particle Image Velocimetry. In Proceedings of the ISFV13-13th international Symposium on Flow Visualization, Nice, France, 1-4 July 2008; pp. 1-10. Parametric study of air curtain door aerodynamics performance based on experiments and numerical simulations. D Qi, S Goubran, L Wang, R Zmeureanu, Build. Environ. 129Qi, D.; Goubran, S.; Wang, L.; Zmeureanu, R. Parametric study of air curtain door aerodynamics performance based on experiments and numerical simulations. Build. Environ. 2018, 129, 65-73. An approach to determine infiltration characteristics of building entrance equipped with air curtains. L Wang, Z Zhong, Energy Build. 75Wang, L.; Zhong, Z. An approach to determine infiltration characteristics of building entrance equipped with air curtains. Energy Build. 2014, 75, 312-320. The effects of an opposing buoyancy force on the performance of an air curtain in the doorway of a building. D Frank, P F Linden, Energy Build. 96Frank, D.; Linden, P.F. The effects of an opposing buoyancy force on the performance of an air curtain in the doorway of a building. Energy Build. 2015, 96, 20-29. On-site assessment of the discharge coefficient of open windows. H Cruz, J C Viegas, Energy Build. 126Cruz, H.; Viegas, J.C. On-site assessment of the discharge coefficient of open windows. Energy Build. 2016, 126, 463-476. Applied Fluid Dynamics Handbook. R D Blevins, New York, Van Nostrand ReinholdBlevins, R.D. Applied Fluid Dynamics Handbook; New York, Van Nostrand Reinhold: 1984. Evaluation of Measurement Data-Guide to the Expression of Uncertainty in Measurement. Joint Committee for Guides in Metrology: 2008. © 2018 by the authors. 100Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BYJCGM 100:2008-Evaluation of Measurement Data-Guide to the Expression of Uncertainty in Measurement; Joint Committee for Guides in Metrology: 2008. © 2018 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY)
[]
[ "Multi-AAV Cooperative Path Planning using Nonlinear Model Predictive Control with Localization Constraints", "Multi-AAV Cooperative Path Planning using Nonlinear Model Predictive Control with Localization Constraints" ]
[ "Amith Manoharan ", "Rajnikant Sharma ", "P B Sujit " ]
[]
[]
In this paper, we solve a joint cooperative localization and path planning problem for a group of Autonomous Aerial Vehicles (AAVs) in GPS-denied areas using nonlinear model predictive control (NMPC). A moving horizon estimator (MHE) is used to estimate the vehicle states with the help of relative bearing information to known landmarks and other vehicles. The goal of the NMPC is to devise optimal paths for each vehicle between a given source and destination while maintaining desired localization accuracy. Estimating localization covariance in the NMPC is computationally intensive, hence we develop an approximate analytical closed form expression based on the relationship between covariance and path lengths to landmarks. Using this expression while computing NMPC commands reduces the computational complexity significantly. We present numerical simulations to validate the proposed approach for different numbers of vehicles and landmark configurations. We also compare the results with EKF-based estimation to show the superiority of the proposed closed form approach.Note to Practitioners: Abstract-The use of AAVs in urban regions is expected to increase with several logistic and healthcare applications. These AAVs depend on GPS for localization, however, in urban regions, due to interference of building structures obtaining accurate localization information is difficult and at times may not be available. This issue hampers the AAV operations. In this paper, we develop a mechanism by which the AAVs use landmarks in the region and also the availability of other vehicles in the regions to localize and achieve the mission. For localization we use MHE and to generate the paths, we use a NMPC method. In order to improve computational speed, we developed an approximate closed form analytical covariance method which is used in the NMPC for covariance calculation. We showed through several simulations that the proposed joint path planning with localization constraints could determine optimal paths to the vehicles while satisfying the localization accuracy. This approach can be used by the UAV industries as an alternative mechanism for localization while determining the paths for the vehicles. The simulation results are promising but further work is required to experimentally demonstrate the proof-of-concept.
null
[ "https://arxiv.org/pdf/2201.09285v1.pdf" ]
246,240,490
2201.09285
3c111cf298eb07136dd060b2625e0dc77e09466d
Multi-AAV Cooperative Path Planning using Nonlinear Model Predictive Control with Localization Constraints Amith Manoharan Rajnikant Sharma P B Sujit Multi-AAV Cooperative Path Planning using Nonlinear Model Predictive Control with Localization Constraints 1Index Terms-Path planningCooperative localizationNonlin- ear model predictive controlUAVs In this paper, we solve a joint cooperative localization and path planning problem for a group of Autonomous Aerial Vehicles (AAVs) in GPS-denied areas using nonlinear model predictive control (NMPC). A moving horizon estimator (MHE) is used to estimate the vehicle states with the help of relative bearing information to known landmarks and other vehicles. The goal of the NMPC is to devise optimal paths for each vehicle between a given source and destination while maintaining desired localization accuracy. Estimating localization covariance in the NMPC is computationally intensive, hence we develop an approximate analytical closed form expression based on the relationship between covariance and path lengths to landmarks. Using this expression while computing NMPC commands reduces the computational complexity significantly. We present numerical simulations to validate the proposed approach for different numbers of vehicles and landmark configurations. We also compare the results with EKF-based estimation to show the superiority of the proposed closed form approach.Note to Practitioners: Abstract-The use of AAVs in urban regions is expected to increase with several logistic and healthcare applications. These AAVs depend on GPS for localization, however, in urban regions, due to interference of building structures obtaining accurate localization information is difficult and at times may not be available. This issue hampers the AAV operations. In this paper, we develop a mechanism by which the AAVs use landmarks in the region and also the availability of other vehicles in the regions to localize and achieve the mission. For localization we use MHE and to generate the paths, we use a NMPC method. In order to improve computational speed, we developed an approximate closed form analytical covariance method which is used in the NMPC for covariance calculation. We showed through several simulations that the proposed joint path planning with localization constraints could determine optimal paths to the vehicles while satisfying the localization accuracy. This approach can be used by the UAV industries as an alternative mechanism for localization while determining the paths for the vehicles. The simulation results are promising but further work is required to experimentally demonstrate the proof-of-concept. Multi-AAV Cooperative Path Planning using Nonlinear Model Predictive Control with Localization Constraints Amith Manoharan, Rajnikant Sharma and P.B. Sujit Abstract-In this paper, we solve a joint cooperative localization and path planning problem for a group of Autonomous Aerial Vehicles (AAVs) in GPS-denied areas using nonlinear model predictive control (NMPC). A moving horizon estimator (MHE) is used to estimate the vehicle states with the help of relative bearing information to known landmarks and other vehicles. The goal of the NMPC is to devise optimal paths for each vehicle between a given source and destination while maintaining desired localization accuracy. Estimating localization covariance in the NMPC is computationally intensive, hence we develop an approximate analytical closed form expression based on the relationship between covariance and path lengths to landmarks. Using this expression while computing NMPC commands reduces the computational complexity significantly. We present numerical simulations to validate the proposed approach for different numbers of vehicles and landmark configurations. We also compare the results with EKF-based estimation to show the superiority of the proposed closed form approach. Note to Practitioners: Abstract-The use of AAVs in urban regions is expected to increase with several logistic and healthcare applications. These AAVs depend on GPS for localization, however, in urban regions, due to interference of building structures obtaining accurate localization information is difficult and at times may not be available. This issue hampers the AAV operations. In this paper, we develop a mechanism by which the AAVs use landmarks in the region and also the availability of other vehicles in the regions to localize and achieve the mission. For localization we use MHE and to generate the paths, we use a NMPC method. In order to improve computational speed, we developed an approximate closed form analytical covariance method which is used in the NMPC for covariance calculation. We showed through several simulations that the proposed joint path planning with localization constraints could determine optimal paths to the vehicles while satisfying the localization accuracy. This approach can be used by the UAV industries as an alternative mechanism for localization while determining the paths for the vehicles. The simulation results are promising but further work is required to experimentally demonstrate the proof-of-concept. Index Terms-Path planning, Cooperative localization, Nonlinear model predictive control, UAVs I. INTRODUCTION U RBAN air mobility (UAM) is expected to have highly automated, cooperative, passenger, and cargo-carrying aerial vehicles in urban areas [1], and the use of autonomous Amith aerial vehicles (AAVs) for various activities are expected to rise substantially in the near future [2]. Cargo delivery drones operate in urban canyons with high-rise buildings and other obstructions, which calls for significant localization accuracy. However, operating in such environments pose an additional challenge in localization since Global navigation satellite systems (GNSS) are unreliable in such scenarios. A solution to this problem is to use alternate localization schemes such as relative localization [3], [4], vision-based methods [5], [6], and ultra-wide-band (UWB) localization schemes [7], [8]. As the urban airspace is expected to contain a large number of AAVs, relative localization between vehicles can also be used in addition to known landmarks localization. Cooperative path planning with localization constraints involves the following components: (i) localization -vehicles estimate their position by using relative measurements obtained with respect to other vehicles or landmarks, and (ii) cooperative path planning -determine optimal paths for each vehicle from a given source to destination. To achieve (ii), the agents must cooperate with each other to generate motion commands that improves the localization accuracy of the entire group while reducing the path length to reach their respected destinations. Several works have studied (i) and (ii) separately. For instance, [9]- [20] address the problem of cooperative localization, while [21]- [24] focus on cooperative path planning problem. However, the collection of works that jointly address cooperative path planning with the localization constraints is limited. Below, we will review some of the works in this domain. Bopardikar et al. [25] presented a graph-based probabilistic roadmap approach to tackle the path planning problem subject to localization constraints. The generated path is a discretized path while we are addressing a continuous path problem. A time-optimal path planner satisfying the covariance bounds was given in [26] using a swarm optimization technique coupled with a rabbit-carrot based path follower. An approach to optimally place landmarks to satisfy localization constraints was proposed in [27]. The algorithm computes an optimal path for the vehicle and the locations where the landmarks should be placed. A localizability constrained path planning method for autonomous vehicles which takes into account the laser range finder (LRF) sensor model of the vehicle is proposed in [28] to maintain a satisfactory level of localizability throughout the path. Kassas et al. [29] present a multi-objective motion planning algorithm in which the vehicle tries to balance the objectives of navigating to the waypoint and reducing its position estimate uncertainty. All the above works are limited to one vehicle only. Urban air mobility calls for improved localization accuracy due to its innate nature involving close structures, narrow pathways, and a large number of vehicles. Moving horizon estimation (MHE) has been suggested as an alternate to EKF for increasing the accuracy of nonlinear estimation problems by [30]- [32]. Erunsal et al. [33] proposed an approach combining NMPC and pose-graph-MHE for 3D formation control of micro aerial vehicles with relative sensing capability. In [34], a decentralized MHE technique is proposed for networked navigation with packet dropouts. In this paper, we extend the work in [35] and propose a joint cooperative localization and path planning framework with MHE for estimating the vehicle position, and NMPC framework for cooperative path planning, and a closed formulation for covariance calculation to predict the uncertainty. This framework uses a nonlinear vehicle model in both the controller and the estimator, which mitigates the linearization errors. The analytical expression used for the covariance calculation speeds up the computations and is derived by exploiting the relationship between the vehicle-landmark path lengths to the localization uncertainty. The proposed approach provides a flexibility, where each vehicle can decide to maintain, lose, or gain connections depending on their covariance estimates.In most of the literature for multi-agent systems, the studies are formulated either as a control problem or an estimation problem [10], [12]- [14], [31]. We propose a method that combines both and looks at the multi-agent problem in a holistic sense. The major contributions of this paper as follows: • A complete framework for control and estimation of multi-vehicle cooperative path planning problem with localization constraints using NMPC and MHE. • An analysis on the relation of path lengths between vehicles and landmarks on the estimation covariance • An approximate closed form analytical expression to compute localization error covariance • Evaluation of the proposed joint cooperative path planning with localization constraints framework through numerical simulations and comparison with EKF-based estimation framework The rest of the paper is organized as follows. The problem formulation is given in Section II. Moving horizon estimation (MHE) is explained in Section III. The derivation of the analytic expression for covariance is given in Section IV. The NMPC formulation is given in Section V. Simulation results are presented in Section VI, and the conclusions are given in Section VII. II. PROBLEM FORMULATION We consider a scenario where a group of AAVs need to navigate from their source location (S) to destination (D), as shown in Fig. 1a. These vehicles need to transit in a GPS-denied area and we assume that any kind of GNSS are not available. In such scenarios, known landmarks or other vehicles in the area can be used for relative localization using range or bearing measurements. This structure involving landmarks and vehicles can be modeled as a dynamic relative position measurement graph (RPMG) [36] with vehicles and landmarks as nodes and connections/measurements as edges. Definition 1. A relative position measurement graph (RPMG) for n v (t) vehicles with n l (t) landmarks is a graph G n l nv (t) {V n l nv (t), E n l nv (t)}, where V n l nv is the node set consisting of n v (t) vehicle nodes and n l (t) landmark nodes (which makes a total of n v (t) + n l (t) nodes), and E n l nv (t) is the edge set representing available relative measurements. The number of edges is denoted by n e (t) = |E n l nv (t)|. Definition 2. A path from a vehicle node to a landmark node is a finite sequence of edges which joins a sequence of distinct vertices between them. Let G {V, E, φ} be a graph. A path φ from vertex i to vertex j is a sequence of edges { 1 , 2 , . . . , n−1 } for which there is a sequence of distinct vertices {ν 1 , ν 2 , . . . , ν n } such that φ {ν 1 , ν n } where ν 1 = i and ν n = j. An example RPMG (G 3 5 with n e = 7) is shown in Fig. 1b. A path from the vehicle 5 to the landmark a is represented by the edge set φ { 6 , 5 , 3 } which can also be represented using the vertices as 5 − 4 − 2 − a. Previous studies show that for cooperative localization to work, each vehicle should have a direct or indirect path to at least two known landmarks [37]. This condition is very limiting in environments with a low number of landmarks. Hence in this paper, we find a relationship between vehicle uncertainty and path length to the landmarks and then use that relationship to formulate and solve an NMPC problem to guarantee that the covariance does not exceed a specified threshold and desired localization accuracy is achieved while performing individual missions. A moving horizon estimation (MHE) scheme is used to estimate the vehicle states. A graphical representation of the proposed solution using NMPC combined with MHE to tackle the cooperative localization and path planning problem is shown in Fig. 2. The optimal path is different from the shortest path since the latter may not satisfy localization constraints. The components of the block diagram are explained in the subsequent sections. In the first step of the control scheme, vehicle states are estimated by the MHE block using available measurements from the sensors (Sec. III). The second step is the calculation of estimation covariances for the NMPC prediction window, Step 1 Step 2 Step which is accomplished by the covariance calculator block that contains the derived analytical expression (Sec. IV). In the third and final step, the NMPC controller computes the control actions for the vehicles (Sec. V). III. MOVING HORIZON ESTIMATION Moving horizon estimation (MHE) uses optimization techniques to determine state trajectories that best fit a series of measurements acquired over a finite time interval. It uses the exact nonlinear models of the available measurements and system dynamics for estimation. Also, there is another advantage of including the state/control constraints in the formulation, which helps in bounding the estimates. Like NMPC, MHE also has three main components, 1) an internal dynamic model of the process, 2) a history of past measurements, and 3) an optimization cost function over the estimation horizon. The model used for estimation is given as: X(k) = f (X(k − 1), ω(k), k) + q(k), (1) z(k) = h (X(k), ω(k), k) + µ(k),(2) where, f (·) and h(·) represent the state model and the observation model respectively. X(k) and z(k) are the system states and measurements at the k th time instant. The vectors q(k) and µ(k) are the process and measurement noises which are assumed to be additive and zero mean white Gaussian noises with covariance Q and Γ respectively. f (·) is defined as f =            x 1 (k) y 1 (k) ψ 1 (k) . . . x nv (k) y nv (k) ψ nv (k)            =            x 1 (k − 1) + T s v cos ψ 1 (k − 1) y 1 (k − 1) + T s v sin ψ 1 (k − 1) ψ 1 (k − 1) + T s ω 1 (k − 1) . . . x nv (k − 1) + T s v cos ψ nv (k − 1) y nv (k − 1) + T s v sin ψ nv (k − 1) ψ nv (k − 1) + T s ω nv (k − 1)            ,(3) where T s is the sampling time used for discretization. Let m be the current time step, N E is the estimation horizon, and we denote τ = m − N E for simplicity. We formulate the moving horizon estimation problem min X J = X τ −X τ 2 P −1 τ + m k=τ h(X k ) − z k 2 Γ −1 ,(4) subject to: X k+1 = f (X k , ω k ), ω ∈ ω − , ω + , whereX is the estimated states, P is the estimation covariance matrix, and Γ is the measurement covariance. It is assumed that each vehicle can measure relative bearing to other vehicles and landmarks that are in the sensor's field-of-view (R s ). Relative bearing from the i th vehicle to the j th vehicle or landmark is given by the measurement model: h(X) = tan −1 y j − y i x j − x i − ψ i .(5) The first term in (4) is known as the arrival cost and it plays an important role in stabilizing the estimator. It penalizes the deviation of the first state in the moving horizon window and its previous estimateX τ . The weighting matrix P is given by [38] P k+1 = Q + ∇F X (P k − P k ∇H T X (∇H X P k ∇H T X + Γ) −1 ∇H X P k )∇F T X , where Q is the state covariance matrix, and ∇F X , ∇H X are the Jacobians of f and h. The second term in (4) penalizes the change in predicted measurements h(X k ) from the actual measurements z k . Now, we look into the stability of the moving horizon estimator. The following assumptions are required for proving the stability result. Assumption 1. The initial state X 0 and the control input ω are such that, for any noise q, the system trajectory X lies in a compact set χ and ω in a compact set U . Assumption 2. The functions f and h are C 2 functions w.r.t X on co(χ) for every ω ∈ U , where co(χ) is the convex closure of χ. Observation map for a horizon N E + 1 can be defined as F N E (X, ω, q) =         h(X τ ) h • f τ (X τ ) · · · h • f m−1 • · · ·f τ (X τ )         ,(6) where • is function composition. Then it is possible to re-write equation (2) as z τ = F N E (X, ω, q) + µ τ ,(7) and modify the cost function as J m (X τ,Xτ ) = X τ −X τ 2 Pτ + F N E (X τ , ω τ , q τ )−z τ 2 Γ , (8) where, Γ = I N E +1 ⊗ Γ −1 , where ⊗ is the Kronecker product. Now, let's consider the following remarks: Remark 1. System (1), (2) is said to be observable in N E + 1 steps if there exists a K-function φ(·) such that φ x 1 − x 2 2 ≤ F N E (x 1 , ω, 0) − F N E (x 2 , ω, 0) 2 , ∀x 1 , x 2 ∈ χ and ∀ω ∈ U N E . Remark 2. If the observability matrix ∂F N E (X,ω,0) ∂X has full rank, then the system is said to be observable in N E + 1 steps with finite sensitivity 1/δ if the K-function φ(·) satisfies the following condition δ = inf x1,x2∈χ;x1 =x2 φ x 1 − x 2 2 x 1 − x 2 2 ≥ 0.(9) Let k f be an upper bound on the Lipschitz constant of f (X, ω) w.r.t X on χ for every ω ∈ U and P is diagonal with P = pI n , p > 0. Let r µ = max µ∈M µ 2 ,(10) where M is a compact set with 0 ∈ M . Stability of the estimator is proved using the results from [38]- [41]. Consider the cost function defined as: J = X τ −X τ 2 Pτ + m k=τ h(X k ) − z k 2 Γ −1 ,(11) then we can state the following theorem [40], [41]. Theorem 1. If the Assumptions 1,2 are satisfied and the Remarks 1,2 hold, then there exists an upper bound defined by X τ −X τ 2 ≤ ζ τ ,(12) where ζ m is found using the equation ζ m+1 = c 1 k f p p + c 2 δ ζ m + c 3 p + c 2 δ r µ ,(13) c 1 , c 2 , and c 3 are positive constants. Let a(p, δ) = c 1 k f p (p + c 2 δ) ,(14) and if p is selected such that a(p, δ) < 1, then the dynamics of (13) is asymptotically stable. Proof. A summary of the proof given by [40] is detailed here for completeness. The proof is based on defining upper and lower bounds on the optimal cost J * m , which is the cost corresponding to an optimal estimateX * τ . First, the upper bound on J * m should be defined. Let us define X o m as the true value of the state X at time m and assume that Γ = I. We have that J * m ≤ X o τ − X * τ 2 Pτ + m k=τ F N E (X o k ) − z k 2 ,(15) which can be modified as J * m ≤ X o τ − X * τ 2 Pτ + C,(16) where C is a positive constant (please see Lemma. 1 from [40]). Next, the upper bound on J * m is defined. We can write F N E (X o τ ) − F N E (X τ ) 2 = [z τ − F N E (X τ )] − [z τ − F N E (X o τ )] 2 , (17) F N E (X o τ ) − F N E (X τ ) 2 ≤ 2 z τ − F N E (X τ ) 2 + 2 z τ − F N E (X o τ )] 2 , (18) z τ − F N E (X τ ) 2 ≥ 1 2 F N E (X o τ ) − F N E (X τ ) 2 − z τ − F N E (X o τ )] 2 .(19) From (16), we can write z τ − F N E (X o τ )] 2 < C.(20) Hence, we obtain z τ − F N E (X τ ) 2 ≥ 1 2 F N E (X o τ ) − F N E (X τ ) 2 − C. By using a similar procedure, we can write X * τ −X τ 2 ≥ 1 2 X o τ −X τ 2 − X o τ − X * τ 2 . Now the upper bound can be defined as J * m ≥ 1 2 X o τ −X * τ 2 Pτ + 1 2 F N E (X o τ ) − F N E (X * τ ) 2 − X o τ − X * τ 2 Pτ − C.(21) Now, by combining the bounds (16) and (21), and rewriting we get 1 2 X o τ −X * τ 2 Pτ + 1 2 F N E (X o τ ) − F N E (X * τ ) 2 ≤ 2 X o τ − X * τ 2 Pτ + 2C. According to the Remarks 1 and 2, the above equation can be written as F N E (X o τ ) − F N E (X * τ ) 2 = φ X o τ −X * τ 2 , and δ X o τ −X * τ 2 ≤ φ X o τ −X * τ 2 . Now, it is possible to define the bound on the estimation error as X o τ −X * τ 2 ≤ 4p p + δ X o τ − X * τ 2 + 4 p + δ C. Using the Lipschitz continuity of f (·), it can be written that X o τ − X * τ 2 = 2k f X o τ −1 −X * τ −1 2 + 2r µ . Hence, X o τ −X * τ 2 ≤ ζ τ . It can also be deduced that if ζ m < ζ m−1 and a(p, δ) < 1, then ζ m tends to β 1−a(p,δ) as m −→ +∞ (please see Theorem 1 from [39]), where β = c3 p+c2δ r µ . The following section presents the derivation of the analytical expression for calculating the covariances using the path information. This result will be later used for predicting covariances for the NMPC cost function. IV. COVARIANCE CALCULATION Consider an example configuration of two vehicles as shown in Fig. 3a(i), where the vehicles are represented by 1 and 2 and two landmarks by a and b. In order to understand how the paths/connections/measurements from a landmark to a vehicle influence the uncertainty of the vehicle states, we consider each landmark separately and analyze. Consider the segment a − 1 − 2 of the graph in Fig. 3a(i), as shown in Fig. 3a(ii). The observability matrix for the system can be written as O = oa1 0 o12 −o12 ,(22) where oa1, o12, and −o12 are the derivatives of the measurements with respect to the vehicle states. For example, oa1 is the derivative of the measurement between the landmark a and the vehicle-1 with respect to the vehicle-1. Since there is no measurement between the landmark a and the vehicle-2, the corresponding entry (O 12 ) is zero. Note that the size of O depends on the number of edges and it may not be a square matrix. Assuming the measurement covariance matrix Γ = I, and zero-mean white Gaussian noise, the observability grammian is defined as O T O and the covariance matrix P is written as P ≤ (O T Γ −1 O) −1 ,(23)≤ 1 oa1 2 1 oa1 2 1 oa1 2 1 o12 2 + 1 oa1 2 .(24) The first element of the P matrix corresponds to the vehicle-1 connecting to the landmark a, hence oa1 (let us discard the square and fraction for easy understanding). The element 1 o12 2 + 1 oa1 2 of the P matrix corresponds to the vehicle-2. Observing that it is connected to the landmark a through vehicle-1, we can see both oa1 and o12 are present in the entry. Next, we consider the section 1 − 2 − b of the graph, as shown in Fig. 3a(iii). The observability and covariance matrices for the system are written as O = 0 ob2 o12 −o12 , P ≤ 1 o12 2 + 1 ob2 2 1 ob2 2 1 ob2 2 1 ob2 2 .(25) The first element of P indicates that vehicle-1 is connected to the landmark b through vehicle-2. The last entry shows that vehicle-2 is directly connected to the landmark b and hence only ob2 is present. Next, we extend similar analysis for a three vehicle configuration as shown in Fig. 3b(i). Consider the section 3−a−1−2 of the graph, shown in Fig. 3b(ii). The observability matrix and covariance matrix P for this configuration are O =   oa1 0 0 o12 −o12 0 0 0 oa3   , P ≤   1 oa1 2 1 oa1 2 0 1 oa1 2 1 o12 2 + 1 oa1 2 0 0 0 1 oa3 2   . The first element of the P matrix contain only oa1 since the vehicle-1 is directly connected to the landmark a. The vehicle-2 is connected to the landmark a through vehicle-1, hence both oa1 and o12 can be seen in the corresponding entry of the P matrix. Since vehicle-3 is directly connected to the landmark, the last element of P contains only oa3 as expected. Now, let us take the section 1 − 2 − b − 3 of the graph, shown in Fig. 3b(iii). The observability and covariance matrices are given by O =   0 ob2 0 o12 −o12 0 0 0 ob3   , P ≤   1 o12 2 + 1 ob2 2 1 ob2 2 0 1 ob2 2 1 ob2 2 0 0 0 1 ob3 2   . Vehicle-2 and vehicle-3 are directly connected to the landmark b. Therefore, the corresponding entries in the P matrix contains only ob2 and ob3. The vehicle-1 is connected to the landmark b through vehicle-2 and this information is clearly reflected in the first entry of P . Using the above example configuration analysis, we can further extend the analysis to a general result with n v vehicle and n l landmarks as shown in Fig. 3c. The landmarks are represented using stars, and the vehicles are represented using triangles. All the vehicles are denoted by i, i + 1, . . . , n v , landmarks as j, j+1, . . . , n l , and edges connecting the vehicles and landmarks as s , s+1 , . . . , ne , where i = j = s = 1. The observability vector associated with an edge/measurement is represented using with the edge number as subscript for simplicity. For example, the observability vector between landmark-j and vehicle-i, oji is represented by s . Now, the following theorem for a general RPMG can be stated. Theorem 2. The covariance associated with the vehicle-i, i = 1, . . . , n v , due to the landmark-j, j = 1, . . . , n l , is given by p ij = s∈S 1 s ,(26) where s = 1, . . . , n e , S is the set of edges that forms a path from vehicle-i to the landmark-j, n v is the number of vehicles, n l is the number of landmarks, and n e is the number of edges in the RPMG. Proof. The covariances associated with each vehicle for a twovehicle-two-landmark configuration is given by equations (24) and (25), followed by three vehicles in (26) and (26). The relation given in Theorem 2 is clearly reflected in the elements of the corresponding covariance matrices. The generalization to n v vehicles and n l landmarks is straightforward from the previous analysis. However, we prove the theorem through contradiction. Consider Fig. 3b(ii). According to Theorem 2, the term/edge 1 oa1 2 should be present in p 2a . Suppose we write the P matrix without that term. The new P matrix and the original P matrix found by observability analysis given by (26) is written sideby-side showing only the element corresponding to p 2a .   · · · · · · · · · · · · 1 o12 2 · · · · · · · · · · · ·   ,   · · · · · · · · · · · · 1 o12 2 + 1 oa1 2 · · · · · · · · · · · ·   .(27) The new O T O matrix for the system found by inverting the first P matrix will be O T O =    oa1 4 −o12 2 +oa1 2 o12 2 oa1 2 o12 2 −oa1 2 0 o12 2 oa1 2 o12 2 −oa1 2 o12 2 oa1 2 −o12 2 +oa1 2 0 0 0 1 oa3 2    ,(28) which contradicts with the O T O matrix derived from equation (26), which is   o12 2 + oa1 2 −o12 2 0 −o12 2 o12 2 0 0 0 oa3 2   .(29) Similarly, if an additional term 1 oa3 2 is present in p 2a , the corresponding O T O matrix for the system will be O T O =    oa1 2 + o12 2 oa3 2 o12 2 +oa3 2 o12 2 oa3 2 o12 2 +oa3 2 0 − o12 2 oa3 2 o12 2 +oa3 2 o12 2 oa3 2 o12 2 +oa3 2 0 0 0 oa3 2    ,(30) which also contradicts with the O T O matrix given in (29). Hence, the relation given in Theorem 2 is always true. The following corollaries can be written from Theorem 2. Corollary 1. If there is more than one path from a landmark to a vehicle, and these paths are numbered from 1 to g ij , where g ij is the total number of paths from the landmark-j to the vehicle-i, then the total covariance of the vehicle is given by p ij = gij κ=1 p κ ij ,(31) where i = 1, . . . , n v , j = 1, . . . , n l , and p κ ij is the covariance of the vehicle-i due to the landmark-j considering only the path-κ. Corollary 2. If there is more than one landmark connected to a vehicle, then the covariance of the vehicle is given by p i = j∈J p ij ,(32) where i = 1, . . . , n v , j = 1, . . . , n l , and J is the set of landmarks connected to the vehicle-i. Proof. The method to calculate covariances for Corollary 1 and 2 is given in Theorem 2. The summation is based on the properties of the information matrix given as follows [42], [43] If X = (X 1 , X 2 , . . . , X n ) and X 1 , X 2 , . . . , X n are independent random variables, then I X (α) = I X1 (α) + I X2 (α) + . . . I Xn (α), where I x (α) is the information matrix defined as I x (α) = E α ∂ ∂α log f (X|α) 2 = Var α ∂ ∂α log f (X|α) . Since f (x|α) = n i=1 f i (x i |α), where f i (·|α) is the pdf of X i , Var ∂ ∂α log f (X|α) = n i=1 Var ∂ ∂α log f i (X i |α) , I X (α) = n i=1 I Xi (α). Since covariance is the inverse of information, we can find the total covariance associated with each vehicle by adding the components from all the paths and landmarks. In the next sub-sections, we show how this information can be used to analyze the evolution of covariance in multi-vehiclelandmark systems with range and bearing measurements. A. Range measurements Consider the configuration given in Fig. 3(i). The vehicle kinematics are defined aṡ x i = v cos ψ 1 , y i = v sin ψ 1 ,(33) where i = 1, 2 with range measurements h 1a = (x 1 − x a ) 2 + (y 1 − y a ) 2 ,(34)h 12 = (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 ,(35)h 2b = (x 2 − x b ) 2 + (y 2 − y b ) 2 ,(36) and we derive the observability matrix using the Lie derivatives [36]. To simplify the representation we denote x 1a = (x 1 − x a ), x 12 = (x 1 − x 2 ), x 2b = (x 2 − x b ), y 1a = (y 1 − y a ), y 12 = (y 1 − y 2 ), y 2b = (y 2 − y b ). Define f L = cos ψ 1 sin ψ 1 cos ψ 2 sin ψ 2 ,(37) and the vehicle kinematics can be represented aṡ X = vf L .(38) The gradient of zero th order Lie derivatives are given as H 1a = x1a R1a y1a R1a 0 0 , H 2b = 0 0 x 2b R 2b y 2b R 2b , H 12 = x12 R12 y12 R12 −(x12) R12 −(y12) R12 , where R (·) is the distance between nodes. The gradient of first order Lie derivatives are given as ∂ ∂X ∂h 1a ∂X · f =      (y1a) 2 Cψ1−(x1a)(y1a)Sψ1 R 3 1a (x1a) 2 Sψ1−(x1a)(y1a)Cψ1 R 3 1a 0 0      , ∂ ∂X ∂h 2b ∂X · f =      0 0 (y 2b ) 2 Cψ2+(−x 2b )(y 2b )Sψ2 R 3 2b (x 2b ) 2 Sψ2+(−y 2b )(x 2b )Cψ2 R 3 2b      , ∂ ∂X ∂h 12 ∂X · f =          −2(y12)S( ∆ψ − 2 )((x12)C( ∆ψ + 2 )+(y12)S( ∆ψ + 2 )) R 3 12 2(x12)S( ∆ψ − 2 )((x12)C( ∆ψ + 2 )+(y12)S( ∆ψ + 2 )) R 3 12 2(y12)S( ∆ψ − 2 )((x12)C( ∆ψ + 2 )+(y12)S( ∆ψ + 2 )) R 3 12 −2(x12)S( ∆ψ − 2 )((x12)C( ∆ψ + 2 )+(y12)S( ∆ψ + 2 )) R 3 12          , where sin and cos are abbreviated as S and C, ∆ψ − = ψ 1 − ψ 2 , and ∆ψ + = ψ 1 + ψ 2 . Observability matrix is formed by using Lie derivatives up to first order as O = ∇L 0 ∇L 1 ,(39) where L 0 and L 1 are the zero th and first order Lie derivatives respectively. The covariance matrix P is found by inverting O T O with the assumption of Γ = I. The standard deviation in x direction for the first vehicle can be found by taking the square root of the first element of the P matrix, σ x1 = P (1, 1). Similarly, the standard deviation in the y direction and the combined position uncertainty can be found as σ y1 = P (2, 2),(41)σ p1 = σ 2 x1 + σ 2 y1 .(42) Since the derived P matrix is very large with several terms, we consider some simplifying assumptions to formulate an approximate relation. All the distance terms were substituted with a single average value. The resulting relation is given as σ p1 = 2 3 + R 2 g csc 2 (ψ 1 − θ g ),(43) where R g , θ g are the average value of distance and LOS angle. It is evident from the relation that the covariance of vehicle depends on the distances from the landmarks and the LOS angles to them. As the distance increases, the covariance also increases. Similarly, σ p of other vehicles can also be found. B. Bearing measurements Let's extend the vehicle model to contain three states. The new model is given asẋ i = v cos ψ i , y 1 = v sin ψ i , ψ i = ω i , where vehicle i = 1, 2. With the same configuration as in Fig. 3(i), define bearing measurement equations as h a1 = arctan y a − y 1 x a − x 1 − ψ 1 ,(44)h b2 = arctan y b − y 2 x b − x 2 − ψ 2 ,(45)h 12 = arctan y 1 − y 2 x 1 − x 2 − ψ 2 ,(46)h 21 = arctan y 2 − y 1 x 2 − x 1 − ψ 1 ,(47) the gradient of zeroth order Lie derivatives are given as H a1 = −(y1−ya) R 2 a1 x1−xa R 2 a1 −1 0 0 0 ,(48)H b2 = 0 0 0 −(y2−y b ) R 2 b2 x2−x b R 2 b2 −1 ,(49)H 12 = y2−y1 R 2 12 x1−x2 R 2 12 0 y1−y2 R 2 12 x2−x1 R 2 12 −1 ,(50)H 21 = y2−y1 R 2 21 x1−x2 R 2 21 −1 y1−y2 R 2 21 x2−x1 R 2 21 0 ,(51) using the geometry, the equations are changed to make it in terms of LOS angles as follows H a1 = sin θa1 Ra1 − cos θa1 Ra1 −1 0 0 0 ,(52)H b2 = 0 0 0 sin θ b2 R b2 − cos θ b2 R b2 −1 ,(53)H 12 = − sin θ12 R12 cos θ12 R12 0 sin θ12 R12 − cos θ12 R12 −1 ,(54)H 21 = sin θ21 R21 − cos θ21 R21 −1 − sin θ21X = Ẋ 1 X 2 = f v v + f ω1 ω 1 + f ω2 ω 2 ,(56) and a similar procedure to the range measurement case is followed to find the first order Lie derivatives, observability grammian and covariance matrix P . The covariance in position of the first vehicle is given by σ 2 p1 = 9 2 R 2 g 1 + R 2 g 2 + R 2 g + 2 cos(2(ψ 1 − θ g )) + (R 2 g + R 4 g ) csc 2 (ψ 1 − θ g ). Similarly, σ p of other vehicles can also be found. This approximate closed form covariance is used in the NMPC as Step 2 in Fig. 2(a). Next section presents the complete NMPC formulation combining the MHE scheme given in Sec. III and the uncertainty results derived from analysis given in Sec. IV. V. NMPC FORMULATION NMPC is a state-of-the-art technique for real-time optimal control. At each time step, the constrained optimization problem is solved based on the plant model for a finite time horizon, and the procedure is repeated with states updated through feedback in the next iteration [44]. Fig. 2 shows the block diagram of the NMPC scheme used in this paper. The optimal control sequence is computed for the prediction horizon τ h from which only the first action is applied to the system at each time step. A point mass kinematic model is considered for the vehicles. We assume that the altitude and velocities of the AAVs remain constant during transit. The general kinematic model is given asẊ =            v cos ψ 1 v sin ψ 1 ω 1 . . . v cos ψ nv v sin ψ nv ω nv            ,(57) where v is the linear velocity, ψ the heading angle, ω the angular velocity, and n v is the number of vehicles. The objective function for the NMPC is defined as min ω1···ωn v ∈PC(t,t+τ h ) J = t+τ h t nv i=1 [C 1i + W i C 2i ] ,(58)subject to:Ẋ = f (X, ω), ω ∈ ω − , ω + , where C 1i = (x i − x Di ) 2 + (y i − y Di ) 2 ,(59) is the cost associated with minimizing the distance between the vehicle and the destination. (x i , y i ) is the position of the i th vehicle and (x Di , y Di ) are their respective destination points. ω − and ω + are the lower and upper bounds of ω, and PC(t, t + τ h ) denotes the space of piece-wise continuous function defined over the time interval [t, t + τ h ]. C 2i is the cost to ensure the estimation covariance is within a bound. It is defined as: C 2i = 0, if λ i ≥ η. (η − λ i ) 2 , otherwise.(60) where η is a tuning parameter related to the number of connections required. Increasing η will result in vehicles moving closer to the landmarks and increase the connections. For satisfactory localization, observability conditions should be satisfied, which require connections with at least two landmarks [37]. Hence, the value of η should be selected as η ≥ 2. The parameter λ i is the second smallest eigenvalue of the Laplacian matrix which is formed as: L i (X) = ∆ i (X) − A i (X),(61) where A i (X) is the adjacency matrix defined similar to [45] as A imn = e −κ(||Rmn||−ρ) Rs−ρ , ||R mn || ≤ R s . 0, ||R mn || > R s .(62) and ∆ i (X) is a diagonal matrix with elements ∆ imm = N n=1 A imn(63) where m, n = 1 to N , and N is the number of nodes of the graph connecting the vehicles and landmarks. κ is a constant which determines the convergence rate of the exponential function, and ρ is used to set a minimum distance between the landmarks and the vehicles to avoid collisions. ||R mn || is the distance between the m th and n th nodes, and R s is the sensor range of the vehicles. This formulation of the adjacency matrices, rather than updating it with binary values, helps drive the vehicles closer to the landmarks than just maintaining the connections by keeping them in the sensor range and reducing the distance between the landmarks and the vehicles help in decreasing the estimation covariance as explained in sec. IV. The maximum value λ i can take is equal to the number of connections of each vehicle, and this insight is used in formulating (60). The weight W i associated with C 2i is defined in the following way W i = W, if 3σ pi ≥ σ c . 0, otherwise.(64) where σ pi is the standard deviation in the estimated position of the i th vehicle and the constant σ c is the specified critical value. This adaptive weight formulation is used to obtain a trade-off between the two objectives. σ pi s are calculated using the expression σ 2 pi = 9 2 R 2 g 1 + R 2 g 2 + R 2 g + 2 cos(2(ψ i − θ g )) + (R 2 g + R 4 g ) csc 2 (ψ i − θ g ), which is explained in detail in section IV-B. The terms C 1 and C 2 of the objective function (58) is normalized as follows: C 1 (t) = C 1 (t) − min(C 1 (t)) max(C 1 (t)) − min(C 1 (t)) ,(65)C 2 (t) = C 2 (t) − min(C 2 (t)) max(C 2 (t)) − min(C 2 (t)) . The NMPC objective function given equation (58) uses the expressions given in equations (59)(60)(64)(65) and equation (66). The NMPC objective function is solved along with state and control constraints. VI. RESULTS AND DISCUSSION Extensive numerical simulations were carried out for validating the proposed scheme using CasADi-Python [46]. We use the total path length and average estimation error as metric for analyzing the performance of the proposed approach. We perform the following analysis (i) effect of horizon length in the NMPC on the path length and estimation error (ii) effect of cooperation (iii) comparison with the approach proposed in [35] and (iv) effect of increasing the number of vehicles in the region to 10. Before presenting the analysis, we will describe the simulation setting. A. Simulation setup We consider an environment of 200m × 200m, where 20 landmarks are randomly placed. Each vehicle starts at a given location and has a desired goal location. The vehicles have a constant velocity of 5 m/s. The angular velocities of the agents are constrained by [−π/2, π/2] rad/s due to the practical considerations on the turn rate of the agents. The value of η is selected as 2, and the values of k and ρ are selected as 5 and 0.5, respectively. The weight W is selected as 10000, sensor range of the vehicles, R s = 50 m, and σ c = 3 m. The measurement noise covariance matrix Γ is selected as a n Γ × n Γ matrix with 0.01 in its diagonals, where n Γ is the number of received measurements. Each time step is 0.1s for all the simulations. The simulations were carried out on a Ubuntu 18.04, Intel i9 workstation with 64GB RAM. B. Effect of NMPC horizon length In NMPC, the horizon length plays a key role between path optimality and computational time. The larger the horizon, better the path obtained at the cost of increased computational time. This effect can be seen in Fig. 4a, where the computation time for prediction horizon of τ h = 1 s is 0.056 s, however, the paths are not optimal. With increase in τ h to 15s, there is significant improvement in the path of vehicle 1 at increased computational time of 0.42s per iteration. With further increase in horizon length to τ h = 40 s, the average time to compute an iteration is 3.43 s, but the obtained path length for the vehicles is near-optimal. The path length for τ h = 25s is close to that obtained with τ h = 40 but takes only 1.2s. Further, we conducted Monte-Carlo simulation to see the effect of placement of landmarks on the vehicle paths. Figure 5 shows the effect of change in landmark placement for different τ h . From the Fig.5(a), we can see that the computation time for τ h = 1s is very less but the vehicle which reaches the destination at the last on average is also high as shown in 5(b). Note that, on average, the times taken by the last agent reaching its goal for τ h =25s and τ h =40s are almost similar, however, τ 2 5 takes far less time. Hence we consider τ h = 25s for the rest of the simulations C. Effect of cooperation One of the main contributions of this paper is to show that with cooperation, the vehicles can jointly determine minimal distance paths to their goal locations while meeting localization accuracy. To show the effect of cooperation, we consider a specific scenario as shown in Fig. 6, where the landmarks are located at the top of the scenario. Two simulations were carried out with five vehicles, in which one scenario involved cooperative vehicles and the other without cooperation. We consider τ h = 25 s, and R s = 30 m. The trajectories taken by the vehicles in both the cases are shown in Fig. 6. It can be seen from Fig. 6(a) that the vehicles with cooperation determined a shorter path since they used adjacent vehicles for localization, whereas the non-cooperative vehicles took a longer path to go near the landmarks for localization, as shown in Fig. 6(b). The estimation errors are a bit high for the cooperative case due to the mutual localization of the vehicles. D. MHE vs EKF The performance of the MHE estimator was compared against a standard EKF formulation to validate the superiority of the proposed scheme. In Fig. 2, the step 1 sub-block of the MHE is replaced by the EKF. The horizon length for the MHE is selected as N e = 20, and all other initial parameters are taken the same for the MHE and the EKF. The estimated trajectories of the vehicles, true trajectories, and the errors in position for both the MHE and the EKF are given in Fig. 7. It can be seen that the estimation errors are less for the MHE, and it is also more stable compared to the EKF. The vehicles' mean square error (MSE) for the MHE is 0.46 m and that with the EKF is 0.79 m. The computation time requirement for the MHE is 1.21 s, while the EKF takes 1.10 s. Although the MHE takes 9% more computational time than the EKF, its accuracy is 72% better than the EKF. E. 10 vehicles To test the performance of the scheme for larger systems, a scenario involving ten vehicles in a 500m × 500m plane with all the vehicles moving at a velocity of 10 m/s was considered. Multiple simulations were carried out with random configurations of the landmarks, initial positions, and goal points. Two example results are shown in Fig. 8. It can be seen that the estimator performance was satisfactory since the actual states and the estimated states are well aligned for all the ten vehicles. The average computation time per iteration was 7.25 s due to the increase in the size of state matrix for 10 vehicles. VII. CONCLUSIONS A nonlinear model predictive control scheme combined with moving horizon estimation was proposed to aid cooperative localization of a group of AAVs in transit. The controller used an approximate analytical expression for calculating the expected covariance of the vehicles through the prediction horizon, which was derived using the insights obtained from analyzing the observability and the path information from the landmark-vehicle graph. The controller determined near optimal paths for the vehicles while satisfying various state and localization constraints. We analyzed through simulations the role of prediction horizon on the optimality of the vehicle paths and the required computation time. The proposed moving horizon estimator also outperformed the EKF with lower estimation error values at a small additional computation time. A comparison was performed between cooperative and noncooperative vehicles to show the significance of cooperation in determining paths under localization constraints. The approached proposed in this paper can be extended in several direction. One potential analysis is to determine how many landmarks are sufficient for a given vehicle to reach the destination meeting localization constraints. Another extension can be to include obstacle avoidance which reaching their destination as part of the problem. Experimental validation is another direction to implement the algorithms in real-world vehicle. Fig. 1 : 1(a) Path planning scenario. (b) Relative position measurement graph with vehicles and landmarks as nodes and measurements as edges. Fig. 2 : 2Block diagram and a graphical representation of the proposed NMPC-MHE control scheme. Fig. 3 : 3(a) Different configurations of two vehicles and two landmarks. (b) Different configuration of the system with 3 vehicles and 2 landmarks (c) Notations for a general multi-vehicle-landmark RPMG. v = cos ψ 1 sin ψ 1 0 cos ψ 2 sin ψ 2 0 , f ω1 = 0 0 1 0 0 0 , f ω2 = 0 0 0 0 0 1 , and the dynamics can be represented aṡ Fig. 4 :Fig. 5 :Fig. 6 : 456The average computational time taken per iteration for different τ h . Monte-Carlo simulation for prediction horizon τ h = 1, 25, 40. (a) Average computational time per iteration (b) Average time taken by all the agents to reach their destinations. Effect of cooperation. (a) Trajectory of vehicles with cooperation. (b) Trajectory of vehicles without cooperation. Fig. 7 :Fig. 8 : 78(a) NMPC solution with EKF based estimation (b) NMPC solution with MHE (c) Error in position of vehicle 1 (d) Error in position of vehicle 2 (e) Error in position of vehicle 3 Trajectories of 10 vehicles using NMPC-MHE scheme. Manoharan is a Graduate Student at IIIT Delhi, New Delhi -110020, India. email: [email protected] Rajnikant Sharma is Assistant Professor at University of Cincinnati, Cincinnati, OH 45221. email: [email protected] P.B. Sujit is Associate Professor at IISER Bhopal, Bhopal -462066, India. email: [email protected] Urban air mobility. Faa, concept of operations v1.0. [OnlineFAA. (2020) Urban air mobility, concept of operations v1.0. [Online]. European drones outlook study. Sesar, SESAR. (2016) European drones outlook study. [Online]. Mobile robot localization using landmarks. M Betke, L Gurvits, IEEE transactions on robotics and automation. 132M. Betke and L. Gurvits, "Mobile robot localization using landmarks," IEEE transactions on robotics and automation, vol. 13, no. 2, pp. 251- 263, 1997. Finding landmarks for mobile robot navigation. S Thrun, Proc. of International Conference on Robotics and Automation. of International Conference on Robotics and AutomationIEEE2S. Thrun, "Finding landmarks for mobile robot navigation," in Proc. of International Conference on Robotics and Automation, vol. 2. IEEE, 1998, pp. 958-963. Vision-only localization. H Lategahn, C Stiller, IEEE Transactions on Intelligent Transportation Systems. 153H. Lategahn and C. Stiller, "Vision-only localization," IEEE Transac- tions on Intelligent Transportation Systems, vol. 15, no. 3, pp. 1246- 1257, 2014. Vision-based simultaneous localization and mapping with two cameras. G.-H Kim, J.-S Kim, K.-S Hong, IEEE/RSJ International Conference on Intelligent Robots and Systems. G.-H. Kim, J.-S. Kim, and K.-S. Hong, "Vision-based simultaneous localization and mapping with two cameras," in IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005, pp. 1671-1676. A uwb based localization system for indoor robot navigation. S Krishnan, P Sharma, Z Guoping, O H Woon, International Conference on Ultra-Wideband. IEEES. Krishnan, P. Sharma, Z. Guoping, and O. H. Woon, "A uwb based localization system for indoor robot navigation," in International Con- ference on Ultra-Wideband. IEEE, 2007, pp. 77-82. Linear bayesian filter based low-cost uwb systems for indoor mobile robot localization. S Zhang, R Han, W Huang, S Wang, Q Hao, IEEE Sensors. S. Zhang, R. Han, W. Huang, S. Wang, and Q. Hao, "Linear bayesian filter based low-cost uwb systems for indoor mobile robot localization," in IEEE Sensors, 2018, pp. 1-4. Study on Cooperative Positioning System -Optimum Moving Strategies for CPS-III. R Kurazume, S Hirose, IEEE International Conference on Robotics and Automation. 4R. Kurazume and S. Hirose, "Study on Cooperative Positioning System -Optimum Moving Strategies for CPS-III," IEEE International Confer- ence on Robotics and Automation, vol. 4, pp. 2896-2903, 1998. Collective localization: a distributed Kalman filter approach to localization of groups of mobile robots. S Roumeliotis, G Bekey, IEEE International Conference on Robotics and Automation. 3S. Roumeliotis and G. Bekey, "Collective localization: a distributed Kalman filter approach to localization of groups of mobile robots," IEEE International Conference on Robotics and Automation, vol. 3, pp. 2958- 2965, 2000. Cooperative localization and control for multi-robot manipulation. J Spletzer, A K Das, R Fierro, C J Taylor, V Kumar, J P Ostrowski, IEEE International Conference on Intelligent Robots and Systems. 2J. Spletzer, A. K. Das, R. Fierro, C. J. Taylor, V. Kumar, and J. P. Ostrowski, "Cooperative localization and control for multi-robot ma- nipulation," IEEE International Conference on Intelligent Robots and Systems, vol. 2, pp. 631-636, 2001. Multirobot Cooperative Localization. A I Mourikis, S I Roumeliotis, IEEE Transactions on Robotics. 224A. I. Mourikis and S. I. Roumeliotis, "Multirobot Cooperative Local- ization," IEEE Transactions on Robotics, vol. 22, no. 4, pp. 666-681, 2006. Distributed maximum a posteriori estimation for multi-robot cooperative localization. E D Nerurkar, S I Roumeliotis, A Martinelli, Proceedings -IEEE International Conference on Robotics and Automation. -IEEE International Conference on Robotics and AutomationE. D. Nerurkar, S. I. Roumeliotis, and A. Martinelli, "Distributed maximum a posteriori estimation for multi-robot cooperative localiza- tion," Proceedings -IEEE International Conference on Robotics and Automation, pp. 1402-1409, 2009. Cooperative localization of multiuavs via dynamic nonparametric belief propagation under gps signal loss condition. J Wan, L Zhong, F Zhang, International Journal of Distributed Sensor Networks. 102562380J. Wan, L. Zhong, and F. Zhang, "Cooperative localization of multi- uavs via dynamic nonparametric belief propagation under gps signal loss condition," International Journal of Distributed Sensor Networks, vol. 10, no. 2, p. 562380, 2014. Vision-based target detection and localization via a team of cooperative uav and ugvs. S Minaeian, J Liu, Y.-J Son, IEEE Transactions on systems, man, and cybernetics: systems. 46S. Minaeian, J. Liu, and Y.-J. Son, "Vision-based target detection and localization via a team of cooperative uav and ugvs," IEEE Transactions on systems, man, and cybernetics: systems, vol. 46, no. 7, pp. 1005- 1016, 2016. Cooperative localization of vehicles without inter-vehicle measurements. M Frohle, C Lindberg, H Wymeersch, IEEE Wireless Communications and Networking Conference. M. Frohle, C. Lindberg, and H. Wymeersch, "Cooperative localization of vehicles without inter-vehicle measurements," in IEEE Wireless Communications and Networking Conference (WCNC), 2018, pp. 1-6. Rangeonly based cooperative localization for mobile robots," in 21st international conference on information fusion (FUSION). C Pierre, R Chapuis, R Aufrère, J Laneurit, C Debain, IEEEC. Pierre, R. Chapuis, R. Aufrère, J. Laneurit, and C. Debain, "Range- only based cooperative localization for mobile robots," in 21st interna- tional conference on information fusion (FUSION). IEEE, 2018, pp. 1933-1939. Multi-robot cooperative localization with range-only measurement by uwb. J Liu, J Pu, L Sun, Y Zhang, Chinese Automation Congress (CAC). IEEEJ. Liu, J. Pu, L. Sun, and Y. Zhang, "Multi-robot cooperative localization with range-only measurement by uwb," in Chinese Automation Congress (CAC). IEEE, 2018, pp. 2809-2813. Ultra-wideband and odometry-based cooperative relative localization with application to multi-uav formation control. K Guo, X Li, L Xie, IEEE transactions on cybernetics. 506K. Guo, X. Li, and L. Xie, "Ultra-wideband and odometry-based cooperative relative localization with application to multi-uav formation control," IEEE transactions on cybernetics, vol. 50, no. 6, pp. 2590- 2603, 2019. Cooperative localization under limited connectivity. J Zhu, S S Kia, IEEE Transactions on Robotics. 356J. Zhu and S. S. Kia, "Cooperative localization under limited connec- tivity," IEEE Transactions on Robotics, vol. 35, no. 6, pp. 1523-1530, 2019. Evolutionary route planner for unmanned air vehicles. C Zheng, L Li, F Xu, F Sun, M Ding, IEEE Transactions on robotics. 214C. Zheng, L. Li, F. Xu, F. Sun, and M. Ding, "Evolutionary route planner for unmanned air vehicles," IEEE Transactions on robotics, vol. 21, no. 4, pp. 609-620, 2005. Health aware stochastic planning for persistent package delivery missions using quadrotors. A A Agha-Mohammadi, N K Ure, J P How, J Vian, IEEE International Conference on Intelligent Robots and Systems. A. A. Agha-Mohammadi, N. K. Ure, J. P. How, and J. Vian, "Health aware stochastic planning for persistent package delivery missions using quadrotors," IEEE International Conference on Intelligent Robots and Systems, pp. 3389-3396, 2014. Planning Paths for Package Delivery in Heterogeneous Multirobot Teams. N Mathew, S L Smith, S L Waslander, IEEE Transactions on Automation Science and Engineering. 124N. Mathew, S. L. Smith, and S. L. Waslander, "Planning Paths for Pack- age Delivery in Heterogeneous Multirobot Teams," IEEE Transactions on Automation Science and Engineering, vol. 12, no. 4, pp. 1298-1308, 2015. Heuristic approaches in robot path planning: A survey. T T Mac, C Copot, D T Tran, R. De Keyser, Robotics and Autonomous Systems. 86T. T. Mac, C. Copot, D. T. Tran, and R. De Keyser, "Heuristic approaches in robot path planning: A survey," Robotics and Autonomous Systems, vol. 86, pp. 13-28, 2016. Multiobjective path planning: Localization constraints and collision probability. S D Bopardikar, B Englot, A Speranzon, IEEE Transactions on Robotics. 313S. D. Bopardikar, B. Englot, and A. Speranzon, "Multiobjective path planning: Localization constraints and collision probability," IEEE Transactions on Robotics, vol. 31, no. 3, pp. 562-577, 2015. Landmarks based path planning for uavs in gpsdenied areas. S Singh, P Sujit, IFAC-PapersOnLine. 491S. Singh and P. Sujit, "Landmarks based path planning for uavs in gps- denied areas," IFAC-PapersOnLine, vol. 49, no. 1, pp. 396-400, 2016. Path planning for unmanned vehicles with localization constraints. K Sundar, S Rathinam, R Sharma, Optimization Letters. 135K. Sundar, S. Rathinam, and R. Sharma, "Path planning for unmanned vehicles with localization constraints," Optimization Letters, vol. 13, no. 5, pp. 993-1009, 2019. A localizability constraint-based path planning method for autonomous vehicles. B Irani, J Wang, W Chen, IEEE Transactions on Intelligent Transportation Systems. 207B. Irani, J. Wang, and W. Chen, "A localizability constraint-based path planning method for autonomous vehicles," IEEE Transactions on Intelligent Transportation Systems, vol. 20, no. 7, pp. 2593-2604, 2018. Uav waypoint opportunistic navigation in gnss-denied environments. J B Rawlings, B R M Bakshi ; Z, Y Kassas, J Yang, J Khalife, Morales, IEEE Transactions on Aerospace and Electronic Systems, 2021. estimation. 29Particle filtering and moving horizonJ. B. Rawlings and B. R. Bakshi, "Particle filtering and moving horizon [29] Z. M. Kassas, Y. Yang, J. Khalife, and J. Morales, "Uav waypoint op- portunistic navigation in gnss-denied environments," IEEE Transactions on Aerospace and Electronic Systems, 2021. estimation," Computers & chemical engineering, vol. 30, no. 10-12, pp. 1529-1541, 2006. An optimization based moving horizon estimation with application to localization of autonomous underwater vehicles. S Wang, L Chen, D Gu, H Hu, Robotics and Autonomous Systems. 6210S. Wang, L. Chen, D. Gu, and H. Hu, "An optimization based moving horizon estimation with application to localization of autonomous un- derwater vehicles," Robotics and Autonomous Systems, vol. 62, no. 10, pp. 1581-1596, 2014. An optimization based approach for relative localization and relative tracking control in multirobot systems. M W Mehrez, G K Mann, R G Gosine, Journal of Intelligent & Robotic Systems. 852M. W. Mehrez, G. K. Mann, and R. G. Gosine, "An optimization based approach for relative localization and relative tracking control in multi- robot systems," Journal of Intelligent & Robotic Systems, vol. 85, no. 2, pp. 385-408, 2017. Decentralized nonlinear model predictive control for 3d formation of multirotor micro aerial vehicles with relative sensing and estimation. I K Erunsal, A Martinoli, R Ventura, International Symposium on Multi-Robot and Multi-Agent Systems. IEEEI. K. Erunsal, A. Martinoli, and R. Ventura, "Decentralized nonlinear model predictive control for 3d formation of multirotor micro aerial ve- hicles with relative sensing and estimation," in International Symposium on Multi-Robot and Multi-Agent Systems. IEEE, 2019, pp. 176-178. Decentralized moving horizon estimation for networked navigation system with packet dropouts. S Liu, G Zhao, Y He, C Gao, 39th Chinese Control Conference (CCC). IEEES. Liu, G. Zhao, Y. He, and C. Gao, "Decentralized moving horizon estimation for networked navigation system with packet dropouts," in 39th Chinese Control Conference (CCC). IEEE, 2020, pp. 3381-3384. Nonlinear model predictive control to aid cooperative localization. A Manoharan, R Sharma, P B Sujit, International Conference on Unmanned Aircraft Systems (ICUAS). IEEEA. Manoharan, R. Sharma, and P. B. Sujit, "Nonlinear model predictive control to aid cooperative localization," in International Conference on Unmanned Aircraft Systems (ICUAS). IEEE, 2019, pp. 26-32. Graph-based observability analysis of bearing-only cooperative localization. R Sharma, R W Beard, C N Taylor, S Quebe, IEEE Transactions on Robotics. 282R. Sharma, R. W. Beard, C. N. Taylor, and S. Quebe, "Graph-based observability analysis of bearing-only cooperative localization," IEEE Transactions on Robotics, vol. 28, no. 2, pp. 522-529, 2011. Observability based control for cooperative localization. R Sharma, International Conference on Unmanned Aircraft Systems. R. Sharma, "Observability based control for cooperative localization," in International Conference on Unmanned Aircraft Systems, 2014, pp. 134-139. Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations. C V Rao, J B Rawlings, D Q Mayne, IEEE transactions on automatic control. 482C. V. Rao, J. B. Rawlings, and D. Q. Mayne, "Constrained state estima- tion for nonlinear discrete-time systems: Stability and moving horizon approximations," IEEE transactions on automatic control, vol. 48, no. 2, pp. 246-258, 2003. A neural state estimator with bounded errors for nonlinear systems. A Alessandri, M Baglietto, T Parisini, R Zoppoli, IEEE Transactions on Automatic Control. 4411A. Alessandri, M. Baglietto, T. Parisini, and R. Zoppoli, "A neural state estimator with bounded errors for nonlinear systems," IEEE Transactions on Automatic Control, vol. 44, no. 11, pp. 2028-2042, 1999. Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes. A Alessandri, M Baglietto, G Battistelli, Automatica. 447A. Alessandri, M. Baglietto, and G. Battistelli, "Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes," Automatica, vol. 44, no. 7, pp. 1753-1765, 2008. Advances in moving horizon estimation for nonlinear systems. A Alessandri, M Baglietto, G Battistelli, V Zavala, 49th IEEE Conference on Decision and Control (CDC). A. Alessandri, M. Baglietto, G. Battistelli, and V. Zavala, "Advances in moving horizon estimation for nonlinear systems," in 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 5681-5688. Fisher information. D Pati, D. Pati. (2016) Fisher information. [Online]. Available: https: //ani.stat.fsu.edu/ ∼ debdeep/Fisher.pdf Fisher information properties. P Zegers, Entropy. 177P. Zegers, "Fisher information properties," Entropy, vol. 17, no. 7, pp. 4918-4939, 2015. Model-based predictive control: a practical approach. J A Rossiter, CRC pressJ. A. Rossiter, Model-based predictive control: a practical approach. CRC press, 2003. Decentralized control of connectivity for multi-agent systems. M C De Gennaro, A Jadbabaie, Proceedings of the 45th IEEE Conference on Decision and Control. the 45th IEEE Conference on Decision and ControlM. C. De Gennaro and A. Jadbabaie, "Decentralized control of con- nectivity for multi-agent systems," in Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 3628-3633. CasADi -A software framework for nonlinear optimization and optimal control. J A E Andersson, J Gillis, G Horn, J B Rawlings, M Diehl, Mathematical Programming Computation. 111J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, "CasADi -A software framework for nonlinear optimization and opti- mal control," Mathematical Programming Computation, vol. 11, no. 1, pp. 1-36, 2019.
[]
[ "Considering human aspects on strategies for designing and managing distributed human computation", "Considering human aspects on strategies for designing and managing distributed human computation" ]
[ "Lesandro Ponciano \nDepartment of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil\n", "Francisco Brasileiro \nDepartment of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil\n", "Nazareno Andrade [email protected] \nDepartment of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil\n", "Lívia Sampaio \nDepartment of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil\n" ]
[ "Department of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil", "Department of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil", "Department of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil", "Department of Computing and Systems\nFederal University of Campina Grande\nAv. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil" ]
[ "Journal of Internet Services and Applications" ]
A human computation system can be viewed as a distributed system in which the processors are humans, called workers. Such systems harness the cognitive power of a group of workers connected to the Internet to execute relatively simple tasks, whose solutions, once grouped, solve a problem that systems equipped with only machines could not solve satisfactorily. Examples of such systems are Amazon Mechanical Turk and the Zooniverse platform. A human computation application comprises a group of tasks, each of them can be performed by one worker. Tasks might have dependencies among each other. In this study, we propose a theoretical framework to analyze such type of application from a distributed systems point of view. Our framework is established on three dimensions that represent different perspectives in which human computation applications can be approached: quality-of-service requirements, design and management strategies, and human aspects. By using this framework, we review human computation in the perspective of programmers seeking to improve the design of human computation applications and managers seeking to increase the effectiveness of human computation infrastructures in running such applications. In doing so, besides integrating and organizing what has been done in this direction, we also put into perspective the fact that the human aspects of the workers in such systems introduce new challenges in terms of, for example, task assignment, dependency management, and fault prevention and tolerance. We discuss how they are related to distributed systems and other areas of knowledge.
10.1186/s13174-014-0010-4
[ "https://arxiv.org/pdf/1506.02200v1.pdf" ]
644,776
1506.02200
ad845f164db218fbbe9ebe400595b3244972088b
Considering human aspects on strategies for designing and managing distributed human computation SpringerCopyright Springer2014 Lesandro Ponciano Department of Computing and Systems Federal University of Campina Grande Av. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil Francisco Brasileiro Department of Computing and Systems Federal University of Campina Grande Av. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil Nazareno Andrade [email protected] Department of Computing and Systems Federal University of Campina Grande Av. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil Lívia Sampaio Department of Computing and Systems Federal University of Campina Grande Av. Aprígio Veloso, 882 -Bloco CO, 58.429-900Campina Grande -PBBrazil Considering human aspects on strategies for designing and managing distributed human computation Journal of Internet Services and Applications LondonSpringer510201410.1186/s13174-014-0010-4Human computationCrowdsourcingDistributed applicationsHuman factors A human computation system can be viewed as a distributed system in which the processors are humans, called workers. Such systems harness the cognitive power of a group of workers connected to the Internet to execute relatively simple tasks, whose solutions, once grouped, solve a problem that systems equipped with only machines could not solve satisfactorily. Examples of such systems are Amazon Mechanical Turk and the Zooniverse platform. A human computation application comprises a group of tasks, each of them can be performed by one worker. Tasks might have dependencies among each other. In this study, we propose a theoretical framework to analyze such type of application from a distributed systems point of view. Our framework is established on three dimensions that represent different perspectives in which human computation applications can be approached: quality-of-service requirements, design and management strategies, and human aspects. By using this framework, we review human computation in the perspective of programmers seeking to improve the design of human computation applications and managers seeking to increase the effectiveness of human computation infrastructures in running such applications. In doing so, besides integrating and organizing what has been done in this direction, we also put into perspective the fact that the human aspects of the workers in such systems introduce new challenges in terms of, for example, task assignment, dependency management, and fault prevention and tolerance. We discuss how they are related to distributed systems and other areas of knowledge. Introduction Many studies have focused on increasing the performance of machine-based computational systems over the last decades. As a result, much progress has been made allowing increasingly complex problems to be efficiently solved. However, despite these advances, there are still tasks that cannot be accurately and efficiently performed even when the most sophisticated algorithms and computing architectures are used [1,2]. Examples of such tasks are those related to natural language processing, image understanding and creativity [3,4]. A common factor in these kinds of tasks is their suitability to human abilities; human beings can solve them with high efficiency and accuracy [1,2]. In the last years, there has emerged a new computing approach that takes advantage of human abilities to execute these kinds of tasks. Such approach has been named Human Computation [1,5]. Applications designed to execute on human computation systems may encompass one or multiple tasks. They are called distributed human computation applications when they are composed of multiple tasks, and each individual task can be performed by a different human being, called worker. In the last years, distributed computing systems have been developed to support the execution of this type of application. They gather a crowd of workers connected to the Internet and manage them to execute application tasks. The precursor of such systems is reCAPTCHA [6]. Currently, there is a broad diversity of distributed human computation applications and distributed systems devoted to execute them, such as: games with a purpose [7], contests sites [8], online labor markets [9], and volunteer thinking systems [10]. In this paper, we focus on online labor markets and volunteer thinking systems. Online labor markets gather a crowd of workers that have a financial motivation [9]. The precursor of this type of system is the Amazon Mechanical Turk platform (mturk.com). Such plaform reports to have more than 400, 000 registered workers [11], and receives between 50, 000 and 400, 000 new tasks to be executed per day (mturk-tracker.com) at the time of writing. Volunteer thinking systems, in turn, gather a crowd of workers willing to execute tasks without any financial compensation [10]. One of the precursors of this type of system is the Zooniverse citizen-science platform (zooniverse.org). Currently, Zooniverse hosts 21 scientific projects and has over one million registered workers. Only Galaxy Zoo, the largest project at Zooniverse, had 50 million tasks performed by 150, 000 workers in a year of operation [12]. Thus, both labor markets and volunteer thinking are large-scale distributed human computation systems. Because the computing units in human computation systems are human beings, both the design and management of applications tap into concepts and theories from multiple disciplines. Quinn and Bederson conducted one of the first efforts to delimit such concepts and theories [1,5]. They present a taxonomy for human computation, highlighting differences and similarities to related concepts, such as collective intelligence and crowdsourcing. Yuen et al., in turn, focus on distinguishing different types of human computation systems and platforms [3,4]. More recently, Kittur et al. built a theoretical framework to analyze future perspectives in developing online labor markets that are attractive and fair to workers [13]. Differently from previous efforts, in this study, we analyze human computation under the perspective of programmers seeking to improve the design of distributed human computation applications and managers seeking to increase the effectiveness of distributed human computation systems. To conduct this study, we propose a theoretical framework that integrates theories about human aspects, design and management (D&M) strategies, and quality of service (QoS) requirements. Human aspects include characteristics that impact workers' ability to perform tasks (e.g., cognitive system and emotion), their interests (e.g., motivation and preferences), and their differences and relations (e.g., individual differences and social behavior). QoS requirements, in turn, are metrics directly related to how application owners measure applications and systems effectiveness. These metrics are typically defined in terms of time, cost, fidelity, and security. Finally, D&M strategies consist of strategies related to how the application is designed and managed. They involve activities such as application composition, task assignment, dependency management, and fault prevention and tolerance. This framework allows us to perform a literature review that expands previous literature reviews to build a vision of human computation focused on distributed systems issues. We emphasize our analysis on three perspectives: 1) findings on relevant human aspects which impact D&M decisions; 2) major D&M strategies that have been proposed to deal with human aspects; and 3) open challenges and how they relate to other disciplines. Besides providing a distributed systems viewpoint of this new kind of computational system, our analysis also puts into perspective the fact that human computation introduces new challenges in terms of effective D&M strategies. Although these challenges are essentially distributed systems challenges, some of them do not exist in machine-based distributed systems, as they are related to human aspects. These challenges call for combining distributed systems design with theories and mechanisms used in other areas of knowledge where there is extensive theory on treating human aspects, such as Cognitive Science, Behavioral Sciences, and Management Sciences. In the following, we briefly describe the human computation ecosystem addressed in this paper. Then, we present our theoretical framework. After that, we analyze the literature in the light of our framework. This is followed by the discussion of challenges and perspectives for future research. Finally, we present our conclusions. Distributed human computation ecosystem The core agents in a distributed human computation ecosystem are: requesters, workers, and platform. Requesters act in the system by submitting human computation applications. An application is a set of tasks with or without dependencies among them. Typically, a human computation task consists of some input data (e.g., image, text) and a set of instructions. There are several types of instructions, such as: transcription of an item content (e.g., reCAPTCHA tasks [6]), classification of an item (e.g., Galaxy Zoo tasks [14]), generation of creative content about an item [15], ranking and matching items [16], etc. Workers are the human beings who act as human computers in the system executing one or more tasks. They generate the task output by performing the instructions upon the received items. After executing a task, the worker provides its output. The application output is an aggregation of the outputs of all their tasks. In paid systems, when a task is performed, the requester may accept the solution if the task was satisfactorily performed; otherwise, she/he can reject it. Workers are paid only when their solutions are accepted. The receiving of tasks to be executed, the provision of their outputs, and the receiving of the payment for the performed tasks occur via a human computation platform. The Platform is a distributed system that acts as a middleware receiving requester applications and managing the execution of their tasks by the workers. Platforms manage several aspects of tasks execution, such as: providing an interface and language for tasks' specification, performing task replication, maintaining a job board with a set of tasks waiting to be performed, controlling the state of each task from the submission up to its completion. Examples of platforms with such characteristics are the online labor markets Amazon Mechanical Turk (mturk.com) and CrowdFlower (crowdflower.com), and the volunteer thinking systems Zooniverse (zooniverse.org) and CrowdCrafting (crowdcrafting.org). Online labor markets also implement functionalities that allow requesters to communicate with workers that performed their tasks, provide feedback about their outputs, and pay for the tasks performed by them. Some studies have analyzed human computation ecosystem. In general, they focus mainly on proposing a taxonomy for the area [1,3,5,17] and discussing platform issues [18,19]. Quinn et al. [1,5] propose a taxonomy for human computation that delimits its similarities and differences when compared to oth-ers fields based on human work, such as: crowdsourcing, social computing, and collective intelligence. Vukovic and Yuen et al. focus on classifying human computation platforms based on their function (e.g., design and innovation), mode (e.g., competition and marketplace platforms), or algorithms [3,17]. Dustdar and Truong [18] and Crouser and Chang [19] focus on hybrid platforms based on the collaboration between machine and human computing units. Dustdar and Truong [18] focused on strategies to provide machine computation and human computation as a service, using a single interface. Crouser and Chang [19] propose a framework of affordances, i.e., properties that are inherent to human and properties that are inherent to machine, so that they complement each other. Differently from these previous efforts, in the present work we focus on analyzing strategies for designing and managing distributed applications onto human computation platforms. Our main focus is not to survey existing human computation platforms, but to analyze D&M strategies that have been proposed to be used in these kind of platforms. Our analysis is based on a theoretical framework built upon theories and concepts from multiple disciplines dealing with (i) human aspects, such as: Motivation Theory [20], Self-determination Theory [21], Sense of Community Theory [22], Human Error Theory [23], Coordination Theory [24] and Human-in-the-Loop literature [25], and (ii) applications design, applications management and QoS aspects, such as: the great principles of computing [26]; application design methodologies [27,28]; taxonomies for application management in grid computing [29], web services [30], and organizations [31]. Theoretical framework Theoretical frameworks have several distinct roles [32]. Most important for us, they allow researchers to look for patterns in observations and to inform designers of relevant aspects for a situation. Our framework is designed to assist the analysis of the diverse aspects related to human computation applications. It is organized in three dimensions which represent different perspectives in which it is possible to approach human computation: QoS requirements, D&M strategies, and human aspects. Each dimension is closely connected to an agent in the human computation ecosystem: QoS requirements are requesters' effectiveness measures; D&M strategies are mainly related to how platforms manage application execution; and human aspects are worker characteristics. Each dimension is composed of a set of factors. Figure 1 provides an overview of the framework. Considering their relations, it is clear that the dimensions are not independent. The definition of D&M strategies is affected by both the QoS requirements and the human aspects. QoS requirements reflect requester objectives that should guide the design of suitable D&M strategies. Human aspects, in turn, consist in workers' characteristics that delimit a restriction space where D&M strategies may act aiming at optimizing QoS requirements. D&M strategies generate a final output whose quality is a measure of their capacity of optimizing QoS requirements taking into account human aspects. In the following we detail our framework by discussing the theories that support its dimensions and their factors. QoS requirements The QoS requirements dimension encompasses a set of quantitative characteristics that indicate requesters' objectives and how they evaluate application effectiveness. QoS requirements have been mainly addressed in two distinct areas: process management for organizations [33], and QoS for software systems [26]. Based on the literature from these areas, we define QoS requirements in terms of the following factors: • Time refers to the urgency to transform an input into an output. It includes response time, and delays on work queues, and a time limit (deadline) for generating an output; • Cost refers to expenditure on the application execution. It is usually divided into enactment and realization cost. Enactment cost concerns expenditure on D&M strategies, and realization cost, expenditure on application execution. • Fidelity reflects how well a task is executed, according to its instructions. Fidelity cannot be described in a universal formula and it is related to specific properties and characteristics that define the meaning of "well executed" [30]. It is a quantitative indicative of accuracy and quality. • Reproducibility refers to obtain similar output when the application is executed at different times and/or by different group of workers taken from the same population [34]. • Security relates to the confidentiality of application tasks and the trustworthiness of resources that execute them. D&M strategies Based on application design and management methodologies in machine-based computation [29,30,35] and in organizations [28,31], we define five factors for the D&M strategies dimension: application composition, incentives and rewards, dependency management, task assignment, output aggregation, and fault prevention and tolerance. Application Composition. It consists of two major activities: problem decomposition and application structuring. Problem decomposition includes the following decisions: 1) tasks granularity, e.g., generating fewer task that require more effort to be executed (coarse-grained) or generating many small tasks that require less effort to be executed (fine-grained); 2) worker interfaces for the tasks, i.e., the interface design of the web page that shows the instructions of the work to be done. Application structuring consists in how to compose the application considering possible dependencies between its tasks. As exemplified in Figures 2 and 3, the two major application structuring patterns are bag-of-tasks and workflow. Bag-of-tasks applications are composed of a set of independent tasks. For example, a group of Human Intelligence Tasks (HITs) in MTurk platform [36]. Workflow applications, in turn, are composed of a set of tasks organized in a sequence of connected steps [37]. Each step is composed by one or more tasks. Independent tasks are usually grouped in the same workflow step. Interdependent tasks, in turn, constitute different workflow steps. Incentives and Rewards. Incentive are put in place when the participants exhibit distinct objectives and the information about them are decentralized [38]. In human computation systems, requesters and workers may have different interests. For example, some workers may be interested in increasing their earnings, while requesters are interested in which tasks are performed with greater accuracy. Incentive strategies are used to align the interests of requesters and workers [39]. They are usually put in place to incentivize workers to exhibit a behavior and achieve a performance level desired by the requester, which includes executing more tasks, improving accuracy and staying longer in the system. Incentives can be broadly divided into non-monetary and monetary. Examples of non-monetary incentives are badges provided as a recognition for workers' achievements, and rank leaderboard for the workers to gauge themselves against peers. Monetary incentives are usually associated with a reward scheme, which defines the conditions for a worker to be rewarded, e.g., providing an output identical to the majority of other workers who perform the task. Game theory is an important theoretical guide to incentivize workers' engagament and effort in human computation systems [40,41]. Dependency Management. It focuses on the coordination between interdependent tasks. A framework of dependencies between tasks in human computation is presented by Minder et al. [42]. It is mainly based on the Coordination Theory [24]. Dependencies among tasks can be broadly divided into four levels: serialization, visibility, cooperation, and temporal. Serialization dependencies specify whether tasks in the application require a serial execution. Such dependencies are usually defined in application structure by routing operations, such as: sequence, parallelism, choice and loops. Visibility dependencies define whether the work performed in a task must be visible to the other tasks (e.g., when a task updates a global variable). Cooperation dependencies, in turn, define which tasks hold a shared object at each time and can perform operations on it without restriction. Finally, temporal dependencies specify whether a set of tasks must perform operations in a particular temporal dependency. Task Assignment. It defines how to choose which worker will execute a task. The strategies can be broadly divided into scheduling, job board, and recommendation. Scheduling is a push assignment; workers receive and execute tasks assigned to them. Scheduling strategies assign tasks to workers trying to optimize one or more QoS requirements. It is usually based on application and/or workers information. Job board, in turn, is a pull assignment; workers use search and browser functionalities to choose the tasks they want to execute. It allows workers to select those tasks they expect to enable them to maximize their metrics, such as: earnings, preferences, and enjoyment. Finally, Recommendation is a hybrid assignment; workers receive a set of tasks and they choose which of them they want to perform. Recommendation is mapped into scheduling when the amount of tasks recommended is 1, and it is mapped into job board when all tasks are recommended. Output Aggregation. It is concerned with aggregating sets of individual task outputs to identify the correct output or to obtain a better output. It is interchangeably called judgment aggregation and crowd consensus. An aggregation function may be implemented in several ways and it may be executed by a machine or a human. A simple example is that of different task outputs that constitute different parts of the application output; thus, the aggregation is only a merge of task outputs. A more sophisticated aggregation function may ask workers to improve available task outputs and generate an application output. Note that output aggregation is an offline procedure, i.e., it is executed after the outputs of all application tasks have already been obtained. There are also online procedures which involve failure detection in each task output, as well as strategies to detect and manage cheating workers. We discuss online procedures in fault tolerance strategies. Fault Prevention and Tolerance. Faults are events which potentially may cause a failure. A failure is a malfunction or incorrect output. Thus, fault prevention consists in avoiding events which may cause failures and fault tolerance consists in identifying and managing failures after they occur. To analyze human error in human computation systems, we join together human error concepts from Human Error Theory [23] and concepts related to the implementation of fault prevention and tolerance in computing system from Human-in-the-Loop [25] and machine-based distributed systems [43] literatures. To execute a task, a human first constructs a mental plan of a sequence of actions that ends with the conclusion of the task [23]. Three types of failures may occur in this process: mistakes, lapses and slips. Mistake is a failure in constructing a suitable plan to execute the task, then the plan is not correct. Lapses and slips are failures in the execution of a correct plan. Lapses occur when the worker forgets to perform one action of the plan. Finally, slips occur when the worker performs incorrectly an action of the plan. A diversity of faults can generate such failures, for example: lack of knowledge or time to execute the task, and stochastic cognitive variability such as variability of attention. Fault prevention strategies usually focus on methodologies for design, testing, and validation of tasks instructions, and testing resources capabilities. Fault tolerance, in turn, consists of four phases: failure detection, damage confinement, failure recovery, and fault treatment. Failure detection consists of identifying the presence of a failure, e.g., identifying that a task output is incorrect. Damage confinement aims at determining the boundaries of failures and preventing their propagation, e.g., identifying which tasks outputs are incorrect and preventing that other tasks make use of these outputs. Failure recovery tries to bring the flow of execution to a consistent state, e.g., re-executing tasks that produced incorrect outputs. Finally, fault treatment involves treating faults to prevent the occurrence of new failures. Human aspects In our context, human aspects are human beings characteristics that determine the way they perform tasks. These aspects have been widely addressed in Psychology studies. They can be broadly divided into the following factors: cognitive system [44,45], motivation [20], preferences [46], social behavior [47], emotion [48,49], individual differences [50,51], and intraindividual changes [52]. Cognitive system. Its function includes several processes of task execution, such as information processing, understanding, and learning. It specifies processes' organization on long-term and short-term memory. Long-term memory is where knowledge is stored. In turn, short-term memory is a working memory used to process information in the sense of organizing, contrasting, and comparing [44]. Humans are able to deal with few items of information simultaneously in their working memory, and any interactions between items held in their working memory also require working memory capacity, reducing the number of items that can be dealt with simultaneously [45]. Cognitive overload occurs when tasks processing exceeds working memory capacity. Motivation. From the motivation theory viewpoint [20], humans are guided by motivation impulses or goals, i.e., the desire to do/obtain new things and to achieve new conditions. Incentive studies explore the way such goals influence human behavior. Considering the self-determination theory [21], motivation is broadly divided into intrinsic and extrinsic. In task execution, intrinsic motivation may consist of workers internal desires to perform a particular task because it gives them pleasure or allows them to develop a particular skill. Extrinsic motivation, in turn, consists of factors external to the workers and unrelated to the task they are executing. Preferences. Humans exhibit personal preferences [46]. Such preferences are explained on the basis of two types of influences: their own past experiences and the experiences of others which are directly observable by them. As an example of workers preferences in task, consider the case where, after feeling bored several times when executing high-time-consuming tasks, workers always choose to execute only low-time-consuming tasks. Social behavior. Sociality means group/community organization to perform activities [53]. In general, communities form and persist because the individual takes advantage of them and thereby serve their interests. Sense of Community theory suggests that members develop sense of community based on membership, influence, integration and fulfillment of needs, and shared emotional connection [22]. This behavior may influence the way community members behave and execute tasks in the system. Emotion. Emotion can be defined as a human complex psychological and physiological state that allows humans to sense whether a given event in their environment is more desirable or less desirable [49]. Emotion concerns, for instance, mood, affection, feeling and opinion. It interacts with and influences other human aspects relevant to task execution effectiveness. For example, it influences cognitive system functions related to perception, learning and reasoning [49]. Intraindividual variability and change. Humans show intraindividual variability and change [52]. Intraindividual variability is a short-term or transient fluctuation characterized by factors as: wobble, inconsistency, and noise. Intraindividual change is a long-term lasting change as a result of learning, development, or aging. Individual differences. Humans show variability between themselves in several factors [50,51], such as decision making and performance. In this study we focus mainly on individual differences in terms of the following three human competencies: knowledge, skills, and abilities. Knowledge refers to an organized body of information applied directly in the execution of a task. Skill refers to the proficiency in tasks execution, usually measured qualitatively and quantitatively. Abilities are those appropriate on-the-job behaviors needed to bring both knowledge and skills in task execution. Instantiating the framework Now we turn to map the literature on human computation by using our theoretical framework. This is done through a literature review focused on analyzing: a) how human computation studies on D&M strategies have dealt with human aspects discussed in our framework to satisfy the QoS requirements of requesters; and b) what are the relevant results regarding these human aspects upon which future D&M strategies decisions can be based. Throughout this section, for each D&M factor, we discuss the human computation studies and extract the major implications for system design related to human factors. Application composition Application composition consists of two major activities: task design and application structuring. Task design impacts on the ability of worker to complete tasks quickly, effectively, and accurately [54]. Poorly designed tasks, which show high cognitive load, can cause fatigue of workers, compromising their understanding of instructions, decreasing their productivity and increasing their errors [55]. This usually occurs because of the limitations of the human working memory. This is the case of tasks where humans are asked to choose the best item among several options [56,57]. To perform this task, humans compare the items and choose the best of them on their working memory. Such kind of task generates a cognitive load, which increases in proportion as the number of items to be compared increases. The higher the cognitive load, the higher error chances. Tasks can also be designed to motivate workers to put more effort in generating correct outputs. Huang et al. show that one way to achieve it is to tell workers explicitly that their work will be used to evaluate the output provided by other workers [58]. Application structuring studies can be broadly divided into static workflows and dynamic workflows. Example of static workflow composition is that used in Soylent [59]. Soylent is a word processor add-in that uses MTurk's workers (mturk.com) to perform text shortening. Soylent implements the Find-Fix-Verify workflow, i.e., find areas of the text that can be shortened, fix and shorten the highlighted area without changing the meaning of the text, and verify if the new text retain its meaning. This task distinction captures human individual differences mainly in terms of the type of tasks workers want to perform, i.e., find, fix, or verify tasks. Static workflows can be optimized. Cascade [60] and Deluge [61] are examples of optimized workflows to taxonomy creation tasks. Example of dynamic workflow composition is that used in Turkomatic [55]. In Turkomatic, workers compose the workflow dynamically on a collaborative planning process. When a worker receives a task, she/he performs it only if it is simple to be executed; otherwise, the worker subdivides it into two other tasks. A problem occurs when workers generate unclear tasks that will be executed by other workers. Other approach is proposed by Lin et al. [62]. They assume that workflow composition results in different output quality when executed by different groups and they propose the availability of multi-ple workflows composition with independent difficulty level and dynamically switch between them to obtain higher accuracy in a given group of workers. Finally, Bozzon et al. propose a system that dynamically controls the composition and execution of the application, and reacts to some specific situations, such as: achievement of a suitable output and identification of a spammer [63]. It allows the system to adapt the application to changes in workers characteristics and behavior. Implications for systems design. We extract two major guidelines from this discussion: 1) application designers must avoid task cognitive overload, in what requiring a small amount of specialized ability, skill and knowledge in a same task can contribute; 2) given that workers in a platform display individual differences, application designers can take advantage of worker diversity by developing applications with different types of tasks, each type requiring a different skill; this may be done either by defining a static composition of tasks that require different skills or by using dynamic composition to adapt to different groups of workers. Incentives and rewards Incentive and reward schemes have been designed to incentivize specific behaviors and maximize requesters' QoS requirements [39,64,65]. Unfortunately, there is no consensus in the literature on the effect of incentives on worker behavior. Its effect seems to vary with the type of task. In some tasks, increasing financial incentives allows one to increase the amount of workers interested in executing the task [8,66,67], but not necessarily the quality of the outputs [67]. In other tasks, quality may be improved by increasing intrinsic motivations [68]. Incentives also relates to other aspects of task design, for example, some studies show that incentives work better when they are placed in the focal position in task worker interface [69], and task fidelity is improved when financial incentives are combined with a task design that asks workers to think about the output provided by other workers for the same task [70]. Besides defining right incentives, requesters must also define a suitable reward scheme. Witkowski et al. analyze a scheme that pays workers only if his/her output agrees with those provided by others, and that penalizes with negative payment in the case his/her output disagrees. They show that such scheme acts as a self selection mechanism that makes workers with lower quality choose not to participate [71]. Rao et al. show that a reward scheme that informs workers that they will be paid if their outputs are similar to the majority motivates workers to reflect more on what other workers would choose. This generates a higher percentage of correct outputs and the obtained outputs are closer to the correct one [72]. Huang et al. analyze three schemes in a group of workers [73]: (i) individual, in which reward depends only on worker performance; (ii) teamwork, in which workers in the group are paid similarly based on the average of their performance; and (iii) competition, in which workers in the group are opponents and they are paid based on their differences of performance. The effectiveness of these schemes tends to vary with other application settings such as social transparency. Implications for systems design. We extract two major guidelines from this discussion: 1) given that the effect of incentives on intrinsic and extrinsic motivations appears to be different in different types of tasks, designers must test how combinations of such incentives contribute towards the desired quality in their specific context; 2) task peformance can be improved by using incentives that motivate workers to reflect more on what output other workers would provide for the same task. Task assignment We broadly divided task assignment strategies into scheduling, job board, and recommendation. Task Scheduling strategies try to optimize some QoS requirements by exploiting information about the affinity between tasks and workers. Scheduling strategies in human computation literature have con-sidered several human aspects. Some strategies consider workers' emotional states allocating tasks that are appropriate to current worker's mood [74]. Heidari and Kearns take into account task difficulty and workers abilities [75]. They analyze the case in which workers can decide between execute one task or forward it. When one worker decides to forward a task, it will be scheduled to a more qualified worker. It generates a forwarding structure on task scheduling that improves outputs' quality. Waterhouse proposes a strategy based on a probabilistic metric of mutual information between workers. The strategy tunes the assignment of tasks to the workers that will increase the amount of information gain [76]. Other thread of scheduling strategies is inspired by the hierarchical structure in today's organizations, exploring individual differences [77,78]. They consider working teams and roles, such as supervisors and workers. Tasks are first assigned to supervisors, who assign them to workers in their team taking into account the qualification and skills of each worker. Skill-based scheduling considers different workers qualification levels and that qualification level increases in the proportion that workers adequately perform more tasks [79]. There are also approaches that use information and contents liked by the worker on social networks to automatically match workers preferences and task requirements [80]. Job Board strategies are used mainly in online labor market platforms, where tasks are usually made available together with their rewards to workers in boards [81]. Job boards allow workers to choose tasks that fit their preferences [82]; thus, task instructions must be clear and attractive to workers [83]. Requesters define job parameters so as to address workers interests and make their tasks attractive to workers. For example, AutoMan adjusts tasks' price and tasks' allocation time to motivate more workers to try executing them [66]. By adjusting these parameters, AutoMan tries to optimize QoS requirements addressing workers' time and financial incentives. Toomim et al. propose a method to characterize workers preferences for some interfaces and tasks [84]. This information is used in future task design. Task Recommendation strategies recommend tasks to workers according to some affinity criteria [85]. The platforms oDesk (odesk.com) and Elance (elance.com) are based on job boards, but they also make use of a recommendation system to inform workers about new jobs that match their skills. We are not aware of studies that evaluate the effectiveness of such functionalities on these platforms. Yi et al. propose a matrix completion approach to infer workers preferences in pairwise comparison tasks [86]. The method may be useful to improve task recommendation strategies. In job board and tasks recommendation approaches, it is also required a mechanism that allows requesters to choose which of the candidate workers will perform the task or which solution will be paid. In research and practice, three strategies that explore these dimensions are: auction, in which the task is allocated to the worker that charges the lowest value (e.g., odesk.com); challenge, in which all workers perform the available tasks, but only the best solution is paid (e.g., topcoder.com [8]); and registration order, in which the task is allocated to the first worker that signed up to run it (e.g., mturk.com [87]). To the best of our knowledge, no study was conducted to compare the performance of these approaches and to indicate in what situations each should be used. Implications for systems design. Two basic guidelines to highlight in this context are: 1) given that most of platforms are based on job boards and that in this environment the effectiveness of a task relies on its ability to gain attention of suitable workers, requesters must provide task descriptions with information that allows workers to easily identify if the task match their skills, preferences and interests; 2) requesters must avoid generating too restrictive tasks in order to take advantage of the diversity and larger quantities of workers that job boards and recommendation strategies give access to. Dependency management We focus on analyzing the dependency management strategies that take into account human aspects, while ensuring temporal, serialization, visibility, and cooperation dependencies. Most studies address only temporal dependencies, in which a set of tasks must be performed in a particular order (e.g., [36,77]) or in a synchronous way (e.g., [88,89]). Serialization dependency studies in human computation have focused maily on applications with loop or without loop. Example of human computation applications without loops are those that deal with planning activities [90]. Such applications usually include a sequence of steps such as: decomposition of the problem into small tasks, execution of each task and aggregation of these partial task outputs to obtain the final output as a result to the problem. This is the case of Turkomatic [55], CrowdForge [91], CrowdPlan [90], and combination of creative design [92]. Application with loops, in turn, include some iterative processes [36] as in Find-Fix-and-Verify [59] and Iterative Improvement [93]. Visibility dependency is common in working group. It usually needs a shared environment that unobtrusively offers up-to-date group context and explicit notification of each user's action when appropriate. Mao et al. [89] and Zhang et al. [94] address visibility dependencies in human computation applications. In their studies, workers try to achieve a global goal. This goal can be, for example, a collaborative itinerary planning [94], a collaborative graph coloring [89]. In these cases, workers can see outputs generated to correlated tasks. Such type of task is usually related to agreement or consensus and the visibility decision may impact both worker behavior as well as the time required to obtain an output [88]. Cooperation dependencies are also related to working group. Mobi [94] and TurkServer [89] allow one to implement applications that contain cooperative tasks. Zhang et al. show that the unified view of the task status allows workers to coordinate and communicate more effectively with one another, allowing them to view and build upon each other's ideas [94]. Platforms such as MTurk maintain workers invisible and not able to communicate with each other. Turkopticon is a tool that allows one to interrupt such invisibility, making possible workers to communicate among themselves [95]. Implications for systems design. The major guideline regarding dependency management is that designers must consider that some degree of visibility and communication between workers may be positive for application performance in terms of time and accuracy. It seems that workers should be allowed: 1) to see the status of tasks that are interdependent with his/her task in order to synchronize its execution with any global task constraint; and 2) to communicate with other workers that are executing tasks interdependent with his/her task in order to improve cooperation. Output aggregation There are a range of output aggregation strategies in human computation, most of them are already discussed by Law and von Ahn [96] and Nguyen et al. [97]. We focus on discussing studies that account for human aspects. An example of comparable outputs aggregation strategy is majority vote [36,66], in which the output of the application is the most frequent task output. This strategy assumes that the majority of the workers assigned to any task are reliable. Majority vote does not perform properly when the error chance in task execution is high. Sheng et al. investigate the impact of the number of task executions on output accuracy and shows that quality of the output is improved by using additional workers only when the workers accuracy is higher than 0.5 [98]. They propose a repeated-labeling technique that selects data points for which application quality should be improved by the acquisition of multiple task outputs. Diverse studies have been devoted to aggregating a set of outputs and obtaining an accurate output taking into account workers expertise and task characteristics. Whitehill et al. consider that an output accuracy depend on the difficulty of the task and expertise of the worker [99]. They proposed Generative Model of Labels, Abilities, and Difficulties (GLAD) which estimates these parameters using Expectation Maximization (EM) probabilistic models and evaluate tasks output in a way that experts' outputs count more. Hovy et al. propose Multi-Annotator Competence Estimation (MACE) which uses EM to identify which annotators are trustworthy and considers this information to predict outputs [100]. Wang et al. propose a recursive algorithm to improve the efficiency of compute EM models in these contexts [101]. Dalvi et al. propose a technique to estimate worker reliabilities and improve output aggregation [102]. The strategy is based on measuring agreement between pairs of workers. Another output aggregation challenge arises in unstructured outputs, e.g., open-ended [56,59] and image annotation tasks [103]. In this case, a way to find the best output is to apply a vote-on-the-best strategy in which workers evaluate the quality of each output or they choose which of them exhibits the highest quality [104]. It exploits individual differences, given that some workers are better at identifying the correct outputs than producing them themselves [56]. When the set of options is too large, it may be difficult for workers choose the best item. An alternative in this case is to develop a second human computation application in which few items are compared in each task and the best item is chosen by tournament (e.g., [56,57]). Other peculiarity of unstructured outputs is that even poor outputs may be useful. For example, other workers can aggregate such poor outputs, and generate a new better output [55]. The quality of the aggregation can also be improved by using estimations of the difficulty level of tasks and skills of workers [103]. Implications for systems design. When developing output aggregation strategies, designers must weigh at least three parameters that impact on the quality of the final output: 1) task cognitive load; 2) the amount of different workers that provided task outputs, i.e., redundancy degree; and 3) the accuracy of each worker that provided the outputs. As in the literature, the value of each of parameters can be obtained in a statistical evaluation, considering that the accuracy of the final output tends to be higher with more accurate estimation of these parameters. Fault prevention and tolerance The prevention of faults in task instructions can be done by using offline and/or online pilot tests [87]. Offline tests are conducted with accessible people that can provide feedback on issues and improvements in the tasks instructions. Online tests, in turn, are driven onto a platform, and they are more realistic than offline tests. In this case, workers may not be accessible to provide feedback about the task instructions, but their outputs can be analyzed to identify problems. The prevention of undesired workers is usually done by using qualification tests [87]. They consist in requiring the execution of a gold standard test that certifies whether the worker has the skills and qualifications required to perform application tasks. Only workers who perform accurately are considered qualified. A downside of this approach is not considering changes in workers behavior after executing the test. Malicious workers usually change their behavior over time [105]. CrowdScape is a system that allows requesters to select workers based on both task output quality and workers' behavioral changes [106]. Studies also have been devoted to fault tolerance which consists in four phases: failure detection, damage confinement, failure recovery, and fault treatment. Failure detection has been made by using: 1) conformance voting, which allows one to detect poorly executed tasks; and 2) timing, which allows one to detect worker evasion, i.e., the worker is assigned to perform a task, but gives up executing and do not deallocate the task. In conformance vote, one worker or a group of workers evaluate whether a task output is correct. When the output is not correct, the task needs to be re-executed by another worker [55]. Timing, in turn, defines a maximum time that a task can remain allocated to a worker; it is expected that a worker provides an output up to this time. If that time expires without an output being provided, the task is deallocated and made available to another worker [59]. Damage Confinement is usually made by using error recovery techniques in each task or workflow step. It prevents that damages propagate to the next workflow step. This propagation occurs, for example, in workflow derailment problems [104], which arises when an error in the task executions prevents the workflow conclusion. Failure Recovery has been made by using majority voting, alternative, and human assessment. These strategies exploit human individual differences by using redundancy of workers. If different and independent workers provide the same output to a task, it increases the confidence that the task is being performed in accordance with its instructions [107]. In majority voting, several workers perform the same task in parallel and the most frequent output is considered correct. In alternative strategies, in turn, a worker executes the task and, if an error occurs, the task is executed again by another worker [55]. In these redundancy-based strategies, the impact of the redundancy degree on output accuracy is highly dependent on the type of task. Increasing the redundancy of workers does not necessarily increase the confidence that the correct output will be obtained [108]. Furthermore, the perception of redundancy by the workers may have a negative effect on their motivation and work quality. The more co-workers working in the same task are perceived by workers, the lower their work quality [109]. This occurs because workers demotivate thinking that their effort does not count for much. Finally, in human assessment strategies, the outputs generated by a worker are evaluated by others. This can be implemented in two ways: arbitration and peer review. In arbitration, two workers independently execute the tasks and another worker evaluates their outputs and solve disagreements. In peer review, the output provided by each worker is reviewed by another worker. Hansen et al. show that in text transcribe tasks, the peer review strategy is significantly more efficient, but not as accurate for certain tasks as the arbitration strategy [110]. Fault Treatment has been made by fixing problems in task design, and by eliminating or reducing the reputation of unskilled or malicious workers. For example, TopCoder [8] maintains a historical track of the number of tasks each worker chooses to execute, but did not conclude. This track is used to estimate the probability that the worker chooses tasks and do not execute it. Ipeirotis et al. propose to separate systematic errors from bias due to, for example, an intraindividual variability such as distraction [111]. This distinction allows also a better estimation of accuracy and reputation of the worker. Such estimation may be used to prevent assigning to a worker tasks for which he/she is not qualified or that he/she will not complete the execution. Another important aspect in fault treatment is to provide feedback to workers about his/her work [112]. It helps workers to learn how to accurately execute the task (intraindividual changes) and avoid errors due to lapses (intraindividual variability) [111] . Implications for systems design. The three major guidelines extracted from this discussion are: 1) designers must test the task worker interface and check workers skills/reputation; to this end pilot tests and qualification tests can be applied; 2) redundancy is the basis of fault tolerance strategies, but requesters must generate tasks that maximize the number of workers capable of executing it, increasing the potential of redundancy of the task; and 3) requesters must provide workers assessment and feedback in order to allow them to learn from tasks they perform incorrectly. In the last section, we analyzed the human computation literature and its implication for design in the light of our theoretical framework. Now we turn to discuss challenges and perspectives in D&M strategies. Although our list is by no means exhaustive, it offers examples of topics in need of further work and directions that appear promising. Table 1 synthesizes the contributions on the relationships between D&M strategies and human aspects identified in the last section. As shown, there are several relationships for which we could not find any study. This state of affairs indicates a large amount of research still to be conducted after mapping the impact of human aspects on D&M effectiveness. Two other issues that still require further understanding are: 1) adequate combinations of D&M strategies; and 2) the impact of D&M strategies on workers' cognition and behavior. Intraindividual Variability [106,111,112] Intraindividual changes [79] [ 105,112] It is intuitive that one D&M strategy may impact on the effectiveness of another. For example, by generating a fine-grained application composition to account for the human cognitive system, one may generate undesired effects: 1) designing tasks too susceptible to cheater workers, which reduces the effectiveness of fault tolerance strategies; or 2) generating a large number of tasks with a too high number of dependencies among them, which may reduce parallelism in task execution and impact on dependency management. More empirical research on how to adequately combine D&M strategies in distributed human computation is still required. Relations between dimensions of the framework Besides the requesters' perspective that tries to understand how to take advantage of human aspects to achieve QoS requirements, studies must also identify possible side-effects of the strategies on workers cognition and behavior. Two cognitive effects that may be relevant to consider are: Framing effect -workers may generate different outputs based on how a task is designed-, and Hawthorne effectworkers may alter their behavior when they know that they are being observed. Two behavioral effects are collusion, an agreement between workers to act similarly (e.g., planning collusion against requesters which submit poorly designed tasks [55]), and sabotage, workers change their behavior to take advantage of the system (e.g., inhibiting competitors in a "maximum observability" audition [8]). Also, there is room for studies focused on workers and on the fair relationship between workers and requesters [114]. Exploring the Interdisciplinarity of D&M strategies improvement Application composition. The main human aspects factors that have been addressed in application composition are cognitive system and motivation/incentives. By taking into account such factors in the context of task execution, human computation application composition is clearly related to the disciplines: ecological interface [115], and goal setting [116]. Ecological interface principles are grounded on how the human cognitive system works and its effects on information understanding and processing. Such principles may support the development of task designs to avoid cognitive overload and improve task execution effectiveness. Goal setting studies, in turn, may help better defining both task instructions and the way their outputs will be evaluated by the requester. Knowledge of such topics and reasoning about their relationships to human computation can help in the formulation of new strategies. Task assignment. Studies on task assignment have mainly taken into account: preferences and individual differences. Two other disciplines that take into account these aspects in task assignment are person-job fit [117] and achievement motivation [118]. The domain of person-job fit research consists on tasks characteristics, worker characteristics, and required outcomes. It emphasizes the fit and matching of workers and tasks in the prediction of both worker and organizational outcomes. Achievement motivation is a motivation for demonstrating high rather than low ability. This motivation influences the tasks a human chooses to perform, i.e., his/her preferences. According to this concept, individuals select tasks they expect to enable them to maximize their chances of demonstrating high ability and avoiding demonstrating low ability. These concepts may inspire tasks scheduling and task recommendation strategies in human computation. Dependency management. Ensuring tasks dependencies and still extracting the greatest potential (optimizing QoS requirement) of a crowd of workers is one of the main challenges of dependency management strategies. Similar challenge has been addressed in at least two other disciplines: work teams [119] and Groupware [120]. Both disciplines focus on group behavior and performance. Work team studies usually focus on group work in an organization not necessarily performed through a computer system. Groupware is generally associated with small groups working collaboratively through a computer system. Experiences on how human aspects are addressed in these disciplines may inspire solutions that consider these factors in human computation. Output aggregation. Two important areas related to output aggregation are Judgment Aggregation [121] and Social choice theory [122]. Judgment aggregation is the subject of a growing body of work in Economics, Political science, Philosophy and related disciplines. It aims at aggregating consistent individual judgments on logically interconnected propositions into a collective judgment on those propositions. In these situations, majority voting cannot ensure an equally consistent collective conclusion. Social choice theory, in turn, is a theoretical framework for analysis of combining individual preferences, and interests to reach a collective decision or social welfare. According to this theory any choice for the entire group should reflect the desires/options of the individual to the extent possible. The studies that have been conducted in these disciplines seem to be related to human computation output aggregation [123,124]. A better mapping of their similarity and differences may help in the reuse and development of new output/judgment aggregation strategies. Fault prevention and tolerance. Besides preventing and tolerating faults, one should also consider how to evaluate system QoS in the presence of human faults. For example, fault tolerance is mainly based on task redundancy, but defining the appropriate level of redundancy is a challenging task. Maintaining a low level of redundancy may not recover failures and maintaining a high level of redundancy can lead to a high financial cost or high volunteer effort to run the entire application. This kind of study has been conducted in other disciplines such as: human aspects evaluation [125] and performability [126]. Human aspects evaluation is an assessment of the conformity between the performance of a worker and its desired performance. Performability, in turn, focuses on modeling and measuring system QoS degradation in the presence of faults. Experiences on performability and human aspects evaluation may be useful to address QoS requirements in the presence of worker faults. Conclusions In this paper, we analyzed the design and management of distributed human computation applications. Our contribution is three-fold: 1) we integrated a set of theories in a theoretical framework for analyzing distributed human computation applications; 2) by using this theoretical framework, we analyzed human computation literature putting into perspective the results in this literature on how to leverage human aspects of workers in D&M strategies in order to satisfy the QoS requirements of requesters; and 3) we highlighted open challenges in human computation and discussed their relationship with other disciplines from a distributed systems viewpoint. Our framework builds on studies in different disciplines to discuss advances and perspectives in a variety of immediate practical needs in distributed human computation systems. Our literature analysis shows that D&M strategies have accounted for some human aspects to achieve QoS requirements. However, it also shows that there are still many unexplored aspects and open challenges. Inevitably, a better understanding of how humans behave in human computation systems and a proper delimitation of the human aspects involved is essential to overcome these challenges. We hope our study inspires both discussion and further research in this direction. Figure 1 1Framework. An overview of the dimensions and factors of the proposed theoretical framework. Figure 2 Figure 3 23Bag of human computation tasks. Example of a bag-of-tasks application structuring. Workflow of human computation tasks. Example of a workflow application structuring. Table 1 1Human aspects factors addressed in D&M strategiesApplication Incentives Dependency Task Output Fault composition and rewards management assignment aggregation tolerance Cognitive System [54-57] [75] [99,103] [111] Motivation [8,58,64-67,69,71,72] [66] [67,68,109] Preferences [59] [81-85] Social behavior [70,73] [88,89,94,95] [80] [66] Emotion [74,113] [111] Individual differences [55,59,62] [8,76-78] [55,56,59,98-100,102] [55,59,104,106,110] AcknowledgementsLesandro Ponciano thanks the support provided by CAPES/Brazil in all aspects of this research. Francisco Brasileiro acknowledges the support received from CNPq/Brazil in all aspects of this research.Competing interestsThe authors declare that they have no competing interests.Authors' contributions Human computation: a survey and taxonomy of a growing field. A J Quinn, B B Bederson, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)Quinn AJ, Bederson BB (2011) Human computation: a survey and taxonomy of a growing field. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). Gaining wisdom from crowds. N Savage, Commun ACM. 55313Savage N (2012) Gaining wisdom from crowds. Commun ACM 55(3):13 A survey of human computation systems. M-C Yuen, L-J Chen, I King, Proceedings of the International Conference on Computational Science and Engineering (CSE). the International Conference on Computational Science and Engineering (CSE)Washington, DCIEEE Computer Society4Yuen M-C, Chen L-J, King I (2009) A survey of human computation systems. In: Proceedings of the International Conference on Computational Science and Engineering (CSE), vol. 4. IEEE Computer Society, Washington, DC, pp 723-728 A survey of crowdsourcing systems. M-C Yuen, I King, K-S Leung, Proceedings of the International Conference on Privacy. the International Conference on PrivacySecurity, Risk; Washington, DCIEEE Computer Societyand Trust (PASSAT)Yuen M-C, King I, Leung K-S (2011) A survey of crowdsourcing systems. In: Proceedings of the International Conference on Privacy, Security, Risk and Trust (PASSAT). IEEE Computer Society, Washington, DC, pp 766-773 Bederson BB A taxonomy of distributed human computation. A J Quinn, University of MarylandTechnical reportQuinn AJ, Bederson BB A taxonomy of distributed human computation. Technical report, Univer- sity of Maryland (2009) recaptcha: Human-based character recognition via web security measures. L Von Ahn, B Maurer, C Mcmillen, D Abraham, M Blum, Science. 3215895Von Ahn L, Maurer B, McMillen C, Abraham D, Blum M (2008) recaptcha: Human-based char- acter recognition via web security measures. Science 321(5895):1465-1468 Designing games with a purpose. L Von Ahn, L Dabbish, Commun ACM. 518von Ahn L, Dabbish L (2008) Designing games with a purpose. Commun ACM 51(8):58-67 Money, glory and cheap talk: analyzing strategic behavior of contestants in simultaneous crowdsourcing contests on topcoder. N Archak, Proceedings of the International World Wide Web Conference. the International World Wide Web ConferenceNew YorkACMArchak N (2010) Money, glory and cheap talk: analyzing strategic behavior of contestants in simultaneous crowdsourcing contests on topcoder.com. In: Proceedings of the International World Wide Web Conference (WWW). ACM, New York, pp 21-30 Analyzing the amazon mechanical turk marketplace. P G Ipeirotis, XRDS. 172Ipeirotis PG (2010) Analyzing the amazon mechanical turk marketplace. XRDS 17(2):16-21 Volunteers' engagement in human computation astronomy projects. L Ponciano, F Brasileiro, R Simpson, A Smith, Comput Sci Eng PP. 99Ponciano L, Brasileiro F, Simpson R, Smith A (2014) Volunteers' engagement in human compu- tation astronomy projects. Comput Sci Eng PP(99):1-12 Who are the crowdworkers?: shifting demographics in mechanical turk. J Ross, L Irani, M Silberman, A Zaldivar, B Tomlinson, Proceedings of the ACM SIGCHI Conference on Human Factors in Computing Systems. the ACM SIGCHI Conference on Human Factors in Computing SystemsNew YorkACMExtended Abstracts (EARoss J, Irani L, Silberman M, Zaldivar A, Tomlinson B (2010) Who are the crowdworkers?: shift- ing demographics in mechanical turk. In: Proceedings of the ACM SIGCHI Conference on Human Factors in Computing Systems, Extended Abstracts (EA) ACM New York, pp 2863-2872 Canfar+ skytree: Mining massive datasets as an essential part of the future of astronomy. N M Ball, American Astronomical Society Meeting Abstracts. Washington, DCAmerican Astronomical Society221Ball NM (2013) Canfar+ skytree: Mining massive datasets as an essential part of the future of astronomy. In: American Astronomical Society Meeting Abstracts, vol. 221. American Astro- nomical Society, Washington, DC The future of crowd work. A Kittur, J V Nickerson, M Bernstein, E Gerber, A Shaw, J Zimmerman, M Lease, J Horton, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMKittur A, Nickerson JV, Bernstein M, Gerber E, Shaw A, Zimmerman J, Lease M, Horton J (2013) The future of crowd work. In: Proceedings of the ACM Conference on Computer-Supported Co- operative Work and Social Computing (CSWC). ACM New York, pp 1301-1318 Galaxy Zoo: morphologies derived from visual inspection of galaxies from the Sloan Digital Sky Survey. C J Lintott, K Schawinski, A Slosar, K Land, S Bamford, D Thomas, M J Raddick, R C Nichol, A Szalay, D Andreescu, P Murray, J Vandenberg, Mon Notices R Astronomical Soc. 389Lintott CJ, Schawinski K, Slosar A, Land K, Bamford S, Thomas D, Raddick MJ, Nichol RC, Szalay A, Andreescu D, Murray P, Vandenberg J (2008) Galaxy Zoo: morphologies derived from visual inspection of galaxies from the Sloan Digital Sky Survey. Mon Notices R Astronomical Soc 389: 1179-1189 99designs: An analysis of creative competition in crowdsourced design. R M De Araújo, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. Palo Alto. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. Palo Altode Araújo RM (2013) 99designs: An analysis of creative competition in crowdsourced design. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. Palo Alto, pp 17-24 Human-powered sorts and joins. A Marcus, E Wu, D Karger, S Madden, R Miller, Proc VLDB Endow. 51Marcus A, Wu E, Karger D, Madden S, Miller R (2011) Human-powered sorts and joins. Proc VLDB Endow 5(1):13-24 Crowdsourcing for enterprises. M Vukovic, Congress on Services -I. IEEE Computer Society. Washington, DCVukovic M (2009) Crowdsourcing for enterprises. In: Congress on Services -I. IEEE Computer Society Washington, DC, pp. 686-692 Virtualizing software and humans for elastic processes in multiple clouds -a service management perspective. S Dustdar, H-L Truong, IJNGC. 32Dustdar S, Truong H-L (2012) Virtualizing software and humans for elastic processes in multiple clouds -a service management perspective. IJNGC 3(2):109-126 An affordance-based framework for human computation and humancomputer collaboration. R J Crouser, R Chang, IEEE Trans Vis Comput Graph. 1812Crouser RJ, Chang R (2012) An affordance-based framework for human computation and human- computer collaboration. IEEE Trans Vis Comput Graph 18(12):2859-2868 A theory of human motivation. A H Maslow, Psychol Rev. 50Maslow AH (1943) A theory of human motivation. Psychol Rev 50: 370-396 Intrinsic Motivation and Self-determination in Human Behavior. E L Deci, R M Ryan, Plenum PressNew YorkDeci EL, Ryan RM (1985) Intrinsic Motivation and Self-determination in Human Behavior. Plenum Press, New York Sense of community. D W Mcmillan, J Commun Psychol. 244McMillan DW (1996) Sense of community. J Commun Psychol 24(4):315-325 Human Error. Reason J. Cambridge University PressReason J (1990) Human Error. Cambridge University Press Cambridge [England], New York The interdisciplinary study of coordination. T Malone, ACM Comput Surv. 261Malone T (1994) The interdisciplinary study of coordination. ACM Comput Surv 26(1):87-119 A framework for reasoning about the human in the loop. L F Cranor, Proceedings of UPSEC. USENIX Association. UPSEC. USENIX AssociationBerkeleyCranor LF (2008) A framework for reasoning about the human in the loop. In: Proceedings of UPSEC. USENIX Association, Berkeley, pp 1-15 Great principles of computing. P J Denning, Commun ACM. 461115Denning PJ (2003) Great principles of computing. Commun ACM 46(11):15 An overview of workflow management: from process modeling to workflow automation infrastructure. D Georgakopoulos, M Hornick, A Sheth, Distrib Parallel Databases. 3Georgakopoulos D, Hornick M, Sheth A (1995) An overview of workflow management: from process modeling to workflow automation infrastructure. Distrib Parallel Databases 3(2):119-153 On structured workflow modelling. B Kiepuszewski, Ahmt Hofstede, C Bussler, CAiSE. LondonSpringer-VerlagKiepuszewski B, Hofstede AHMT, Bussler C (2000) On structured workflow modelling. In: CAiSE. Springer-Verlag, London, pp 431-445 A taxonomy of workflow management systems for grid computing. J Yu, R Buyya, J Grid Comput. 33Yu J, Buyya R (2005) A taxonomy of workflow management systems for grid computing. J Grid Comput 3(3):171-200 Modeling quality of service for workflows and web service processes. J Cardoso, J Miller, A Sheth, J Arnold, J Web Semant. 1Cardoso J, Miller J, Sheth A, Arnold J (2002) Modeling quality of service for workflows and web service processes. J Web Semant 1:281-308 Dynamic work distribution in workflow management systems: How to balance quality and performance. A Kumar, Wmp Van Der Aalst, Emw Verbeek, J Manage Inf Syst. 183Kumar A, Van Der Aalst WMP, Verbeek EMW (2002) Dynamic work distribution in workflow management systems: How to balance quality and performance. J Manage Inf Syst 18(3):157-193 Taxonomy and theory in computer supported cooperative work. J Grudin, S Poltrock, Handbook of Organizational Psychology. OxfordOxford University PressGrudin J, Poltrock S (2012) Taxonomy and theory in computer supported cooperative work. In: Handbook of Organizational Psychology. Oxford University Press, Oxford, pp 1323-1348 W Van Der Aalst, K M Van Hee, Workflow Management: Models, Methods, and Systems. Cooperative Information Systems Series. Cambridge, Massachusetts, United StatesMit PressVan der Aalst W, van Hee KM (2004) Workflow Management: Models, Methods, and Systems. Cooperative Information Systems Series. Mit Press, Cambridge, Massachusetts, United States Human computation must be reproducible. P Paritosh, Proc.of CrowdSearch. .of CrowdSearchParitosh P (2012) Human computation must be reproducible. In: Proc.of CrowdSearch. pp 20-25 Running bag-of-tasks applications on computational grids: The mygrid approach. W Cirne, D Paranhos, L Costa, E Santos-Neto, F Brasileiro, J Sauvé, F A Silva, C O Barros, C Silveira, Proceedings of the International Conference on Parallel Processing. the International Conference on Parallel ProcessingWashington, DCIEEE Computer SocietyCirne W, Paranhos D, Costa L, Santos-Neto E, Brasileiro F, Sauvé J, Silva FA, Barros CO, Sil- veira C (2003) Running bag-of-tasks applications on computational grids: The mygrid approach. In: Proceedings of the International Conference on Parallel Processing. IEEE Computer Society, Washington, DC, pp 407-416 TurKit: Human Computation Algorithms on Mechanical Turk. G Little, L B Chilton, M Goldman, R C Miller, Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). the ACM Symposium on User Interface Software and Technology (UIST)New YorkACMLittle G, Chilton LB, Goldman M, Miller RC (2010) TurKit: Human Computation Algorithms on Mechanical Turk. In: Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). ACM, New York, pp 57-66 Flexible social workflows: Collaborations as human architecture. C Dorn, R N Taylor, S Dustdar, Internet Comput IEEE. 162Dorn C, Taylor RN, Dustdar S (2012) Flexible social workflows: Collaborations as human archi- tecture. Internet Comput IEEE 16(2):72-77 The Theory of Incentives: the Principal-agent Model. J-J Laffont, D Martimort, Princeton University PressPrinceton, New JerseyLaffont J-J, Martimort D (2009) The Theory of Incentives: the Principal-agent Model. Princeton University Press, Princeton, New Jersey Incentives and rewarding in social computing. O Scekic, H-L Truong, S Dustdar, Commun ACM. 566Scekic O, Truong H-L, Dustdar S (2013) Incentives and rewarding in social computing. Commun ACM 56(6):72-82 Game theory and incentives in human computation systems. A Ghosh, Handbook of Human Computation. Michelucci P.New YorkSpringerGhosh A (2013) Game theory and incentives in human computation systems. In: Michelucci P. (ed.) Handbook of Human Computation. Springer, New York, pp 725-742 The role of game theory in human computation systems. S Jain, D C Parkes, Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP) ACM. the ACM SIGKDD Workshop on Human Computation (HCOMP) ACMJain S, Parkes DC (2009) The role of game theory in human computation systems. In: Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP) ACM, pp 58-61 Crowdlang -first steps towards programmable human computers for general computation. P Minder, A Bernstein, Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). the ACM SIGKDD Workshop on Human Computation (HCOMP)Minder P, Bernstein A (2011) Crowdlang -first steps towards programmable human computers for general computation. In: Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP) Upper Saddle River 44. Simon HA (1990) Invariants of human behavior. P Jalote, Annu Rev Psychol. 411Prentice-HallFault Tolerance in Distributed SystemsJalote P (1994) Fault Tolerance in Distributed Systems. Prentice-Hall, Upper Saddle River 44. Simon HA (1990) Invariants of human behavior. Annu Rev Psychol 41(1):1-19 Cognitive architecture and instructional design. J Sweller, Jjgv Merrienboer, Fgwc Paas, Educ Psychol Rev. 10Sweller J, Merrienboer JJGV, Paas FGWC (1998) Cognitive architecture and instructional design. Educ Psychol Rev 10: 251-296 The dynamics of preference formation. A Kapteyn, T Wansbeek, J Buyze, Econ Lett. 11Kapteyn A, Wansbeek T, Buyze J (1978) The dynamics of preference formation. Econ Lett 1(1):93-98 The evolution of social behavior. R D Alexander, Annu Rev Ecol Evol Syst. 51Alexander RD (1974) The evolution of social behavior. Annu Rev Ecol Evol Syst 5(1):325-383 The emerging field of emotion regulation: An integrative review. J J Gross, Rev Gen Psychol. 23Gross JJ (1998) The emerging field of emotion regulation: An integrative review. Rev Gen Psychol 2(3):271-299 Emotion, cognition, and behavior. R J Dolan, Science. 2985596Dolan RJ (2002) Emotion, cognition, and behavior. Science (New York, N.Y.) 298(5596):1191- 1194 Individual differences in cognition, affect, and performance: Behavioral, neuroimaging, and molecular genetic approaches. R Parasuraman, Y Jiang, NeuroImage. 591Parasuraman R, Jiang Y (2012) Individual differences in cognition, affect, and performance: Be- havioral, neuroimaging, and molecular genetic approaches. NeuroImage 59(1):70-82 Individual differences in rational thought. K Stanovich, R West, J Exp Psychol Gen. 1272Stanovich K, West R (1998) Individual differences in rational thought. J Exp Psychol Gen 127(2):161-188 Cognitive performance inconsistency: Intraindividual change and variability. N Ram, P Rabbitt, B Stollery, J R Nesselroade, Psychol Agin. 204Ram N, Rabbitt P, Stollery B, Nesselroade JR (2005) Cognitive performance inconsistency: In- traindividual change and variability. Psychol Agin 20(4):623-633 Foundations of Social Theory. J Coleman, Harvard, Cambridge, Massachusetts, United StatesColeman J (1990) Foundations of Social Theory. Harvard, Cambridge, Massachusetts, United States Evaluating and improving the usability of mechanical turk for low-income workers in india. S Khanna, A Ratan, J Davis, W Thies, Proceedings of the ACM Annual Symposium on Computing for Development. the ACM Annual Symposium on Computing for DevelopmentNew YorkACMKhanna S, Ratan A, Davis J, Thies W (2010) Evaluating and improving the usability of mechan- ical turk for low-income workers in india. In: Proceedings of the ACM Annual Symposium on Computing for Development (ACM DEV). ACM, New York, pp 1-10 Collaboratively Crowdsourcing Workflows with Turkomatic. A Kulkarni, M Can, B Hartmann, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMKulkarni A, Can M, Hartmann B (2012) Collaboratively Crowdsourcing Workflows with Turko- matic. In: Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). ACM, New York, pp 1003-1012 Beyond independent agreement: A tournament selection approach for quality assurance of human computation tasks. Y-A Sun, S Roy, G Little, Proceedings of the AAAI Workshop on Human Computation (HCOMP). the AAAI Workshop on Human Computation (HCOMP)Palo Alto, CA, USAAAAISun Y-A, Roy S, Little G (2011) Beyond independent agreement: A tournament selection approach for quality assurance of human computation tasks. In: Proceedings of the AAAI Workshop on Human Computation (HCOMP), p. 113-118. AAAI, Palo Alto, CA, USA Max algorithms in crowdsourcing environments. P Venetis, H Garcia-Molina, K Huang, N Polyzotis, Proceedings of the International World Wide Web Conference. the International World Wide Web ConferenceNew YorkACMVenetis P, Garcia-Molina H, Huang K, Polyzotis N (2012) Max algorithms in crowdsourcing envi- ronments. In: Proceedings of the International World Wide Web Conference (WWW). ACM, New York, pp 989-998 Enhancing reliability using peer consistency evaluation in human computation. S-W Huang, W-T Fu, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMHuang S-W, Fu W-T (2013) Enhancing reliability using peer consistency evaluation in human computation. In: Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). ACM, New York, pp 639-648 Soylent: a word processor with a crowd inside. M S Bernstein, G Little, R C Miller, B Hartmann, M S Ackerman, D R Karger, D Crowell, K Panovich, Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). the ACM Symposium on User Interface Software and Technology (UIST)New YorkACMBernstein MS, Little G, Miller RC, Hartmann B, Ackerman MS, Karger DR, Crowell D, Panovich K (2010) Soylent: a word processor with a crowd inside. In: Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). ACM, New York, pp 313-322 Cascade: Crowdsourcing taxonomy creation. L B Chilton, G Little, D Edge, D S Weld, J A Landay, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMChilton LB, Little G, Edge D, Weld DS, Landay JA (2013) Cascade: Crowdsourcing taxonomy creation. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). ACM, New York, pp 1999-2008 Crowdsourcing multi-label classification for taxonomy creation. J Bragg, Mausam, D S Weld, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoBragg J, Mausam, Weld DS (2013) Crowdsourcing multi-label classification for taxonomy cre- ation. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 25-33 Dynamically switching between synergistic workflows for crowdsourcing. C H Lin, Mausam, D S Weld, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI. the AAAI Conference on Artificial Intelligence (AAAI). AAAIPalo AltoLin CH, Mausam, Weld DS (2012) Dynamically switching between synergistic workflows for crowdsourcing. In: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI, Palo Alto, pp 87-93 Reactive crowdsourcing. A Bozzon, M Brambilla, S Ceri, A Mauri, Proceedings of the International World Wide Web Conference (WWW) International World Wide Web Conferences Steering Committee (IW3C2). the International World Wide Web Conference (WWW) International World Wide Web Conferences Steering Committee (IW3C2)GenevaBozzon A, Brambilla M, Ceri S, Mauri A (2013) Reactive crowdsourcing. In: Proceedings of the International World Wide Web Conference (WWW) International World Wide Web Conferences Steering Committee (IW3C2), Geneva, pp 153-164 Truthful incentives in crowdsourcing tasks using regret minimization mechanisms. A Singla, A Krause, Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2). the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2)GenevaSingla A, Krause A (2013) Truthful incentives in crowdsourcing tasks using regret minimization mechanisms. In: Proceedings of the International World Wide Web Conference (WWW). Interna- tional World Wide Web Conferences Steering Committee (IW3C2), Geneva, pp 1167-1177 Pricing mechanisms for crowdsourcing markets. Y Singer, M Mittal, Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2). the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2)GenevaSinger Y, Mittal M (2013) Pricing mechanisms for crowdsourcing markets. In: Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2), Geneva, pp 1157-1166 Automan: a platform for integrating human-based and digital computation. D W Barowy, C Curtsinger, E D Berger, A Mcgregor, SIGPLAN Not. 4710Barowy DW, Curtsinger C, Berger ED, McGregor A (2012) Automan: a platform for integrating human-based and digital computation. SIGPLAN Not 47(10):639-654 Financial incentives and the "performance of crowds. W Mason, D J Watts, Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). the ACM SIGKDD Workshop on Human Computation (HCOMP)New YorkACMMason W, Watts DJ (2009) Financial incentives and the "performance of crowds". In: Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). ACM, New York, pp 77-85 An assessment of intrinsic and extrinsic motivation on task performance in crowdsourcing markets. J Rogstadius, V Kostakos, A Kittur, B Smus, J Laredo, M Vukovic, Proceedings of the International Conference on Weblogs and Social Media (ICWSM). AAAI. the International Conference on Weblogs and Social Media (ICWSM). AAAIPalo AltoRogstadius J, Kostakos V, Kittur A, Smus B, Laredo J, Vukovic M (2011) An assessment of intrin- sic and extrinsic motivation on task performance in crowdsourcing markets. In: Proceedings of the International Conference on Weblogs and Social Media (ICWSM). AAAI, Palo Alto, pp 321-328 Labor allocation in paid crowdsourcing: Experimental evidence on positioning, nudges and prices. D Chandler, J J Horton, Proceedings of the AAAI Workshop on Human Computation (HCOMP). AAAI. the AAAI Workshop on Human Computation (HCOMP). AAAIPalo AltoChandler D, Horton JJ (2011) Labor allocation in paid crowdsourcing: Experimental evidence on positioning, nudges and prices. In: Proceedings of the AAAI Workshop on Human Computation (HCOMP). AAAI, Palo Alto, pp 14-19 Designing incentives for inexpert human raters. A D Shaw, J J Horton, D L Chen, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMShaw AD, Horton JJ, Chen DL (2011) Designing incentives for inexpert human raters. In: Pro- ceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Comput- ing (CSWC). ACM, New York, pp 275-284 Dwelling on the negative: Incentivizing effort in peer prediction. J Witkowski, Y Bachrach, P Key, D C Parkes, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoWitkowski J, Bachrach Y, Key P, Parkes DC (2013) Dwelling on the negative: Incentivizing effort in peer prediction. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 190-197 What will others choose? how a majority vote reward scheme can improve human computation in a spatial location identification task. H Rao, S-W Huang, W-T Fu, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoRao H, Huang S-W, Fu W-T (2013) What will others choose? how a majority vote reward scheme can improve human computation in a spatial location identification task. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 130-137 Don't hide in the crowd!: Increasing social transparency between peer workers improves crowdsourcing outcomes. S-W Huang, W-T Fu, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMHuang S-W, Fu W-T (2013) Don't hide in the crowd!: Increasing social transparency between peer workers improves crowdsourcing outcomes. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). ACM, New York, pp 621-630 The emergence of affective crowdsourcing. R Morris, Proceedings of the CHI Workshop on Crowdsourcing and Human Computation. the CHI Workshop on Crowdsourcing and Human ComputationNew York, NY, USAACMMorris R (2011) The emergence of affective crowdsourcing. In: Proceedings of the CHI Workshop on Crowdsourcing and Human Computation. ACM, New York, NY, USA Depth-workload tradeoffs for workforce organization. H Heidari, M Kearns, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoHeidari H, Kearns M (2013) Depth-workload tradeoffs for workforce organization. In: Proceed- ings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 60-68 Pay by the bit: An information-theoretic metric for collective human judgment. T P Waterhouse, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMWaterhouse TP (2013) Pay by the bit: An information-theoretic metric for collective human judg- ment. In: Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). ACM, New York, pp 623-638 Platemate: crowdsourcing nutritional analysis from food photographs. J Noronha, E Hysen, H Zhang, K Z Gajos, Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). the ACM Symposium on User Interface Software and Technology (UIST)New YorkACMNoronha J, Hysen E, Zhang H, Gajos KZ (2011) Platemate: crowdsourcing nutritional analysis from food photographs. In: Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). ACM, New York, pp 1-12 Crowdsourcing tasks to social networks in bpel4people. D Schall, B Satzger, H Psaier, World Wide Web. 171Schall D, Satzger B, Psaier H (2012) Crowdsourcing tasks to social networks in bpel4people. World Wide Web, 17(1):1-32 Stimulating skill evolution in market-based crowdsourcing. B Satzger, H Psaier, D Schall, S Dustdar, BPM. LondonSpringer-VerlagSatzger B, Psaier H, Schall D, Dustdar S (2011) Stimulating skill evolution in market-based crowd- sourcing. In: BPM. Springer-Verlag, London, pp 66-82 Pick-a-crowd: Tell me what you like, and i'll tell you what to do. D E Difallah, G Demartini, P Cudré-Mauroux, Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2). the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2)GenevaDifallah DE, Demartini G, Cudré-Mauroux P (2013) Pick-a-crowd: Tell me what you like, and i'll tell you what to do. In: Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2), Geneva, pp 367-377 Task search in a human computation market. L B Chilton, J J Horton, R C Miller, S Azenkot, Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). the ACM SIGKDD Workshop on Human Computation (HCOMP)New YorkACMChilton LB, Horton JJ, Miller RC, Azenkot S (2010) Task search in a human computation market. In: Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP) ACM New York, pp 1-9 Analyzing crowd workers in mobile pay-for-answer qa. U Lee, J Kim, Yi E Sung, J Gerla, M , Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMLee U, Kim J, Yi E, Sung J, Gerla M (2013) Analyzing crowd workers in mobile pay-for-answer qa. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI) ACM New York, pp 533-542 Crowdsourcing a hit: Measuring workers' pre-task interactions on microtask markets. J T Jacques, P O Kristensson, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoJacques JT, Kristensson PO (2013) Crowdsourcing a hit: Measuring workers' pre-task interactions on microtask markets. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 86-93 Utility of human-computer interactions: toward a science of preference measurement. M Toomim, T Kriplean, C Pörtner, J Landay, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMToomim M, Kriplean T, Pörtner C, Landay J (2011) Utility of human-computer interactions: to- ward a science of preference measurement. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI) ACM New York, pp 2275-2284 Towards task recommendation in micro-task markets. V Ambati, S Vogel, J G Carbonell, Proceedings of the AAAI Workshop on Human Computation (HCOMP). AAAI. the AAAI Workshop on Human Computation (HCOMP). AAAIPalo AltoAmbati V, Vogel S, Carbonell JG (2011) Towards task recommendation in micro-task markets. In: Proceedings of the AAAI Workshop on Human Computation (HCOMP). AAAI, Palo Alto, pp 80-83 Inferring users' preferences from crowdsourced pairwise comparisons: A matrix completion approach. J Yi, Jin R Jain, S Jain, A K , Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoYi J, Jin R, Jain S, Jain AK (2013) Inferring users' preferences from crowdsourced pairwise com- parisons: A matrix completion approach. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 207-215 Opportunities for crowdsourcing research on amazon mechanical turk. J J Chen, N J Menezes, A D Bradley, T North, Proceedings of the CHI Workshop on Crowdsourcing and Human Computation. the CHI Workshop on Crowdsourcing and Human ComputationNew YorkACMChen JJ, Menezes NJ, Bradley AD, North T (2011) Opportunities for crowdsourcing research on amazon mechanical turk. In: Proceedings of the CHI Workshop on Crowdsourcing and Human Computation. ACM, New York, pp 1-4 Experiments in social computation. M Kearns, Commun ACM. 5510Kearns M (2012) Experiments in social computation. Commun ACM 55(10):56-67 Human Computation and Multiagent Systems: An Algorithmic Perspective. A Mao, D C Parkes, A D Procaccia, H Zhang, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI. the AAAI Conference on Artificial Intelligence (AAAI). AAAIPalo AltoMao A, Parkes DC, Procaccia AD, Zhang H (2011) Human Computation and Multiagent Systems: An Algorithmic Perspective. In: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI Palo Alto, pp 1-6 Towards large-scale collaborative planning: Answering high-level search queries using human computation. E Law, H Zhang, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI. the AAAI Conference on Artificial Intelligence (AAAI). AAAIPalo AltoLaw E, Zhang H (2011) Towards large-scale collaborative planning: Answering high-level search queries using human computation. In: Proceedings of the AAAI Conference on Artificial Intelli- gence (AAAI). AAAI Palo Alto, pp 1210-1215 Crowdforge: Crowdsourcing complex work. A Kittur, B Smus, S Khamkar, Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). the ACM Symposium on User Interface Software and Technology (UIST)New YorkACMKittur A, Smus B, Khamkar S (2011) Crowdforge: Crowdsourcing complex work. In: Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). ACM, New York, pp 43-52 Cooks or cobblers?: crowd creativity through combination. L Yu, J V Nickerson, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMYu L, Nickerson JV (2011) Cooks or cobblers?: crowd creativity through combination. In: Pro- ceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). ACM, New York, pp 1393-1402 Decision-theoretic control of crowd-sourced workflows. P Dai, Mausam, D S Weld, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI. the AAAI Conference on Artificial Intelligence (AAAI). AAAIPalo AltoDai P, Mausam, Weld DS (2010) Decision-theoretic control of crowd-sourced workflows. In: Pro- ceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI, Palo Alto, pp 1168- 1174 Human computation tasks with global constraints. H Zhang, E Law, R Miller, K Gajos, D Parkes, E Horvitz, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMZhang H, Law E, Miller R, Gajos K, Parkes D, Horvitz E (2012) Human computation tasks with global constraints. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). ACM, New York, pp 217-226 Turkopticon: Interrupting worker invisibility in amazon mechanical turk. L C Irani, M S Silberman, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). the SIGCHI Conference on Human Factors in Computing Systems (CHI)New YorkACMIrani LC, Silberman MS (2013) Turkopticon: Interrupting worker invisibility in amazon mechan- ical turk. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI). ACM, New York, pp 611-620 Human Computation: An Integrated Approach to Learning from the Crowd. E Law, L Von Ahn, Synthesis Lectures on Artificial Intelligence and Machine Learning Series. Morgan & Claypool. Law E, von Ahn L (2011) Human Computation: An Integrated Approach to Learning from the Crowd. In: Synthesis Lectures on Artificial Intelligence and Machine Learning Series. Morgan & Claypool, San Rafael, CA, United States An Evaluation of Aggregation Techniques in Crowdsourcing. Qvh Nguyen, T Nguyen Thanh, Lam Ngoc, T Aberer, K , Proceedings of the International Conference on Web Information Systems Engineering (WISE). the International Conference on Web Information Systems Engineering (WISE)New YorkSpringerNguyen QVH, Nguyen Thanh T, Lam Ngoc T, Aberer K (2013) An Evaluation of Aggregation Techniques in Crowdsourcing. In: Proceedings of the International Conference on Web Informa- tion Systems Engineering (WISE). Springer, New York, pp 1-15 Get another label? improving data quality and data mining using multiple, noisy labelers. V S Sheng, F Provost, P G Ipeirotis, Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD). the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD)New YorkACMSheng VS, Provost F, Ipeirotis PG (2008) Get another label? improving data quality and data min- ing using multiple, noisy labelers. In: Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD). ACM, New York, pp 614-622 Whose vote should count more: Optimal integration of labels from labelers of unknown expertise. J Whitehill, P Ruvolo, Fan Wu, T Bergsma, J Movellan, J , Advances in Neural Information Processing Systems. Red HookCurran Associates, IncWhitehill J, Ruvolo P, fan Wu T, Bergsma J, Movellan J (2009) Whose vote should count more: Optimal integration of labels from labelers of unknown expertise. In: Advances in Neural Infor- mation Processing Systems. Curran Associates, Inc., Red Hook, pp 2035-2043 Learning whom to trust with mace. D Hovy, T Berg-Kirkpatrick, A Vaswani, E Hovy, Proceedings of the Conference of the North American Chapter of the Association of Computational Linguistics, Human Language Technologies (NAACL-HLT). the Conference of the North American Chapter of the Association of Computational Linguistics, Human Language Technologies (NAACL-HLT)StroudsburgAssociation for Computational LinguisticsHovy D, Berg-Kirkpatrick T, Vaswani A, Hovy E (2013) Learning whom to trust with mace. In: Proceedings of the Conference of the North American Chapter of the Association of Computa- tional Linguistics, Human Language Technologies (NAACL-HLT). Association for Computational Linguistics, Stroudsburg, pp 1120-1130 Recursive fact-fnding: A streaming approach to truth estimation in crowdsourcing applications. D Wang, T Abdelzaher, L Kaplan, C C Aggarwal, Proceedings of the International Conference on Distributed Computing Systems (ICDCS). the International Conference on Distributed Computing Systems (ICDCS)Washington, DCIEEE Computer SocietyWang D, Abdelzaher T, Kaplan L, Aggarwal CC (2013) Recursive fact-fnding: A streaming ap- proach to truth estimation in crowdsourcing applications. In: Proceedings of the International Conference on Distributed Computing Systems (ICDCS). IEEE Computer Society, Washington, DC, pp 530-539 Aggregating crowdsourced binary ratings. N Dalvi, A Dasgupta, R Kumar, V Rastogi, Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2). the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2)GenevaDalvi N, Dasgupta A, Kumar R, Rastogi V (2013) Aggregating crowdsourced binary ratings. In: Proceedings of the International World Wide Web Conference (WWW). International World Wide Web Conferences Steering Committee (IW3C2), Geneva, pp 285-294 Hotspotting -a probabilistic graphical model for image object localization through crowdsourcing. M Salek, Y Bachrach, P Key, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI. the AAAI Conference on Artificial Intelligence (AAAI). AAAIPalo AltoSalek M, Bachrach Y, Key P (2013) Hotspotting -a probabilistic graphical model for image ob- ject localization through crowdsourcing. In: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI, Palo Alto, pp 1156-1162 The complexity of crowdsourcing: Theoretical problems in human computation. A Kulkarni, Proceedings of the CHI 2011 Workshop on Crowdsourcing and Human Computation. the CHI 2011 Workshop on Crowdsourcing and Human ComputationNew YorkACMKulkarni A (2011) The complexity of crowdsourcing: Theoretical problems in human computa- tion. In: Proceedings of the CHI 2011 Workshop on Crowdsourcing and Human Computation. ACM, New York, pp 1-4 How Much Spam Can You Take? An Analysis of Crowdsourcing Results to Increase Accuracy. J Vuurens, Apd Vries, C Eickhoff, Proceedings of the ACM SIGIR Workshop on Crowdsourcing for Information Retrieval (CIR). the ACM SIGIR Workshop on Crowdsourcing for Information Retrieval (CIR)New YorkACMVuurens J, Vries APD, Eickhoff C (2011) How Much Spam Can You Take? An Analysis of Crowdsourcing Results to Increase Accuracy. In: Proceedings of the ACM SIGIR Workshop on Crowdsourcing for Information Retrieval (CIR). ACM, New York, pp 48-55 Crowdscape: interactively visualizing user behavior and output. J Rzeszotarski, A Kittur, Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). the ACM Symposium on User Interface Software and Technology (UIST)New YorkACMRzeszotarski J, Kittur A (2012) Crowdscape: interactively visualizing user behavior and output. In: Proceedings of the ACM Symposium on User Interface Software and Technology (UIST). ACM, New York, pp 55-62 The anatomy of a large-scale human computation engine. S Kochhar, S Mazzocchi, P Paritosh, Proceedings of the Acm Sigkdd Workshop on Human Computation (HCOMP). the Acm Sigkdd Workshop on Human Computation (HCOMP)New YorkACMKochhar S, Mazzocchi S, Paritosh P (2010) The anatomy of a large-scale human computation engine. In: Proceedings of the Acm Sigkdd Workshop on Human Computation (HCOMP). ACM, New York, pp 10-17 On the verification complexity of group decision-making tasks. O Amir, Y Shahar, Y Gal, L Ilani, Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI. the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAIPalo AltoAmir O, Shahar Y, Gal Y, Ilani L (2013) On the verification complexity of group decision-making tasks. In: Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP). AAAI, Palo Alto, pp 2-8 Co-worker transparency in a microtask marketplace. P Kinnaird, L Dabbish, S Kiesler, H Faste, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMKinnaird P, Dabbish L, Kiesler S, Faste H (2013) Co-worker transparency in a microtask market- place. In: Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). ACM, New York, pp 1285-1290 Quality control mechanisms for crowdsourcing: Peer review, arbitration, and expertise at familysearch indexing. D L Hansen, P J Schone, D Corey, M Reid, J Gehring, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMHansen DL, Schone PJ, Corey D, Reid M, Gehring J (2013) Quality control mechanisms for crowdsourcing: Peer review, arbitration, and expertise at familysearch indexing. In: Proceed- ings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). ACM, New York, pp 649-660 Quality management on amazon mechanical turk. P G Ipeirotis, F Provost, J Wang, Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). the ACM SIGKDD Workshop on Human Computation (HCOMP)New YorkACMIpeirotis PG, Provost F, Wang J (2010) Quality management on amazon mechanical turk. In: Pro- ceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). ACM, New York, pp 64-67 Shepherding the crowd yields better work. S Dow, A Kulkarni, S Klemmer, B Hartmann, Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC)New YorkACMDow S, Kulkarni A, Klemmer S, Hartmann B (2012) Shepherding the crowd yields better work. In: Proceedings of the ACM Conference on Computer-Supported Cooperative Work and Social Computing (CSWC). ACM, New York, pp 1013-1022 Affective computing: challenges. R W Picard, Int J Hum-Comput St. 591-2Picard RW (2003) Affective computing: challenges. Int J Hum-Comput St 59(1-2):55-64 Sellers' problems in human computation markets. M S Silberman, J Ross, L Irani, B Tomlinson, Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). the ACM SIGKDD Workshop on Human Computation (HCOMP)Silberman MS, Ross J, Irani L, Tomlinson B (2010) Sellers' problems in human computation markets. In: Proceedings of the ACM SIGKDD Workshop on Human Computation (HCOMP). Coping with human errors through system design: implications for ecological interface design. J Rasmussen, K Vicente, Int J Man-Mach Stud. 315Rasmussen J, Vicente K (1989) Coping with human errors through system design: implications for ecological interface design. Int J Man-Mach Stud 31(5):517-534 Building a practically useful theory of goal setting and task motivation. A 35-year odyssey. E A Locke, G P Latham, Am Psychol. 579Locke EA, Latham GP (2002) Building a practically useful theory of goal setting and task motiva- tion. A 35-year odyssey. Am Psychol 57(9):705-717 Person-job Fit: A Conceptual Integration, Literature Review, and Methodological Critique. J R Edwards, Charlottesville, Virginia, United StatesUniversity of VirginiaEdwards JR (1990) Person-job Fit: A Conceptual Integration, Literature Review, and Methodolog- ical Critique. University of Virginia, Charlottesville, Virginia, United States Achievement motivation: Conceptions of ability, subjective experience, task choice, and performance. J G Nicholls, Psychol Rev. 913Nicholls JG (1984) Achievement motivation: Conceptions of ability, subjective experience, task choice, and performance. Psychol Rev 91(3):328-346 Handbook of Organizational Behavior. J R Hackman, Lorsch J.Prentice-HallNew JerseyThe design of work teamsHackman JR (1987) The design of work teams. In: Lorsch J. (ed.) Handbook of Organizational Behavior. Prentice-Hall, New Jersey, pp 315-342 An algorithm for distributed groupware applications. A Karsenty, M Beaudouin-Lafon, Proceedings of the International Conference on Distributed Computing Systems (ICDCS). the International Conference on Distributed Computing Systems (ICDCS)New YorkIEEE Computer SocietyKarsenty A, Beaudouin-Lafon M (1993) An algorithm for distributed groupware applications. In: Proceedings of the International Conference on Distributed Computing Systems (ICDCS). IEEE Computer Society, New York, pp 195-202 Arrow's theorem in judgment aggregation. F Dietrich, C List, Soc Choice Welfare. 291Dietrich F, List C (2007) Arrow's theorem in judgment aggregation. Soc Choice Welfare 29(1):19- 33 A Taylor, A M Pacelli, Mathematics and Politics: Strategy, Voting, Power, and Proof. New YorkSpringerTaylor A, Pacelli AM (2008) Mathematics and Politics: Strategy, Voting, Power, and Proof. Springer, New York Better human computation through principled voting. A Mao, A D Procaccia, Y Chen, Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI. the AAAI Conference on Artificial Intelligence (AAAI). AAAIPalo AltoMao A, Procaccia AD, Chen Y (2013) Better human computation through principled voting. In: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI). AAAI, Palo Alto, pp 1142-1148 Deliberative democracy and the discursive dilemma. P Pettit, Phil Issues. 35s1Pettit P (2001) Deliberative democracy and the discursive dilemma. Phil Issues 35(s1):268-299 A framework for human factors evaluation. A Whitefield, F Wilson, J Dowell, Behav Inform Technol. 101Whitefield A, Wilson F, Dowell J (1991) A framework for human factors evaluation. Behav Inform Technol 10(1):65-79 Handbook of Performability Engineering. K B Misra, SpringerLondonMisra KB (2008) Handbook of Performability Engineering. Springer, London
[]
[ "Random Latin squares and Sudoku designs genera- tion", "Random Latin squares and Sudoku designs genera- tion" ]
[ "Roberto Fontana \nDept.of Mathematical Sciences\nPolitecnico di Torino\nTorinoItaly\n" ]
[ "Dept.of Mathematical Sciences\nPolitecnico di Torino\nTorinoItaly" ]
[]
Uniform random generation of Latin squares is a classical problem. In this paper we prove that both Latin squares and Sudoku designs are maximum cliques of properly defined graphs. We have developed a simple algorithm for uniform random sampling of Latin squares and Sudoku designs. It makes use of recent tools for graph analysis. It has been implemented using SAS.
10.1214/14-ejs913
[ "https://arxiv.org/pdf/1305.3697v1.pdf" ]
88,520,785
1305.3697
5568bbd7bd173ea2b63d492cdfa0f7327589d2a6
Random Latin squares and Sudoku designs genera- tion 16 May 2013 Roberto Fontana Dept.of Mathematical Sciences Politecnico di Torino TorinoItaly Random Latin squares and Sudoku designs genera- tion 16 May 2013 Uniform random generation of Latin squares is a classical problem. In this paper we prove that both Latin squares and Sudoku designs are maximum cliques of properly defined graphs. We have developed a simple algorithm for uniform random sampling of Latin squares and Sudoku designs. It makes use of recent tools for graph analysis. It has been implemented using SAS. Introduction Generating uniformly distributed random Latin squares is a relevant topic. Already in 1933, F. Yates (Yates (1933)) wrote ... it would seem theoretically preferable to choose a square at random from all the possible squares of given size. The widely used algorithm for generating random Latin squares of a given order is (Jacobson and Matthews (1996)). It is based on a proper set of moves that connect all the squares and make the distribution of visited squares approximately uniform. In this paper we present a new approach that is based on the equivalence between Latin squares and maximum cliques of a graph. This approach is also valid for Sudoku designs. The paper is organized as follows. In Section 2 the equivalence between Latin squares (Sudoku designs) and maximum cliques of a suitable graph is demonstrated. Section 3 describes an algorithm for generating uniformly distributed random Latin squares and Sudoku designs. The corresponding SAS code is available in the supporting material. Concluding remarks are made in Section 4. Latin squares and Sudoku designs are maximum cliques Latin squares A Latin square of order n is an n × n matrix L n in which each of n distinct symbols appear n times, once in each row and one in each column. For the sake of simplicity we consider the integers 1, 2, . . . , n as symbols. We denote by L n [., c], c = 1, . . . , n (L n [r, .], r = 1, . . . , n) the columns (resp. the rows) of L n and by L n the set of all the Latin squares of order n. For example, a Latin square of order 4, L 4 ∈ L 4 , is L 4 =     1 2 3 4 4 1 2 3 3 4 1 2 2 3 4 1     (1) E-mail: [email protected] We can look at a Latin square L n = [ℓ rc ; r, c = 1, . . . , n] as a set of n disjoint permutation matrices, following the approach recently adopted in (Dahl (2009)) and (Fontana (2011)). For each symbol s, s = 1, . . . , n we consider the n × n matrix P (s) = [p (s) rc ; r, c = 1, . . . , n] where p (s) rc = 1 if ℓ rc = s 0 otherwise .(2) Given a permutation matrix P the corresponding permutation π = (π 1 , . . . , π n ) of (1, . . . , n) is defined as π = P 1 n where 1 n is the n × 1 column vector whose elements are all equal to 1. Viceversa, given a permutation π = (π 1 , . . . , π n ) of (1, . . . , n) the corresponding permutation matrix P = [p rc ; r, c = 1, . . . , n] is defined as p rc = 1 if c = π r 0 otherwise(3) We denote by φ the function that transform a permutation π of (1, . . . , n) into a permutation matrix P = φ(π) according to Equation 3. For the Latin square L 4 ∈ L 4 of Equation 4, the permutation matrix P (2) is P (2) =     0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0    (4) and the corresponding permutation π (2) of (1, 2, 3, 4) is π (2) = (2, 3, 4, 1) . It immediately follows that a Latin square of order n can be written as L n = P (1) + 2P (2) + . . . + nP (n)(5) where P (s) , s = 1, . . . , n are mutually disjoint permutation matrices. Two permutation matrices P (s) and P (t) are disjoint if and only if p (s) rc p (t) rc = 0 for each r, c ∈ {1, . . . , n}. Equivalently two permutations π (s) and π (t) are disjoint if and only if π (s) (r) = π (t) (r) for r = 1, . . . , n. Without loss of generality, as we will explain below, let us suppose that P (1) = I n where I n is the n × n identity matrix. The permutation π (1) corresponding to P (1) is the identity permutation ι n , π (1) ≡ ι n = (1, . . . , n). Let us denote by P n the set of all the permutations of {1, . . . , n} and, given π ∈ P n , by L π n ⊂ L n the set of all the Latin squares of order n for which P (1) = φ(π) and P (2) < . . . < P (n) where P (s) < P (t) or, equivalently, π (s) < π (t) means that (π (s) 1 , . . . , π (s) n ) < lex (π (t) 1 , . . . , π (t) n ). The symbol "< lex " denotes the standard lexicographic order, (a 1 , . . . , a n ) < lex (b 1 , . . . , b n ) ⇔ ∃m > 0 ∀i < m a i = b i and a m < b m . For simplicity we will write "<" in place of "< lex ". As it will become clear later on, any order between permutations can be chosen. Let us consider L ιn n , the set of all the Latin squares of order n for which P (1) = I n and P (2) < . . . < P (n) . As L ιn n is built, we can generate all the Latin squares of order n, L n ∈ L n , considering (a) all the (n − 1)! permutations (s 2 , . . . , s n ) of the symbols 2, . . . , n and assigning them to the permutation matrices P (2) , . . . P (n) I n + s 2 P (2) + . . . + s n P (n) (b) all the n! sets L π n where π ∈ P n is a permutation of (1, . . . , n). We observe that L π n contains all the Latin squares that are generated permuting the columns of a Latin square L n of L ιn n L π n = {[L n [., π 1 ]| . . . |L n [., π n ]] : L n ∈ L ιn n } It follows that in order to generate a random Latin square L n it is sufficient: (a) to generate a random Latin square L (1) n ∈ L ιn n ; (b) to generate L We observe that the number #L n of Latin squares of order n is #L n = n!(n − 1)!#L ιn n (6) To generate a Latin square L n ∈ L ιn n we have to build n − 1 permutation matrices P (s) , s = 2, . . . , n, P (2) < . . . < P (n) , that are mutually disjoint and that are disjoint with I n . In the language of permutations, we have to build n − 1 derangements δ (s) , s = 2, . . . , n of (1, . . . , n), δ (2) < . . . < δ (n) such that δ (s) (r) = δ (t) (r), r = 1, . . . , n for each s, t ∈ {2, . . . , n}, s = t. Let D n ⊂ P n be the set of all the derangements of (1, . . . , n): we denote by d n the number of derangements of (1, . . . , n), d n = #D n . Let G n = (V n , E n ) be the undirected graph whose set of vertices V n is the set of derangements D n and whose set of edges E n contains all the couples of derangements (δ (i) , δ (j) ), i < j such that δ (i) (r) = δ (j) (r), r = 1, . . . , n. The following theorem holds. Theorem 1. The Latin squares L n of order n of L ιn n are the ordered cliques C n−1 of size n − 1 of G n = (V n , E n ) C n−1 = (δ (2) , . . . , δ (n) ), δ (2) < . . . < δ (n) C n−1 are the largest cliques of G n . Proof. A Latin square L n ∈ L ιn n can be written as L n = I n + 2P (2) + . . . + nP (n) where P (s) is the permutation matrix corresponding to the derangement δ (s) = P (s) 1 n , s = 2, . . . , n and δ (2) < . . . < δ (n) . The derangements δ (s) are disjoint. It follows that {δ (2) , . . . , δ (n) } is a clique of G n . Viceversa, given the clique {δ (2) , . . . , δ (n) } with δ (2) < . . . < δ (n) we build L ⋆ n = I n + 2φ(δ (2) ) + . . . + nφ(δ (n) ) where φ is defined in Equation 3 . It is immediately evident that L ⋆ n = L n . Finally, Latin squares correspond to the largest cliques because it is evident that it is not possible to find a set of m > n derangements of {1, . . . , n} that are disjoint. Sudoku designs For the definition of Sudoku designs we refer to Bailey et al. (2008) In 1956, W. U. Behrens (Behrens (1956)) introduced a specialisation of Latin squares which he called gerechte. The n×n grid is partitioned into n regions, each containing n cells of the grid; we are required to place the symbols 1, ..., n into the cells of the grid in such a way that each symbol occurs once in each row, once in each column, and once in each region. The row and column constraints say that the solution is a Latin square, and the last constraint restricts the possible Latin squares. By this point, many readers will recognize that solutions to Sudoku puzzles are examples of gerechte designs, where n = 9 and the regions are the 3 × 3 subsquares. (The Sudoku puzzle was invented, with the name number place, by Harold Garns in 1979.) Analogously to Latin squares, we describe a Sudoku design in terms of Sudoku permutation matrices, Dahl (2009) andFontana (2011). Let us define the regions in which the matrix is divided. We will refer to regions as boxes. Let us consider a n × n matrix, where n = p 2 and p is a positive integer. Its row and column positions (i, j) are coded with the integer from 0 to p 2 − 1. We define boxes B k,m , k, m = 0, . . . , p − 1 as the following sets of positions B k,m = {(i, j) : kp ≤ i < (k + 1)p, mn ≤ j < (m + 1)n} It follows that any n × n matrix A can be partitioned into submatrices A km corresponding to boxes B k,m . An n × n matrix S n is a Sudoku, if in each row, in each column and in each box, each of the integers 1, . . . , n appears exactly once. We denote by S n the set of all the n × n Sudokus. In Sudoku literature, the set of boxes B b,m , m = 0, . . . , p − 1 constitutes the b th band, b = 0, . . . , p − 1, while the set of boxes B k,s , k = 0, . . . , p − 1 constitutes the s th stack, s = 0, . . . , p − 1. Let us define a Sudoku permutation matrixP , referred to as an S-matrixP , as a permutation matrix of order n which has exactly one ′′ 1 ′′ in each submatrix P k,m corresponding to boxes B k,m , k, m = 0, . . . , p − 1. Let us denote byP n ⊂ P n the set of all Sudoku permutations. An n × n Sudoku S n identifies n matricesP (i) , i = 1, . . . , n, whereP (i) is the S-matrix corresponding to the positions occupied by the integer i. It follows that a Sudoku S n ∈ S n can be written as S n =P (1) + 2P (2) + . . . + nP (n)(7) We observe thatP (1) , . . . ,P (n) are mutually disjoint and that Equation (7) is the analogous of Equation (5) for Sudoku designs. We observe that the identity permutation ι n is not a Sudoku permutation, apart from the trivial n = 2 case. We can easily generate S-matrices. Let us define a more compact representation of an S-matrix S, by building the p × p matrix S ⋆ , whose elements are the only possible position, within each box, where S is equal to 1. Among the S-matrices we can define S ⋆ 0,n whose elements (S ⋆ 0,n ) km , k, m = 1, . . . , p are (S ⋆ 0,n ) km = (m, k). For the n = 4 case we obtain S ⋆ 0,4 = (1, 1) (2, 1) (1, 2) (2, 2) . and the corresponding S-matrix S 0,4 is S 0,n =     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     .(9) We will denote by σ 0,n the permutation corresponding to the S-matrix S 0,n , σ 0,n = S 0,n 1 n . It will play the role of the identity permutation ι n for Sudoku designs. All the S-matrices can be generated by permuting the rows within each band and the columns within each stack. It follows that the total number of S-matrices is p! 2p , Dahl (2009). LetG n = (Ṽ n ,Ẽ n ) be the undirected graph whose set of verticesṼ n is the set of derangements of σ 0,n that are also S-permutations, briefly Sudoku-derangements, and whose set of edgesẼ n contains all the couple of Sudoku-derangements (δ (i) ,δ (j) ), i < j such that δ (i) (r) =δ (j) (r), r = 1, . . . , n. Theorem 1 holds if we replace the graph G n with the graphG n . We observe thatG n is a subgraph of G n . Let us denote by S σ0,n n the set of all the Sudokus of order n for whichP (1) = S 0,n and P (2) < . . . <P (n) . As S (b) all the p! 2p sets S σ n where σ is a Sudoku permutation of (1, . . . , n). For the total number of Sudokus of order n we get Equation (10) which is the equivalent of Equation (6): #S n = (n − 1)!p! 2p #S σ0,n n (10) 3. An algorithm for random sampling Latin squares The algorithm takes n as input and gives L n , a random Latin square of order n, as output. The main steps of the algorithm are as follows. (a) Build the undirected graph G n = (V n , E n ); (i) generate V n ≡ D n , the set of all the derangements δ (i) , i = 1, . . . , d n of {1, . . . , n}; (ii) generate E n , the set of all the edges corresponding to all the couples of derangements (δ (i) , δ (j) ), i < j such that δ (i) (r) = δ (j) (r), r = 1, . . . , n. (b) generate all the largest cliques of G n (c) randomly extract one of the largest clique and order its vertices lexicographically. Let use denote this ordered clique by C n−1 = (δ (2) , . . . , δ (n) ). The corresponding Latin square is L (1) n = I n + 2φ(δ (2) ) + . . . + nφ(δ (n) ) (d) randomly choose one permutation σ = (s 2 , . . . , s n ) of (2, . . . , n) and generate L (2) n = I n + s 2 φ(δ (2) ) + . . . + s n φ(δ (n) ) (e) randomly choose one permutation γ of (1, . . . , n) and generate L n permuting the columns of L (2) n according to γ. We describe the algorithm for n = 5. (a) We generate D 5 taking all the permutations δ of (1, . . . , 5) such that δ(r) = r, r = 1, . . . , 5. D 5 contains 44 derangements. We denote by δ (i) , i = 1, . . . , 44 the elements of D 5 . (b) We generate E 5 considering all the 44 2 = 946 couples of derangements (δ (s) , δ (t) ), δ (s) < δ (t) such that δ (s) (r) = δ (t) (r), r = 1, . . . , 5. We find 276 edges. The graph G 5 , generated using the function tkplot of the R package igraph, (Csardi and Nepusz (2006)), is shown in Figure 1. (c) We use the function largest.cliques of the R package igraph, (Csardi and Nepusz (2006)), to get all the largest cliques of D 0,5 . Equivalently we can use the Optnet procedure of SAS/OR, (sas (2012)) or Cliquer, Niskanen andÖstergård (2003). We find 56 cliques of size 4. (d) We randomly choose one clique C 4 and we order it lexicographically C 4 = (δ (11) , δ (17) , δ (23) , δ (37) ) where δ (11) = (2, 5, 4, 3, 1), δ (17) = (3, 4, 5, 1, 2), δ (23) = (4, 1, 2, 5, 3) and δ (37) = (5, 3, 1, 2, 4). The corresponding Latin square is L (1) 5 = I n + 2φ(δ (2) ) + 3φ(δ (17) ) + 4φ(δ (30) ) + 5φ(δ (36) ), that is Table 1. Number of derangements of (1, . . . , n), n ≤ 9 0 1 2 3 4 5 6 7 8 9 1 0 1 2 9 44 265 1854 14833 133496 Sudoku designs The algorithm remains the same apart from the substitution of the graph G n withG n and by limiting the permutations of step (e) to the permutations of the rows within band and of the columns within stacks. For example, for the case n = 2 2 we find three maximum cliques ofG 4 . By randomly choosing one permutation of the symbols 2, 3, 4 among the six available, and one S-matrix to be used asP (1) , among the sixteen available, we can randomly generate one Sudoku from the total of 288, (#S 4 = 288). Computational aspects We ran the algorithm using a standard laptop (CPU Intel Core i7-2620M CPU 2.70 GHz 2.70 GHz, RAM 8 Gb). We were able to solve the problems corresponding to the orders up to n = 7 for which we found 16, 942, 080 cliques. For n = 7 we used Cliquer Niskanen andÖstergård (2003) to find all the cliques. Taking into account symbol and column permutations our algorithm was able to extract uniformly at random a Latin square of order 7 among all the order 7 Latin squares that are 7!6!16, 942, 080 = 61, 479, 419, 904, 000. For Latin squares the number of nodes of G n coincides with the number of derangements (see Table 1). For Sudoku designs the numbers of Sudoku-derangements are 7 for n = 4 and 17, 972 for n = 9. If n becomes large with respect to the available computational resources it is possible to replace the graph G n (G n ) with a random subgraph A k n (Ã k n ) of it, where k denotes the number of the selected nodes. We point out that, if we take one clique at random from those of the subgraph A k n (Ã k n ), the distribution from which we are sampling is not uniform. Anyhow this approach can be useful to select the starting point of the algorithm described in Jacobson and Matthews (1996) that is based on moves between different designs. We experimented this approach for the 9 × 9 Sudoku, which is the most common structure for the popular Sudoku puzzle. We randomly chose 809 Sudoku derangements among the 17, 972 available. The subgraph has 112, 579 edges. Its largest cliques have dimensions equal to 8 and are 73. By randomly choosing one clique, one permutation of the symbols 2, . . . , 9 and one Sudoku matrix we can generate the Sudoku S 9 ∈ S 9 S 9 =               1 3 4 5 7 6 2 9 8 8 7 2 1 4 9 6 3 5 6 9 5 3 2 8 1 7 4 7 1 9 8 5 3 4 2 6 2 8 6 7 1 4 9 5 3 4 5 3 6 9 2 8 1 7 3 4 1 9 6 7 5 8 2 5 2 8 4 3 1 7 6 9 9 6 7 2 8 5 3 4 1              (13) It is worth noting that recent advances in software for huge graph analysis (millions of nodes) make it possible to manage problems that are extremely interesting from a practical point of view. Conclusion This paper presented a simple algorithm for uniform random sampling from the population of Latin squares and Sudoku designs. The algorithm is based on the largest cliques of proper graphs and has been implemented in SAS. The code exports the graph in a format that can be used by other software, like Cliquer, Niskanen andÖstergård (2003). The algorithm could be run using the entire graph G n up to an order n equal to 7 on a standard pc. Future research will aim at testing the algorithm for higher orders. Recent advances in graph analytics on huge graph such those arising in social sciences (see e.g. Shao et al. (2012) for an overview on the subject), make this objective feasible and challenging at the same time. generate L n by a random permutation of the columns of L , we can generate all the Sudokus of order n = p 2 considering (a) all the (n − 1)! permutations (s 2 , . . . , s n ) of the symbols 2, . . . , n and assigning them to the permutation matricesP (2) , . . .P (n) I n + s 2P (2) + . . . + s nP (n) Fig. 1 . 1we finally get L 5 by randomly choosing one permutation σ for the symbols 2, . . . , 5, σ = (4, 3, 2, 5), and one permutation γ for the columns 1, . . . , 5, γ = (The graph G5 OR(R) 12.1 User's Guide: Network Optimization Algorithms. SAS. SAS/OR(R) 12.1 User's Guide: Network Optimization Algorithms. Cary, NC, United States. Sudoku, Gerechte Designs, Resolutions, Affine Space, Spreads, Reguli, and Hamming Codes. R A Bailey, P J Cameron, R Connelly, Amer. Math. Monthly. 1155Bailey, R. A., P. J. Cameron, and R. Connelly (2008). Sudoku, Gerechte Designs, Resolu- tions, Affine Space, Spreads, Reguli, and Hamming Codes. Amer. Math. Monthly 115 (5), 383-404. Feldversuchsanordnungen mit verbessertem ausgleich der bodenunterschiede. W Behrens, Zeitschrift für Landwirtschaftliches Versuchsund Untersuchungswesen. 2Behrens, W. (1956). Feldversuchsanordnungen mit verbessertem ausgleich der bodenunter- schiede. Zeitschrift für Landwirtschaftliches Versuchsund Untersuchungswesen 2, 176- 193. The igraph software package for complex network research. InterJournal Complex Systems. G Csardi, T Nepusz, 1695Csardi, G. and T. Nepusz (2006). The igraph software package for complex network research. InterJournal Complex Systems, 1695. Permutation matrices related to sudoku. G Dahl, Linear Algebra and its Applications. 4308-9Dahl, G. (2009). Permutation matrices related to sudoku. Linear Algebra and its Applica- tions 430 (8-9), 2457 -2463. Fractions of permutations. an application to sudoku. R Fontana, Journal of Statistical Planning and Inference. 14112Fontana, R. (2011). Fractions of permutations. an application to sudoku. Journal of Sta- tistical Planning and Inference 141 (12), 3697-3704. Generating uniformly distributed random latin squares. M T Jacobson, P Matthews, Journal of Combinatorial Designs. 46Jacobson, M. T. and P. Matthews (1996). Generating uniformly distributed random latin squares. Journal of Combinatorial Designs 4 (6), 405-437. Cliquer User's Guide: Version 1. S Niskanen, P R Östergård, Helsinki University of TechnologyNiskanen, S. and P. R.Östergård (2003). Cliquer User's Guide: Version 1.0. Helsinki University of Technology. Managing and mining large graphs: Systems and implementations. B Shao, H Wang, Y Xiao, Proceedings of the 2012 international conference on Management of Data. the 2012 international conference on Management of DataACMShao, B., H. Wang, and Y. Xiao (2012). Managing and mining large graphs: Systems and implementations. In Proceedings of the 2012 international conference on Management of Data, pp. 589-592. ACM. The formation of latin squares for use in field experiments. F Yates, Empire Journal of Experimental Agriculture. 13Yates, F. (1933). The formation of latin squares for use in field experiments. Empire Journal of Experimental Agriculture 1 (3), 235-244.
[]
[ "Experimentally-realizable PT phase transitions in reflectionless quantum scattering", "Experimentally-realizable PT phase transitions in reflectionless quantum scattering" ]
[ "Micheline B Soley \nDepartment of Chemistry\nUniversity of Wisconsin-Madison\n1101 University Ave53706MadisonWI\n\nDepartment of Physics\nUniversity of Wisconsin-Madison\n1150 University Ave53706MadisonWI\n\nYale Quantum Institute\nYale University\nPO Box 20833406520New HavenCTUSA\n\nDepartment of Chemistry\nYale University\n225 Prospect St06520New HavenCTUSA\n", "Carl M Bender \nDepartment of Physics\nWashington University\n63130St. LouisMOUSA\n", "A Douglas Stone \nYale Quantum Institute\nYale University\nPO Box 20833406520New HavenCTUSA\n\nDepartment of Applied Physics\nYale University\n06520New HavenCTUSA\n" ]
[ "Department of Chemistry\nUniversity of Wisconsin-Madison\n1101 University Ave53706MadisonWI", "Department of Physics\nUniversity of Wisconsin-Madison\n1150 University Ave53706MadisonWI", "Yale Quantum Institute\nYale University\nPO Box 20833406520New HavenCTUSA", "Department of Chemistry\nYale University\n225 Prospect St06520New HavenCTUSA", "Department of Physics\nWashington University\n63130St. LouisMOUSA", "Yale Quantum Institute\nYale University\nPO Box 20833406520New HavenCTUSA", "Department of Applied Physics\nYale University\n06520New HavenCTUSA" ]
[]
A class of above-barrier quantum-scattering problems is shown to provide an experimentallyaccessible platform for studying PT -symmetric Schrödinger equations that exhibit spontaneous PT symmetry breaking despite having purely real potentials. These potentials are one-dimensional, inverted, and unstable and have the form V (x) = −|x| p (p > 0), terminated at a finite length or energy to a constant value as x → ±∞. The signature of unbroken PT symmetry is the existence of reflectionless propagating states at discrete real energies up to arbitrarily high energy. In the PTbroken phase, there are no such solutions. In addition, there exists an intermediate mixed phase, where reflectionless states exist at low energy but disappear at a fixed finite energy, independent of termination length. In the mixed phase exceptional points (EPs) occur at specific p and energy values, with a quartic dip in the reflectivity in contrast to the quadratic behavior away from EPs. PT -symmetry-breaking phenomena have not been previously predicted in a quantum system with a real potential and no reservoir coupling. The effects predicted here are measurable in standard cold-atom experiments with programmable optical traps. The physical origin of the symmetrybreaking transition is elucidated using a WKB force analysis that identifies the spatial location of the above-barrier scattering.
null
[ "https://export.arxiv.org/pdf/2209.05426v1.pdf" ]
252,199,616
2209.05426
d5b9b0eec6e1efee42d2a4619e1c92806b24c197
Experimentally-realizable PT phase transitions in reflectionless quantum scattering Micheline B Soley Department of Chemistry University of Wisconsin-Madison 1101 University Ave53706MadisonWI Department of Physics University of Wisconsin-Madison 1150 University Ave53706MadisonWI Yale Quantum Institute Yale University PO Box 20833406520New HavenCTUSA Department of Chemistry Yale University 225 Prospect St06520New HavenCTUSA Carl M Bender Department of Physics Washington University 63130St. LouisMOUSA A Douglas Stone Yale Quantum Institute Yale University PO Box 20833406520New HavenCTUSA Department of Applied Physics Yale University 06520New HavenCTUSA Experimentally-realizable PT phase transitions in reflectionless quantum scattering A class of above-barrier quantum-scattering problems is shown to provide an experimentallyaccessible platform for studying PT -symmetric Schrödinger equations that exhibit spontaneous PT symmetry breaking despite having purely real potentials. These potentials are one-dimensional, inverted, and unstable and have the form V (x) = −|x| p (p > 0), terminated at a finite length or energy to a constant value as x → ±∞. The signature of unbroken PT symmetry is the existence of reflectionless propagating states at discrete real energies up to arbitrarily high energy. In the PTbroken phase, there are no such solutions. In addition, there exists an intermediate mixed phase, where reflectionless states exist at low energy but disappear at a fixed finite energy, independent of termination length. In the mixed phase exceptional points (EPs) occur at specific p and energy values, with a quartic dip in the reflectivity in contrast to the quadratic behavior away from EPs. PT -symmetry-breaking phenomena have not been previously predicted in a quantum system with a real potential and no reservoir coupling. The effects predicted here are measurable in standard cold-atom experiments with programmable optical traps. The physical origin of the symmetrybreaking transition is elucidated using a WKB force analysis that identifies the spatial location of the above-barrier scattering. Above-barrier reflection is a fundamental quantum effect in which particles with sufficient energy to surmount a potential-energy barrier are nonetheless reflected backwards. It can be seen as complementary to the phenomenon of quantum tunneling, in which a particle that lacks sufficient energy to overcome a barrier is nonetheless able to pass through the barrier with some probability. Both effects arise from the fundamental wave character of the quantum wavefunction, which cannot change abruptly with small changes in the potential. For a typical potential barrier with maximum energy V 0 , a quantum particle will reflect with high probability when its energy E is slightly higher than V 0 , and at higher energies the reflection probability decreases rapidly on average. However, it has been known for quite some time that for certain shapes of potential this decrease is monotonic, so that unit transmission is only achieved at infinite energy. For other shapes the decrease is nonmonotonic, and there exist discrete energies near V 0 at which perfect above-barrier transmission is possible. In this work we elucidate this difference for a large class of potential shapes by relating it to the existence of broken or unbroken parity-time (PT ) symmetry in the corresponding Schrödinger equation. Experiments testing the predictions of our approach are feasible using cold atomic beams and current atomic-trap technology. Restricting our discussion to one-dimensional potentials V (x), PT symmetry refers to Schrödinger equations and boundary conditions that map into themselves under combined x → −x and complex conjugation. This PT -symmetry condition is less restrictive than the Hermiticity condition imposed in conventional quantum mechanics and allows for complex potentials with imaginary parts antisymmetric around the origin. Essentially all research on this topic has focused on Schrödinger equations with complex potentials. Here we treat Schrödinger equations with real potentials that have only PT symmetry and are not Hermitian due to the reflectionless boundary conditions. Because investigations of PT symmetry in quantum mechanics have typically involved complex potentials, which involve imaginary terms that cannot be implemented in closed quantum systems, there have been few experimental demonstrations of phenomena related to PT symmetry in quantum mechanics. Those few have been restricted to open systems coupled to reservoirs that are treated statistically and introduce phenomenological terms into the Schrödinger equation [1][2][3]. In contrast, there has been an intensive experimental study of PT symmetry and its breaking in systems described by classical wave equations, where imaginary terms representing loss and gain are introduced into the equations to represent coupling to reservoirs, but these tend to be more controllable and relatively easy to fabricate and measure. Examples span classical electromagnetism [4][5][6][7][8][9], acoustics [10,11], electronics [12][13][14][15], and mechanical systems [16]. In several cases these classical PT -symmetric systems have shown potential utility for applications in laser technology [17,18], sensing [19,20], and wireless power transfer [21,22]. Our work builds off the work in Refs. [23,24], which studied PT -symmetry phenomena in a class of quantum systems with complex potentials of the form V (x) = x 2 (ix) and V (x) = x 4 (ix) ( real), all of which satisfy the PT -symmetry condition. Importantly, for even integer values of , this set includes purely real "upside-arXiv:2209.05426v1 [quant-ph] 12 Sep 2022 down" PT -symmetric potentials (specifically V (x) = −x 2 , −x 4 , −x 6 , −x 8 , . . .). It was shown that for these real potentials with even integer power p ≥ 4 there exist discrete weakly-bound states for real energies E i > 0, but for p = 2 there are no real-energy solutions, only complex-energy ones. It was realized that real energies correspond to a kind of reflectionless-scattering state [25], although due to the unbounded-below nature of the potentials, the particles would be moving arbitrarily fast as they approached ±∞, causing the wavefunctions to oscillate infinitely fast and invalidating the assumptions of standard scattering theory. This paper proposes a clear way to probe this quantum physics in an experimentallyrealizable setup. Here we show that the reflectionless states of a continuous class of truncated upside-down real potentials show all the characteristics of PT -symmetric systems; specifically the presence of unbroken, mixed, and broken phases and spontaneous symmetry-breaking transitions at exceptional points, despite the absence of any imaginary terms in the potential. We quantitatively confirm the presence of weakly-bound states above such potentials with energies given precisely by those predicted for the integer cases considered in Ref. [26]. This constitutes the first theoretical prediction of such PT -symmetry phenomena in experimentally-realizable quantum scattering systems governed by the Schrödinger equation. Here we consider potentials of the form V (x) = −|x| p for p ∈ R, truncated to constant energy outside of the domain −L ≤ x ≤ L (see Fig. 1); we consider energy truncation instead of length truncation in the Appendix. These potentials are real and parity symmetric and are nonanalytic at the origin for noneven p values. To yield a continuously differentiable potential at the truncation point, we introduce a smoothed potential V (x, w), where the real parameter w determines the sharpness of the truncation (see Methods). V (x, w) reproduces the original upside-down PT -symmetric potentials in a finite region close to x = 0, where the eigenmode amplitude is expected to be greatest. Motivated by a different truncation used in ref. [26], our approach is qualitatively different, as it generalizes the original potentials to noneveninteger powers and truncates them to yield a standard quantum-scattering geometry. The nonanalytic behavior distinguishes this class of P-and T -symmetric real potentials from the aforementioned PT -symmetric potentials V (x) = x 2 (ix) and V (x) = x 4 (ix) , which can be continued in the complex plane, whereas V (x) = −|x| p cannot. Note that these noneven-integer potentials are analytic in p, but not in x; hence, the eigenvalues are analytic functions of p, but the corresponding Schrödinger equation is not analytic in x. We first verify that the reflectionless states of these truncated V (x) = −V 0 |x| p potentials obey PT symmetry. We introduce V 0 to clarify the choice of units. Since the potential is real, it is both P-and T -symmetric, and the problem superficially appears to be Hermitian; however, we shall see that the symmetry is reduced by the ). The solid red line illustrates our truncation of these potentials for p = 4 to length L = 2 and smoothing parameter w = 1000. We study below the resulting abovebarrier scattering phenomena and their relation to the corresponding unbounded, infinite system. reflectionless boundary conditions. We start with the one-dimensional Schrödinger equation 0 = − 2 2m φ (x) + [V (x) − E] φ(x), where m is the mass of the relevant quantum particle. We introduce the length scale x 0 ≡ (E/V 0 ) 1/p , where for values of p for which the infinite potential has bound states we choose E = E 0 , the ground-state energy, so that V (x) = −E 0 |x| p . (In the case of no bound states E can be chosen arbitrarily.) Henceforth, x denotes the position in units of x 0 and we take energy units with E 0 = 1 to recover V (x) = −|x| p . Assuming that the potential and kinetic energy are of the same order and hence of order E 0 , we find that x 0 ∼ (h 2 /mV 0 ) 1 p+2 , E 0 ∼ V 0 x p 0 . Now we look for a solution with a right-moving wave only, which satisfies the boundary conditions φ (−L) ∼ ikφ(−L), φ (L) ∼ ikφ(L) (w → 0), where k ≡ 2mE/ . These conditions map into leftmoving waves under either P or T separately, but map into themselves under the product PT . The well-known implication is that any solutions must either have a real energy or must occur in complex-conjugate pairs. If the former holds for all solutions, the PT symmetry is said to be unbroken; if the latter holds, it is broken. If both types of solutions exist, the spectrum is mixed. For the infinite-length and depth potentials studied in Ref. [24], the real solutions are peaked at the origin and decay weakly [φ(x) ∼ 1/x as x → ±∞ for the case p = 4]. This contrasts strongly with the standard exponentiallydecaying states of real attractive potentials. For the finite potentials (after truncation) and the scattering geometry, the reflectionless above-barrier states propagate as plane waves at infinity and are not square integrable. A recent general theory of reflectionless scattering modes (RSMs) [27,28] predicts an infinite set of discrete solutions for generic potentials with PT symmetry that also occur at real energies or in complex-conjugate pairs. We conjecture that these propagating reflectionless solutions will mimic closely the weakly-bound states of the unbounded potentials in the vicinity of the origin and will occur at similar energies. The hypothesis is verified by applying the RSM theory to quantum mechanics to determine accurately the reflectionless scattering modes. Until now, this theory has been applied to electromagnetic/optical systems [27,28], but the method is general and can be applied to quantum-scattering systems in any number of dimensions for scattering of both free-space and guided waves. We derive a specifically quantum formalism for the RSM theory in Appendix A. An attractive feature of the theory is that it calculates directly the discrete complex spectrum of reflectionless energies (referred to as R-zeros). There is no need to solve the scattering problem and search for zero reflection. The R-zero spectrum is similar to the more familiar spectrum of complex energy resonances, which satisfy purely outgoing boundary conditions; but the R-zero energies are distinct from the resonances. A sketch of the RSM/R-zero theory for general scattering geometries is given in the Methods. The general theory simplifies for the current case of a one-dimensional (two-channel) geometry, and the reflectionless energies can be found by a simple modification of the method of perfectly-matched layers, complex scaling [29], or complex absorbing potentials [30], which we use here. Real-energy solutions correspond to steadystate harmonic scattering and thus imply a zero of the reflection coefficient at the correct input energy. Complex solutions do not give zero reflection under uniform harmonic excitation but, if the R-zero is isolated and near the real axis, there is a narrow dip in the reflection below the background near the real part of its energy. Hence we can study these R-zero spectra and the corresponding eigenfunctions for the truncated potentials introduced and compare their properties to the infinite unbounded potentials studied previously. We find striking agreement between the two systems, bringing the novel predictions of the previous theory into experimental reach. As shown in Fig. 2(a) and Appendix B, the low-energy RSM energies for p = 4 are indeed real and agree with the bound-state energies found for the infinite system with 7-8 digits of accuracy for sufficiently large L. Even for the shortest truncation length L considered, the groundstate energy is accurately found, and higher eigenenergies converge to known infinite system values with increasing length L. Once the incident energies are known, these results are easily tested and confirmed by quantumscattering calculations, as detailed in Appendix C. Similar convergence is found with results obtained by energy truncation at |V max | (Appendix B); see also more general scattering results in Fig. 3 (b). Not only are the energies accurately predicted [26], but also the eigenfunctions in the interaction region perfectly mimic the behavior predicted for the infinite system. As shown in Fig. 2(b), the eigenfunctions are symmetric around the origin and exhibit the predicted 1/x decay with three-digit accuracy. Thus, the intriguing weaklybound states of the infinite potential are directly observable in the truncated system. Note the rapid increase in the spatial oscillation frequency of the eigenfunction as the particle accelerates toward the asymptotic region, where the force vanishes. RSM theory also allows us to search for a PT quantum phase transition within the class of potentials V (x) = −|x| p as p is varied, a phenomenon suggested by earlier work on complex PT potentials cited above [23,24]. In Fig. 3(a) we plot the low-lying real-energy eigenvalues of this class of potential as p is varied between 2 ≤ p ≤ 8. The PT symmetry implies that as p is varied, eigenvalues cannot disappear, but can meet at certain parameter values and then generically move into the complex plane as conjugate pairs. We find that for p ≥ 4 there are discrete real energies up to a high energy, above which the effect of our truncation becomes visible. This indicates that the behavior found for the infinite system at the discrete integers p = 4, 6, 8 (infinite number of real-energy bound states) is generic and continuous in p. In the interval 2 < p < 4 we see clearly a series of eigenvalue intersections, which happen at higher p for higher energies. These are exceptional points (EPs), where two distinct eigenfunctions coalesce before the energies become complex (we only plot real energies here). In the interval 2 < p < 4 there are only a finite number of real reflectionless energies, corresponding to partially-broken PT symmetry, similar but distinct from the behavior of the real eigenenergies of the complex potentials [24], V (x) = x 2 (ix) . Finally, for p ≤ 2, there are no real reflectionless energies, and the reflection coefficient simply decays monotonically with increasing energy. This monotonically-decreasing, nonresonant behavior of scattering above the inverted harmonic potential has been known since the early days of quantum mechanics [31]. Now we can appreciate that the behavior changes qualitatively for any sharper polynomial barrier, and that the regime p ≤ 2 has fully-broken PT symmetry, with no real-energy reflectionless states. As for p = 4, our results for p = 2 agree very well with the known results for the infinite potentials for sufficiently long truncation length L (see Appendix C). Quantum-scattering results shown in Fig. 3(b) dramatically confirm the identification of a distinct transition between the fully-broken PT phase (p ≤ 2) and the partially-broken phase (2 < p ≤ 4). Up to p = 2, we see a slower than exponential, but monotonic, decay of the reflection coefficient as a function of energy, without any deep dips. At p = 2, the asymptotic decay is exponential, and above it the decay is nonmonotonic with deep dips at the predicted reflection zeros (resolved only to a finite depth due to the finite accuracy of the numerics). Note also that the peak reflection for p = 6, after the first zero, has a reflection coefficient of ∼ 0.10, which should be easily measurable in experiments. a E 0 =1.477 E 1 =6.003 E 2 =11.802 E 3 =18.459 E 4 =25. The final signature of this quantum phase transition occurs for scattering at energies near the EPs (Fig. 4(a)), at which two RSMs meet and then separate as they move into the complex plane. The general theory [27,32] predicts that the reflection coefficients at EPs will vanish like (E − E EP ) 4 , and not quadratically as for isolated RSMs. We confirm this behavior in Fig. 4(b) and in additional related results in Appendix C. This straightforward measurement of the shape of the reflection dip away from and at an EP will provide a "smoking-gun" confirmation of the presence of a quantum PT phase transition in this system. The PT transition we identify here is driven by the increasing strength of above-barrier scattering as the variation of the potential becomes more rapid, and not by increasing loss/gain or differential loss, as in many prior studies. To analyze further how the physics changes with the shape of the potential we employ an approach to identify the origin of above-barrier reflection, which we term the WKB force analysis [33,34]. This method has been used previously to improve the accuracy of calculations of above-barrier reflection [33] and to reduce unphysical errors in quantum-mechanical simulations [34]; here we use the technique to determine the spatial origin of above-barrier reflection in these quantum potentials and the effect of truncation. The method begins by observing that WKB wavefunctions, although approximate solutions of the Schrödinger equation, cannot capture above-barrier reflection in 1D. Starting with a positive momentum solution at −∞, the local WKB momentum p(x, E) ≡ 2m(E − V (x)) can never change sign if E > V max . In addition, it can be shown that given a potential V (x), the WKB wavefunctions exactly satisfy a Schrödinger equation to which, in addition to V (x), one adds an energy-dependent potential correction V WKB (x, E) of order 2 of the form V WKB (x,E) = − 2 5 32m V (x) E − V (x) 2 + V (x) 8m (E − V (x)) . If we employ the reflectionless WKB states in the actual potential V (x) to calculate above-barrier reflection using the distorted-wave Born approximation, the quantum reflection from the potential V (x) at energy E arises solely from the additional scattering induced by V WKB (x, E). Thus, V WKB (x, E) reveals the spatial location of abovebarrier scattering; V WKB (x, E) is localized in a region around the potential maximum and vanishes far from the origin as 1/x 2 , even if V (x) extends to infinity; thus, it can be used to analyze the infinite-length upside-down potentials V (x) = −|x| p , as well as the truncated scattering potentials introduced here. First, we calculate V WKB (x, E) for upside-down infinite-length potentials, to look for a signature of the quantum phase transition. We study the shape of V WKB (x, E, p) as p is varied with E = 1.477 (the ground state for p = 4). As shown in Fig. 5(a) and Appendix D, there is a qualitative change in V WKB with p, and this change is insensitive to the value of E. For p ≤ 2, V WKB has a single peak at the origin, which is divergent due to nonanalytic behavior at the origin for p < 2. (Note that V WKB depends on V and V , at least one of which di- In (a) results indicate no RSMs exist as p → 2 from above. The single remaining RSM for p < 3.44 is pushed asymptotically to infinity as p → 2, in agreement with the result for complex potentials [24]. For p ≤ 2 we find no RSMs, corresponding to a fully-broken PT phase. There is a mixed phase region for 2 < p ≤ 4; and many RSMs for p ≥ 4, with the number increasing with the increasing truncation length, indicating an unbroken PT phase. The lower energy RSMs agree closely with the known energies of the weakly-bound states of the infinite-length potentials for even p (red circles). (b) Quantum-scattering results show monotonic decay of the reflectance with energy for p ≤ 2 and nonmonotonic behavior for p > 2 with deep dips at the energies of the RSMs. For p > 2 there is a second reflection peak after the first dip, which is substantial (∼ 0.10) for p = 6, 7.8. verges for p < 2.) At p = 2, V (x) is analytic and for small x the peak of V WKB is an inverted parabola. However, for p > 2 the peak splits into two maxima, symmetrically displaced from the origin, creating a small central well in V WKB , and this well becomes deeper as p → 4, where it creates an approximately parabolic "trap." For p > 4, the trap gets flatter at the center, making the scattering more localized at x ≈ ±1, while V (x) itself tends to a (semi-infinite in height) square potential well, which has an infinite set of reflectionless above-barrier resonances. We now analyze V WKB for the truncated potential V (x) = −|x| 4 at E = 1.477. In general, due to its local nature, V WKB is completely unchanged within the truncation length L, but the truncation does lead to a rapid spike at L, after which V WKB = 0 (as opposed to the 1/x 2 decay for the infinite barrier). The V WKB for the truncated p = 4 potential with lengths L = 2 and 5 are shown in Fig. 5(b) and on a larger scale in Appendix D. The size of the spikes in V WKB decreases as the truncation length is increased, indicating that the truncated potential reproduces the behavior of the infinite-length potentials at low energies. In summary, the occurrence of a PT symmetrybreaking phase transition in the spectrum of abovebarrier reflectionless quantum resonances of a class of real quantum-scattering potentials brings this physics within the reach of standard cold trapped-atom experiments in engineered potentials, generated by techniques such as rapidly scanned lasers [35], Digital Micromirror Devices (DMD) [36,37], intensity masks [38], and holographic methods [39][40][41][42]. Since this spectrum is robust to smooth changes in the potential, these phenomena will be relatively insensitive to experimental nonidealities, such as noise in the engineering of the power-law potential or introduction of additional polynomial terms (as we confirm in Appendix E). In addition, given the ubiquity of quantum reflection in near-threshold quantum systems, the occurrence of such phenomena may not be limited to the V (x) = −|x| p potentials, but instead may be present in a wide range of quantum systems and possible quantum technologies. Our results here should motivate a renewed search for PT -symmetry behaviors in fundamental quantum systems and development of possible PT -symmetric quantum innovations in fields such as atomic-analogue lasing, quantum transport theory, and quantum sensing. METHODS Potential smoothing We determine the reflectionless scattering modes of the V (x) = −|x| p potential in terms of the smoothed potentials V (x) = −x 4 f (x, w, L) − L 4 f (−x, w, L) , f (x, w, L) = 1 1 + e −w(x+L) + 1 1 + e w(x−L) − 1, where f (x, w, L) is the smoothing function with sharpness parameter w and truncation length L. Note this class of potentials presents a distinct quantummechanical analysis from quantum-field-theoretical analysis of positive V (x) = |x| p potentials [43], as these are upside-down and unbounded (before truncation) and feature no traditional bound states. Quantum-scattering calculations Numerical quantum-scattering calculations are performed for the right-moving scattering solution to determine the reflectance ratio |R/T | and subsequently the reflection coefficient |R| 2 as detailed in Appendix C. The truncation length is accounted for implicitly both through the specification of the region in which the scattering wavefunction is determined and the Neumann boundary condition. For increased numerical stability, quantum-scattering calculations near exceptional points are determined via explicit integration with Dormand-Prince coefficients. RSM calculations According to RSM theory, under general conditions, for a finite-range potential, there exist eigensolutions in which the wave is incident in a given set of incoming scattering channels and exits through the complementary channels with zero reflection into the incoming channels. These R-zero modes are found by considering the linear N × N scattering matrix, S, which satisfies β = S (E) α, where α is the vector of incoming wave amplitudes and β is the vector of outgoing wave amplitudes. Choosing n < N incoming channels, let R in ∈ C n×n be the submatrix of the S-matrix that connects the n nonzero entries of the input wave vector α to the n corresponding entries of β. An R-zero exists at (possibly complex) energy E if there exists an n-component input α in at that energy for which all the corresponding n outputs are zero, R in (E) α in = 0, implying that det [R in (E)] = 0. The RSM theory demonstrates that under general conditions a countably infinite spectrum of R-zeros exists at complex energies for each choice of input space; these spectra are distinct, but similar in many ways, to the complex spectrum of resonances familiar in quantum scattering. As noted, if the system has PT symmetry, then these energies are either real or come in complex-conjugate pairs. RSMs are determined by imposing quadratic complex absorbing potentials to enforce boundary conditions for the right-moving solution and solving for the eigenenergies of the resulting non-Hermitian Schrödinger equation via exact diagonalization. Here the truncation length L and truncation rapidity w are included explicitly. Both, as well as the strength and position of the complex absorbing potential and the choice of position space grid, are converged to yield the RSMs to the desired accuracy. WKB force potential The WKB force potential is determined akin to the W (x) potential of ref. [33] or the WKB correction potential of ref. [34]. In contrast to previous works, the potential is considered alone without summation with either the original potential or additional terms (e. g., the Coulomb potential) in order to directly pinpoint sources of quantum reflection in position space. All 1 E − E λ + i 2 Γ λ Γ 1/2 µc × e −i(φc+φ c ) , where U cc is the collision matrix element that connects the incoming c and outgoing c channels as a function of the energy E, δ cc is the Kronecker delta, φ c and φ c define the relative phase of the channels, and E λ is the eigenenergy of the resonance wavefunctions X λ . The coupling between the the resonance wavefunctions X λ and the channel wavefunctions ϕ c at a dividing surface S is given by the reduced-width amplitude γ λc = 2 /(2M c a c ) dSϕ c X λ for system mass M c and channel radius a c . The coupling matrix elements Γ λ = c Γ λc are given by Γ 1/2 µc = γ µc (2P c ) 1/2 for the incoming wave at the channel radius I c , the outgoing wave at channel radius O c , and the derived term P = (IO) −1 . We proceed by recognizing that the scattering matrix is equivalent to the collision matrix S = U and the matrix form of the delta function is the identity matrix I N,cc ≡ δ cc . We then define the background scattering in terms of the relative phase of the channels S 0,cc ≡ e −i(φc+φ c ) . Together, this yields S cc = I cc − i λµ Γ 1/2 λc 1 E − E λ + i 2 Γ λ Γ 1/2 µc S 0,cc . To reexpress the fractional term, we define a column coupling vector in terms of the coupling matrix elements d λc ≡ Γ 1/2 λc , where the coupling to the input channel c is the complex conjugate of the coupling to the output channel c . The overall coupling to a given resonance is then d † λ d λ = c |Γ 1/2 λc | 2 = c Γ λc = Γ λ , such that the denominator of the fractional term in the Breit-Wigner cross section is E − E λ + i 2 Γ λ = E − E λ + i 2 d † λ d λ . Te numerator is independent of µ, so the fractional term in the Breit-Wigner cross section is λµ Γ 1/2 λc 1 E − E λ + i 2 Γ λ Γ 1/2 µc = λ Γ 1/2 λc 1 E − E λ + i 2 d † λ d λ µ Γ 1/2 µc = λ d λ 1 E − E λ + i 2 d † λ d λ µ d † µ cc . Let the coupling matrix be defined by the coupling vectors as D = d λ1 d λ2 · · · d λn T . This gives the result λ d λ 1 E − E λ + i 2 d † λ d λ µ d † µ = D 1 E − E λ + i 2 D † D D † and the scattering matrix S(E) = I − iD 1 E − E λ + i 2 D † D D † S 0 , where E λ is now an M × M diagonal matrix (the same shape as D † D) of the M (theoretically infinite in number) resonance energies. Finally, to express the Breit-Wigner cross section in temporal coupled-mode theory form, we identify the system Hamiltonian. The closed Hamiltonian is the resonance Hamiltonian H close = E λ as the resonances completely describe the behavior of the system within the aforementioned surface S. The effective Hamiltonian is then the sum of the resonance Hamiltonian and its coupling to external channels H eff = E λ + i 2 D † D. This yields the temporal coupled-mode form of the Breit-Wigner cross section S (E) = I N − iD 1 E − H eff D † S 0 . Proof of an infinite number of R-zero modes To prove that there are an infinite number R-zeros defined as the energies E, where the reflection matrix R in (E) is zero, we find the energies at which the inverse reflection matrix R in (E) is infinite. Let F be a filtering matrix that eliminates all but the incoming channels and F be a filtering matrix that eliminates all but the outgoing channels. By definition, the filtering matrix and its complex conjugate yield FF † = I Nin , which reduces the dimension of the channel space from N to N in (that is F ij = δ ij for i ≤ N in , j ≤ N in ). Also, by definition FF † = I Nout . The reflection matrix is then R in (E) = FS (E) F † = F I N − iD 1 E − H eff D † S 0 F † . Repeated application of the filtering matrix yields no change because a projection onto a projection yields the same matrix), so we are free to employ a second round of filtering matrices R in (E) = F I N − iD 1 E − H eff D † F † FS 0 F † = I Nin − iFD 1 E − H eff D † F † S 0,in . The inverse is then determined according to the equation for the inverse of a matrix product (AB) −1 = B −1 A −1 , which gives R −1 in (E) = S −1 0,in I Nin − iFD 1 E − H eff D † F † −1 . The second term is evaluated via the Woodbury matrix identity [50] (A + UCV) −1 = A −1 − A −1 U C −1 + VA −1 U −1 VA −1 , where A = I Nin , U = FD, C = −i/ (E − H eff ), and V = D † F † . The above results give R −1 in (E) = S −1 0,in I Nin − (FD) × − i E − H eff −1 + iD † F † FD −1 D † F † = S −1 0,in I Nin + i (FD) × 1 E − H eff − iD † F † FD D † F † . To determine the coupling matrices in reduced-channel form, we define the filtered coupling matrices D in ≡ FD and D out ≡FD and express the effective Hamiltonian H eff in terms of its components, which yields R −1 in (E) = S −1 0,in × I Nin + iD in 1 E − H close + i 2 D † D − iD † F † FD D † in . Subsequent application of the filtering matrix identity F † F +F †F = I N gives the complete inverse reflection matrix R −1 in (E) = S −1 0,in I N,in + iD in 1 E − H eff D † in , H eff ≡ H close + i 2 D † in D in − i 2 D † out D out , in direct analogy to temporal coupled-mode theory. We now consider the number of R-zeros entailed by the expression. Since the background scattering S 0 amounts to a phase factor, R −1 in has no poles and does not play a role in the existence of R-zeros. The RSMs are therefore wholly determined by poles of the effective Hamiltonian H eff . When there are an infinite number of resonances E λ , there are an infinite number of eigenvalues of the resonance H close and effective H eff Hamiltonians. Thus, there are an infinite number of solutions for which E − H eff = 0. In this case, the inverse reflection matrix is singular, so there are an infinite number of singularities in the inverse reflection matrix R −1 in (E) and an infinite number of R-zeros in the reflection matrix R in (E). Relationship between applications of RSM theory to optics and quantum mechanics RSM theory in quantum mechanics is fundamentally different from RSM theory in optics. The susceptibility ε(x) multiplies the eigenvalue in the Maxwell equation ∇ 2 + (x)ω 2 c 2 E(x) = 0, but the potential V (x) in the Schrödinger equation − 2 2m ∇ 2 + V (x) − E ψ(x) = 0 does not. This special property of quantum mechanics accounts for the unique above-barrier quantum reflection effect examined in this paper. Appendix B: Truncation Lengths and Energy Bounds As shown in Fig. 2(a), Fig. 6, and Fig. 7, the RSMs of the truncated V (x) = −|x| p potential accurately reproduce the known analytic eigenenergies of the weaklybound states of the V (x) = −x 4 , −x 6 , and −x 8 potentials for sufficiently large truncation length L and energy bound |V max |. Similar to the RSMs as a function of truncation length, arbitrarily higher-lying eigenenergies are determined as the energy bound is increased, with the eigenenergies for p = 4 most sensitive to changes in the energy bound. Appendix C: Quantum-Scattering Calculations We support our RSM theory calculations with traditional quantum-scattering techniques following Ref. [26]. According to the exact quantum numerical approach, the scattering wavefunction is divided into three regions: (I) x < L, (II) −L ≤ x ≤ L, and (III) x > L. In Region I the wavefunction is expressed as a sum of incident and reflected waves at energy E ψ(x) = e i √ Ex + Re −i √ Ex , and in Region III the wavefunction is expressed only in terms of the amplitude-T transmitted wave T exp ix √ E . The scattering wavefunction is determined by integrating the time-dependent Schrödinger equation from the lower limit of region III y III (L) = 1, y III (L) = i √ E to the upper limit of region I y I (−L) = e −2iL √ E /T + R/T, y I (−L) = i √ Ee −2iL √ E /T − iR √ E/T, where y (x) is the scaled scattering wavefunction The reflectance follows as y (x) ≡ ψ (x) e −iL √ E /T.|R| 2 = |R/T | 1 + |R/T | 2 2 . We consider the quantum-scattering results for the class of truncated V (x) = −|x| p potentials. As shown in Fig. 8, the quantum-scattering results for the p = 2 potential indicate an absence of R-zeros in the energy range studied for sufficiently large L, in agreement with the results of RSM theory and the known monotonic decay of the reflection coefficient [31] and absence of weaklybound states in the infinite-length V (x) = −x 2 potential [24]. As shown in Fig. 4(b) and Fig. 9, the quantumscattering calculations also demonstrate signatures of exceptional points, such as the transition from a linear reflectance ratio near the RSM to a quadratic reflectance ratio at the exceptional point (this is associated with enhanced sensitivity [20]). RSMs are robust to various types of experimental errors. We show that the RSMs of the truncated p = 4 potential reproduce the weakly-bound-state energies of the upsidedown, PT -symmetric p = 4 potential in the presence of four types of error. The effect of random noise that conserves PT symmetry is modeled as a sum of sinusoidal random Fourier coefficients of the form: V R (x) = −(x 4 + n R g(x))f (x, w, L) − L 4 f (−x, w, L) , g(x) = N i=1 a i cos 2π ω i x ,(E1) where f is the smoothing function (see Methods), N = 50 is the number of random Fourier components, a i ∈ [−1, 1] is a random amplitude, ω i ∈ [0.1, 5] is a random frequency, and n R is the strength of the random noise. Parity-breaking random noise is likewise modeled with sinusoidal random Fourier-coefficient noise of the form g(x) = N i=1 a i sin 2π ω i x + φ i (E2) where ω i ∈ [0.1, 5] is a random frequency. Experimental errors that result in polynomial variation of the potential are modeled with an additive negative quadratic potential term V NQ (x) = −(x 4 + n NQ x 2 )f (x, w, L) − (L 4 + n NQ L 2 )f (−x, w, L) ,(E3) or a positive quadratic term V PQ (x) = −(x 4 − n PQ x 2 )f (x, w, L) − (L 4 − n PQ L 2 )f (−x, w, L) ,(E4) where n NQ and n PQ are the the negative and positive quadratic term strengths. Figure 11 indicates that the RSMs of the model with symmetric random noise closely agree with the known weakly-bound states. The lowest-lying eigenenergies agree with the RSMs for all noise strength terms considered, and higher eigenenergies are reproduced by further decreasing the noise strength. As shown in Fig. 11, the R-zeros accurately reproduce the weakly-bound-state energies even in the presence of parity-breaking random noise. As expected for R-zeros in time-reversal-symmetric systems, the R-zeros lie off the real axis. The real parts of the R-zeros closely agree with the known RSMs of the exact PT -symmetric potential. Even when random noise causes significant variation in the potential near the origin, the low-lying eigenenergies are accurately recovered for noise strengths that vary over orders of magnitude. Higher energies are recovered as the noise strengths are decreased. In addition, the distance from the real axis scales according to a power law with the parity-breaking noise strength, such that symmetry breaking does not significantly affect the RSMs for sufficiently small errors. Likewise, additive quadratic terms that significantly change the form of the potential yield near-reflectionless scattering modes that closely agree with the low-lying exact weakly-bound-state energies for a wide range of quadratic term strengths (see Fig. 13). And, higher-lying energies are determined accurately for smaller quadratic term strengths. real part and (c) amplitude of the imaginary part of its R-zeros (with imaginary parts lower than threshold value ε = 3) for varying nR. The real part of the R-zero in the presence of the noise (colored lines with points) closely matches the noise-free result (horizontal gray lines, Ei for i = 0 − 5) over several orders of magnitude in strength term nR, and the imaginary part of the R-zero (colored lines with points to corresponding Ei for i = 0 − 5) due to parity breaking approaches zero as nR is decreased. Results are averaged over three random sets of parameters ai, ωi, and φi. FIG. 1 . 1Upside-down, unbounded-below, PT -symmetric potential −x 4 (dashed blue line) is shown as a member of the larger class of purely real potentials V (x) = −|x| p for p ∈ R (dashed lines indicate instances p = 2 [orange], 3.25 [yellow], and 6.5 [purple] FIG. 2 . 2(a) Reflectionless scattering modes (RSMs) of the truncated V (x) = −x 4 potential are found to have real energies that converge to the exact analytic weakly-bound-state energies Ei of the corresponding infinite-length potential (gray horizontal lines, i = 0 − 4 shown) for sufficient length L. (b) The scattering wavefunction corresponding to RSMs of the truncated V (x) = −x 4 potential (real part, black solid line; absolute value, dashed orange line) for E = 1.477. Note that the modulus of the wavefunction exhibits parity symmetry, and the real part has an accelerating oscillatory behavior away from the origin. The envelope obeys an asymptotic power law in precise agreement with predictions for the weakly bound states of the infinite-length V (x) = −x 4 potential, fit by ψenv = 0.0228x −1.000 (dashed red line). FIG . 3. (a) RSM spectrum of the class of truncated V (x) = −|x| p potentials and (b) quantum-scattering results for same. coefficient near the lowest-energy exceptional point FIG. 4. (a) Close-up of PT -symmetry transitions at exceptional points in the RSM spectrum for p ≈ 3.44, EEP = 8.4 and p ≈ 3.94, EEP = 20.2. Left inset shows RSMs at p = 3.4 below the lowest-energy exceptional point, which indicates a single real RSM accompanied by complex-conjugate RSMs in the broken PT phase prior to reaching the two lowest-energy exceptional points. Right inset demonstrates the unbroken to broken PT transition near the lowest-energy exceptional point, with a transition from complex-conjugate pairs of RSMs to a single real RSM. (b)The scattering near these EPs exhibits the anomalous quartic lineshape of the reflectance dip at E = EEP for a reflectionless EP, (red line, shown for EEP = 8.4). In contrast an isolated RSM will have a quadratic behavior at the dip, as shown for p = 4.0, ERSM = 6.0 (blue circle). Other signatures in the reflectance ratio transitions at at EPs are detailed in Appendix C. FIG. 5 . 5WKB force analysis potentials at E = 1.477 for (a) the V (x) = −|x| p potential as a function of p and (b) the truncated V (x) = −|x| 4 and infinite-length V (x) = −x 4 potentials as a function of the truncation length L. The WKB force potential for varying p values indicates a quantum phase transition in reflection at p = 2 (solid blue line), in accordance with the quantum-scattering results and reflectionless scattering mode spectra. The WKB force analysis potentials for the truncated potential with varying truncation lengths (solid colored lines) compared to the WKB force analysis potential for the infinitelength potential (dashed red line) suggest reflection due to truncation reduces as L increases. Extended images are provided in Appendix D. This expression allows the ratio of the reflection and transmission coefficients to be calculated in terms of the wavefunction value at x = −L |R/T | = 1 2 |y (−L) + iy (−L) / √ E|. Similarly, the reflection ratio of the left-moving wave ψ = Re i √ Ex + e −i √ Ex with boundary conditions y I (−L) = 1, y I (−L) = −i √ E is |R/T | = 1 2 |y III (L) − iy III (L) / √ E|. FIG. 6 .FIG. 8 . 68RSMs of the truncated V (x) = −|x| p potential for (a) p = 6 and (b) p = 8 (colored lines with points) agree with the weakly-bound state energies Ei for i = 0 − 2 of the infinite-length potentials (gray horizontal lines) for longer truncation lengths L. V (x)=-|x 8 potential FIG. 7. RSMs of the truncated V (x) = −|x| p potential (colored lines with points) are found to reproduce the analytic weaklybound state energies Ei for i = 0 − 2 of the infinite-length (a) V (x) = −x 4 , (b) −x 6 , and (c) −x 8 potentials (gray horizontal lines) for sufficiently large energy bound |Vmax|. Quantum-scattering theory results for the truncated V (x) = −x 2 reproduce the RSM theory results, as the system accurately reproduces the lack of real eigenenergies of the infinite-length potential within an arbitrarily wide energy domain as L increases (e.g. no R-zeros or RSMs). For the truncated potential with L = 200 (red line), no reflectionless scattering modes occur in the energy range E ∈ [3.0, 6.0], in agreement with the infinite-length result. FIG. 9 . 9Transitions from a linear (blue line) to quadratic reflectance ratio (red line) at the (a) lowest-energy and (b) secondlowest energy exceptional points. The nonexceptional point scaling is illustrated for the RSM near p = 4, E1 = 6.00. As expected, the truncation yields a shift in the measured RSMs (non-EP, p = 4, E0 = 5.97 for L = 15; lowest-energy EP p = 3.4, E0 = 8.50 for L = 50; and second-lowest-energy EP p = 3.8, E0 = 22.68 for L = 50). potentials for V (x)=-|x 4 for varying L FIG. 10. WKB force potentials at E = 1.477 for (a) the infinite-length V (x) = −|x| p potential for varying p and (b) the truncated V (x) = −|x| 4 (colored lines), extending the magnitude and breadth shown in Fig. 5. The height of the twin peaks about the origin in the V (x) = −|x| p is seen to increase as p increases beyond p = 4, in agreement with the deeper dips in the reflection coefficient at the RSM energies at higher p values. For the L values depicted, the peaks are found to be largest for L = 2 (green medium-long dashed line), followed by L = 5 (yellow medium dashed line) and L = 8 (red fine dashed line), in agreement with the prediction that anomalous quantum reflection due to truncation decreases as L increases. FIG. 11. (a) Close-up near the origin of a truncated V (x) = −|x| 4 potential (dashed black line) with symmetric random noise Eq. (E1) of strength nR = 0.01 (solid red line) and its (b) RSMs as a function of noise strength nR. The RSMs for the noisy potential (colored lines with points) are found to closely agree with those without noise (horizontal gray lines, Ei for i = 0 − 5) over several orders of magnitude of nR. The RSMs are averaged for three random sets of parameters ai and ωi. FIG. 12 . 12(a) Detail of truncated V (x) = −|x| 4 potential (dashed black line) with parity-breaking random noise Eq. (E2) (solid red line) near the origin (noise strength nR = 0.01 pictured) and the (b) . 13. (a) Detail of truncated V (x) = −|x| 4 potential near the origin with additive quadratic terms Eq. (E3) and Eq. (E4). Potentials are shown for a positive quadratic term with strength nPQ = 0.1 (thick solid red line), negative quadratic term with strength nNQ = 0.1 (medium solid blue line), and no additive term (thin dashed black line). The RSMs in the presence of additive quadratic terms (colored lines with points) reproduce those of the potential with no additive term (horizontal gray lines for Ei with i = 0 − 5) over a broad range of negative nNQ and positive nP Q quadratic term strengths. RSMs are shown as a function of the term strength (b) nNQ and (c) nPQ. calculations are evaluated in Wolfram Mathematica 12.3.1.0. The authors thank E. J. Heller for stimulating conversations. MBS acknowledges financial support from the Yale Quantum Institute Postdoctoral Fellowship. ADS and CMB acknowledge financial support from the Simons Collaboration on Extreme Wave Phenomena Based on Symmetries. CMB also acknowledges financial support from the Alexander von Humboldt Foundation and the UK Engineering and Physical Sciences Research Council (EPSRC) grant at King's College London.ACKNOWLEDGMENTS Amplitude of the imaginary part of R-zeros for parity-breaking noiseE 0 =1.477 E 1 =6.003 E 2 =11.802 E 3 =18.459 E 4 =25.792 E 5 =33.694 Appendix A: Quantum RSM TheoryTo emphasize the applicability of RSM theory to quantum mechanics, we prove that an infinite number of Rzeros exist in suitable quantum-scattering systems. This adapts the temporal coupled-mode theory proof in optics[27,28]to quantum mechanics via the Breit-Wigner approximation to multichannel quantum R-matrix theory and generalizes the recently identified phenomenological link between temporal coupled-mode theory and Breit-Wigner cross sections[44,45]to the nonoverlappingresonance case.Relationship between temporal coupled-mode theory and quantum R-matrix theoryWe begin by developing an expression for the Breit-Wigner inelastic cross section that assumes the form of the temporal coupled-mode theory formulawhere S(ω) is the scattering matrix as a function of the frequency ω, D is the coupling matrix, S 0 is the background scattering matrix, H eff is the effective Hamiltonian, and H close is the Hamiltonian associated with the closed scattering region (the resonances)[46,47]. The initial expression is given by the Breit-Wigner approximation to the inelastic cross section in level-matrixformat multichannel quantum R-matrix theory[48,49]U cc (E) = δ cc − i λµ Γ 1/2 λcAppendix D: Extended WKB Force PotentialsThe behavior of the WKB force potential for the V (x) = −|x| p potential is depicted with extended domain and range inFig. 10(a). The force potentials support the identified quantum phase transition at p = 2, at which the central peak in the force potential bifurcates. In addition, the force potentials indicate that quantum reflection increases for values above p = 4, reaching peak heights above one for p = 7.8, one hundred near p = 50, and one thousand near p = 150, which supports the quantumscattering finding that deeper energy dips occur at RSM energies for larger p values.The extended image of the WKB force potential as a function of the truncation length L provided inFig. 10(b)further supports the use of the truncated V (x) = −|x| 4 potential with sufficiently long L to examine scattering from the infinite-length V (x) = −x 4 potential. For truncation lengths beyond the main region of reflection from the infinite-length potential (approximately |x| ≤ 3), the largest magnitude of the force potential decreases as the truncation length L increases. This finding agrees with the prediction that quantum reflection is minimized where the wavelength is short compared to the rate of change of the potential with position. In addition, the truncated and infinite-length WKB force potentials are found to agree in the domain |x| < L. Since the force potentials tend towards zero in the large-x limit, truncated potentials with sufficiently long L capture a large degree of the quantum reflection due to the infinite-length potential.Appendix E: Robustness of RSM PredictionsTo demonstrate that RSMs of the truncated V (x) = −|x| p potentials are effective for experimental measurements of PT -symmetry behavior, we demonstrate the . K F Zhao, M Schaden, Z Wu, Phys. Rev. A. 8142903K. F. Zhao, M. Schaden, and Z. Wu, Phys. Rev. A 81, 042903 (2010). . S Bittner, B Dietz, U Günther, H L Harney, M Miski-Oglu, A Richter, F Schäfer, Phys. Rev. Lett. 10824101S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski- Oglu, A. Richter, and F. Schäfer, Phys. Rev. Lett. 108, 024101 (2012). . C Zheng, L Hao, G L Long, Philos. Trans. R. Soc. A. 37120120053C. Zheng, L. Hao, and G. L. Long, Philos. Trans. R. Soc. A 371, 20120053 (2013). . A Guo, G J Salamo, D Duchesne, R Morandotti, M Volatier-Ravat, V Aimez, G A Siviloglou, D N Christodoulides, Phys. Rev. Lett. 10393902A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009). . C E Rüter, K G Makris, R El-Ganainy, D N Christodoulides, M Segev, D Kip, Nat. Phys. 6192C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010). . L Feng, M Ayache, J Huang, Y.-L Xu, M.-H Lu, Y.-F Chen, Y Fainman, A Scherer, Science. 333729L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.- F. Chen, Y. Fainman, and A. Scherer, Science 333, 729 (2011). . A Regensburger, C Bersch, M.-A Miri, G Onishchukov, D N Christodoulides, U Peschel, Nature. 488167A. Regensburger, C. Bersch, M.-A. Miri, G. On- ishchukov, D. N. Christodoulides, and U. Peschel, Na- ture 488, 167 (2012). . L Xiao, T Deng, K Wang, Z Wang, W Yi, P Xue, Phys. Rev. Lett. 126230402L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue, Phys. Rev. Lett. 126, 230402 (2021). . B Peng, Ş K Özdemir, F Lei, F Monifi, M Gianfreda, G L Long, S Fan, F Nori, C M Bender, L Yang, Nat. Phys. 10394B. Peng, Ş. K.Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014). . C Shi, M Dubois, Y Chen, L Cheng, H Ramezani, Y Wang, X Zhang, Nat. Commun. 71C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, Nat. Commun. 7, 1 (2016). . Y Aurégan, V Pagneux, Phys. Rev. Lett. 118174301Y. Aurégan and V. Pagneux, Phys. Rev. Lett. 118, 174301 (2017). . N M Chtchelkatchev, A A Golubov, T I Baturina, V M Vinokur, Phys. Rev. Lett. 109150405N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, Phys. Rev. Lett. 109, 150405 (2012). . J Schindler, A Li, M C Zheng, F M Ellis, T Kottos, Phys. Rev. A. 8440101J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kot- tos, Phys. Rev. A 84, 040101 (2011). . N Bender, S Factor, J D Bodyfelt, H Ramezani, D N Christodoulides, F M Ellis, T Kottos, Phys. Rev. Lett. 110234101N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Phys. Rev. Lett. 110, 234101 (2013). . W Cao, C Wang, W Chen, S Hu, H Wang, L Yang, X Zhang, Nature Nanotechnol. 1W. Cao, C. Wang, W. Chen, S. Hu, H. Wang, L. Yang, and X. Zhang, Nature Nanotechnol. , 1 (2022). . C M Bender, B K Berntson, D Parker, E Samuel, Am. J. Phys. 81173C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013). . L Feng, Z J Wong, R.-M Ma, Y Wang, X Zhang, Science. 346972L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Science 346, 972 (2014). . H Hodaei, M.-A Miri, M Heinrich, D N Christodoulides, M Khajavikhan, Science. 346975H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Science 346, 975 (2014). . P.-Y Chen, J Jung, Phys. Rev. Appl. 564018P.-Y. Chen and J. Jung, Phys. Rev. Appl. 5, 064018 (2016). . Z.-P Liu, J Zhang, Ş K Özdemir, B Peng, H Jing, X.-Y Lü, C.-W Li, L Yang, F Nori, Y Liu, Phys. Rev. Lett. 117110802Z.-P. Liu, J. Zhang, Ş. K.Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y. Liu, Phys. Rev. Lett. 117, 110802 (2016). . S Assawaworrarit, X Yu, S Fan, Nature. 546387S. Assawaworrarit, X. Yu, and S. Fan, Nature 546, 387 (2017). . S Assawaworrarit, S Fan, Nat. Electron. 3273S. Assawaworrarit and S. Fan, Nat. Electron. 3, 273 (2020). . C M Bender, S Boettcher, Phys. Rev. Lett. 805243C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). . C M Bender, Rep. Prog. Phys. 70947C. M. Bender, Rep. Prog. Phys. 70, 947 (2007). . Z Ahmed, C M Bender, M V Berry, J. Phys. A: Math. Gen. 38627Z. Ahmed, C. M. Bender, and M. V. Berry, J. Phys. A: Math. Gen. 38, L627 (2005). . C M Bender, M Gianfreda, Phys. Rev. A. 552118C. M. Bender and M. Gianfreda, Phys. Rev. A 5, 052118 (2018). . W R Sweeney, C W Hsu, A D Stone, Phys. Rev. A. 10263511W. R. Sweeney, C. W. Hsu, and A. D. Stone, Phys. Rev. A 102, 063511 (2020). . A D Stone, W R Sweeney, C W Hsu, K Wisal, Z Wang, Nanophotonics. 10343A. D. Stone, W. R. Sweeney, C. W. Hsu, K. Wisal, and Z. Wang, Nanophotonics 10, 343 (2021). . N Moiseyev, Phys. Rep. 302212N. Moiseyev, Phys. Rep. 302, 212 (1998). . J Muga, J Palao, B Navarro, I Egusquiza, Phys. Rep. 395357J. Muga, J. Palao, B. Navarro, and I. Egusquiza, Phys. Rep. 395, 357 (2004). . E C Kemble, Phys. Rev. 48549E. C. Kemble, Phys. Rev. 48, 549 (1935). . W R Sweeney, C W Hsu, S Rotter, A D Stone, Phys. Rev. Lett. 12293901W. R. Sweeney, C. W. Hsu, S. Rotter, and A. D. Stone, Phys. Rev. Lett. 122, 093901 (2019). . N T Maitra, E J Heller, Phys. Rev. A. 544763N. T. Maitra and E. J. Heller, Phys. Rev. A 54, 4763 (1996). . M B Soley, K N Avanaki, E J Heller, Phys. Rev. A. 10341301M. B. Soley, K. N. Avanaki, and E. J. Heller, Phys. Rev. A 103, L041301 (2021). . K Henderson, C Ryu, C Maccormick, M Boshier, New J. Phys. 1143030K. Henderson, C. Ryu, C. MacCormick, and M. Boshier, New J. Phys. 11, 043030 (2009). . G Gauthier, I Lenton, N M Parry, M Baker, M Davis, H Rubinsztein-Dunlop, T Neely, Optica. 31136G. Gauthier, I. Lenton, N. M. Parry, M. Baker, M. Davis, H. Rubinsztein-Dunlop, and T. Neely, Optica 3, 1136 (2016). . N Navon, R P Smith, Z Hadzibabic, Nat. Phys. 1N. Navon, R. P. Smith, and Z. Hadzibabic, Nat. Phys. , 1 (2021). . D R Scherer, C N Weiler, T W Neely, B P Anderson, Phys. Rev. Lett. 98110402D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, Phys. Rev. Lett. 98, 110402 (2007). . S Bergamini, B Darquié, M Jones, L Jacubowiez, A Browaeys, P Grangier, JOSA B. 211889S. Bergamini, B. Darquié, M. Jones, L. Jacubowiez, A. Browaeys, and P. Grangier, JOSA B 21, 1889 (2004). . V Boyer, R Godun, G Smirne, D Cassettari, C Chandrashekar, A Deb, Z Laczik, C Foot, Phys. Rev. A. 7331402V. Boyer, R. Godun, G. Smirne, D. Cassettari, C. Chan- drashekar, A. Deb, Z. Laczik, and C. Foot, Phys. Rev. A 73, 031402 (2006). . M Pasienski, B Demarco, Opt. Express. 162176M. Pasienski and B. DeMarco, Opt. Express 16, 2176 (2008). . A L Gaunt, Z Hadzibabic, Sci. Rep. 21A. L. Gaunt and Z. Hadzibabic, Sci. Rep. 2, 1 (2012). . C M Bender, K A Milton, S S Pinsky, L SimmonsJr, J. Math. Phys. 301447C. M. Bender, K. A. Milton, S. S. Pinsky, and L. Sim- mons Jr, J. Math. Phys. 30, 1447 (1989). . R E Hamam, A Karalis, J D Joannopoulos, M Soljačić, Phys. Rev. A. 7553801R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, Phys. Rev. A 75, 053801 (2007). J D Joannopoulos, S G Johnson, J N Winn, R D Meade, Photonic Crystals: Molding the Flow of Light. Princeton University2nd ed.J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008). . S Fan, W Suh, J D Joannopoulos, JOSA A. 20569S. Fan, W. Suh, and J. D. Joannopoulos, JOSA A 20, 569 (2003). Quantum Electron. W Suh, Z Wang, S Fan, 401511W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Elec- tron. 40, 1511 (2004). . A M Lane, R G Thomas, Rev. Mod. Phys. 30257A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30, 257 (1958). Brief review of the R-Matrix theory. L C Leal, SAMMY. L. C. Leal, Brief review of the R-Matrix theory, SAMMY (1999). . W W Hager, Rev, 31221W. W. Hager, SIAM Rev. 31, 221 (1989).
[]
[ "Theory of the power spectrum of spin-torque nanocontact vortex oscillators", "Theory of the power spectrum of spin-torque nanocontact vortex oscillators" ]
[ "Joo- Kim ", "T Devolder ", "\nInstitut d'Electronique Fondamentale\nUMR 8622\nUniv. Paris-Sud\n91405OrsayFrance\n", "\nCNRS\n91405OrsayFrance\n" ]
[ "Institut d'Electronique Fondamentale\nUMR 8622\nUniv. Paris-Sud\n91405OrsayFrance", "CNRS\n91405OrsayFrance" ]
[]
Spin-transfer torques in magnetic nanocontacts can lead to self-sustained magnetization oscillations that involve large-amplitude gyrotropic vortex motion. This dynamics consists of a steady state orbit around the nanocontact, which is made possible because the intrinsic magnetic damping is compensated by spin torques. In this article, we present an analytical theory of the power spectrum of these oscillations based on a rigid-vortex model. The appearance of vortex oscillations in nanocontacts is not associated with a Hopf bifurcation: there is no critical current and the only precondition for steady-state oscillations at finite currents is the existence of a vortex in the system, in contrast with conventional spin-torque oscillators that involve large-angle magnetization precession. The oscillation frequency is found to depend linearly on the applied current and inversely proportional to the orbital radius. By solving the associated Langevin problem for the vortex dynamics, the lineshape and linewidth for the power spectrum are also obtained. Under typical experimental conditions, a Lorentzian lineshape with a current-independent linewidth is predicted. Good quantitative agreement between the theory and recent experiments is shown.
null
[ "https://arxiv.org/pdf/1007.3859v1.pdf" ]
118,469,169
1007.3859
53a33d98268e237cf27af78fac755bb31535dadb
Theory of the power spectrum of spin-torque nanocontact vortex oscillators 22 Jul 2010 Joo- Kim T Devolder Institut d'Electronique Fondamentale UMR 8622 Univ. Paris-Sud 91405OrsayFrance CNRS 91405OrsayFrance Theory of the power spectrum of spin-torque nanocontact vortex oscillators 22 Jul 2010(Dated: July 23, 2010)numbers: 7570Kw7575-c8575-d Spin-transfer torques in magnetic nanocontacts can lead to self-sustained magnetization oscillations that involve large-amplitude gyrotropic vortex motion. This dynamics consists of a steady state orbit around the nanocontact, which is made possible because the intrinsic magnetic damping is compensated by spin torques. In this article, we present an analytical theory of the power spectrum of these oscillations based on a rigid-vortex model. The appearance of vortex oscillations in nanocontacts is not associated with a Hopf bifurcation: there is no critical current and the only precondition for steady-state oscillations at finite currents is the existence of a vortex in the system, in contrast with conventional spin-torque oscillators that involve large-angle magnetization precession. The oscillation frequency is found to depend linearly on the applied current and inversely proportional to the orbital radius. By solving the associated Langevin problem for the vortex dynamics, the lineshape and linewidth for the power spectrum are also obtained. Under typical experimental conditions, a Lorentzian lineshape with a current-independent linewidth is predicted. Good quantitative agreement between the theory and recent experiments is shown. I. INTRODUCTION A number of novel phenomena in magnetization dynamics are made possible by spin-transfer torques. These torques arise from the transfer of spin angular momentum between a spin-polarized electrical current and local magnetic moments, 1,2 and are most apparent in nanoscale structures in which current densities are large. An interesting example of current-driven dynamics are self-sustained oscillations of magnetization, whereby the torques imparted on the local moments act in part to compensate the intrinsic dissipation associated with the magnetization dynamics. As a result of this compensation, the system can attain a self-sustained oscillatory state in which the moments precess freely. These selfsustained oscillations have been observed in a variety of geometries and multilayer compositions, [3][4][5][6][7][8][9][10][11][12][13] and have drawn interest for both their fundamental nature and potential applications as nanoscale radio-frequency oscillators. One class of spin-torque driven oscillations involve magnetic solitons, such as domain walls 14,15 and vortices. 6,13,16,17 Self-sustained oscillations involving vortices were first brought to light in an experiment on spin-valve nanopillars. 6 In this study, the physical dimensions and aspect ratio of the active magnetic "free" layer of the spin-valve were chosen such that a magnetic vortex ground state is favorable. It was shown that steady-state vortex oscillations could be induced by spin torques with relatively high quality factors, a dynamical behavior that differs fundamentally from the transient or resonant response studied in other works. [18][19][20] Since then, other experiments have shown that similar behavior can be observed in magnetic tunnel junctions. 13 A more striking example of self-sustained vortex oscillations has been observed in magnetic nanocontacts. 9,11 In this geometry, large current densities are attained by channeling electron flow through a magnetoresistive multilayer stack via a metallic point contact, which is typically tens of nanometers in radius. 4 In contrast to nanopillars where the physical ge-ometry confines the vortex motion, the magnetic free layer in a nanocontact structure is a continuous film with much larger lateral dimensions (typically tens of microns), so translational invariance for the vortex is restored within the film plane in the absence of structural or magnetic defects. A confining potential in this medium arises, however, from the Zeeman energy due to the Oersted-Ampère fields associated with the applied current through the nanocontact. If we imagine the nanocontact to be an infinitely long cylindrical conductor, then the Oersted-Ampère fields generated by the current flow follow a circular pattern in the film plane, which shares the same cylindrical symmetry as the magnetic vortex. Because of this, the Zeeman energy due to these fields acts as a confining potential for the vortex around the nanocontact, with the potential minimum being centered on the nanocontact (if this is perfectly circular). The resulting dynamics in this geometry, as revealed by micromagnetics simulations, 11 is a steady state motion of the vortex that involves a large orbit around the nanocontact, as shown schematically in Fig. 1. By virtue of such large orbital motion, the free layer magnetization underneath the contact undergoes full rotations, which in turn results in a large time-varying component of the magnetoresistance. The observation of sub-gigahertz GMR variations in experimental nanocontact systems has been attributed to such vortex dynamics. 9,11,[21][22][23][24][25][26] Nanocontact vortex oscillations represent a dynamical state associated with the current flow, because spatial confinement for the vortex is absent when no current is applied (aside from defects that can pin the vortex). Furthermore, the existence itself of a vortex is not guaranteed in the absence of currents, because the vortex state bears a higher cost in magnetic energy than the uniform ground state. For these reasons, vortex oscillations in nanocontacts differ fundamentally from their counterparts in nanopillars, where the physical geometry guarantees both the vortex state and a confining potential that allow for damped or resonant dynamics. This distinction means that existing theories of spin-torque nano-oscillators, which describe a current-driven Hopf bifurcation between a damped oscillatory state and a limit cycle, [27][28][29][30][31][32] or other theoretical works describing current-driven vortex dynamics in magnetic dots, [33][34][35][36][37][38] cannot be applied directly to the nanocontact system. In this article, we present an analytical theory of the power spectrum of current-driven vortex oscillations in magnetic nanocontacts. While some elements of this theory have been reported in a prior publication, 11 a number of experimental observations since have required an extension of that earlier work. In particular, the revised theory presented here goes toward explaining two key experimental facts. First, it is now well established that vortex oscillations in in-plane magnetized systems can be nucleated and sustained without any applied magnetic fields. [22][23][24] This observation cannot be explained by the earlier theory in which a spin-polarization component perpendicular to the film plane, which is expected to be negligibly small in the absence of applied perpendicular fields, is necessary for sustaining oscillations. 11 While this component is known to be important for nanopillars, 13 the reproducible nature of the zero-field result for the nanocontact system suggests a different spin-transfer mechanism is predominant for sustaining oscillations. Second, the quality factor of the nanocontact vortex oscillations (i.e., the ratio between the oscillation frequency and the spectral linewidth) is typically an order of magnitude lower than that for nanopillar excitations. This is not immediately intuitive, since high quality factors have been observed for vortex oscillations in nanopillars. 16 Finally, we derive an analytical form for the Zeeman energy associated with the Oersted-Ampère field generated by the applied currents and show that it varies linearly with the radial distance for large amplitude oscillations. This linear dependence was used in earlier work, but it was only found empirically from numerical calculations. This paper is organized as follows. In Section II, we present the geometry, model and equations of motion in the limit of the rigid vortex approximation used. There, we derive the Thiele equation describing the vortex dynamics. In Section III, we consider the large amplitude limit of steady-state vortex motion in which the orbit is far away outside the nanocontact. In this limit, we derive some simplified equations of motion and analytical solutions for the oscillation radius and frequency. In Section IV, we examine the stochastic dynamics in the large orbit limit and derive analytical forms for the spectral linewidth. Some discussion of the results obtained and comparison with experimental data and other work are presented in Section V. A summary of key results is given in Section VI. (a) (b) (c) t 1 t 2 t 3 t 4 t V GMR t 1 t 2 t 3 t 4 Vortex II. MODEL The theory we present describes the magnetization dynamics of the free magnetic layer in a spin-valve stack, as shown in Fig. 2a. We do not account for any coupling that may appear between the free and reference magnetic layers, and the dynamics of the latter are ignored. The geometry we consider is defined in Fig. 2a. In our notation, z represents the axis perpendicular to the film plane, a the nanocontact radius, and d the free layer thickness. The magnetization dynamics is governed by the Landau-Lifshitz equation of motion, with an additional phenomenological Gilbert damping term and a spin-torque term Γ st , ∂m ∂t + |γ 0 |m × H eff = α m × ∂m ∂t + Γ st ,(1) where m is a unit vector representing the magnetization orientation, γ 0 = µ 0 |γ| is the gyromagnetic constant, µ 0 H eff = −∇ M E is the effective field, and α is the Gilbert damping constant. There are two main sources of spin torques that are relevant in this geometry, Γ st = Γ CPP + Γ CIP (see Fig. 3). The first involves torques related to electron transport perpendicular to the film plane (CPP), which is associated with the CPP magnetoresistance of the spin valve and affects only the magnetization dynamics within the nanocontact region, The unit vector p represents the magnetization orientation of the reference layer and the scalar function P accounts for the angular dependence of spin transfer, which depends strongly on the layer thicknesses and material parameters of the multilayer stack. 2,39 I is the applied current representing electron flow and Γ CPP = −σ 1 I P(m · p) m × (p × m).(2)σ 1 = P 1 e γ M s d 1 πa 2(3) is the CPP spin-transfer efficiency, where P 1 is the spin polarization of the CPP current, and M s is the saturation magnetization of the free layer. Under this convention, I > 0 represents the flow of spins (electrons) from the free to the reference layer along the negative z direction. The second spin torque contribution originates from electron transport within the film plane (CIP). Such in-plane currents arise in experimental systems because the electrical pads that connect to the bottom electrode are usually located at a significant lateral distance from the nanocontact. Because electrical currents flowing in a ferromagnetic metal are also spin-polarized, spin-torques arise in regions in which a spatial gradient in magnetization occurs. These CIP torques can be separated into an adiabatic and a nonadiabatic component, respectively, 40,41 Γ CIP = −(u · ∇)m + β m × [(u · ∇)m],(4) which arise from different spin transport regimes relative to the spatial magnetization gradients. In this description, the spin torques are more conveniently parametrized by an effective spin drift current, u = P 2 2e γ M s j(r),(5) where j is the current density of electron flow in the film plane and P 2 represents the spin polarization of this CIP current. 42 In order to keep the ensuing calculations tractable a simple model for electron flow is used, which follows from the schematic illustration presented in Fig. 3b. The applied current is taken to flow uniformly into the cylindrical region underneath the nanocontact in the free layer, and the subsequent electron flow within the film plane has a uniform density normal to the surface of this cylinder. A similar argument is applied to the flow within the nanocontact. Within this approximation, the current density in the film plane is purely radial from the nanocontact, j(r) =r I 2πad ×        r/a r < a a/r r ≥ a(6) In what follows, we consider the dynamics of a single magnetic vortex in the free layer. We will not seek to describe the nucleation process here as it is beyond the scope of this work. (A recent experiment has shown that vortices and vortex oscillations can be initiated in a reproducible way in such nanocontact systems 43 ). Instead, we will focus on how the vortex responds under the influence of the spin torques described above. A crucial assumption we make is that the spatial profile of the vortex remains constant throughout its motion. While the vortex core can deform significantly during the course of its motion, 44 and that such changes can be account for by spin-wave theories or through the inclusion of high-order time derivatives in the equations of motion, 45 such effects will not be treated here for the sake of simplicity. Let (Θ, Φ) represent the magnetization orientation in polar coordinates. In the rigid vortex approximation, the spatial profile can be parametrized entirely in terms of the vortex core position in the film plane, X = (X, Y), for which the origin is taken to be the center of the nanocontact. We consider a vortex with an angular variation of magnetization in the film plane of the form Φ(x, y; X, Y) = n tan −1 y − Y x − X + π 2 ,(7) where n = ±1 is the topological charge (n = 1 for a vortex, n = −1 for an antivortex). The component of magnetization perpendicular to the film plane, which describes the vortex core profile, is given by a function of the form Θ = Θ( r − R ), where r and R denote the position radial vectors in polar coordinates, i.e., r − R 2 = (x − X) 2 + (y − Y) 2 . The exact functional form we choose for the vortex core results only in small quantitative differences in the resulting dynamics. For the sake of consistency for the remainder of this paper, we use a modified version of the Usov ansatz for the vortex core profile that accounts for the uniform tilt angle, Θ 0 , of the free layer magnetization out of the film plane, 37 cos Θ(r; R) =            b 2 + (2 cos Θ 0 − 1) r − R 2 b 2 + r − R 2 p, r − R < b 0, r − R ≥ b (8) where b is the vortex core radius and p = ±1 is the core polarization, which describes the orientation of the core magnetization relative to the normal of the film plane. An illustration of the core profile for three tilt angles Θ 0 is given in Fig. 2b. The tilt angle is important to describe the magnetization profile under fields applied perpendicularly to the film plane. A detailed comparison between simulated core profiles, this and other ansatz, are given in Ref. 37. The dynamics of the rigid-vortex is derived following the method of collective coordinates. 46 We elevate the core position to a dynamic variable, X → X(t), which allows us to express the spatial magnetization profile as Θ = Θ[x − X(t)], with an analogous expression for Φ. The ensuing dynamics is then generated using the spin Lagrangian L = dV L(Θ[x − X(t)], Φ[x − X(t)]).(9) By integrating over the Lagrangian density, L, with the chosen magnetization profile for the vortex, we generate the equation of motion for each (generalized) coordinate ξ in the usual way, d dt ∂L ∂ξ − ∂L ∂ξ = F ξ,nc ,(10) where F ξ,nc represents the nonconservative forces such as damping and spin-torques. The conservative part of the Lagrangian dynamics, i.e., the left-hand side of (10), is given by the spin Berry phase term, L B = (M s /γ)Φ(1 − cos Θ) , and the magnetic energy density that plays the role of a potential, U = U(Θ, Φ). The Euler-Lagrange equations become G × dX dt + ∂U ∂X = F nc .(11) The first term on the left hand side is the gyrotropic term, where the gyrovector G is defined by G = M s γ dV sin Θ(∇Φ × ∇Θ)(12) and is equal to G = Gẑ, where G = 2π(p − cos Θ 0 )ndM s /γ. The second term is obtained directly from the magnetic energy density U = dV U. If we assume that the lateral dimensions of the film are much larger than the vortex core, the magnetostatic energy, in addition to the usual exchange and crystalline anisotropy energies, becomes independent of the vortex core position. Translational symmetry in the film plane is therefore only broken by the Oersted-Ampère field generated by the electrical current applied through the contact, constituting the sole contribution to the potential U. For the sake of simplicity, we assume that this Oersted-Ampère field is equivalent to that produced by an infinitely long current-carrying cylindrical conductor, which allows the Zeeman energy associated with this field to be written as U Z = −µ 0 M s H I dV f (r) sin Θ(cos φ cos Φ + sin φ sin Φ),(13) where H I = |I|/(2πa) is the magnitude of the Oersted-Ampère field at the edge of the contact, (r, φ) denotes the spatial variables (x, y) in polar coordinates. The function f (r) describes the variation of the magnitude of the Oersted field as a function of radial distance r in the film plane, f (r) =        r/a r < a a/r r ≥ a(14) The nonconservative forces we consider are Gilbert damping and spin torques, as discussed above. In a Lagrangian formalism, the force due to Gilbert damping F G can be included by means of a Rayleigh dissipation function W, F ξ,G = − ∂W ∂ξ = − αM s 2γ ∂ ∂ξ dV Θ 2 + sin 2 ΘΦ 2 ,(15) which leads to F G = −α ← → D · dX dt ,(16) where α is the Gilbert damping constant and ← → D is the damping dyadic, ← → D = M s γ dV ∇Θ ⊗ ∇Θ + sin 2 Θ∇Φ ⊗ ∇Φ .(17) With the core profile chosen, this term is computed to be ← → D = D ← → I , where ← → I is the 2 × 2 identity matrix, D = πM s d 2 + sin 2 Θ 0 ln(L/b) /γ, and L is lateral size of the free layer. To describe the CPP spin torques, we first identify the relevant forces using the Landau-Lifshitz equation (1) and (2), F Θ,CPP = 0,(18)F Φ,CPP = −σ CPP I M s γ P(m · p) sin 2 Θ.(19) By applying the chain rule, we find that the CPP torques can be divided into two components, F CPP = F CPP,|| +F CPP,⊥ , which describe contributions from the in-plane and perpendicular-toplane components, respectively, of the spin polarization unit vector p, F CPP,|| = σ CPP I P M s γ NC dV p || ∇Θ sin Φ +(20)1 2 ∇Φ cos 2Θ sin Φ , F CPP,⊥ = −σ CPP I P M s γ NC dV p ⊥ sin 2 Θ ∇Φ,(21) where the volume integration is limited to the region of the free layer underneath the nanocontact (NC). Without loss of generality, we have assumed that the in-plane spin polarization component is along the x axis. To account for the CIP torques, we note that the adiabatic and nonadiabatic terms can be derived by generalizing the time derivatives to convectional derivatives using the spin-drift velocity u, in both the Landau-Lifshitz and Lagrangian formulations. The adiabatic component can be obtained from the generalized Berry phase term, 47 ∂Φ ∂t (1 − cos Θ) → ∂ ∂t + u · ∇ Φ (1 − cos Θ),(22) which leads to a term that resembles the gyrotropic force, F ad = − M s γ dV sin Θ(∇Φ × ∇Θ) × u,(23) while the nonadiabatic term requires a proportionality factor β/α, i.e., we make the substitution ∂ ∂t → ∂ ∂t + β α u · ∇(24) for the time derivatives in the Rayleigh dissipation function, which reflects the fact that Galilean invariance is not generally present for magnetic dissipative processes in real systems. 48,49 As expected, the resulting form for the nonadiabatic term is similar to Gilbert damping, F n−ad = − βM s γ dV ∇Θ ⊗ ∇Θ + sin 2 Θ∇Φ ⊗ ∇Φ · u.(25) III. STEADY-STATE OSCILLATIONS This section is devoted to the description of the large-orbit vortex motion in the steady state. It begins with a brief discussion on the initial transient dynamics leading to the steady state, where it is shown that the parallel component of the CPP torques play a major role in driving the vortex out of the nanocontact. Next, a simple form for the Zeeman energy due to the Oersted-Ampère field is found. This result, combined with the CIP torques in the large-amplitude limit, leads to equations of motion that are solved to give the steady state orbital radius and frequency. For the initial transient dynamics, the small-amplitude limit is considered in which the vortex is assumed to remain within the nanocontact area and close to its center. In this limit, the CPP torques can be readily evaluated. For the purposes of illustrating the qualitative behavior in this limit, it is assumed that P = 1 and no external magnetic field is applied so that the tilt magnetization angle can be ignored, i.e. Θ 0 = π/2. The parallel component F CPP,|| is nonvanishing only within the vortex core where the gradient in the polar angle, ∇Θ, and cos(Θ) terms are nonvanishing. As such, it suffices to limit the integration in (21) over only the vortex core region, which leads to F CPP,|| = σ 1 I M s γ (πbd p) ln 2î + π − 2 4ĵ .(26) This force is independent of the vortex position and acts to drive the vortex out of the nanocontact area. A good estimate of the perpendicular component of the CPP torques can be obtained by neglecting the core contribution using the approximation sin 2 Θ ≈ 1. Within this approximation, the integral in (21) reduces to a simple integral of the quantity ∇Φ, which yields (27) In contrast to the parallel component, the perpendicular CPP spin torques acts to drive a gyrotropic motion for the vortex, with a magnitude that is inversely proportional to the radial vortex distance from the nanocontact center. The combination of the two CPP spin torques gives a force acting on the vortex that leads to a spiraling motion of the vortex out of the nanocontact area. F CPP,⊥ = σ 1 I M s γ πa 2 2 1 X 2 + Y 2 −Yî + Xĵ + O(b 2 ). For the remainder of this section, we will assume that the steady-state regime is attained after such transient dynamics. We focus on the large amplitude motion of the vortex around the nanocontact, which is relevant for describing the power spectrum of the steady-state oscillations as observed experimentally. We will not concern ourselves with the transient dynamics associated with the nucleation process; this is beyond the scope of this paper and will be treated elsewhere. The Zeeman energy due to the Oersted-Ampère fields can be computed without difficulty if contributions from the vortex core are neglected; we take Θ = Θ 0 in (13) and retain only the magnetization variation in Φ. Let (r, φ) and (R, ϕ) represent the spatial variables (x, y) and (X, Y) in polar coordinates, respectively. We proceed by separating the integration in Eq. 13 into two parts. The first part involves integrating over the contact area, which is found to be U Z,in = sin(Θ 0 ) 2π 0 dφ a 0 dr r 2 a r − R cos φ r 2 − 2rR cos φ + R 2 , = sin(Θ 0 ) 4|a − R| 9a (2a 2 − R 2 )E(ψ) + (a + R) 2 K(ψ) ,(28) where ψ ≡ −4aR/(a−R) 2 , and K(ψ) = π/2 0 dz (1−ψ sin 2 z) −1/2 and E(ψ) = π/2 0 dz (1 − ψ sin 2 z) 1/2 are elliptic integrals. The second integral, which runs over the region outside the contact, is infinite. Nevertheless, we can identify the infinite background term from the indefinite integral 2π 0 dφ dr a(r − R cos φ) r 2 − 2rR cos φ + R 2 = 4a|r − R|E − 4rR (r − R) 2 ,(29) by noting that E[−4rR/(r − R) 2 ] = π/2 as r → ∞, from which it is deduced that U Z,out = µ 0 M s H I d sin(Θ 0 ) 2πaR + 4a|a − R|E(ψ) ,(30) with ψ as defined above. In the large amplitude limit the elliptic integrals can be expanded in a power series in terms of a/R, which allows the total Zeeman energy U Z = U Z,in +U Z,out to be expressed as U Z ≈ µ 0 M s H I d sin Θ 0 (4πaR), = (2µ 0 M s d sin Θ 0 ) |I|R ≡ κ(Θ 0 ) |I| R.(31) This linear dependence on the radial distance of the vortex was found empirically through numerical calculations in previous work, 11 but we provide a solid basis for this functional form here. The linear approximation is very good for R/a ≥ 1, which can be seen in Fig. 4 where a comparison with the exact solution is presented. For in-plane magnetized systems, such as the CoFe or Permalloy films studied in experiment, 9,11,[22][23][24] we assert that the CPP spin torques can be neglected in the absence of any applied magnetic fields. Under these conditions, both in-plane and perpendicular components of the CPP torques should vanish for the following reasons. First, in the absence of any perpendicular fields, the p ⊥ component should be vanishingly small because the magnetization of the reference layer, which is either a hard ferromagnetic material or is exchange biased by an antiferromagnet, lies entirely in the film plane. This perpendicular component is important, however, for dynamics in large perpendicular applied fields. 11,13 Second, if the vortex core is sufficiently far from the nanocontact, the magnetization gradient ∇Θ vanishes and cos 2Θ ≃ cos π = 0 within the nanocontact region. As a result the parallel component also vanishes, which can be seen by inspecting Eq. 21. We are therefore led to the conclusion that CIP spin torques are the dominant mechanism for self-sustained vortex oscillations in the large orbit limit under zero or low applied fields. Let σ 2 represent the spin-torque efficiency for the CIP component in this limit, σ 2 ≡ P 2 e γ M s d 1 4πa 2 ,(32) such that the spin-current drift velocity can be expressed as (r ≥ a), u(r) = σ 2 I a 2 rr .(33) With this definition, σ 2 I retains units of angular frequency, in accordance with the convention adopted in spin torque oscillation theory. By substituting this functional dependence for the adiabatic torques into (23) above and integrating over the region outside the nanocontact, we find the simple and appealing result F ad = G × u(X).(34) Similarly, the nonadiabatic torques (25) under the same assumptions are found to be F n−ad = βD u(X). The full equation of motion for the current-driven vortex in the large amplitude limit is therefore given by F ad F Z F n-ad v F G (a) (b)(35)G × dX dt − u(X) + D α dX dt − β u(X) + ∂U ∂X = 0. (36) The role of each term is shown schematically in Fig. 5. The conic structure of the Zeeman potential leads to a force that pulls the vortex towards the nanocontact center, while the nonadiabatic torque counteracts this force pushing the vortex radially outwards. The adiabatic torque acts as a "boost" along the circular trajectory of the vortex orbit, while Gilbert damping acts to oppose this motion. We are interested in the orbital motion of the vortex around the nanocontact, so it is more convenient to work with the equations of motion for the radial (R) and angular (ϕ) variables, which can be obtained from (36) by constructing the appropriate combinations of the coupled differential equations. We find ∂R ∂t = − καD G 2 |I| + σ 2 Ia 2 R ,(37)∂ϕ ∂t = (α − β)D G σ 2 Ia 2 R 2 + κ G |I| R .(38) The stationary solution is obtained by setting the time variation in the radial variable to zero, i.e., ∂ t R = 0. From this condition, a solution to the stationary orbit radius R(t) = R 0 is found, R 0 = G 2 σ 2 a 2 καD sgn(I), = P 2e 1 αµ 0 M s d (p − cos Θ 0 ) 2 (2 + sin 2 Θ 0 ln[L/b]) sin Θ 0 .(39) There are a number of important points worth noting here. First, a physical solution for the radius of the steady-state orbit, R 0 > 0, exists only for one current polarity, I > 0, which corresponds to the case where electrons flow outward from the nanocontact. Intuitively, we can understand this as an outward "pressure" exerted by the spin torques that counterbalances the Gilbert damping and therefore prevents the vortex spiraling into the contact center, which corresponds to the position of energy minimum in the Zeeman energy. Second, the orbital radius is independent of the applied current and the nanocontact size, and depends only on material parameters. As such, it also follows that there is no critical current for self-sustained oscillations, in stark contrast to nanopillar spin-torque oscillators that involve large amplitude spin waves. The present theory therefore predicts the existence of vortex oscillations for any value of the applied electron current I > 0, provided a vortex is already present in the nanocontact system. The onset of oscillations in this scenario is therefore be determined by a threshold for vortex nucleation, rather than a supercritical Hopf bifurcation as in the case for nanopillar oscillators. Indeed, the experimental observation that vortex oscillations persist below an onset current, after having been nucleated above this current, 9,25 supports this idea. From the stationary condition for the radial dynamics, the frequency of the vortex oscillation can be evaluated directly by substituting R 0 for R in (38). If we assume that the nonadiabatic torque is similar in magnitude to the adiabatic torque, β ≃ α, which is found to be a good approximation for transitional metal ferromagnets, 50,51 the first term on the right hand side of (38) can be neglected and the expression for the frequency reduces to the simpler form ω ≡ ∂ t ϕ ≃ κ|I| GR 0 = µ 0 γ π sin Θ 0 p − cos Θ 0 |I| R 0 ,(40) where |I|/R 0 plays the role of an effective magnetic field, with the sense of rotation is given by the vortex core polarization p. This is also an appealing result because it allows a quantitative estimate of the vortex orbital radius to be obtained readily from experiment: By measuring the slope of the frequency versus current curve in zero applied field (Θ 0 = π/2), a measure of R 0 can be obtained directly. IV. POWER SPECTRUM The power spectrum measured in experiment is related to voltage variations v(t) associated with the time-varying change in magnetoresistance in the nanocontact region. There are two contributions to the magnetoresistance: one from CPP currents, the other from CIP currents. Because the vortex orbital motion leads to the largest time-varying variation in magnetization in the contact region, the larger contribution to the total magnetoresistance is expected to come from the CPP component. Let ∆V represent the total CPP magnetoresistance for the spin valve. The time-varying component of interest can be written as v(t) = 1 2 ∆V NC d 3 x m(t) · p.(41) It is assumed that the fixed layer magnetization remains static and is uncoupled to the vortex dynamics in the free layer to simplify the calculations, although in practice this may need to be accounted for. To simplify the calculations, it will also be assumed that the fixed layer magnetization is uniform in the nanocontact region. As such, the voltage calculation amounts to averaging over the in-plane component of magnetization of the free layer over the point-contact region. Without loss of r 2 − 2rR cos(φ − ϕ) + R 2 . (42) This integral can be solved using a power-series expansion in the variable r/R < 1 and summing over all terms. We find v(t) = − 1 4 πa 2 d (∆V) sin ϕ 2 F 1 − 1 2 ; 1 2 ; 2; (a/R) 2 ,(43) where 2 F 1 is a hypergeometric function which can expressed by the integral 2 F 1 (a; b; c; z) = Γ(b) Γ(c)Γ(c − b) 1 0 dt t b−1 (1 − t) c−b−1 (1 − tz) a .(44) A plot of the variation of the magnetoresistance signal magnitude as a function of orbit radius is shown in Fig. 6. For small amplitude motion, the GMR signal exhibits a linear variation as a function in the radial distance. In the limit of large amplitude motion, the hypergeometric function can be approximated by 2 F 1 − 1 2 ; 1 2 ; 2; (a/R) 2 ≃ 1 − a 2 /(8R 2 ), which leads to a simple expression of the time-varying magnetoresistance signal v(t) ≃ V 0        1 − 1 2 a 2R(t) 2        e iϕ(t) ,(45) where V 0 ≡ πa 2 d∆V/2. This result indicates that the vortex nanocontact oscillator can be considered as a phase oscillator to a very good approximation, with radial fluctuations that decrease like 1/R 2 . As such the power spectrum of the voltage fluctuations S (ω), S (ω) = ∞ −∞ dt K(t)e −iωt ,(46) which is the Fourier transform of the voltage autocorrelation function K(t) = v(t)v * (0) , is well described by the phase variance of the oscillator, ∆ϕ 2 , K(t) = V 2 0 exp i ϕ(t) exp − 1 2 ∆ϕ 2 (t) ,(47) where ∆ϕ 2 (t) = ϕ(t) 2 − ϕ(t) 2 . Once the phase variance is known, the lineshape and linewidth of the power spectrum can be obtained directly. The power spectrum of any oscillator is broadened at finite temperatures by thermal noise. We can describe the influence of such stochastic processes on the oscillator dynamics by including additional noise terms in the equations of motion (36), which we can express symbolically as the set of coupled Langevin equations 52,53 dX dt = v(X) + √ q η(t).(48) Note that the noise contribution enters as an additive term, rather than a multiplicative process (such as a random field in the Landau-Lifshitz equation); it has been shown elsewhere that such an additive noise is adequate for of vortex dynamics. 52,53 η = (η X , η Y ) is a two-component vector that represents a Gaussian white-noise forcing, which possesses the spectral properties η i (t) = 0; η i (t)η j (t ′ ) = 2δ i j δ(t − t ′ ),(49) with q representing the thermal noise amplitude. The choice of q = αk B T γ M s d(50) ensures that the fluctuation-dissipation theorem is satisfied in the absence of spin-transfer torques. Since we are interested in describing the fluctuations about the steady-state orbital motion, we linearize the Langevin equations about the stationary orbit R 0 , R(t) = R 0 + r(t), which allows us to obtain from (48), ∂r ∂t = − σ 2 Ia 2 R 2 0 r + q 1 − r R 0 + √ q η R (t),(51) where η R,ϕ possess the same spectral properties as η X,Y . By analogy with spin torque oscillator theory, we identify a restoration rate Γ r = σ 2 Ia 2 /R 2 0 , which describes the rate at which fluctuations in the orbital radius are damped out. The radial fluctuations are independent of the phase fluctuations and are driven by white thermal noise, subject to a spurious drift. By using material parameters relevant for typical experiments, it is straightforward to show that the term in q is small compared with the additive noise term proportional to √ q, which allows the Langevin equation in r(t) to be reduced to the simpler Ornstein-Uhlenbeck process, ∂r ∂t = −Γ r r + √ q η R (t).(52) The formal solution to this equation, r(t) = c exp(−Γ r t) + √ q exp(−Γ r t) t dτ η R (τ) exp(Γ r τ), where c is a constant, allows the two-time autocorrelation function for the radial fluctuation to be computed directly. In the limit of large times (at which the initial correlations are forgotten), we recover the known result r(t)r(t ′ ) = q Γ r exp Γ r |t − t ′ | .(53) The linearized Langevin equation for the phase dynamics is ∂ϕ ∂t = κ|I| GR 0 1 − r(t) R 0 + √ q η ϕ (t),(54) where we have neglected a cross term in r(t)η ϕ (t). In contrast to the radial dynamics, which is uncoupled from the phase fluctuations, the phase variable is coupled to the radial fluctuations. As a result, the phase dynamics is driven by the radial fluctuations r(t), which appears as colored noise, in additional to the additive white noise proportional √ qη ϕ (t). From the formal solution to this differential equation, and by using the spectral noise properties for η, we obtain the phase variance to be 1 2 ∆ϕ(t) 2 = q R 2 0               1 + κ|I| GR 0 1 Γ r 2        |t| − κ|I| GR 0 2 1 − e −Γ r t Γ 3 r        .(55) We can identify a nonlinearity parameter ν, ν ≡ κ|I| GR 0 1 Γ r = G αD ,(56) along with a "linear" linewidth parameter ∆ω 0 = q/R 2 0 , which allows the expression for the phase variance to be simplified to the form, 1 2 ∆ϕ(t) 2 = ∆ω 0 1 + ν 2 |t| − ν 2 Γ r 1 − e −Γ r t .(57) This expression is identical to the phase variance for a spin torque oscillator, which describes an inhomogeneous broadening of the spectral line due to radial (amplitude) fluctuations. 32 Two limiting cases appear as a result of this inhomogeneous broadening. In the "low" temperature limit in which the coherence time, τ c , of oscillations is much longer than the inverse of the restoration rate, Γ r τ c ≫ 1, the exponential term can be neglected and the phase variance is proportional to |t|. As such, the power spectrum in this limit is described by a Lorentzian lineshape with a full width at half maximum (FWHM) of ∆ω LT = ∆ω 0 1 + ν 2 = q R 2 0 1 + ν 2 .(58) Through the linear dependence of ∆ω 0 on the noise parameter q, the linewidth in this limit varies linearly as a function of temperature and is inversely proportional to the square of the steady-state radius R 0 . Because the orbital radius is independent of the applied current, the linewidth is also independent of the current in the low temperature limit, which is in stark contrast to conventional spin torque oscillators for which the supercriticality, which describes the ratio between the applied and threshold currents, is a primordial factor. For the present case the linewidth is determined primarily by material parameters. In the opposite "high" temperature limit in which the coherence time is much shorter than the inverse of the restoration rate, Γ r τ c ≪ 1, the exponential function in the phase variance can be expanded in a power series to give, 1 2 ∆ϕ(t) 2 ≃ ∆ω 0 |t| + 1 2 Γ r ν 2 t 2 .(59) If the nonlinearity ν is sufficiently large, the linear term in |t| can be neglected and the power spectrum is described by a Gaussian lineshape with a FWHM of ∆ω HT = 2 √ 2 ln 2 |ν| ∆ω 0 Γ r ≈ 2.35 |ν|a √ q σ 2 I R 2 0 .(60) In this limit, the temperature dependence of the linewidth is of the form T 1/2 , which is consistent with inhomogeneous broadening due to phase-amplitude coupling. 32 Furthermore, the linewidth also acquires a square-root dependence on the applied current through the restoration rate, which is markedly different from the low temperature case. This gives a direct experimental means of identifying the temperature regime by measuring the current dependence of the linewidth. V. DISCUSSION AND COMPARISON TO EXPERIMENTAL DATA The fundamental premise of this theory is that sub-GHz voltage oscillations observed in experimental nanocontact systems are due to the orbital motion of a single magnetic vortex. The quasi-linear current dependence of the oscillation frequency provides the strongest evidence to support this hypothesis to date. However, there remain open questions concerning how the single vortex state is attained from an initially uniform magnetic ground state. This crucial issue is a problem of topology and can be discussed in terms of the Skyrmion number Q. 54 A vortex or an antivortex possesses a half-integer Skyrmion charge of Q = np/2, where n is the winding number, with n = 1 for vortices and n = −1 for antivortices, and p is the core polarization. As such, a system with a single vortex possesses a charge of Q = 1/2, while the uniform state has Q = 0. 55 In this light, the single vortex and uniform magnetic states are in different topological sectors, which means some physical process that does not conserve topological charge needs to occur during the nucleation. It is interesting to note that a vortex-antivortex pair with parallel core polarizations is in the same topological sector as the uniform state, while a vortex-antivortex pair with opposite core polarizations possesses Q = ±1. The latter case is interesting because it admits a rotating solution; 56,57 in the absence of damping, such a vortex-antivortex pair (or "dipole") rotates about its centerof-mass with a frequency of ω d = 4γA/(M s l 2 ), where l is the separation between the vortices and A is the exchange stiffness. Indeed, the possible existence of such rotating vortex dipoles in a nanocontact geometry were seen in a recent numerical study by Berkov and Gorn. 58 While such a solution is theoretically appealing because a uniform magnetic state is conserved far from the nanocontact, it is not immediately apparent how dynamics involving vortex dynamics can account for experimental observations. First, there is no obvious mechanism by which a linear current dependence of the oscillation frequency can be obtained. As we have shown in a recent study, 43 the Oersted-Ampère field gives rise to a Zeeman potential that drives the vortex pair apart, which would lead to a decrease in the oscillation frequency with current. Second, a rotating dipole pair possesses a Skyrmion charge of Q = ±1, which is also topologically distinct from the uniform ground state. Therefore, as for the single vortex case, some charge non-conserving process needs to take place. As discussed briefly in Section III, we assert that in-plane spin-torques, rather than perpendicular-to-plane torques, should play a dominant role in the vortex dynamics in magnetic nanocontact systems under zero or low-applied fields. This hypothesis is strongly supported by a number of recent experiments in which vortex oscillations have been reproducibly initiated in the absence of any applied magnetic field. 22,23,43 Under such conditions the magnetization orientation of the reference magnetic layer should lie entirely in the film plane, so there is no reason to expect a large contribution from the F CPP,⊥ term, which is necessary for the existence of self-sustained oscillations. 11 This point is well-illustrated by a recent experiment on vortex oscillations in magnetic nanopillars in which large out-of-plane fields are required. 13 While the dominant CIP adiabatic torques lead to the same functional form as the F CPP,⊥ term in the reduced equations of motion, there is a crucial difference between the two: CIP torques act independently of the vortex core polarization, while the existence of vortex oscillations under CPP torques depend on relative orientation of the core polarization with respect to the perpendicular spin torque component. 11,13 Therefore, vortex core reversal would not restrict self-sustained oscillations under CIP torques but would lead to a damped oscillatory regime under purely CPP torques. In this light, it would be interesting to see whether such core reversal processes could be detectable in a time-resolved experiment. The hypothesis of self-sustained oscillations driven by CIP torques might explain the ubiquitous presence of higher harmonics in the power spectrum. In this picture, the magnitude of the spin-torques due to lateral currents in the film plane determines the shape of the vortex orbit. A uniform radial flow has been assumed in the present work as a matter of simplicity, but the actual current distribution in an experimental system would certainly be more complex. As such, one could expect elliptical orbits to result, which would lead to significant contributions to the harmonic content of the power spectrum. An important test of the model described here is the slope of the frequency versus current relation, ∂ω ∂|I| = κ GR 0 = κ 2 αD G 3 σ 2 a 2 ,(61) which, as discussed previously, is a material-dependent parameter. It is a quantity that can readily be extracted from experimental measurements and represents a robust characterization of the oscillator properties because the oscillation frequency, as opposed to other spectral parameters such as the linewidth or power, is a stable physical parameter that depends solely on the vortex dynamics and is independent of thermal fluctuations. By using the experimentally determined values of µ 0 M s = 1.56 T, d = 3.5 nm, α = 0.013, L = 10 µm from Ref. 22, and by assuming a spin polarization of P = 0.5 and b = 10 nm, we find a theoretical value of ∂ω/∂|I| = 4.4 MHz/mA, which is within a factor of two of the observed slope of 7.4 MHz/mA. 22,23 For the low-field vortex oscillation studies of Refs. 9 and 21, a frequency slope of ≈ 30 MHz/mA is predicted (by assuming µ 0 M s = 0.8 T, d = 5 nm, α = 0.01, L = 10, P = 0.5, and b = 10 nm). This is a factor of three larger than the observed slope of ≈ 10 MHz/mA. While some uncertainty exists in the spin polarization P of the applied currents and the vortex core radius b in these experiments, the agreement between theory and experiment nevertheless remains relative good on a quantitative level for a simple model with no adjustable parameters. Another important point of comparison is the spectral linewidth. Indeed, understanding the underlying physics governing linewidths is crucial for any potential oscillator applications. From the experiment described in Ref. 22, it was shown that the autocorrelation function for the measured highfrequency voltage signals can be well-described by a decaying exponential function, v(t)v(0) ∝ exp (−|t|/τ c ), where τ c is the characteristic coherence time of the voltage oscillations. As discussed in the previous section, this form for the autocorrelation function indicates that the spectral lineshape is Lorentzian, with a FWHM given by ∆ω = 2/τ c . For an applied current of 18.7 mA in zero applied field (Θ 0 = π/2), a coherence time of 140 ns was observed. 22 From the results obtained for the low-temperature limit in the previous section, the present theory predicts the current-independent value of τ c ≈ 70 ns with the same experimental parameters, which is in reasonable agreement with the experimental value. In a similar manner, a current-independent linewidth of ∆ f ≈ 2 MHz is predicted for the experiment of Ref. 21, which is within a factor of three of the experimentally measured FWHM of 0.78 MHz for the oscillation mode at 128 MHz. The time-domain analysis performed for this particular experiment shows that the dominant contribution to the spectral linewidth is due to phase noise, so the present theory is applicable. The discrepancies between the theoretical and experimental values of the frequency and linewidth might originate from the way the Oersted-Ampère field and spin torques are computed in this theory. For instance, the spatial profile of the Oersted-Ampère field has been computed by approximating the current flow through the nanocontact with the flow through an infinite cylindrical wire. On the other hand, it has also been assumed that a large component of this current flows laterally from the nanocontact because of how the contact pads are located. These assumptions are not compatible with one another, and as a consequence both the spin-torques and the Oersted-Ampère field are overestimated in the theory. A simple way of correcting these estimates is to include ad-hoc correction factors κ 0 ≤ 1 and σ 0 ≤ 1 into the definitions of the Zeeman energy and spin-torque parameters, respectively, κ ′ = κ 0 κ, σ ′ 2 = σ 0 σ 2 .(62) Following this line of reasoning, the frequency versus current slope and Lorentzian linewidth also acquire these correction factors, ∂ω ∂|I| ′ = κ 2 0 σ 0 ∂ω ∂|I| , ∆ω ′ L = κ 0 σ 0 2 ∆ω L ,(63) but with different functional forms. By setting the ratios between the theoretical and experimental values of the frequency and linewidth to unity, it is found empirically that κ 0 ≈ 0.4 and σ 0 ≈ 0.3. The empirical value of κ 0 suggests that a halfinfinite cylindrical wire, for which one would expect a correction factor of 0.5, gives a better approximation for computing the Oersted-Ampère field. It would be interesting to see whether more detailed finite-element calculations of the current flow in realistic geometries would give such correction factors. VI. SUMMARY In summary, a theory of the power spectrum of currentdriven vortex oscillations in magnetic nanocontacts has been presented. The theory is based on a rigid-vortex model and the equations of motion describing the vortex dynamics have been derived and solved in the steady-state limit. In contrast to conventional spin torque nano-oscillators that involve largeangle magnetization precession, the self-oscillatory state in the nanocontact system is found to exist with the only condition being the existence of a vortex in the system. As such, the onset of oscillations does not involve a Hopf bifurcation and therefore no critical current is predicted. It is found that spin-torques due to current flow in the plane of the free magnetic layer are crucial for the existence of self-sustained oscillations. The oscillation frequency is found to vary linearly as a function of current, in accordance with experimental observations, with the function form ω = κ|I| GR 0 ,(40) where the orbital radius R 0 is current-independent and depends only on material parameters, R 0 = G 2 σ 2 a 2 καD sgn(I).(39) In the low-temperature limit in which the coherence time of the oscillations is greater than the restoration rate of the radial fluctuations, it is found that the power-spectrum is described by a Lorentzian lineshape with a linewidth (FWHM) of ∆ω LT = ∆ω 0 (1 + ν 2 ), where ∆ω 0 is a "linear" linewidth that is proportional to the temperature and ν is a "nonlinearity" parameter that represents the ratio between the magnitude of the damping dyadic and the gyrovector, ν = αD/G. At higher temperatures, the lineshape is inhomogeneously broadened by fluctuations in the orbital radius and a Gaussian lineshape is predicted, with a current-dependent linewidth that varies like √ I. FIG. 2 . 2(Color online) (a) Geometry of the nanocontact in the film plane of the free magnetic layer and system of coordinates used. The nanocontact makes a circular cross section with the free layer with radius a and centered at the origin. (X, Y) denotes the position of the vortex core in the film plane. (b) Profile of vortex core for different magnetization tilt angles Θ 0 . FIG. 3 . 3(Color online) Sources of spin-torques in the free ferromagnetic layer (F1), with the arrows indicating current flow. (a) Current perpendicular-to-plane (CPP) and (b) current in-plane (CIP) torques both contribute to vortex dynamics. The hashed regions indicate the regions in which the spin-torques act. NC denotes the nanocontact, N the metallic spacer, and F2 the ferromagnetic reference layer. online) Comparison between the exact form and linear approximation of Zeeman energy, U Z , as a function of scaled radial distance R/a, where a is the point contact radius. The inset represents a zoom for R/a ≤ 1. FIG. 5 . 5(Color online) Schematic diagram illustrating the different force terms in Eq. 36. (a) Static forces: Zeeman potential (F G ), adiabatic (F ad ), and non-adiabatic (F n−ad ) torques, for I > 0. (b) Gilbert damping, F G , associated with the vortex motion v. FIG. 6 . 6(Color online) Variation of the GMR voltage signal as a function of vortex radial distance, in normalized units. The variation, computed by numerical integration, is compared with the analytical result and simple asymptotic forms for small (|v|/V 0 ≃ R/a) and large amplitude (Eq. 45) orbits. generality, it suffices to consider the x component of magnetization in the free layer, sin φ − R sin ϕ ACKNOWLEDGMENTSThe authors thank R. E. Camley, M. Pufall, M. Keller, G. Hrkac, and C. Chappert for stimulating discussions. This work was partly supported by the European Communities pro-gram "Structuring the ERA", under Contract No. MRTN-CT-2006-035327 SPINSWITCH, the French National Research Agency (ANR), under contract no. VOICE PNANO-09-P231-36, and by the local government of Région Ile-de-France within the "C'Nano IdF" program. . L Berger, Phys. Rev. B. 549353L. Berger, Phys. Rev. B, 54, 9353 (1996). . J C Slonczewski, J. Magn. Magn. Mater. 1591J. C. Slonczewski, J. Magn. Magn. Mater., 159, L1 (1996). . S I Kiselev, J C Sankey, I N Krivorotov, N C Emley, R J Schoelkopf, R A Buhrman, D C Ralph, 10.1038/nature01967Nature. 425380S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em- ley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature (London), 425, 380 (2003). . W H Rippard, M R Pufall, S Kaka, S E Russek, T J Silva, Phys. Rev. Lett. 9227201W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett., 92, 027201 (2004). . Q Mistral, J.-V Kim, T Devolder, P Crozat, C Chappert, J A Katine, M J Carey, K Ito, Appl. Phys. Lett. 88192507Q. Mistral, J.-V. Kim, T. Devolder, P. Crozat, C. Chappert, J. A. Katine, M. J. Carey, and K. Ito, Appl. Phys. Lett., 88, 192507 (2006). . V S Pribiag, I N Krivorotov, G D Fuchs, P M Braganca, O Ozatay, J C Sankey, D C Ralph, R A Buhrman, 10.1038/nphys619Nat. Phys. 3498V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat. Phys., 3, 498 (2007). . D Houssameddine, U Ebels, B Delaet, B Rodmacq, I Firastrau, F Ponthenier, M Brunet, C Thirion, J P Michel, L Prejbeanu-Buda, M C Cyrille, O Redon, B Dieny, 10.1038/nmat1905Nat. Mater. 6447D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Fi- rastrau, F. Ponthenier, M. Brunet, C. Thirion, J. P. Michel, L. Prejbeanu-Buda, M. C. Cyrille, O. Redon, and B. Dieny, Nat. Mater., 6, 447 (2007). . O Boulle, V Cros, J Grollier, L G Pereira, C Deranlot, F Petroff, G Faini, J Barnas, A Fert, 10.1038/nphys618Nat. Phys. 3492O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. De- ranlot, F. Petroff, G. Faini, J. Barnas, and A. Fert, Nat. Phys., 3, 492 (2007). . M R Pufall, W H Rippard, M L Schneider, S E Russek, Phys. Rev. B. 75140404M. R. Pufall, W. H. Rippard, M. L. Schneider, and S. E. Russek, Phys. Rev. B, 75, 140404 (2007). . A M Deac, A Fukushima, H Kubota, H Maehara, Y Suzuki, S Yuasa, Y Nagamine, K Tsunekawa, D D Djayaprawira, N Watanabe, 10.1038/nphys1036Nat. Phys. 4803A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y. Suzuki, S. Yuasa, Y. Nagamine, K. Tsunekawa, D. D. Djayaprawira, and N. Watanabe, Nat. Phys., 4, 803 (2008). . Q Mistral, M Van Kampen, G Hrkac, J.-V Kim, T Devolder, P Crozat, C Chappert, L Lagae, T Schrefl, Phys. Rev. Lett. 100257201Q. Mistral, M. van Kampen, G. Hrkac, J.-V. Kim, T. Devolder, P. Crozat, C. Chappert, L. Lagae, and T. Schrefl, Phys. Rev. Lett., 100, 257201 (2008). . S Cornelissen, L Bianchini, G Hrkac, M Op De Beeck, L Lagae, J.-V Kim, T Devolder, P Crozat, C Chappert, T Schrefl, Europhys. Lett. 8757001S. Cornelissen, L. Bianchini, G. Hrkac, M. Op de Beeck, L. La- gae, J.-V. Kim, T. Devolder, P. Crozat, C. Chappert, and T. Schrefl, Europhys. Lett., 87, 57001 (2009). . A Dussaux, B Georges, J Grollier, V Cros, A V Khvalkovskiy, A Fukushima, M Konoto, H Kubota, K Yakushiji, S Yuasa, K A Zvezdin, K Ando, A Fert, 10.1038/ncomms1006Nat. Commun. 18A. Dussaux, B. Georges, J. Grollier, V. Cros, A. V. Khvalkovskiy, A. Fukushima, M. Konoto, H. Kubota, K. Yakushiji, S. Yuasa, K. A. Zvezdin, K. Ando, and A. Fert, Nat. Commun., 1, 8 (2010). . J He, S Zhang, 10.1063/1.2719646Appl. Phys. Lett. 90142508J. He and S. Zhang, Appl. Phys. Lett., 90, 142508 (2007). . A Bisig, L Heyne, O Boulle, M Kläui, Appl. Phys. Lett. 95162504A. Bisig, L. Heyne, O. Boulle, and M. Kläui, Appl. Phys. Lett., 95, 162504 (2009). . V S Pribiag, G Finocchio, B J Williams, D C Ralph, R A Buhrman, 10.1103/PhysRevB.80.180411Phys. Rev. B. 80180411V. S. Pribiag, G. Finocchio, B. J. Williams, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B, 80, 180411 (2009). . A Yamaguchi, K Motoi, H Miyajima, A Hirohata, T Yamaoka, T Uchiyama, Y Utsumi, Appl. Phys. Lett. 95122506A. Yamaguchi, K. Motoi, H. Miyajima, A. Hirohata, T. Yamaoka, T. Uchiyama, and Y. Utsumi, Appl. Phys. Lett., 95, 122506 (2009). . V Novosad, M Grimsditch, K Y Guslienko, P Vavassori, Y Otani, S D Bader, 10.1103/PhysRevB.66.052407Phys. Rev. B. 6652407V. Novosad, M. Grimsditch, K. Y. Guslienko, P. Vavassori, Y. Otani, and S. D. Bader, Phys. Rev. B, 66, 052407 (2002). . V Novosad, F Y Fradin, P E Roy, K S Buchanan, K Y Guslienko, S D Bader, Phys. Rev. B. 7224455V. Novosad, F. Y. Fradin, P. E. Roy, K. S. Buchanan, K. Y. Gus- lienko, and S. D. Bader, Phys. Rev. B, 72, 024455 (2005). . K S Buchanan, P E Roy, M Grimsditch, F Y Fradin, K Y Guslienko, S D Bader, V Novosad, 10.1038/nphys173Nat. Phys. 1172K. S. Buchanan, P. E. Roy, M. Grimsditch, F. Y. Fradin, K. Y. Gus- lienko, S. D. Bader, and V. Novosad, Nat. Phys., 1, 172 (2005). . M W Keller, A B Kos, T J Silva, W H Rippard, M R Pufall, 10.1063/1.3133356Appl. Phys. Lett. 94193105M. W. Keller, A. B. Kos, T. J. Silva, W. H. Rippard, and M. R. Pufall, Appl. Phys. Lett., 94, 193105 (2009). . T Devolder, J.-V Kim, P Crozat, C Chappert, M Manfrini, M Van Kampen, W Van Roy, L Lagae, G Hrkac, T Schrefl, 10.1063/1.3170234Appl. Phys. Lett. 9512507T. Devolder, J.-V. Kim, P. Crozat, C. Chappert, M. Manfrini, M. van Kampen, W. van Roy, L. Lagae, G. Hrkac, and T. Schrefl, Appl. Phys. Lett., 95, 012507 (2009). . M Manfrini, T Devolder, J.-V Kim, P Crozat, N Zerounian, C Chappert, W Van Roy, L Lagae, G Hrkac, T Schrefl, 10.1063/1.3263727Appl. Phys. Lett. 95192507M. Manfrini, T. Devolder, J.-V. Kim, P. Crozat, N. Zerounian, C. Chappert, W. van Roy, L. Lagae, G. Hrkac, and T. Schrefl, Appl. Phys. Lett., 95, 192507 (2009). . M Van Kampen, L Lagae, G Hrkac, T Schrefl, J.-V Kim, T Devolder, C Chappert, J. Phys. D: Appl. Phys. 42245001M. van Kampen, L. Lagae, G. Hrkac, T. Schrefl, J.-V. Kim, T. De- volder, and C. Chappert, J. Phys. D: Appl. Phys., 42, 245001 (2009). T Devolder, J.-V Kim, M Manfrini, G Hrkac, P Crozat, L Lagae, T Schrefl, C Chappert, Proc. SPIE. SPIE7398739808T. Devolder, J.-V. Kim, M. Manfrini, G. Hrkac, P. Crozat, L. La- gae, T. Schrefl, and C. Chappert, Proc. SPIE, 7398, 739808 (2009). . A Ruotolo, V Cros, B Georges, A Dussaux, J Grollier, C Deranlot, R Guillemet, K Bouzehouane, S Fusil, A Fert, Nat. Nanotech. 4528A. Ruotolo, V. Cros, B. Georges, A. Dussaux, J. Grollier, C. Der- anlot, R. Guillemet, K. Bouzehouane, S. Fusil, and A. Fert, Nat. Nanotech., 4, 528 (2009). . S M Rezende, F M De Aguiar, A Azevedo, Phys. Rev. Lett. 9437202S. M. Rezende, F. M. de Aguiar, and A. Azevedo, Phys. Rev. Lett., 94, 037202 (2005). . A N Slavin, P Kabos, IEEE Trans. Magn. 411264A. N. Slavin and P. Kabos, IEEE Trans. Magn., 41, 1264 (2005). . J.-V Kim, Phys. Rev. B. 73174412J.-V. Kim, Phys. Rev. B, 73, 174412 (2006). . V Tiberkevich, A Slavin, J.-V Kim, Appl. Phys. Lett. 91192506V. Tiberkevich, A. Slavin, and J.-V. Kim, Appl. Phys. Lett., 91, 192506 (2007). . J.-V Kim, V Tiberkevich, A N Slavin, Phys. Rev. Lett. 10017207J.-V. Kim, V. Tiberkevich, and A. N. Slavin, Phys. Rev. Lett., 100, 017207 (2008). . V S Tiberkevich, A N Slavin, J.-V Kim, Phys. Rev. B. 7892401V. S. Tiberkevich, A. N. Slavin, and J.-V. Kim, Phys. Rev. B, 78, 092401 (2008). . B A Ivanov, C E Zaspel, Phys. Rev. Lett. 99247208B. A. Ivanov and C. E. Zaspel, Phys. Rev. Lett., 99, 247208 (2007). . Y Liu, H He, Z Zhang, Appl. Phys. Lett. 91242501Y. Liu, H. He, and Z. Zhang, Appl. Phys. Lett., 91, 242501 (2007). . Y.-S Choi, S.-K Kim, K.-S Lee, Y.-S Yu, Appl. Phys. Lett. 93182508Y.-S. Choi, S.-K. Kim, K.-S. Lee, and Y.-S. Yu, Appl. Phys. Lett., 93, 182508 (2008). . J.-H Moon, D.-H Kim, M H Jung, K.-J Lee, Phys. Rev. B. 79134410J.-H. Moon, D.-H. Kim, M. H. Jung, and K.-J. Lee, Phys. Rev. B, 79, 134410 (2009). . Y Gaididei, V P Kravchuk, D D Sheka, Int. J. Quantum Chem. 11083Y. Gaididei, V. P. Kravchuk, and D. D. Sheka, Int. J. Quantum Chem., 110, 83 (2009). . A V Khvalkovskiy, J Grollier, A Dussaux, K A Zvezdin, V Cros, 10.1103/PhysRevB.80.140401Phys. Rev. B. 80140401A. V. Khvalkovskiy, J. Grollier, A. Dussaux, K. A. Zvezdin, and V. Cros, Phys. Rev. B, 80, 140401 (2009). . J Xiao, A Zangwill, M D Stiles, Phys. Rev. B. 70172405J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B, 70, 172405 (2004). . S Zhang, Z Li, Phys. Rev. Lett. 93127204S. Zhang and Z. Li, Phys. Rev. Lett., 93, 127204 (2004). . A Thiaville, Y Nakatani, J Miltat, Y Suzuki, Europhys. Lett. 69990A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett., 69, 990 (2005). Note that P 1 is largely determined by the spin-dependent transport properties across the entire multilayer stack. while P 2 depends largely on the transport properties of the magnetic free layerNote that P 1 is largely determined by the spin-dependent transport properties across the entire multilayer stack, while P 2 depends largely on the transport properties of the magnetic free layer. Vortex nucleation in spin-torque nanocontact oscillators. T Devolder, J.-V Kim, M Manfrini, W Van Roy, L Lagae, C Chappert, Appl. Phys. Lett. to appear inT. Devolder, J.-V. Kim, M. Manfrini, W. van Roy, L. Lagae, and C. Chappert, "Vortex nucleation in spin-torque nanocontact oscil- lators," to appear in Appl. Phys. Lett. (2010). . E G See, A Vansteenkiste, K W Chou, M Weigand, M Curcic, V Sackmann, H Stoll, T Tyliszczak, G Woltersdorf, C H Back, G Schutz, B Van Waeyenberge, 10.1038/nphys1231Nat. Phys. 5332See, e.g., A. Vansteenkiste, K. W. Chou, M. Weigand, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, G. Woltersdorf, C. H. Back, G. Schutz, and B. Van Waeyenberge, Nat. Phys., 5, 332 (2009). . F G Mertens, A R Bishop, Lecture Notes in Physics. P. L. Christiansen, M. P. Sørensen, and A. C. Scott542Springer-Verlagin Nonlinear Science at the Dawn of the 21st CenturyF. G. Mertens and A. R. Bishop, in Nonlinear Science at the Dawn of the 21st Century, Lecture Notes in Physics, Vol. 542, edited by P. L. Christiansen, M. P. Sørensen, and A. C. Scott (Springer- Verlag, 2000) pp. 137-170. R Rajaraman, Solitons and Instantons. AmsterdamNorth-Holland1st ed.R. Rajaraman, Solitons and Instantons, 1st ed. (North-Holland, Amsterdam, 1982). . J Shibata, G Tatara, H Kohno, Phys. Rev. Lett. 9476601J. Shibata, G. Tatara, and H. Kohno, Phys. Rev. Lett., 94, 076601 (2005). . S E Barnes, S Maekawa, Phys. Rev. Lett. 95107204S. E. Barnes and S. Maekawa, Phys. Rev. Lett., 95, 107204 (2005). . Y Le Maho, J.-V Kim, G Tatara, Phys. Rev. B. 79174404Y. Le Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B, 79, 174404 (2009). . M Hayashi, L Thomas, Y B Bazaliy, C Rettner, R Moriya, X Jiang, S S P Parkin, Phys. Rev. Lett. 96197207M. Hayashi, L. Thomas, Y. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett., 96, 197207 (2006). . C Burrowes, A P Mihai, D Ravelosona, J.-V Kim, C Chappert, L Vila, A Marty, Y Samson, F G Sanchez, L D Buda-Prejbeanu, I Tudosa, E E Fullerton, J P Attané, Nat. Phys. 617C. Burrowes, A. P. Mihai, D. Ravelosona, J.-V. Kim, C. Chap- pert, L. Vila, A. Marty, Y. Samson, F. G. Sanchez, L. D. Buda- Prejbeanu, I. Tudosa, E. E. Fullerton, and J. P. Attané, Nat. Phys., 6, 17 (2010). . T Kamppeter, F G Mertens, E Moro, A Sánchez, A R Bishop, 10.1103/PhysRevB.59.11349Phys. Rev. B. 5911349T. Kamppeter, F. G. Mertens, E. Moro, A. Sánchez, and A. R. Bishop, Phys. Rev. B, 59, 11349 (1999). . T Kamppeter, F G Mertens, A Sánchez, A R Bishop, F Dominguez-Adame, N Grønbech-Jensen, Eur. Phys. J. B. 7607T. Kamppeter, F. G. Mertens, A. Sánchez, A. R. Bishop, F. Dominguez-Adame, and N. Grønbech-Jensen, Eur. Phys. J. B, 7, 607 (1999). The Skyrmion number is a topological charge and is defined by Q = −(1/4π) d 2 x ∂(cos Θ, Φ)/∂(x, y). See Ref. 55 for a detailed discussion. The Skyrmion number is a topological charge and is defined by Q = −(1/4π) d 2 x ∂(cos Θ, Φ)/∂(x, y). See Ref. 55 for a detailed discussion. . O A Tretiakov, O Tchernyshyov, Phys. Rev. B. 7512408O. A. Tretiakov and O. Tchernyshyov, Phys. Rev. B, 75, 012408 (2007). Dynamics of vortexantivortex pairs in ferromagnets. S Komineas, N Papanicolaou, arXiv:0712.3684v1condmat.mes-hallS. Komineas and N. Papanicolaou, "Dynamics of vortex- antivortex pairs in ferromagnets," arXiv:0712.3684v1 [cond- mat.mes-hall] (2007). . S Komineas, 10.1103/PhysRevLett.99.117202Phys. Rev. Lett. 99117202S. Komineas, Phys. Rev. Lett., 99, 117202 (2007). . D V Berkov, N L Gorn, 10.1103/PhysRevB.80.064409Phys. Rev. B. 8064409D. V. Berkov and N. L. Gorn, Phys. Rev. B, 80, 064409 (2009).
[]
[ "Congruences for sums involving products of three binomial coefficients", "Congruences for sums involving products of three binomial coefficients" ]
[ "Zhi-Hong Sun [email protected] \nSchool of Mathematics and Statistics\nHuaiyin Normal University Huaian\n223300JiangsuP.R. China\n" ]
[ "School of Mathematics and Statistics\nHuaiyin Normal University Huaian\n223300JiangsuP.R. China" ]
[]
Let p > 3 be a prime, and let a be a rational p-adic integer, using WZ method we establish the congruences modulo p 3 for p−1 k=0
null
[ "https://arxiv.org/pdf/2202.05077v2.pdf" ]
246,706,322
2202.05077
86496f29a3cbca6f094fd582c6143b1c78bf96de
Congruences for sums involving products of three binomial coefficients 14 Feb 2022 Zhi-Hong Sun [email protected] School of Mathematics and Statistics Huaiyin Normal University Huaian 223300JiangsuP.R. China Congruences for sums involving products of three binomial coefficients 14 Feb 2022arXiv:2202.05077v2 [math.NT] Let p > 3 be a prime, and let a be a rational p-adic integer, using WZ method we establish the congruences modulo p 3 for p−1 k=0 a k −1 − a k 2k k w(k) 4 k , where w(k) = 1, 1 k + 1 , 1 (k + 1) 2 , 1 (k + 1) 3 , 1 2k − 1 , 1 k + 2 , 1 k + 3 , k, k 2 , k 3 , 1 a + k , 1 a + k − 1 . Introduction For a ∈ Z and given odd prime p let ( a p ) denote the Legendre symbol. For positive integers a, b and n, if n = ax 2 + by 2 for some integers x and y, we briefly write that n = ax 2 + by 2 . Let p > 3 be a prime. In 1987, Beukers [B] conjectured a congruence equivalent to p−1 k=0 2k k 3 64 k ≡ 4x 2 − 2p (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 0 (mod p 2 ) if p ≡ 3 (mod 4). This congruence was proved by several authors including Ishikawa [I](p ≡ 1 (mod 4)), van Hamme [H](p ≡ 3 (mod 4)) and Ahlgren [A]. Combining the results in [LR], [S10] and [T3], in [S14] the author stated that (1.1) Let p > 3 be a prime. In 2003, Rodriguez-Villegas [RV] posed 22 conjectures on supercongruences modulo p 2 . In particular, the following congruences are equivalent to conjectures due to Rodriguez-Villegas: p−1 k=0 2k k 3 64 k ≡          4x 2 − 2p − p 24xp−1 k=0 2k k 2 3k k 108 k ≡ 4x 2 − 2p (mod p 2 ) if p = x 2 + 3y 2 ≡ 1 (mod 3), 0 (mod p 2 ) if p ≡ 2 (mod 3), p−1 k=0 2k k 2 4k 2k 256 k ≡ 4x 2 − 2p (mod p 2 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), 0 (mod p 2 ) if p ≡ 5, 7 (mod 8), p 3 p−1 k=0 2k k 3k k 6k 3k 12 3k ≡ 4x 2 − 2p (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 0 (mod p 2 ) if p ≡ 3 (mod 4). These conjectures have been solved by Mortenson [M] and Zhi-Wei Sun [Su]. In 2018, J.C. Liu [Liu] conjectured the congruences for in terms of p−adic Gamma functions. In [S11], the author conjectured that where [a] is the greatest integer not exceeding a. It is easy to see that (see [S4-S7]) Thus, a natural and general problem is to determine p−1 k=0 a k −1−a k 2k k 1 4 k modulo p 3 , where p > 3 is a prime and a is a rational p-adic integer. p−1 k=0 2k k 2 3k k 108 k ≡        4x 2 − 2p − p 2 4x 2 (mod p 3 ) if p = x 2 + 3y 2 ≡ 1 (mod 3),− 1 2 k = 2k k (−4) k , − 1 3 k − 2 3 k = 2k k 3k k 27 k , For a prime p let Z p be the set of rational numbers whose denominator is not divisible by p. For a ∈ Z p let a p be determined by a p ∈ {0, 1, . . . , p − 1} and a ≡ a p (mod p). In [S7], the author showed that for any given odd prime p and a ∈ Z p , p−1 k=0 a k −1 − a k 2k k 1 4 k ≡ 0 (mod p 2 ) for a p ≡ 1 (mod 2). For an odd prime p and x ∈ Z p , the p-adic Gamma function Γ p (x) is defined by Γ p (0) = 1, Γ p (n) = (−1) n k∈{1,2,...,n−1} p∤k k for n = 1, 2, 3, . . . and Γ p (x) = lim n∈{0,1,...} |x−n|p→0 Γ p (n). In [PTW], Pan, Tauraso and Wang established a result equivalent to (1.5) p−1 k=0 a k −1 − a k 2k k 1 4 k ≡          Γ p ( 1 2 ) 2 Γ p ( 2+a 2 ) 2 Γ p ( 1−a 2 ) 2 (mod p 3 ) if 2 | a p , a ′ (a ′ + 1) 4 p 2 Γ p ( 1 2 ) 2 Γ p ( 2+a 2 ) 2 Γ p ( 1−a 2 ) 2 (mod p 3 ) if 2 ∤ a p , where a ′ = (a − a p )/p. But they only gave the proof for the case 2 | a p , and their method is somewhat complicated. In the case 2 | a p , (1.5) was first conjectured by Liu in [Liu]. Using (1.5) and Jacobi sums, recently Mao [Mao] proved (1.2) and (1.3). We also note that Guo [Guo] established three congruences modulo p 3 concerning the sums in (1.2)-(1.4) via q-congruences. For example, Guo showed that for any prime p ≡ 5 (mod 6), In [Su], Z.W. Sun showed that p−1 k=0 2k k 3 64 k (k + 1) ≡ 4x 2 − 2p (mod p 2 ) for p = x 2 + 4y 2 ≡ 1 (mod 4), p−1 k=0 2k k 2 3k k 108 k (k + 1) ≡ 4x 2 − 2p (mod p 2 ) for p = x 2 + 3y 2 ≡ 1 (mod 3), p−1 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ 4x 2 − 2p (mod p 2 ) for p = x 2 + 2y 2 ≡ 1, 3 (mod 8), p 3 p−1 k=0 2k k 3k k 6k 3k 1728 k (k + 1) ≡ 4x 2 − 2p (mod p 2 ) for p = x 2 + 4y 2 ≡ 1 (mod 4). In [T3], Tauraso showed that (p−1)/2 k=0 2k k 3 64 k (k + 1) 3 ≡          8 − 24p 2 Γ p ( 1 4 ) 4 ≡ 8 + 6p 2 x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 8 − 384 Γ p ( 1 4 ) 4 ≡ 8 − 96 2 p−1 (1 + 2p + (3 − 1 2 E p−3 )p 2 ) p−3 2 p−3 4 2 (mod p 3 ) if 4 | p − 3, where {E n } are the Euler numbers given by E 2n−1 = 0, E 0 = 1, E 2n = − n k=1 2n 2k E 2n−2k (n = 1, 2, 3, . . .). In [S11], [S13] and [S14], the author posed numerous conjectures on congruences modulo p 3 for the sums p−1 k=0 w(k) 2k k 3 m k , p−1 k=0 w(k) 2k k 2 3k k m k , p−1 k=0 w(k) 2k k 2 4k 2k m k , p−1 k=0 w(k) 2k k 3k k 6k 3k m k , where m is an integer not divisible by p and w(k) ∈ {1, 1 k+1 , 1 2k−1 , 1 (k+1) 2 , k, k 2 , k 3 }. Let p > 3 be a prime and a ∈ Z p . Inspired by the above work, in this paper, using WZ method we establish the congruences modulo p 3 for p−1 k=0 a k −1 − a k 2k k w(k) 4 k , where w(k) ∈ 1, 1 k + 1 , 1 (k + 1) 2 , 1 (k + 1) 3 , 1 2k − 1 , 1 k + 2 , 1 k + 3 , k, k 2 , k 3 , 1 a + k , 1 a + k − 1 . Our approach is natural and elementary. As consequences, taking a = − 1 2 , − 1 3 , − 1 4 , − 1 6 we deduce many congruences modulo p 3 and so solve some conjectures in [S8,S10,S11]. For instance, for p ≡ 1 (mod 4) and so p = x 2 + 4y 2 we prove that (p−1)/2 k=0 2k k 3 64 k (k + 1) ≡ 4x 2 − 2p (mod p 3 ), (p−1)/2 k=0 2k k 3 64 k (k + 2) ≡ 52 27 x 2 − 26 27 p − p 2 27x 2 (mod p 3 ), p−1 k=0 2k k 3 64 k (2k − 1) ≡ −2x 2 + p + p 2 4x 2 (mod p 3 ), p−1 k=0 2k k 3 64 k (2k − 1) 2 ≡ 2x 2 − p − p 2 2x 2 (mod p 3 ), for p ≡ 1 (mod 3), p > 7 and so p = x 2 + 3y 2 we show that p−2 k=0 2k k 2 3k k 108 k (k + 1) ≡ 4x 2 − 2p (mod p 3 ), p−2 k=0 2k k 2 3k k 108 k (k + 1) 2 ≡ 8x 2 − 4p + 9p 2 8x 2 (mod p 3 ) p−2 k=0 2k k 2 3k k 108 k (k + 1) 3 ≡ 9 − 2x 2 + p + 117p 2 16x 2 (mod p 3 ), p−3 k=0 2k k 2 3k k 108 k (k + 2) ≡ 29 15 x 2 − 29 30 p − 3p 2 80x 2 (mod p 3 ), p−4 k=0 2k k 2 3k k 108 k (k + 3) ≡ 32 25 x 2 − 16 25 p − 99p 2 2800x 2 (mod p 3 ), p−1 k=0 2k k 2 3k k 108 k (2k − 1) ≡ − 20 9 x 2 + 10 9 p + p 2 4x 2 (mod p 3 ), p−1 k=0 k 2k k 2 3k k 108 k ≡ − 8 9 x 2 + 4 9 p + p 2 9x 2 (mod p 3 ), p−1 k=0 k 2 2k k 2 3k k 108 k ≡ 32 243 x 2 − 16 243 p − 17p 2 486x 2 (mod p 3 ), p−1 k=0 k 3 2k k 2 3k k 108 k ≡ 1 10935 16x 2 − 8p + 113p 2 2x 2 (mod p 3 ). We also pose some challenging conjectures on the congruences modulo p 3 for the sum (p−1)/2 k=0 w(k) ( 2k k ) 3 m k , where m ∈ {1, −8, 16, −64, 256, −512, 4096} and w(k) ∈ { 1 k+2 , 1 k+3 , 1 (k+1) 2 , 1 (k+2) 2 , 1 (k+1) 3 }. In addition to the above notation, throughout this paper we use the following notations. Let H 0 = H (2) 0 = 0. For n ≥ 1 let H n = 1 + 1 2 + · · · + 1 n and H (2) n = 1 + 1 2 2 + · · · + 1 n 2 . For an odd prime and a ∈ Z p set q p (a) = (a p−1 − 1)/p and R 1 (p) = (2p + 2 − 2 p−1 ) (p − 1)/2 [p/4] 2 , R 2 (p) = (5 − 4(−1) p−1 2 ) 1 + (4 + 2(−1) p−1 2 )p − 4(2 p−1 − 1) − p 2 [p/8] k=1 1 k p−1 2 [ p 8 ] 2 , R 3 (p) = 1 + 2p + 4 3 (2 p−1 − 1) − 3 2 (3 p−1 − 1) (p − 1)/2 [p/6] 2 , R 7 (p) = (p−1)/2 k=0 2k k 3 k + 1 . Let S n (a) = n−1 k=0 a k −1 − a k 2k k 1 4 k (n = 1, 2, 3, . . .), and let {U n } be the sequence given by U 2n−1 = 0, U 0 = 1, U 2n = −2 n k=1 2n 2k U 2n−2k (n = 1, 2, 3, . . .). The congruence for p−1 k=0 a k −1−a k 2k k 1 4 k modulo p 3 For k = 0, 1, 2, . . . set F (a, k) = a k −1 − a k 2k k 1 4 k and G(a, k) = (a + 2)(2a + 3) k 4 k−1 (a + 1 + k) 2k − 1 k − 1 a + 1 k − 1 −3 − a k − 1 . It is easy to check that (a + 2) 2 F (a + 2, k) − (a + 1) 2 F (a, k) = G(a, k + 1) − G(a, k). Thus, (a + 2) 2 S n (a + 2) − (a + 1) 2 S n (a) = (a + 2) 2 n−1 k=0 F (a + 2, k) − (a + 1) 2 n k=0 F (a, k) = n−1 k=0 (G(a, k + 1) − G(a, k)) = G(a, n) − G(a, 0) = G(a, n). That is, (2.1) (a + 2) 2 S n (a + 2) − (a + 1) 2 S n (a) = (a + 2)(2a + 3) n 4 n−1 (a + 1 + n) 2n − 1 n − 1 a + 1 n − 1 −3 − a n − 1 . Lemma 2.1. Let p > 3 be a prime, a ∈ Z p , a ≡ −1 (mod p) and a ′ = (a − a p )/p. Then (a + 2) 2 S p (a + 2) − (a + 1) 2 S p (a) ≡    1 a + 1 + 1 a + 2 a ′ (a ′ + 1)p 3 (mod p 4 ) if a p < p − 2, (a + 2)p (mod p 3 ) if a p = p − 2. Proof. From (2.1) we see that (a + 2) 2 S p (a + 2) − (a + 1) 2 S p (a) = (a + 2)(2a + 3) p 4 p−1 (a + 1 + p) 2p − 1 p a + 2 − 1 p − 1 −a − 2 − 1 p − 1 . We first assume a p < p − 2. Then a + 2 − a + 2 p = a − a p = a ′ p. From [S9, Lemma 2.2] we know that (2.2) a + 2 − 1 p − 1 −a − 2 − 1 p − 1 ≡ a ′ (a ′ + 1)p 2 (a + 2) 2 (mod p 3 ). Hence, (a + 2) 2 S p (a + 2) − (a + 1) 2 S p (a) ≡ (a + 2)(2a + 3) p 4 p−1 (a + 1 + p) 2p − 1 p − 1 a ′ (a ′ + 1)p 2 (a + 2) 2 ≡ 1 a + 1 + 1 a + 2 a ′ (a ′ + 1)p 3 (mod p 4 ). Now we assume that a p = p − 2. Then a + 2 = p(a ′ + 1). Appealing to [S12, (2.6)], a + 2 − 1 p − 1 −a − 2 − 1 p − 1 = p − 1 + a ′ p p − 1 p − 1 − (a ′ + 2)p p − 1 ≡ (1 + a ′ pH p−1 )(1 − (a ′ + 2)pH p−1 ) ≡ 1 (mod p 2 ). Hence, (a + 2) 2 S p (a + 2) − (a + 1) 2 S p (a) ≡ (a + 2)(2a + 3) p 4 p−1 (a + 1 + p) 2p − 1 p − 1 ≡ (a + 2)p (mod p 3 ). This completes the proof. Lemma 2.2. Let p be an odd prime and t ∈ Z p . Then p−1 k=0 pt k −1 − pt k 2k k 1 4 k ≡ 1 − 2t(2 p−1 − 1) + (2t 2 + t)(2 p−1 − 1) 2 (mod p 3 ). Proof. Clearly, p−1 k=0 pt k −1 − pt k 2k k 1 4 k = 1 + p−1 k=1 pt pt − k · ((−1) 2 − p 2 t 2 )((−2) 2 − p 2 t 2 ) · · · ((−k) 2 − p 2 t 2 ) k! 2 · 2k k 4 k ≡ 1 − p−1 k=1 pt(pt + k) k 2 · 2k k 4 k (mod p 3 ). By [S8,Remark 3.1] or [T1], (2.3) p−1 k=1 2k k k · 4 k = p−1 k=1 (−1) k k −1/2 k ≡ 2q p (2) − pq p (2) 2 (mod p 2 ). Taking x = 1 4 in [T2, (9)] and then applying [S2,Theorem 4.1] we see that (2.4) p−1 k=1 2k k k 2 · 4 k ≡ 4 p−1 k=1 1 k 2 · 2 k ≡ −2q p (2) 2 (mod p). It then follows that p−1 k=0 pt k −1 − pt k 2k k 1 4 k ≡ 1 − pt(2q p (2) − pq p (2) 2 ) − p 2 t 2 (−2q p (2) 2 ) (mod p 3 ), which yields the result. Lemma 2.3. Let p be an odd prime, n ∈ {1, 2, . . . , p−1 2 } and t ∈ Z p . Then p−1 2 + pt n ≡ p−1 2 n 1 − 2pt n k=1 1 2k − 1 + 2p 2 t t n k=1 1 2k − 1 2 − (t + 1) n k=1 1 (2k − 1) 2 (mod p 3 ). Proof. For m ∈ {n + 1, . . . , p − 1} we see that (2.5) m + pt n = (pt + m)(pt + m − 1) · · · (pt + m − (n − 1)) n! ≡ m n 1 + pt m−n+1≤k≤m 1 k + p 2 t 2 m−n+1≤i<j≤m 1 ij = m n 1 + pt(H m − H m−n ) + 1 2 p 2 t 2 (H m − H m−n ) 2 − (H (2) m − H (2) m−n ) (mod p 3 ). For given positive integer r we have (p−1)/2 k=1 1 k r − (p−1)/2−n k=1 1 k r = n k=1 1 p−(2k−1) 2 r = 2 r n k=1 (p + 2k − 1) r (p 2 − (2k − 1) 2 ) r ≡ (−2) r n k=1 (2k − 1) r + rp(2k − 1) r−1 (2k − 1) 2r = (−2) r n k=1 1 (2k − 1) r + (−2) r rp n k=1 1 (2k − 1) r+1 (mod p 2 ). Hence, H p−1 2 − H p−1 2 −n ≡ −2 n k=1 1 2k − 1 − 2p n k=1 1 (2k − 1) 2 (mod p 2 ), (p−1)/2 k=1 1 k 2 − (p−1)/2−n k=1 1 k 2 ≡ 4 n k=1 1 (2k − 1) 2 (mod p). Now, from the above we deduce that p−1 2 + pt n ≡ p−1 2 n 1 + pt − 2 n k=1 1 2k − 1 − 2p n k=1 1 (2k − 1) 2 + 1 2 p 2 t 2 − 2 n k=1 1 2k − 1 2 − 4 n k=1 1 (2k − 1) 2 (mod p 3 ), which yields the result. Theorem 2.1. Let p be an odd prime, a ∈ Z p , a ≡ 0, −1 (mod p) and a ′ = (a− a p )/p. If 2 | a p and a p = 2n, then S p (a) ≡ (a−1)/2 n 2 a/2 n 2 1 − 2a ′ (2 p−1 − 1) + a ′ (2a ′ + 1)(2 p−1 − 1) 2 ) ≡ (p − 1)/2 n 2 1 + p (2a ′ + 2)H 2n − (2a ′ + 1)H n − 2a ′ q p (2) + p 2 2 2a ′ q p (2) 2 + (2a ′ + 2)H 2n − (2a ′ + 1)H n − 2a ′ q p (2) 2 + 1 2 (2a ′ 2 − 1)H (2) n + 2(1 − a ′ 2 )H (2) 2n (mod p 3 ). If 2 ∤ a p , then S p (a) ≡ 4 a p −1 · a ′ (a ′ + 1)p 2 a 2 a p −1 a p −1 2 2 ≡ a ′ (a ′ + 1)p 2 a 2 (p−1)/2 ( a p −1)/2 2 (mod p 3 ). Proof. Set a ′ = (a − a p )/p. We first assume that 2 | a p and a p = 2n. From Lemma 2.1 we see that S p (a) ≡ (a − 1) 2 a 2 S p (a − 2) ≡ (a − 1) 2 a 2 · (a − 3) 2 (a − 2) 2 S p (a − 4) ≡ · · · ≡ (a − 1) 2 (a − 3) 2 · · · (a − 2n + 1) 2 a 2 (a − 2) 2 · · · (a − 2n + 2) 2 S p (a − 2n) = n k=1 ( a+1 2 − k) 2 ( a+2 2 − k) 2 · S p (a − 2n) = (a−1)/2 n 2 a/2 n 2 S p (a − 2n) (mod p 3 ). By Lemma 2.2, S p (a − 2n) = S p (a ′ p) ≡ 1 − 2a ′ (2 p−1 − 1) + a ′ (2a ′ + 1)(2 p−1 − 1) 2 (mod p 3 ). Thus, S p (a) ≡ (a − 1)/2 n 2 a/2 n −2 1 − 2a ′ pq p (2) + a ′ (2a ′ + 1)p 2 q p (2) 2 (mod p 3 ). Since (1 + bp + cp 2 )(1 − bp + (b 2 − c)p 2 ) ≡ 1 (mod p 3 ), appealing to (2.5) we get a/2 n −1 ≡ n + a ′ p/2 n −1 ≡ 1 + 1 2 a ′ pH n + 1 8 a ′ 2 p 2 (H 2 n − H (2) n ) −1 ≡ 1 − 1 2 a ′ pH n + 1 8 a ′ 2 p 2 (H 2 n + H (2) n ) (mod p 3 ). By Lemma 2.3, a−1 2 n = (−1) n − a−1 2 + n − 1 n = (−1) n p−1 2 − a ′ +1 2 p n ≡ (−1) n p−1 2 n 1 + (a ′ + 1)p n k=1 1 2k − 1 + (a ′ + 1)p 2 a ′ + 1 2 n k=1 1 2k − 1 2 + 1 − a ′ 2 n k=1 1 (2k − 1) 2 (mod p 3 ). Therefore, (−1) n a−1 2 n a 2 n −1 p−1 2 n −1 ≡ 1 + (a ′ + 1)p n k=1 1 2k − 1 + (a ′ + 1)p 2 a ′ + 1 2 n k=1 1 2k − 1 2 + 1 − a ′ 2 n k=1 1 (2k − 1) 2 × 1 − 1 2 a ′ pH n + 1 8 a ′ 2 p 2 (H 2 n + H (2) n ) ≡ 1 + p (a ′ + 1) n k=1 1 2k − 1 − a ′ 2 H n + p 2 1 8 a ′ 2 H 2 n + 1 8 a ′ 2 H (2) n + (a ′ + 1) 2 2 n k=1 1 2k − 1 2 + 1 − a ′ 2 2 n k=1 1 (2k − 1) 2 − 1 2 a ′ (a ′ + 1)H n n k=1 1 2k − 1 = 1 + p (a ′ + 1) n k=1 1 2k − 1 − a ′ 2 H n + p 2 1 8 a ′ H n − 2(a ′ + 1) n k=1 1 2k − 1 2 + 1 8 a ′ 2 H (2) n + 1 − a ′ 2 2 n k=1 1 (2k − 1) 2 (mod p 3 ). Note that (1 + bp + cp 2 ) 2 ≡ 1 + 2bp + (b 2 + 2c)p 2 (mod p 3 ). From the above we derive that a−1 2 n 2 a 2 n −2 p−1 2 n −2 ≡ 1 + p 2(a ′ + 1) n k=1 1 2k − 1 − a ′ H n + p 2 2 (a ′ + 1) n k=1 1 2k − 1 − a ′ 2 H n 2 + 1 4 a ′ 2 H (2) n + (1 − a ′ 2 ) n k=1 1 (2k − 1) 2 = 1 + p 2(a ′ + 1)(H 2n − 1 2 H n ) − a ′ H n + p 2 2 (a ′ + 1)(H 2n − 1 2 H n ) − a ′ 2 H n 2 + 1 4 a ′ 2 H (2) n + (1 − a ′ 2 )(H (2) 2n − 1 4 H (2) n = 1 + p((2a ′ + 2)H 2n − (2a ′ + 1)H n ) + p 2 1 2 ((2a ′ + 2)H 2n − (2a ′ + 1)H n ) 2 + (1 − a ′ 2 )H (2) 2n + 2a ′ 2 − 1 4 H (2) n (mod p 3 ) and so S p (a) ≡ (p − 1)/2 n 2 (1 − 2a ′ pq p (2) + a ′ (2a ′ + 1)p 2 q p (2) 2 ) × 1 + p((2a ′ + 2)H 2n − (2a ′ + 1)H n ) + p 2 1 2 ((2a ′ + 2)H 2n − (2a ′ + 1)H n ) 2 + (1 − a ′ 2 )H (2) 2n + 2a ′ 2 − 1 4 H (2) n ≡ (p − 1)/2 n 2 1 + p(−2a ′ q p (2) + 2(a ′ + 1)H 2n − (2a ′ + 1)H n ) + p 2 a ′ q p (2) 2 + 1 2 (2a ′ + 2)H 2n − (2a ′ + 1)H n − 2a ′ q p (2) 2 + (1 − a ′ 2 )H (2) 2n + 2a ′ 2 − 1 4 H (2) n (mod p 3 ), which yields the result in the case 2 | a p . Now assume that 2 ∤ a p . By Lemma 2.1, for a p ≤ p − 4, S p (a) ≡ (a + 2) 2 (a + 1) 2 S p (a + 2) ≡ (a + 2) 2 (a + 1) 2 · (a + 4) 2 (a + 3) 2 S p (a + 4) ≡ · · · ≡ p− a p 2 −2 k=0 (a + 2k + 2) 2 (a + 2k + 1) 2 · S p (p + a − a p − 2) = p− a p 2 −2 k=0 (a + 2k + 1) 2 (a + 2k + 2) 2 (a + 2k + 1) 4 · S p (p + a − a p − 2) = (a + 1) 2 (a + 2) 2 · · · (p + a − a p − 3) 2 (p + a − a p − 2) 2 2 4( p− a p 2 −1) ( a+1 2 ( a+1 2 + 1) · · · ( a+1 2 + p− a p 2 − 2)) 4 × S p (p + a − a p − 2) (mod p 3 ). By Lemmas 2.1 and 2.2, ((a ′ + 1)p − 1) 2 S p ((a ′ + 1)p − 2) ≡ (a ′ + 1) 2 p 2 S p ((a ′ + 1)p) − (a ′ + 1)p 2 ≡ (a ′ + 1) 2 p 2 − (a ′ + 1)p 2 = a ′ (a ′ + 1)p 2 (mod p 3 ). Hence, S p (p + a − a p − 2) = S p ((a ′ + 1)p − 2) ≡ a ′ (a ′ + 1)p 2 (mod p 3 ). Now, from the above we deduce that S p (a) ≡ a ′ (a ′ + 1)p 2 · ( a p + 1) 2 ( a p + 2) 2 · · · (p − 3) 2 (p − 2) 2 2 4( p− a p 2 −1) ( a p +1 2 ( a p +1 2 + 1) · · · ( p−3 2 )) 4 = a ′ (a ′ + 1)p 2 · (p − 2)! 2 · a p −1 2 ! 4 a p ! 2 · 2 2(p− a p )−4 · p−3 2 ! 4 = a ′ (a ′ + 1)p 2 · ( p−1 2 ) 4 − 1 4 H (2) p+1 4 + 3 2 H (2) p+1 2 ≡ (p − 1)/2 (p + 1)/4 2 1 + p 2 p + 1 − 2q p (2) + pq p (2) 2 + q p (2) + p 2 2 − q p (2) 2 + 2 p + 1 − 2q p (2) + pq p (2) 2 + q p (2) 2 − 1 4 16 (p + 1) 2 − 4E p−3 + 3 2 · 4 (p + 1) 2 ≡ (p − 1)/2 (p − 3)/4 2 1 + p(2 − q p (2)) + p 2 ((1 − q p (2)) 2 + 1 2 E p−3 ) ≡ (p + 1) 2 2 p−1 (p − 1)/2 (p − 3)/4 2 + p 2 2 (p − 1)/2 (p − 3)/4 2 E p−3 (mod p 3 ). This proves the lemma. Theorem 2.2 ([S14, Conjecture 5.4]). Let p be a prime with p > 3. Then p−1 k=0 2k k 3 64 k (2k − 1) ≡          −2x 2 + p + p 2 4x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 1 4 2 + p 2 E p−3 (p + 1) 2 2 p−1 (p − 1)/2 (p − 3)/4 2 + p 2 2 (p−1)/2 (p−3)/4 2 (mod p 3 ) if 4 | p − 3. Proof. Note that (2.6) 1 2 k − 3 2 k = 1 + 2k 1 − 2k − 1 2 k 2 = − 1 − 2 2k − 1 2k k 2 1 16 k . We then have (2.7) 2 p−1 k=0 2k k 3 64 k (2k − 1) + p−1 k=0 2k k 3 64 k = − p−1 k=0 1 2 k − 3 2 k 2k k 1 4 k = −S p 1 2 . Now, applying (1.1) and Lemma 2.5 yields the result. The congruence for p−2 k=0 a k −1−a k 2k k 1 4 k (k+1) mod- ulo p 3 Suppose that F (a, k) and R(a, k) satisfy R(a, 0) = 0 and F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k) (k = 0, 1, 2, . . .). For any positive integer n we then have (3.1) n−1 k=0 F (a, k) = n−1 k=0 (F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k)) = F (a, n)R(a, n) − F (a, 0)R(a, 0) = F (a, n)R(a, n). Using the WZ method and Zeilberger's algorithm one can easily find R(a, k) by Maple. See [PWZ]. Lemma 3.1. For any positive integer n and real number a = 0 we have n−1 k=0 a k −1 − a k 2k k 1 4 k (k + 1) = S n (a) + a + 1 a S n (a + 1) + F (a, n)R(a, n), where F (a, n) = a n −1 − a n 1 n + 1 − 1 − a + 1 a a + 1 n −2 − a n 2n n 1 4 n , R(a, n) = − 2n 2 (n + 1) n 2 + 2(a + 1) 2 n + (a + 1) 2 . Proof. It is easy to check that R(a, 0) = 0, F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). Thus the result follows from (3.1). Lemma 3.2. Let p be an odd prime, a ∈ Z p , a ≡ 0, −1 (mod p) and a ′ = (a − a p )/p. Then a p−1 −1−a p−1 2(p−1) p−1 4 p−1 · p ≡ − a ′ (a ′ + 1) a(a + 1) p 2 (mod p 3 ). Proof. Since a p−1 = a a+1−p a−1 p−1 and 2(p−1) p−1 = p 2p−1 2p−1 p−1 , we see that a p−1 −1−a p−1 2(p−1) p−1 4 p−1 · p = a (a + 1 − p)(2p − 1) · 4 p−1 a − 1 p − 1 −1 − a p − 1 2p − 1 p − 1 . By [S9, Lemma 2.2], a−1 p−1 −1−a p−1 ≡ a ′ (a ′ +1) a 2 p 2 (mod p 3 ). Thus, the result follows. Theorem 3.1. Let p be an odd prime, a ∈ Z p , a ≡ 0, −1 (mod p) and a ′ = (a− a p )/p. Then p−1 k=0 a k −1 − a k 2k k 1 4 k (k + 1) + a ′ (a ′ + 1) a(a + 1) p 2 ≡ p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) ≡ S p (a) + a + 1 a S p (a + 1) (mod p 3 ). Thus, for 2 | a p we have p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) ≡ (p − 1)/2 a p /2 2 1 + p (2a ′ + 2)H a p − (2a ′ + 1)H a p 2 − 2a ′ q p (2) + p 2 2 2a ′ q p (2) 2 + (2a ′ + 2)H a p − (2a ′ + 1)H a p 2 − 2a ′ q p (2) 2 + 1 2 (2a ′ 2 − 1)H (2) a p 2 + 2(1 − a ′ 2 )H (2) a p + p 2 a ′ (a ′ + 1) a(a + 1) (p − 1)/2 a p /2 −2 (mod p 3 ); for 2 ∤ a p and a p = p − 2 we have p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) ≡ a + 1 a (p − 1)/2 ( a p + 1)/2 2 1 + p (2a ′ + 2)H a p +1 − (2a ′ + 1)H a p +1 2 − 2a ′ q p (2) + p 2 2 2a ′ q p (2) 2 + (2a ′ + 2)H a p +1 − (2a ′ + 1)H a p +1 2 − 2a ′ q p (2) 2 + 1 2 (2a ′ 2 − 1)H (2) a p+1 2 + 2(1 − a ′ 2 )H (2) a p +1 + p 2 a ′ (a ′ + 1) (a + 1) 2 · (p − 1)/2 ( a p + 1)/2 −2 (mod p 3 ). Proof. Since a + 1 p = a p + 1 + a ′ p p ≡ a ′ (mod p), −2 − a p = p − 2 − a p − (a ′ + 1)p p ≡ −(a ′ + 1) (mod p), taking n = p in Lemma 3.1 gives p−1 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − S p (a) − a + 1 a S p (a + 1) ≡ a + 1 a a + 1 p −2 − a p 2p p 1 4 p · 2p 2 (a + 1) 2 ≡ − a ′ (a ′ + 1) a(a + 1) p 2 (mod p 3 ). Now, applying Lemma 3.2 and Theorem 2.1 yields the remaining part. Theorem 3.2. Let p be an odd prime. Then p−1 2 k=0 2k k 3 64 k (k + 1) ≡              4x 2 − 2p (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − (p + 1) 2 2 p−1 (p − 1)/2 (p − 3)/4 2 − p 2 (p − 1)/2 (p − 3)/4 −2 + 1 2 (p − 1)/2 (p − 3)/4 2 E p−3 (mod p 3 ) if 4 | p − 3. Proof. Note that p | 2k k for p 2 < k < p. Taking a = − 1 2 in Theorem 3.1 gives (p−1)/2 k=0 2k k 3 64 k (k + 1) ≡ p−2 k=0 2k k 3 64 k (k + 1) ≡ p−1 k=0 2k k 3 64 k − S p 1 2 (mod p 3 ). Now, applying (1.1) and Lemma 2.5 yields the result. Lemma 3.3. Let p > 3 be a prime. Then H [ p 3 ] ≡ − 3 2 q p (3) + 3 4 pq p (3) 2 − p p 3 U p−3 (mod p 2 ), H [ 2p 3 ] ≡ − 3 2 q p (3) + 3 4 pq p (3) 2 + 2p p 3 U p−3 (mod p 2 ), H (2) [ p 3 ] ≡ −H (2) [ 2p 3 ] ≡ 3 p 3 U p−3 (mod p). Proof. The first congruence was given in [S3,Theorem 3.2]. By [S3,Theorem 3.2], [2p/3] k=1 (−1) k−1 k ≡ 3p( p 3 )U p−3 (mod p 2 ). Thus, H [ 2p 3 ] = H [ p 3 ] + [2p/3] k=1 (−1) k−1 k ≡ − 3 2 q p (3) + 3 4 pq p (3) 2 + 2p p 3 U p−3 (mod p 2 ). By [S3, Theorem 3.3], H (2) [ p 3 ] ≡ 3 p 3 U p−3 (mod p). To complete the proof, we note that H (2) [ 2p 3 ] = p−1 k=1 1 k 2 − [p/3] k=1 1 (p − k) 2 ≡ −H (2) [ p 3 ] (mod p). Theorem 3.3. Let p be a prime with p > 3. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−2 k=0 2k k 2 3k k 108 k (k + 1) ≡ 4x 2 − 2p (mod p 3 ). For p ≡ 2 (mod 3) we have p−2 k=0 2k k 2 3k k 108 k (k + 1) ≡ −2 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 − 1 2 p 2 (p − 1)/2 (p − 5)/6 −2 (mod p 3 ). Proof. For p = x 2 + 3y 2 ≡ 1 (mod 3) we have 2 3 p = p+2 3 ≡ 1 (mod 2) and 2 3 − 2 3 p = − p 3 . Thus, from Theorem 2.1 and the well known fact (p−1)/2 (p−1)/6 ≡ 2x( x 3 ) (mod p) (see [BEW]) we obtain S p 2 3 ≡ − 2 9 p 2 4 9 (p−1)/2 (p−1)/6 2 ≡ − p 2 8x 2 (mod p 3 ). Taking a = − 1 3 in Theorem 3.1 and then applying the above and (1.2) yields p−2 k=0 2k k 2 3k k 108 k (k + 1) ≡ S p − 1 3 − 2S p 2 3 ≡ 4x 2 − 2p − p 2 4x 2 − 2 − p 2 8x 2 = 4x 2 − 2p (mod p 3 ). Now assume that p ≡ 2 (mod 3). By Lemma 3.3, 2 3 H 2(p+1) 3 + 1 3 H p+1 3 ≡ 2 3 1 2(p + 1)/3 − 3 2 q p (3) + 3 4 pq p (3) 2 + 2p p 3 U p−3 + 1 3 1 (p + 1)/3 − 3 2 q p (3) + 3 4 pq p (3) 2 − p p 3 U p−3 ≡ 2 − 3 2 q p (3) + p − 2 + 3 4 q p (3) 2 − U p−3 (mod p 2 ). From Lemma 3.3 we also have H (2) p+1 3 = 9 (p + 1) 2 + H (2) [ p 3 ] ≡ 9 − 3U p−3 (mod p), H (2) 2(p+1) 3 = 9 4(p + 1) 2 + H (2) [ 2p 3 ] ≡ 9 4 + 3U p−3 (mod p). Set a = − 1 3 . Then a p = 2p−1 3 and a ′ = − 2 3 . From Theorem 3.1 and the above, p−2 k=0 2k k 2 3k k 108 k (k + 1) ≡ −2 (p − 1)/2 (p + 1)/3 2 1 + p 2 3 H 2(p+1) 3 + 1 3 H p+1 3 + 4 3 q p (2) + p 2 2 − 4 3 q p (2) 2 + 2 3 H 2(p+1) 3 + 1 3 H p+1 3 + 4 3 q p (2) 2 − 1 18 H (2) p+1 3 + 10 9 H (2) 2(p+1) 3 − 1 2 p 2 (p − 1)/2 (p + 1)/3 −2 ≡ −2 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 − 2 + 3 4 q p (3) 2 − U p−3 + p 2 2 − 4 3 q p (2) 2 + 2 + 4 3 q p (2) − 3 2 q p (3) 2 − 1 18 (9 − 3U p−3 ) + 10 9 9 4 + 3U p−3 − 1 2 p 2 (p − 1)/2 (p − 5)/6 −2 ≡ −2 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 − 1 2 p 2 (p − 1)/2 (p − 5)/6 −2 (mod p 3 ). This completes the proof. Theorem 3.4. Let p > 3 be a prime. Then p−2 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ 4x 2 − 2p (mod p 3 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), − 1 3 R 2 (p) (mod p 2 ) if p ≡ 5, 7 (mod 8). Proof. Taking a = − 1 4 or − 3 4 in Theorem 3.1 gives p−2 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ S p − 1 4 − 3S p 3 4 ≡ S p − 3 4 − 1 3 S p 1 4 (mod p 3 ). For p = x 2 + 2y 2 ≡ 1 (mod 8) we see that 3 4 p = p+3 4 ≡ 1 (mod 2) and 3 4 − 3 4 p = − p 4 . By Theorem 2.1 and the fact that (p−1)/2 (p−1)/8 ≡ 2(−1) [BEW]), p−1 8 + x−1 2 x (mod p) (seeS p 3 4 ≡ − 1 4 (1 − 1 4 )p 2 ( 3 4 ) 2 (p−1)/2 (p−1)/8 2 ≡ − p 2 12x 2 (mod p 3 ). Recall that S p (− 1 4 ) = p−1 k=0 ( 2k k ) 2 ( 4k 2k ) 256 k ≡ 4x 2 − 2p − p 2 4x 2 (mod p 3 ). We then get p−2 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ S p − 1 4 − 3S p 3 4 ≡ 4x 2 − 2p − p 2 4x 2 − 3 − p 2 12x 2 = 4x 2 − 2p (mod p 3 ). For p = x 2 + 2y 2 ≡ 3 (mod 8) we see that 1 4 p = p+1 4 ≡ 1 (mod 2) and 1 4 − 1 4 p = − p 4 . By Theorem 2.1 and the fact that (p−1)/2 (p−3)/8 ≡ 2(−1) p+5 8 + x−1 2 x (mod p) (see [BEW]), we deduce that S p 1 4 ≡ − 1 4 (1 − 1 4 )p 2 (− 1 4 ) 2 (p−1)/2 (p−3)/8 2 ≡ − 3p 2 4x 2 (mod p 3 ). Recall that S p (− 3 4 ) = p−1 k=0 ( 2k k ) 2 ( 4k 2k ) 256 k ≡ 4x 2 − 2p − p 2 4x 2 (mod p 3 ). We then get p−2 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ S p − 3 4 − 1 3 S p 1 4 ≡ 4x 2 − 2p − p 2 4x 2 − 1 3 − 3p 2 4x 2 = 4x 2 − 2p (mod p 3 ). For p ≡ 5 (mod 8), taking a = − 1 4 , a p = p−1 4 and a ′ = − 1 4 in Theorem 3.1 and then applying the fact that H [ p 4 ] ≡ −3q p (2) (mod p) yields p−2 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ −3 (p − 1)/2 (p + 3)/8 2 1 + p 3 2 H p+3 4 − 1 2 H p+3 8 + 1 2 q p (2) ≡ −3 3p + 1 p + 3 2 (p − 1)/2 (p − 5)/8 2 1 + p 3 2 4 p + 3 − 3q p (2) − 1 2 8 p + 3 + H [ p 8 ] + 1 2 q p (2) ≡ − 1 3 − 16 9 p (p − 1)/2 (p − 5)/8 2 1 + p 2 3 − 4q p (2) − 1 2 H [ p 8 ] ≡ − 1 3 R 2 (p) (mod p 2 ). For p ≡ 7 (mod 8), taking a = − 3 4 , a p = p−3 4 and a ′ = − 1 4 in Theorem 3.1 and then applying the fact that H [ p 4 ] ≡ −3q p (2) (mod p) (see [S12,(2.4)]) yields p−2 k=0 2k k 2 4k 2k 256 k (k + 1) ≡ − 1 3 (p − 1)/2 (p + 1)/8 2 1 + p 3 2 H p+1 4 − 1 2 H p+1 8 + 1 2 q p (2) ≡ − 1 3 · 9 (p − 1)/2 (p − 7)/8 2 1 + p 3 2 4 p + 1 − 3q p (2) − 1 2 8 p + 1 + H [ p 8 ] + 1 2 q p (2) ≡ − 1 3 · 9 (p − 1)/2 (p − 7)/8 2 1 + p 2 − 4q p (2) − 1 2 H [ p 8 ] = − 1 3 R 2 (p) (mod p 2 ). Putting all the above together proves the theorem. Theorem 3.5. Let p be a prime with p > 3. Then p−2 k=0 2k k 3k k 6k 3k 1728 k (k + 1) ≡                  ( p 3 )(4x 2 − 2p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 1 5 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, −5 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. For p = x 2 +4y 2 ≡ 1 (mod 4), taking a = − 1 6 in Theorem 3.1 and then applying Theorem 2.1 yields p−2 k=0 2k k 3k k 6k 3k 1728 k (k + 1) ≡ S p − 1 6 + 1 − 1 6 − 1 6 S p − 1 6 + 1 ≡ S p − 1 6 ≡ p 3 (4x 2 − 2p) (mod p 2 ). For p ≡ 7 (mod 12), taking a = − 1 6 , a p = p−1 6 and a ′ = − 1 6 in Theorem 3.1 and then applying the fact that H [ p 6 ] ≡ −2q p (2) − 3 2 q p (3) (mod p) (see [S12,(2.4)]) yields p−2 k=0 2k k 3k k 6k 3k 1728 k (k + 1) ≡ −5 (p − 1)/2 (p + 5)/12 2 1 + p 5 3 H p+5 6 − 2 3 H p+5 12 + 1 3 q p (2) ≡ − 1 5 (p − 1)/2 [p/12] 2 1 + p 5 3 6 p + 5 − 2q p (2) − 3 2 q p (3) − 2 3 12 p + 5 + H [ p 12 ] + 1 3 q p (2) ≡ − 1 5 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ). For p ≡ 11 (mod 12), taking a = − 5 6 , a p = p−5 6 and a ′ = − 1 6 in Theorem 3.1 and then applying the fact that H [ p 6 ] ≡ −2q p (2) − 3 2 q p (3) (mod p) yields p−2 k=0 2k k 3k k 6k 3k 1728 k (k + 1) ≡ 1 − 5 6 − 5 6 (p − 1)/2 (p + 1)/12 2 1 + p 5 3 H p+1 6 − 2 3 H p+1 12 + 1 3 q p (2) ≡ −5 (p − 1)/2 [p/12] 2 1 + p 5 3 6 p + 1 − 2q p (2) − 3 2 q p (3) − 2 3 12 p + 1 + H [ p 12 ] + 1 3 q p (2) ≡ −5 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ). This completes the proof. Remark 3.1 Let p > 5 be a prime. In the case p ≡ 1 (mod 4), Theorem 3.5 is equivalent to a result due to Z.W. Sun [Su]. In [S14], the author made a conjecture equivalent to p 3 p−2 k=0 2k k 3k k 6k 3k 12 3k (k + 1) ≡ 4x 2 − 2p (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 3 5 R 1 (p) (mod p 2 ) if p ≡ 3 (mod 4). For the conjectures concerning Theorems 3.2-3.4 see [S14,Conjectures 5.4,5.11 and 5.16]. The congruence for p−1 k=0 a k −1−a k 2k k k 4 k modulo p 3 Lemma 4.1. For any positive integer n and real number a we have n−1 k=0 a k −1 − a k 2k k k 4 k − (a(a + 1)S n (a) − (a + 1) 2 S n (a + 1)) = F (a, n)R(a, n), where F (a, n) = a n −1 − a n (n − a(a + 1)) + (a + 1) 2 a + 1 n −2 − a n 2n n 1 4 n , R(a, n) = 2n 3 n 2 − 2(a + 1) 2 n − (a + 1) 2 . Proof. It is easy to check that R(a, 0) = 0 and F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). Thus the result follows from (3.1). Theorem 4.1. Let p be an odd prime, a ∈ Z p , a ≡ −1 (mod p) and a ′ = (a − a p )/p. Then p−1 k=0 a k −1 − a k 2k k k 4 k ≡ a(a + 1)S p (a) − (a + 1) 2 S p (a + 1) + a ′ (a ′ + 1) a + 1 p 3 (mod p 4 ). Moreover, for a ≡ 0 (mod p), p−1 k=0 a k −1 − a k 2k k k 4 k ≡ a(a + 1) 2S p (a) − p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) (mod p 3 ). Proof. Since a p = a p + a ′ p p ≡ a ′ (mod p), −1 − a p = p − 1 − a p − (a ′ + 1)p p ≡ −(a ′ + 1) (mod p), a + 1 p = a p + 1 + a ′ p p ≡ a ′ (mod p), −2 − a p = p − 2 − a p − (a ′ + 1)p p ≡ −(a ′ + 1) (mod p), putting n = p in Lemma 4.1 gives p−1 k=0 a k −1 − a k 2k k k 4 k − (a(a + 1)S p (a) − (a + 1) 2 S p (a + 1)) ≡ − a(a + 1) a p −1 − a p + (a + 1) 2 a + 1 p −2 − a p 2p p 1 4 p · − 2p 3 (a + 1) 2 ≡ (−a(a + 1)a ′ (−a ′ − 1) + (a + 1) 2 a ′ (−a ′ − 1)) − p 3 (a + 1) 2 = a ′ (a ′ + 1) a + 1 p 3 (mod p 4 ). This together with Theorem 3.1 yields p−1 k=0 a k −1 − a k 2k k k 4 k + a(a + 1) p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) ≡ 2a(a + 1)S p (a) (mod p 3 ). This completes the proof. Theorem 4.2. Let p be a prime with p > 3. Then p−1 k=0 k 2k k 3 64 k ≡                  −x 2 + p 2 + p 2 8x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − (p + 1) 2 2 p+1 (p − 1)/2 (p − 3)/4 2 + p 2 4 (p − 1)/2 (p − 3)/4 −2 − p 2 8 (p − 1)/2 (p − 3)/4 2 E p−3 (mod p 3 ) if 4 | p − 3. Proof. Taking a = − 1 2 in Theorem 4.1 yields p−1 k=0 k 2k k 3 64 k ≡ − 1 4 S p − 1 2 + S p 1 2 (mod p 3 ). Now applying (1.1) and Lemma 2.5 gives the result. Theorem 4.3. Let p > 3 be a prime. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−1 k=0 k 2k k 2 3k k 108 k ≡ − 8 9 x 2 + 4 9 p + p 2 9x 2 (mod p 3 ). For p ≡ 2 (mod 3) we have p−1 k=0 k 2k k 2 3k k 108 k ≡ p 2 9 (p − 1)/2 (p − 5)/6 −2 − 4 9 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 (mod p 3 ). Proof. We first assume that p = x 2 + 3y 2 ≡ 1 (mod 3). Taking a = − 1 3 and a ′ = − 1 3 in Theorem 4.1 and then applying (1.2) and Theorem 3.3 yields p−1 k=0 k 2k k 2 3k k 108 k ≡ − 1 3 · 2 3 2 4x 2 − 2p − p 2 4x 2 − 4x 2 − 2p = − 8 9 x 2 + 4 9 p + p 2 9x 2 (mod p 3 ). Now we assume that p ≡ 2 (mod 3). Taking a = − 1 3 and a ′ = − 2 3 in Theorem 4.1 and then applying (1.2) and Theorem 3.3 yields p−1 k=0 k 2k k 2 3k k 108 k ≡ − 1 3 · 2 3 − p 2 (p − 1)/2 (p − 5)/6 −2 + 2 (p − 1)/2 (p − 5)/6 2 × 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 + p 2 2 (p − 1)/2 (p − 5)/6 −2 = p 2 9 (p − 1)/2 (p − 5)/6 −2 − 4 9 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 (mod p 3 ). This completes the proof. Theorem 4.4. Let p be an odd prime. Then p−1 k=0 k 2k k 2 4k 2k 256 k ≡      − 3 4 x 2 + 3 8 p + 3p 2 32x 2 (mod p 3 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), − 1 16 R 2 (p) (mod p 2 ) if p ≡ 5,256 k ≡ − 3 16 2 4x 2 − 2p − p 2 4x 2 − (4x 2 − 2p) = − 3 4 x 2 + 3 8 p + 3p 2 32x 2 (mod p 3 ). For p ≡ 2 (mod 3), from the above and Theorem 3.4 we deduce that p−1 k=0 k 2k k 2 4k 2k 256 k ≡ − 3 16 2 · 0 + 1 3 R 2 (p) = − 1 16 R 2 (p) (mod p 2 ). The proof is now complete. Theorem 4.5. Let p > 3 be a prime. Then p−1 k=0 k 2k k 3k k 6k 3k 12 3k ≡                  ( p 3 )(− 5 9 x 2 + 5 18 p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 1 36 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, − 25 36 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. Taking a = − 1 6 in Theorem 4.1 we get (4.1) p−1 k=0 k 2k k 3k k 6k 3k 12 3k ≡ − 5 36 2 p−1 k=0 2k k 3k k 6k 3k 12 3k − p−2 k=0 2k k 3k k 6k 3k 12 3k (k + 1) (mod p 3 ). For p = x 2 + 4y 2 ≡ 1 (mod 4), applying Theorem 3.5 we get p−1 k=0 k 2k k 3k k 6k 3k 12 3k ≡ − 5 36 2 p 3 (4x 2 − 2p) − p 3 (4x 2 − 2p) = p 3 − 5 9 x 2 + 5 18 p (mod p 2 ). For p ≡ 3 (mod 4) we have p−1 k=0 ( 2k k )( 3k k )( 6k 3k ) 12 3k ≡ 0 (mod p 2 ). Thus, from (4.1) and Theorem 3.5 we deduce the result in this case. = −(2a 2 + 2a + 1)S n (a) − 2(a + 1) 2 S n (a + 1) + F (a, n)R(a, n), where F (a, n) = a n −1 − a n 1 2n − 1 + (2a 2 + 2a + 1) + 2(a + 1) 2 a + 1 n −2 − a n 2n n 1 4 n , R(a, n) = − 2n 3 n 2 + 2(a + 1) 2 n − (a + 1) 2 . Proof. It is easy to check that R(a, 0) = 0 and F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). Thus the result follows from (3.1). Theorem 5.1. Let p be an odd prime, a ∈ Z p , a ≡ −1 (mod p) and a ′ = (a − a p )/p. ≡ −(2a 2 + 2a + 1)S p (a) − 2(a + 1) 2 S p (a + 1) − 2(2a + 1) a ′ (a ′ + 1) a + 1 p 3 (mod p 4 ). Proof. By the proof of Theorem 4.1, a p −1 − a p ≡ a + 1 p −2 − a p ≡ −a ′ (a ′ + 1) (mod p). Taking n = p in Lemma 5.1 and then applying the above yields p−1 k=0 a k −1 − a k 2k k 1 4 k (2k − 1) + (2a 2 + 2a + 1)S p (a) + 2(a + 1) 2 S p (a + 1) ≡ −a ′ (a ′ + 1) 1 2p − 1 + 2a 2 + 2a + 1 + 2(a + 1) 2 2p p 1 4 p · 2p 3 (a + 1) 2 ≡ −a ′ (a ′ + 1)(2a(a + 1) + 2(a + 1) 2 ) 2p − 1 p − 1 1 4 p−1 · p 3 (a + 1) 2 ≡ −2(2a + 1) a ′ (a ′ + 1) a + 1 p 3 (mod p 4 ). This proves the theorem. Corollary 5.1. Let p be an odd prime, a ∈ Z p and a ≡ 0, −1 (mod p). Then p−1 k=0 a k −1 − a k 2k k 1 4 k (2k − 1) ≡ −S p (a) − 2a(a + 1) p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) (mod p 3 ). Proof. This is immediate from Theorems 3.1 and 5.1. Theorem 5.2. Let p be an odd prime. Then p−1 k=0 2k k 3 64 k (2k − 1) 2 ≡          2x 2 − p − p 2 2x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 3(p + 1) 2 2 p p−1 2 p−3 4 2 + 3 4 p 2 p−1 2 p−3 4 2 E p−3 − p 2 2 p−1 2 p−3 4 −2 (mod p 3 ) if 4 | p − 3. Proof. Taking a = 1 2 in Theorem 5.1 and then applying Lemma 2.1 gives p−1 k=0 1 2 k − 3 2 k 2k k 1 4 k (2k − 1) ≡ − 2 · 1 4 + 2 · 1 2 + 1 S p 1 2 − 2 1 2 + 1 2 S p 1 2 + 1 ≡ − 5 2 S p 1 2 − 1 2 S p − 1 2 (mod p 3 ). Since 1/2 k −3/2 k = −(1 + 2 2k−1 ) 2k k 2 1 16 k by (2.6), applying (2.7) and the above we see that 2 p−1 k=0 2k k 3 64 k (2k − 1) 2 = − p−1 k=0 2k k 3 64 k (2k − 1) − p−1 k=0 1 2 k − 3 2 k 2k k 1 4 k (2k − 1) ≡ 1 2 S p 1 2 + S p − 1 2 + 5 2 S p 1 2 + 1 2 S p − 1 2 = 3S p 1 2 + S p − 1 2 (mod p 3 ). Now, applying (1.1) and Lemma 2.5 yields the result. Theorem 5.3. Let p > 3 be a prime. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−1 k=0 2k k 2 3k k 108 k (2k − 1) ≡ − 20 9 x 2 + 10 9 p + p 2 4x 2 (mod p 3 ). For p ≡ 2 (mod 3) we have p−1 k=0 2k k 2 3k k 108 k (2k − 1) ≡ − 8 9 p−1 2 p−5 6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 + 5 18 p 2 p−1 2 p−5 6 −2 (mod p 3 ). Proof. Taking a = − 1 3 in Corollary 5.1 and noting that a ′ = − 1 3 or − 2 3 gives p−1 k=0 2k k 2 3k k 108 k (2k − 1) ≡ − p−1 k=0 2k k 2 3k k 108 k + 4 9 p−2 k=0 2k k 2 3k k 108 k (k + 1) (mod p 3 ).256 k (2k − 1) ≡      − 5 2 x 2 + 5 4 p + p 2 4x 2 (mod p 3 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), − 1 8 R 2 (p) (mod p 2 ) if p ≡ 5, 7 (mod 8). Proof. Taking a = − 1 4 in Corollary 5.1 and noting that a ′ = − 1 4 or − 3 4 gives p−1 k=0 2k k 2 4k 2k 256 k (2k − 1) ≡ − p−1 k=0 2k k 2 4k 2k 256 k + 3 8 p−2 k=0 2k k 2 4k 2k 256 k (k + 1) (mod p 3 ).12 3k (2k − 1) ≡                  ( p 3 )(− 26 9 x 2 + 13 9 p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 1 18 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, − 25 18 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. Taking a = − 1 6 in Corollary 5.1 and noting that a ′ = − 1 6 or − 5 6 gives p−1 k=0 2k k 3k k 6k 3k 12 3k (2k − 1) ≡ − p−1 k=0 2k k 3k k 6k 3k 12 3k + 5 18 p−2 k=0 2k k 3k k 6k 3k 12 3k (k + 1) (mod p 3 ). Now, applying Theorem 3.5 yields the result. Remark 5.1 For the conjectures concerning Theorems 5.2-5.5 see [S14,Conjectures 5.4,5.7,5.11 and 5.16]. The congruence for p−1 k=0 a k −1−a k 2k k k 2 4 k modulo p 3 Lemma 6.1. For any positive integer n and real number a we have n−1 k=0 a k −1 − a k 2k k 1 4 k 3k 2 − (2a 2 + 2a − 1)k − a(a + 1) = 2n 3 a n −1 − a n 2n n 1 4 n . Proof. Set F (a, k) = a k −1 − a k 2k k 1 4 k 3k 2 − (2a 2 + 2a − 1)k − a(a + 1) , G(a, k) = 2k 3 a k −1 − a k 2k k 1 4 k . It is easy to check that G(a, 0) = 0 and F (a, k) = G(a, k+1)−G(a, k). Thus n−1 k=0 F (a, k) = G(a, n) − G(a, 0) = G(a, n). Theorem 6.1. Let p be an odd prime, a ∈ Z p and a ′ = (a − a p )/p. Then p−1 k=0 a k −1 − a k 2k k 1 4 k 3k 2 − (2a 2 + 2a − 1)k − a(a + 1) ≡ −a ′ (a ′ + 1)p 3 (mod p 4 ). Proof. Taking n = p in Lemma 6.1 and noting that a p ≡ a ′ (mod p), −1−a p ≡ −(a ′ + 1) (mod p) and 2p p 1 4 p = 1 2·4 p−1 2p−1 p−1 ≡ 1 2 (mod p) yields the result. From Theorems 4.1 and 6.1 we deduce the following result. Theorem 6.2. Let p be an odd prime, a ∈ Z p , a ≡ 0, −1 (mod p) and a ′ = (a− a p )/p. Then p−1 k=0 a k −1 − a k 2k k k 2 4 k ≡ 2 3 a 2 (a + 1) 2 S p (a) − 2a(a + 1) − 1 3 (a + 1) 2 S p (a + 1) + (2a 2 + a − 2)a ′ (a ′ + 1) 3(a + 1) p 3 (mod p 4 ). Theorem 6.3. Let p be a prime with p > 3. Then p−1 k=0 k 2 2k k 3 64 k ≡                  1 6 x 2 − p 12 − p 2 24x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), (p + 1) 2 2 p+2 (p − 1)/2 (p − 3)/4 2 − p 2 24 (p − 1)/2 (p − 3)/4 −2 + p 2 16 (p − 1)/2 (p − 3)/4 2 E p−3 (mod p 3 ) if 4 | p − 3. Proof. Taking a = − 1 2 in Theorem 6.1 gives p−1 k=0 k 2 2k k 3 64 k ≡ − 1 2 p−1 k=0 k 2k k 3 64 k − 1 12 p−1 k=0 2k k 3 64 k (mod p 3 ). Now, appealing to (1.1) and Theorem 4.2 we deduce the result. Theorem 6.4. Let p > 3 be a prime. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−1 k=0 k 2 2k k 2 3k k 108 k ≡ 32 243 x 2 − 16 243 p − 17p 2 486x 2 (mod p 3 ). For p ≡ 2 (mod 3) we have p−1 k=0 k 2 2k k 2 3k k 108 k ≡ − 4 243 p 2 (p − 1)/2 (p − 5)/6 −2 + 52 243 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 (mod p 3 ). Proof. Taking a = − 1 3 in Theorem 6.1 yields p−1 k=0 k 2 2k k 2 3k k 108 k ≡ − 13 27 p−1 k=0 k 2k k 2 3k k 108 k − 2 27 p−1 k=0 2k k 2 3k k 108 k (mod p 3 ). Now applying (1.2) and Theorem 4.3 deduces the result. Theorem 6.5. Let p be a prime with p > 3. Then p−1 k=0 k 2 2k k 2 4k 2k 256 k ≡      3 32 x 2 − 3 64 p − 7p 2 256x 2 (mod p 3 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), 11 384 R 2 (p) (mod p 2 ) if p ≡ 5, 7 (mod 8). Proof. Taking a = − 1 4 in Theorem 6.1 yields p−1 k=0 k 2 2k k 2 4k 2k 256 k ≡ − 11 24 p−1 k=0 k 2k k 2 4k 2k 256 k − 1 16 p−1 k=0 2k k 2 4k 2k 256 k (mod p 3 ). This together with (1.3) and Theorem 4.4 deduces the result. Theorem 6.6. Let p > 3 be a prime. Then p−1 k=0 k 2 2k k 3k k 6k 3k 12 3k ≡                  ( p 3 )( 25 486 x 2 − 25 972 p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 23 1944 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, 575 1944 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p Proof . Taking a = − 1 6 in Theorem 6.1 we get p−1 k=0 k 2 2k k 3k k 6k 3k 12 3k ≡ − 23 54 p−1 k=0 k 2k k 3k k 6k 3k 12 3k − 5 108 p−1 k=0 2k k 3k k 6k 3k 12 3k (mod p 3 ). This together with Theorem 4.5 yields the result. Remark 6.1 In [S4] the author established the congruence for p−1 k=0 k 2 ( 2k k ) 3 64 k modulo p 2 , where p is a prime with p > 5. For the conjectures concerning Theorems 6.4-6.6 see [S13,Conjectures 2.7,2.11 and 2.16]. The congruence for p−1 k=0 a k −1−a k 2k k k 3 4 k modulo p 3 Lemma 7.1. For any positive integer n and real number a we have n−1 k=0 a k −1 − a k 2k k 4 k 15k 3 − (4a 2 (a + 1) 2 − a(a + 1) + 1)k − a(a + 1)(2a(a + 1) − 1) = 2n 3 (3n + 2a(a + 1) − 4) a n −1 − a n 2n n 1 4 n . Proof. Set F (a, k) = a k −1 − a k 2k k 1 4 k 15k 3 − (4a 2 (a + 1) 2 − a(a + 1) + 1)k − a(a + 1)(2a(a + 1) − 1) , G(a, k) = 2k 3 (3k + 2a(a + 1) − 4) a k −1 − a k 2k k 1 4 k . It is easy to check that G(a, 0) = 0 and F (a, k) = G(a, k+1)−G(a, k). Thus n−1 k=0 F (a, k) = G(a, n) − G(a, 0) = G(a, n). Theorem 7.1. Let p be an odd prime, a ∈ Z p and a ′ = (a − a p )/p. Then p−1 k=0 k 3 a k −1 − a k 2k k 4 k × 15k 3 − (4a 2 (a + 1) 2 − a(a + 1) + 1)k − a(a + 1)(2a(a + 1) − 1) ≡ (4 − 2a(a + 1))a ′ (a ′ + 1)p 3 (mod p 4 ). Proof. Taking n = p in Lemma 7.1 and noting that a p ≡ a ′ (mod p), −1−a p ≡ −(a ′ + 1) (mod p) and 2p p 1 4 p = 1 2·4 p−1 2p−1 p−1 ≡ 1 2 (mod p) yields the result. Theorem 7.2. Let p > 5 be a prime, a ∈ Z p , a ≡ −1 (mod p) and a ′ = (a − a p )/p. Then p−1 k=0 k 3 a k −1 − a k 2k k 1 4 k ≡ a 2 (a + 1) 2 (2a + 1) 2 15 S p (a) − (a + 1) 2 (4a 2 (a + 1) 2 − a(a + 1) + 1) 15 S p (a + 1) + 4a 4 + 6a 3 − a 2 + a + 5 15(a + 1) a ′ (a ′ + 1)p 3 (mod p 4 ). Proof. This is immediate from Theorems 4.1 and 7.1. Theorem 7.3. Let p be a prime with p > 5. Then p−1 k=0 k 3 2k k 3 64 k ≡          p 2 160x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − (p + 1) 2 40 · 2 p−1 p−1 2 p−3 4 2 − p 2 80 p−1 2 p−3 4 2 E p−3 (mod p 3 ) if 4 | p − 3. Proof. Putting a = − 1 2 in Theorem 7.1 gives Remark 7.1 Theorem 7.3 can be easily deduced from a result due to Tauraso [T3], although he did not give the details of its proof. Theorem 7.4. Let p > 5 be a prime. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−1 k=0 k 3 2k k 2 3k k 108 k ≡ 1 10935 16x 2 − 8p + 113p 2 2x 2 (mod p 3 ). For p ≡ 2 (mod 3) we have p−1 k=0 k 3 2k k 2 3k k 108 k ≡ − 2 10935 p 2 (p − 1)/2 (p − 5)/6 −2 − 92 2187 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 (mod p 3 ). Proof. Putting a = − 1 3 in Theorem 7.1 we obtain k 3 2k k 3k k 6k 3k 12 3k ≡                  ( p 3 )( 5 2187 x 2 − 5 4374 p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 17 6912 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, − 425 6912 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. Putting a = − 1 6 in Theorem 7.1 we deduce that p−1 k=0 k 3 2k k 3k k 6k 3k 12 3k ≡ 197 2430 p−1 k=0 k 2k k 3k k 6k 3k 12 3k + 23 1944 p−1 k=0 2k k 3k k 6k 3k 12 3k (mod p 3 ). Now applying Theorem 4.5 yields the result. Remark 7.2 In [S4] the author established the congruence for p−1 k=0 k 3 ( 2k k ) 3 64 k modulo p 2 , where p is a prime with p > 5. See also [T3]. For the conjectures concerning Theorems 7.4-7.6 see [S13, Conjectures 2.7, 2.11 and 2.16]. The congruence for 1 4 k 1 k + 2 − 1 3(k + 1) + 1 3(a − 1)(a + 2) = F (a, n)R(a, n), where F (a, n) = a n −1 − a n 2n n 1 4 n 1 n + 2 − 1 3(n + 1) + 1 3(a − 1)(a + 2) , R(a, n) = − 2n 2 (n + 2) n 2 + (2a(a + 1) − 1)n + a(a + 1) . Proof. It is easy to check that R(a, 0) = 0 and F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). Thus the result follows from (3.1). Lemma 8.2. Let p > 3 be a prime, a ∈ Z p , a ≡ 0, ±1, −2 (mod p) and a ′ = (a − a p )/p. Then a p−2 −1−a p−2 2(p−2) p−2 4 p−2 · p ≡ 2a ′ (a ′ + 1) 3a(a + 1)(a − 1)(a + 2) p 2 (mod p 3 ). Proof. It is clear that a p − 2 = a(p − 1) (a + 1 − p)(a + 2 − p) a − 1 p − 1 , −1 − a p − 2 = p − 1 1 − a − p −1 − a p − 1 , 2(p − 2) p − 2 = p(p − 1) 2 (2p − 1)(2p − 2)(2p − 3) 2p − 1 p − 1 . Thus, applying [S9, Lemma 2.2] we get a p−2 −1−a p−2 2(p−2) p−2 4 p−2 · p = a(p − 1) (a + 1 − p)(a + 2 − p) · p − 1 1 − a − p a − 1 p − 1 −1 − a p − 1 × 4(p − 1) 2 (2p − 1)(2p − 2)(2p − 3) · 4 p−1 2p − 1 p − 1 ≡ a (a + 1)(a + 2)(1 − a) · a ′ (a ′ + 1) a 2 p 2 · 4 −6 = 2a ′ (a ′ + 1) 3a(a + 1)(a − 1)(a + 2) p 2 (mod p 3 ). This proves the lemma. Theorem 8.1. Let p > 3 be a prime, a ∈ Z p and a ≡ 0, ±1, −2 (mod p). Then p−3 k=0 a k −1 − a k 2k k 1 4 k (k + 2) ≡ 1 3 p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − 1 3(a − 1)(a + 2) S p (a) ≡ a 2 + a − 3 3(a − 1)(a + 2) S p (a) + a + 1 3a S p (a + 1) (mod p 3 ). Proof. Set a ′ = (a − a p )/p. Taking n = p in Lemma 8.1 and noting that a p −1−a p ≡ −a ′ (a ′ + 1) (mod p) and 2p p 1 4 p ≡ 1 2 (mod p) yields p−1 k=0 a k −1 − a k 2k k 1 4 k 1 k + 2 − 1 3(k + 1) + 1 3(a − 1)(a + 2) ≡ a ′ (a ′ + 1) 3(a − 1)(a + 2) p 2 (mod p 3 ). Appealing to Lemmas 3.2, 8.2 and Theorem 3.1, p−3 k=0 a k −1 − a k 2k k 1 4 k (k + 2) ≡ p−1 k=0 a k −1 − a k 2k k 1 4 k (k + 2) − 2a ′ (a ′ + 1) 3a(a + 1)(a − 1)(a + 2) p 2 ≡ 1 3 p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − a ′ (a ′ + 1) a(a + 1) p 2 − 1 3(a − 1)(a + 2) S p (a) + a ′ (a ′ + 1) 3(a − 1)(a + 2) p 2 − 2a ′ (a ′ + 1) 3a(a + 1)(a − 1)(a + 2) p 2 = 1 3 p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − 1 3(a − 1)(a + 2) S p (a) ≡ 1 3 S p (a) + a + 1 a S p (a + 1) − 1 3(a − 1)(a + 2) S p (a) (mod p 3 ). This yields the result. Theorem 8.2. Let p > 3 be a prime. Then p−1 2 k=0 2k k 3 64 k (k + 2) ≡                  52 27 x 2 − 26 27 p − p 2 27x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − (p + 1) 2 3 · 2 p−1 (p − 1)/2 (p − 3)/4 2 − p 2 13 27 (p − 1)/2 (p − 3)/4 −2 + 1 6 (p − 1)/2 (p − 3)/4 2 E p−3 (mod p 3 ) if p ≡ 3 (mod 4). Proof. Since p | 2k k for p 2 < k < p, taking a = − 1 2 in Theorem 8.1 gives 108 k (k + 2) ≡ 29 15 x 2 − 29 30 p − 3p 2 80x 2 (mod p 3 ). For p ≡ 2 (mod 3) we have p−3 k=0 2k k 2 3k k 108 k (k + 2) ≡ − 2 3 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 − 29 120 p 2 (p − 1)/2 (p − 5)/6 −2 (mod p 3 ). Proof. Set a = − 1 3 . Then 3(a − 1)(a + 2) = − 20 3 . By Theorem 8. 256 k (k + 2) ≡      17 35 (4x 2 − 2p) − 4 105 p 2 (mod p 3 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), − 1 9 R 2 (p) (mod p 2 ) if p ≡ 5, 7 (mod 8). Proof. Set a = − 1 4 . Then 3(a − 1)(a + 2) = − 105 16 . By Theorem 8. 1728 k (k + 2) ≡                  113 231 ( p 3 )(4x 2 − 2p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 1 15 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, − 5 3 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. Set a = − 1 6 . Then 3(a − 1)(a + 2) = − 77 12 . By Theorem 8.1, where F (a, n) = a n −1 − a n p−3 k=0 2k k 3k k 6k 3k 1728 k (k + 2) ≡ 1 3 p−2 k=0 2k k 3k k1 (n + 1) 2 − 2 − 2a 2 + 2a − 1 a 2 a + 1 n −2 − a n 2n n 1 4 n , R(a, n) = − 2(n + 1) 2 ((2a − 1)(n + 1) 2 + (2 − 3a)(n + 1) + a − 1) (2a − 1)(n + 1) 3 + (4a 3 + 6a 2 − a)(n + 1) 2 + a 2 (n + 1) − a 2 (a + 2) . Proof. It is easy to check that R(a, 0) = 0 and F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). Thus the result follows from (3.1). Theorem 9.1. Let p > 3 be a prime, a ∈ Z p , a ≡ 0, −1, −2 (mod p) and a ′ = (a − a p )/p. Then p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 2 ≡ 2S p (a) + 2a 2 + 2a − 1 a 2 S p (a + 1) + 4a 3 + 6a 2 − 3a + 2 a 3 (a + 1)(a + 2) a ′ (a ′ + 1)p 3 (mod p 4 ). Proof. Using [S9,Lemma 2.2] we see that (9.1) a p − 1 −1 − a p − 1 = a a + 1 − p a − 1 p − 1 −a − 1 p − 1 ≡ a ′ (a ′ + 1) a(a + 1) p 2 (mod p 3 ). Also, a+1 p−1 −2−a p−1 ≡ a p +1 p p−2− a p p−1 ≡ 0 (mod p) and 2(p−1) p−1 1 4 p−1 = p 2p−1 2p−1 p−1 1 4 p−1 ≡ −p (mod p 2 ) . Thus, taking n = p − 1 in Lemma 9.1 and then applying the above and Lemma 3.2 we see that p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 2 ≡ 2S p−1 (a) + 2a 2 + 2a − 1 a 2 S p−1 (a + 1) + a ′ (a ′ + 1) a(a + 1) · (−p) · 2(a − 1)p 2 a 2 (a + 2) ≡ 2S p (a) + 2 a ′ (a ′ + 1) a(a + 1) p 3 + 2a 2 + 2a − 1 a 2 S p (a + 1) + 2a 2 + 2a − 1 a 2 · a ′ (a ′ + 1) (a + 1)(a + 2) p 3 − 2(a − 1)a ′ (a ′ + 1) a 3 (a + 1)(a + 2) p 3 = 2S p (a) + 2a 2 + 2a − 1 a 2 S p (a + 1) + 4a 3 + 6a 2 − 3a + 2 a 3 (a + 1)(a + 2) a ′ (a ′ + 1)p 3 (mod p 4 ). This proves the theorem. Theorem 9.2. Let p > 3 be a prime, a ∈ Z p and a ≡ 0, −1, −2 (mod p). Then p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 2 ≡ 2 − 1 a(a + 1) p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) + 1 a(a + 1) S p (a) (mod p 3 ). Proof. By Theorems 9.1 and 3.1, p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 2 ≡ 2S p (a) + 2a 2 + 2a − 1 a 2 S p (a + 1) ≡ 2S p (a) + 2a 2 + 2a − 1 a 2 · a a + 1 p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − S p (a) = 2a(a + 1) − 1 a(a + 1) p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) + 1 a(a + 1) S p (a) (mod p 3 ). This proves the theorem. Theorem 9.3. Let p > 3 be a prime. Then p−1 2 k=0 2k k 3 64 k (k + 1) 2 ≡                  8x 2 − 4p + p 2 x 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 6(p + 1) 2 2 p−1 (p − 1)/2 (p − 3)/4 2 − 2p 2 (p − 1)/2 (p − 3)/4 −2 −3p 2 (p − 1)/2 (p − 3)/4 2 E p−3 (mod p 3 ) if 4 | p − 3. Proof. Since p | 2k k for p 2 < k < p, taking a = − 1 2 in Theorem 9.2 gives p−1 2 k=0 2k k 3 64 k (k + 1) 2 ≡ 6 p−1 2 k=0 2k k 3 64 k (k + 1) − 4 p−1 2 k=0 2k k 3 64 k (mod p 3 ). Now applying (1.1) and Theorem 3.2 yields the result. Theorem 9.4. Let p be a prime with p > 3. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−2 k=0 2k k 2 3k k 108 k (k + 1) 2 ≡ 8x 2 − 4p + 9p 2 8x 2 (mod p 3 ). For p ≡ 2 (mod 3) we have p−2 k=0 2k k 2 3k k 108 k (k + 1) 2 ≡ −13 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 − p 2 (p − 1)/2 (p − 5)/6 −2 (mod p 3 ). Proof. Taking 256 k (k + 1) 2 ≡      8x 2 − 4p + 4p 2 3x 2 (mod p 3 ) if p = x 2 + 2y 2 ≡ 1, 3 (mod 8), − 22 9 R 2 (p) (mod p 2 ) if p ≡ 5, 7 (mod 8). Proof. Taking 1728 k (k + 1) 2 ≡                  ( p 3 )(8x 2 − 4p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), − 46 25 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, −46 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. Taking 64 k (k+1) 2 modulo p 2 can be deduced from [T3,(4)]. In [S14] the author conjectured the congruences modulo p 2 for the sums in Theorem 9.4-9.6. = − 2 a(a + 1) + (2a + 1) 2 a(a + 1) S n (a) + 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) S n (a + 1) + F (a, n)R(a, n), where F (a, n) = a n −1 − a n 1 (n + 1) 3 − (2a + 1) 2 a(a + 1) 2n n 1 4 n − 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) a + 1 n −2 − a n 2n n 1 4 n , R(a, n) = − 2(n + 1) 3 ((4a 2 − 2a + 1)(n + 1) 2 + (−6a 2 + 3a − 2)(n + 1) + 3a 2 − a + 1) (4a 2 − 2a + 1)(n + 1) 4 + (8a 4 + 12a 3 + a)(n + 1) 3 + a 3 (n + 1) − a 3 (a + 2) . Proof. Using Maple it is easy to check that F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). On the other hand, F (a, 0)R(a, 0) = 1 − (2a + 1) 2 a(a + 1) − 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) × − 2a 2 7a 4 + 11a 3 + 4a 2 − a + 1 = 2 a(a + 1) . Thus, n−1 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 3 − (2a + 1) 2 a(a + 1) S n (a) − 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) S n (a + 1) = n−1 k=0 F (a, k) = n−1 k=0 (F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k)) = F (a, n)R(a, n) − F (a, 0)R(a, 0) = F (a, n)R(a, n) − 2 a(a + 1) . This proves the lemma. Theorem 10.1. Let p > 3 be a prime, a ∈ Z p and a ≡ 0, −1, −2 (mod p). Then p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 3 ≡ − 2 a(a + 1) + (2a + 1) 2 a(a + 1) S p (a) + 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) S p (a + 1) ≡ − 2 a(a + 1) + 4a 2 (a + 1) 2 − a(a + 1) + 1 a 2 (a + 1) 2 p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) + 2a(a + 1) − 1 a 2 (a + 1) 2 S p (a) (mod p 3 ). Proof. Clearly a p−1 −1−a p−1 ≡ 0 (mod p 2 ) and p | 2(p−1) p−1 . Hence, taking n = p − 1 in Lemma 10.1 yields p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 3 ≡ − 2 a(a + 1) + (2a + 1) 2 a(a + 1) S p−1 (a) + 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) S p−1 (a + 1). By Lemma 3.2, S p−1 (a) ≡ S p (a) (mod p 3 ) and S p−1 (a + 1) ≡ S p (a + 1) (mod p 3 ). Now, from the above and Theorem 3.1 we get p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) 3 ≡ − 2 a(a + 1) + (2a + 1) 2 a(a + 1) S p (a) + 4a 2 (a + 1) 2 − a(a + 1) + 1 a 3 (a + 1) × a a + 1 p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − S p (a) (mod p 3 ). This yields the result. Theorem 10.2. Let p be a prime with p > 3. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we have p−2 k=0 2k k 2 3k k 108 k (k + 1) 3 ≡ 9 − 2x 2 + p + 117p 2 16x 2 (mod p 3 ).1728 k (k + 1) 3 ≡                  72 5 − 16 5 ( p 3 )(4x 2 − 2p) (mod p 2 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), 72 5 − 1576 125 p−1 2 [ p 12 ] 2 1 + p 2 5 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 7, 72 5 − 1576 5 p−1 2 [ p 12 ] 2 1 + p 2 − 3q p (2) − 5 2 q p (3) − 2 3 H [ p 12 ] (mod p 2 ) if 12 | p − 11. Proof. Putting a = − 1 6 in Theorem 10.1 gives 3a(a + 1) − 10 15(a − 1)(a + 2) S n (a) + (a + 1)(3a(a + 1) − 16) 15a(a − 2)(a + 3) S n (a + 1) + G(a, n), p−2 k=0 2k k 3k k where G(a, n) = 2n 2 N (a, n) 15a(a 2 − 1)(a 2 − 4)(a + 3)(n + 1)(n + 2) a + 1 n −1 − a n 2n n 1 4 n and N (a, n) = (3a 4 + 12a 3 − 13a 2 − 50a + 32)n 2 + (3a 4 + 12a 3 − 29a 2 − 82a + 96)n + 64. Proof. Set F (a, n) = a n −1 − a n 2n n 1 4 n 1 n + 3 − 3a(a + 1) − 10 15(a − 1)(a + 2) − (a + 1)(3a(a + 1) − 16) 15a(a − 2)(a + 3) · a + 1 + n a + 1 − n . Using Maple it is easy to check that F (a, k) = G(a, k+1)−G(a, k). Note that a+1 n −2−a n = a+1+n a+1−n a n −1−a n . We then have n−1 k=0 a k −1 − a k 2k k 1 4 k (k + 3) − 3a(a + 1) − 10 15(a − 1)(a + 2) S n (a) + (a + 1)(3a(a + 1) − 16) 15a(a − 2)(a + 3) S n (a + 1) = n−1 k=0 F (a, k) = n−1 k=0 (G(a, k + 1) − G(a, k)) = G(a, n) − G(a, 0) = G(a, n). This proves the lemma. Lemma 11.2. Let p > 5 be a prime, a ∈ Z p , a ≡ 0, ±1, ±2, −3 (mod p) and a ′ = (a − a p )/p. Then p−3 4 p−3 · p ≡ − 32a ′ (a ′ + 1)p 2 15a(a 2 − 1)(a 2 − 4)(a + 3) (mod p 3 ). Proof. It is easy to see that a p − 3 −1 − a p − 3 = (p − 2) 2 (a + 3 − p)(2 − a − p) a p − 2 −1 − a p − 2 2(p − 3) p − 3 = (p − 2) 2 (2p − 4)(2p − 5) 2(p − 2) p − 2 . Thus, appealing to Lemma 8.2 we obtain a p−3 −1−a p−3 2(p−3) p−3 4 p−3 · p = 4(p − 2) 4 (a + 3 − p)(2 − a − p)(2p − 4)(2p − 5) · a p−2 −1−a p−2 2(p−2) p−2 4 p−2 · p ≡ 16 5(a + 3)(2 − a) · 2a ′ (a ′ + 1) 3a(a + 1)(a − 1)(a + 2) p 2 = − 32a ′ (a ′ + 1)p 2 15(a − 2)(a − 1)a(a + 1)(a + 2)(a + 3) (mod p 3 ). This proves the lemma. Theorem 11.1. Let p > 5 be a prime and a ∈ Z p with a ≡ 0, ±1, ±2, −3 (mod p). Then 2k k 1 4 k (k + 3) + 32a ′ (a ′ + 1)p 2 15a(a 2 − 1)(a 2 − 4)(a + 3) = 3a(a + 1) − 10 15(a − 1)(a + 2) S p (a) + (a + 1)(3a(a + 1) − 16) 15a(a − 2)(a + 3) S p (a + 1) + G(a, p) + 32a ′ (a ′ + 1)p 2 15a(a 2 − 1)(a 2 − 4)(a + 3) ≡ 3a(a + 1) − 10 15(a − 1)(a + 2) S p (a) + (a + 1)(3a(a + 1) − 16) 15a(a − 2)(a + 3) S p (a + 1) ≡ 3a(a + 1) − 10 15(a − 1)(a + 2) S p (a) + 3a(a + 1) − 16 15(a − 2)(a + 3) p−2 k=0 a k −1 − a k 2k k 1 4 k (k + 1) − S p (a) (mod p 3 ), which yields the result. Theorem 11.2. Let p > 5 be a prime. Then − 3q p (3) − 2q p (2)q p (3) + = (a − 1) 2 + 1 2(a − 1) 3 S n (a) + 1 2(a − 1) S n (a − 1) + F (a, n)R(a, n), where F (a, n) = a n −1 − a n 1 a − 1 + n − (a − 1) 2 + 1 2(a − 1) 3 − 1 2(a − 1) a − 1 n −a n 2n n 1 4 n , R(a, n) = − 2n 3 n 2 + (2a 2 − 2a + 1)n + a(a − 1) . Proof. It is easy to check that R(a, 0) = 0 and F (a, k) = F (a, k + 1)R(a, k + 1) − F (a, k)R(a, k). Thus the result follows from (3.1). Theorem 12.1. Let p be an odd prime and a ∈ Z p with a ≡ 0, ±1 (mod p). Then On the other hand, taking n = p in Lemma 12.1 and then applying Theorem 3.1 yields the remaining part. ( 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 (mod 4), mod p 3 )if p ≡ 3 (mod 4). 2 (mod p 3 ) if p = x 2 + 4y 2 ≡ 1 ( 2 , where p is a prime with p > 5. For the conjectures concerning Theorems 4.3-4.5 see[S13, Conjectures 2.7, 2.11 and 2.16]. For any positive integer n and real number a we have . For any positive integer n and real number a = 1, (mod p 3 ). = For any positive integer n and real number a = 0 we have 2S n (a) + 2a 2 + 2a − 1 a 2 S n (a + 1) + F (a, n)R(a, n), For any positive integer n and real number a = 0, Theorem 3.5 yields the result. Theorem 10.5. Let p be an odd prime. Then k (mod p 3 ). Now applying (1.1) and Theorem 3.2 yields the result. For any positive integer n and real number a = 0, ±1, ±2, Set a ′ = (a − a p )/p. From Lemmas 3. = S p (a) + S p (a − 1) ≡ S p p (a) (mod p 3 ). Theorem 9.5. Let p > 3 be a prime. Thena = − 1 3 in Theorem 9.2 yields p−2 k=0 2k k 2 3k k 108 k (k + 1) 2 ≡ 13 2 p−2 k=0 2k k 2 3k k 108 k (k + 1) − 9 2 p−1 k=0 2k k 2 3k k 108 k (mod p 3 ). Now applying (1.2) and Theorem 3.3 yields the result. p−2 k=0 2k k 2 4k 2k Let p be a prime with p > 3. Thena = − 1 4 in Theorem 9.2 yields p−2 k=0 2k k 2 4k 2k 256 k (k + 1) 2 ≡ 22 3 p−2 k=0 2k k 2 4k 2k 256 k (k + 1) − 16 3 p−1 k=0 2k k 2 4k 2k 256 k (mod p 3 ). Now applying (1.3) and Theorem 3.4 yields the result. Theorem 9.6. p−2 k=0 2k k 3k k 6k 3k Now applying Theorem 3.5 yields the result.Remark 9.1 Let p > 5 be a prime. The congruence fora = − 1 6 in Theorem 9.2 yields p−2 k=0 2k k 3k k 6k 3k 1728 k (k + 1) 2 ≡ 46 5 p−2 k=0 2k k 3k k 6k 3k 1728 k (k + 1) − 36 5 p−1 k=0 2k k 3k k 6k 3k 1728 k (mod p 3 ). p−1 2 k=0 ( 2k k ) 3 For p ≡ 2 (mod 3) we have Theorem 10.4. Let p be a prime with p > 5. Thenp−2 k=0 2k k p−2 k=0 2k k 3k k 6k 3k ] (mod p 2 ) if 12 | p − 11. 3k k k (k + 1) (mod p 3 ). Now applying Theorem 3.5 deduces the result.12. The congruence for p−1 k=0 a k −1−a k 2k k 1 4 k (a−1+k) modulo p 3 Lemma 12.1. For any positive integer n and real number a = 0, 1 we have n−1 k=0 a k −1 − a k 2k k 1 4 k (a − 1 + k) 16 k (k + 1) 3 ≡ 2 − 32R 3 (p) − 4p (mod p 2 ).Conjecture 13.3. Let p be a prime with p > 5. 108 k (k + 1) 3 ≡ 9 − 115 2 (p − 1)/2 (p − 5)/6 2 1 + p 2 + 4 3 q p (2) − 3 2 q p (3) + p 2 1 + 8 3 q p (2) + 2 9 q p (2) 2 − 3q p (3) − 2q p (2)q p (3) + 15 8 q p (3) 2 + 3 4 U p−3 + 1 4 p 2 (p − 1)/2 (p − 5)/6 − 32a ′ (a ′ + 1)p 2 15a(a 2 − 1)(a 2 − 4)(a + 3) (mod p 3 ).Let G(a, n) be given in Lemma 11.1. Thenwe see thatProof. Since p | 2k k for p 2 < k < p, taking a = − 1 2 in Theorem 11.1 givesNow applying (1.1) and Theorem 3.2 yields the result. Theorem 11.3. Let p be a prime with p > 7. For p ≡ 1 (mod 3) and so p = x 2 + 3y 2 we haveProof.TakingProof.TakingProof. Taking a = − 1 6 in Theorem 11.1 givesTheorem 12.2. Let p > 3 be a prime. ThenProof. Taking a = − 1 2 in Theorem 12.1 yieldsThis together with (1.1) and Theorem 3.2 yields the result.Some challenging conjecturesCalculations by Maple suggests the following conjectures: Conjecture 13.1. Let p be a prime with p > 7.(i) If p ≡ 1, 2, 4 (mod 7) and so p = x 2 + 7y 2 , then(ii) If p ≡ 3, 5, 6 (mod 7), thenConjecture 13.2. Let p be a prime with p > 5.(i) If p ≡ 1 (mod 3) and so p = x 2 + 3y 2 , theny 2 + 3958 16875 p + 169p 2 33750y 2 (mod p 3 ),Conjecture 13.4. Let p be a prime with p > 5. (i) If p ≡ 1, 3 (mod 8) and so p = x 2 + 2y 2 , then(ii) If p ≡ 5, 7 (mod 8), thenConjecture 13.5. Let p be a prime with p > 5.(i) If p ≡ 1 (mod 4) and so p = x 2 + 4y 2 , then(ii) If p ≡ 3 (mod 4), thenConjecture 13.6. Let p be a prime with p > 5.(i) If p ≡ 1 (mod 3) and so p = x 2 + 3y 2 , then(ii) If p ≡ 2 (mod 3), thenConjecture 13.7. Let p be a prime with p > 7.(i) If p ≡ 1, 2, 4 (mod 7) and so p = x 2 + 7y 2 , then(ii) If p ≡ 3, 5, 6 (mod 7), thenConjecture 13.8. Let p > 7 be a prime. Then (−144) k (k + 1) 2 ≡ − 40 3 y 2 + 2 3 p − 13p 2 18y 2 (mod p 3 ) for p = x 2 + 3y 2 ≡ 1 (mod 3).Remark 13.1 There are many conjectures similar to Conjectures 13.1-13.9. One may consult[S14,] and make similar conjectures by Maple. Gaussian hypergeometric series and combinatorial congruences. S Ahlgren, Symbolic computation, number theory, special functions, physics and combinatorics. Gainesville, FI; Kluwer, Dordrecht4S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, in: Symbolic computation, number theory, special functions, physics and combina- torics (Gainesville, FI, 1999), pp. 1-12, Dev. Math., Vol. 4, Kluwer, Dordrecht, 2001. Another congruence for the Apéry numbers. F Beukers, J. Number Theory. 25F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25(1987), 201-210. B C Berndt, R J Evans, K S Williams, Gauss and Jacobi Sums. New YorkWileyB.C. Berndt, R.J. Evans and K.S. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998. Some q-analogues of supercongruences for truncated 3 F 2 hypergeometric series. V J W Guo, 10.1007/s11139-021-00478-9Ramanujan J. V.J.W. Guo, Some q-analogues of supercongruences for truncated 3 F 2 hypergeo- metric series, Ramanujan J., 2021, https://doi.org/10.1007/s11139-021-00478-9. Proof of a conjecture of Beukers on Apéry numbers. L Van Hamme, Proceedings of the Conference on p-adic Analysis. N. De Grande-De Kimpe and L. van Hamme, ed., Houthalenthe Conference on p-adic AnalysisBrusselsVrije Univ. BrusselL. van Hamme, Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the Conference on p-adic Analysis (N. De Grande-De Kimpe and L. van Hamme, ed., Houthalen, 1987), pp. 189-195, Vrije Univ. Brussel, Brussels, 1986. Super congruence for the Apéry numbers. T Ishikawa, Nagoya Math. J. 118T. Ishikawa,Super congruence for the Apéry numbers, Nagoya Math. J. 118(1990), 195-202. Supercongruences involving p−adic Gamma functions. J.-C Liu, Bull. Aust. Math. Soc. 98J.-C. Liu, Supercongruences involving p−adic Gamma functions, Bull. Aust. Math. Soc. 98(2018), 27-37. Some supercongruences occurring in truncated hypergeometric series. L Long, R Ramakrishna, Adv. Math. 290L. Long and R. Ramakrishna, Some supercongruences occurring in truncated hy- pergeometric series, Adv. Math. 290(2016), 773-808. On some congruences of binomial coefficients modulo p 3 with applications, preprint, Researchgate. G.-S Mao, Doi:10.13140/RG.2.2.12033.17766G.-S. Mao, On some congruences of binomial coefficients modulo p 3 with applica- tions, preprint, Researchgate, Doi:10.13140/RG.2.2.12033.17766. Supercongruences for truncated n+1 F n hypergeometric series with applications to certain weight three newforms. E Mortenson, Proc. Amer. Math. Soc. 133E. Mortenson, Supercongruences for truncated n+1 F n hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc.133(2005), 321-330. H Pan, R Tauraso, C Wang, arXiv:1909.08183v3A local-global theorem for p-adic supercongruences. H. Pan, R. Tauraso and C. Wang, A local-global theorem for p-adic supercongru- ences, arXiv:1909.08183v3. . M Petkovšek, H S Wilf, D Zeilberger, A = B , A K Peters, WellesleyM. Petkovšek, H.S. Wilf and D. Zeilberger, A = B, A K Peters, Wellesley, 1996. F Rodriguez-Villegas, ; , Hypergeometric families of Calabi-Yau manifolds. Yui, NorikoToronto, ON; Providence, RIAmer. Math. Soc38Calabi-Yau Varieties and Mirror SymmetryF. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds,in: Calabi-Yau Varieties and Mirror Symmetry (Yui, Noriko (ed.) et al., Toronto, ON, 2001), 223-231, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003. Z H Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials. 105Z.H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math. 105(2000), 193-223. Congruences involving Bernoulli and Euler numbers. Z H Sun, J. Number Theory. 128Z.H. Sun, Congruences involving Bernoulli and Euler numbers, J. Number Theory 128(2008), 280-312. Identities and congruences for a new sequence. Z H Sun, Int. J. Number Theory. 8Z.H. Sun, Identities and congruences for a new sequence, Int. J. Number Theory 8(2012), 207-225. Z H Sun, Congruences concerning Legendre polynomials II. 133Z.H. Sun, Congruences concerning Legendre polynomials II, J. Number Theory 133(2013), 1950-1976. Z H Sun, Congruences involving 2k. Z. H. Sun, Congruences involving 2k . J. Number Theory. 133k , J. Number Theory 133(2013), 1572- 1595. Legendre polynomials and supercongruences. Z H Sun, Acta Arith. 159Z. H. Sun, Legendre polynomials and supercongruences, Acta Arith. 159(2013), 169-200. Generalized Legendre polynomials and related supercongruences. Z H Sun, J. Number Theory. 143Z. H. Sun, Generalized Legendre polynomials and related supercongruences, J. Number Theory 143(2014), 293-319. Super congruences concerning Bernoulli polynomials. Z H Sun, Int. J. Number Theory. 11Z. H. Sun, Super congruences concerning Bernoulli polynomials, Int. J. Number Theory 11(2015), 2393-2404. Z H Sun, Supercongruences involving Bernoulli polynomials. 12Z. H. Sun, Supercongruences involving Bernoulli polynomials, Int. J. Number The- ory 12(2016), 1259-1271. Super congruences for two Apéry-like sequences. Z H Sun, J. Difference Equ. Appl. 24Z.H. Sun, Super congruences for two Apéry-like sequences, J. Difference Equ. Appl. 24(2018), 1685-1713. Congruences involving binomial coefficients and Apéry-like numbers. Z H Sun, Publ. Math. Debrecen. 96Z.H. Sun, Congruences involving binomial coefficients and Apéry-like numbers, Publ. Math. Debrecen 96(2020), 315-346. Supercongruences and binary quadratic forms. Z H Sun, Z.H. Sun, Supercongruences and binary quadratic forms, Acta Arith. 199(2021), 1-32. Z H Sun, arXiv:2111.04538v2New conjectures involving binomial coefficients and Apéry-like numbers. Z.H. Sun, New conjectures involving binomial coefficients and Apéry-like numbers, arXiv:2111.04538v2. Supercongruences involving Apéry-like numbers and binomial coefficients. Z H Sun, AIMS Math. 7Z.H. Sun, Supercongruences involving Apéry-like numbers and binomial coeffi- cients, AIMS Math. 7(2022), 2729-2781. On sums involving products of three binomial coefficients. Z W Sun, Acta Arith. 156Z.W. Sun, On sums involving products of three binomial coefficients, Acta Arith. 156(2012), 123-141. Congruences involving alternating multiple harmonic sums, Electronic. R Tauraso, J. Combin. 1711R. Tauraso, Congruences involving alternating multiple harmonic sums, Elec- tronic J. Combin. 17(2010), R16, 11pp. Some congruences for central binomial sums involving Fibonacci and Lucas numbers. R Tauraso, 16.5.4J. Integer Sequences. 1910R. Tauraso, Some congruences for central binomial sums involving Fibonacci and Lucas numbers, J. Integer Sequences 19(2016), Article 16.5.4, 10pp. A supercongruence involving cubes of Catalan numbers. R Tauraso, Integers. 206R. Tauraso, A supercongruence involving cubes of Catalan numbers, Integers 20(2020), A44, 6pp.
[]
[ "On a hypergraph Turán problem of Frankl *", "On a hypergraph Turán problem of Frankl *" ]
[ "Peter Keevash ", "Benny Sudakov " ]
[]
[]
Let C (2k) rIntroductionGiven an r-uniform hypergraph F, the Turán number ex(n, F) of F is the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. Determining these numbers is one of the main challenges in Extremal Combinatorics. For ordinary graphs (the case r = 2) a rich theory has been developed, initiated by Turán in 1941, who solved the problem for complete graphs. He also posed the question of finding ex(n, K (r) s ) for complete hypergraphs with s > r > 2, but to this day not one single instance of this problem has been solved. It seems hard even to determine the Turán density, which for general F is defined as π(F) = lim n→∞ ex(n, F)/ n r . The problem of finding the *
10.1007/s00493-005-0042-2
[ "https://arxiv.org/pdf/math/0211179v1.pdf" ]
9,535,200
math/0211179
c396d97ad5c93c5387f8a1a4aa03b6f53066b0af
On a hypergraph Turán problem of Frankl * 11 Nov 2002 Peter Keevash Benny Sudakov On a hypergraph Turán problem of Frankl * 11 Nov 2002 Let C (2k) rIntroductionGiven an r-uniform hypergraph F, the Turán number ex(n, F) of F is the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. Determining these numbers is one of the main challenges in Extremal Combinatorics. For ordinary graphs (the case r = 2) a rich theory has been developed, initiated by Turán in 1941, who solved the problem for complete graphs. He also posed the question of finding ex(n, K (r) s ) for complete hypergraphs with s > r > 2, but to this day not one single instance of this problem has been solved. It seems hard even to determine the Turán density, which for general F is defined as π(F) = lim n→∞ ex(n, F)/ n r . The problem of finding the * be the 2k-uniform hypergraph obtained by letting P 1 , · · · , P r be pairwise disjoint sets of size k and taking as edges all sets P i ∪ P j with i = j. This can be thought of as the 'k-expansion' of the complete graph K r : each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain C (2k) 3 can be obtained by partitioning V = V 1 ∪V 2 and taking as edges all sets of size 2k that intersect each of V 1 and V 2 in an odd number of elements. Let B (2k) n denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. We prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no C (2k) 3 has at most as many edges as B (2k) n . Sidorenko has given an upper bound of r−2 r−1 for the Turán density of C (2k) r for any r, and a construction establishing a matching lower bound when r is of the form 2 p + 1. In this paper we also show that when r = 2 p + 1, any C The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal-Katona theorem and properties of Krawtchouck polynomials. numbers ex(n, F) when r > 2 is notoriously difficult, and exact results on hypergraph Turán numbers are very rare (see [3,9] for surveys). In this paper we obtain such a result for a sequence of hypergraphs introduced by Frankl. Let C (2k) r be the 2k-uniform hypergraph obtained by letting P 1 , · · · , P r be pairwise disjoint sets of size k and taking as edges all sets P i ∪ P j with i = j. This can be thought of as the 'k-expansion' of the complete graph K r : each vertex has been replaced with a set of size k. The Turán problem for C (2k) 3 was first considered by Frankl [2], who determined the density π(C (2k) 3 ) = 1/2. Frankl obtained a large C (2k) 3 -free hypergraph on n vertices by partitioning an n-element set V into 2 parts V 1 , V 2 and taking those edges which intersect each part V i in an odd number of elements. When the parts have sizes n 2 ± t we denote this hypergraph by B (2k) (n, t). To see that it is C (2k) 3 -free, consider any P 1 , P 2 , P 3 that are pairwise disjoint sets of k vertices. Then |V 1 ∩ P i | and |V 1 ∩ P j | have the same parity for some pair ij, so P i ∪ P j is not an edge. Let t * be chosen to maximise the number of edges in B (2k) (n, t), and denote any hypergraph obtained in this manner by B n . Frankl [2] conjectured that the maximum number of edges in a C The proof of this theorem falls naturally into two parts. The first stage is to prove a 'stability' version, which is that any hypergraph with close to the maximum number of edges looks approximately like some B (2k) (n, t). Armed with this, we can analyse any imperfections in the structure and show that they must lead to a suboptimal configuration, so that the optimum is indeed achieved by the construction. This strategy was also used recently in [4] to prove the conjecture of Sós on the Turán number of the Fano plane, so this seems to be a useful tool for developing the Turán theory of hypergraphs. For general r, Sidorenko [8] showed that the Turán density of C (2k) r is at most r−2 r−1 . This is a consequence of Turán's theorem applied to an auxiliary graph G constructed from a 2k-uniform hypergraph H; the vertices of G are the k-tuples of vertices of H, and two k-tuples P 1 ,P 2 are adjacent if P 1 ∪ P 2 is an edge of H. He also gave a construction for a matching lower bound when r is of the form 2 p + 1, which we now describe. Let W be a vector space of dimension p over the field GF (2), i.e. the finite field with 2 elements {0, 1}. Partition a set of vertices V as w∈W V w . Given t and a t-tuple of vertices X = x 1 · · · x t with x i ∈ V w i let ΣX = t 1 w i . Define a 2k-uniform hypergraph H, where a 2k-tuple X is an edge iff ΣX = 0. Observe that this doesn't contain a copy of C (2k) r . Indeed, if P 1 , · · · , P r are disjoint k-tuples then there is some i = j with ΣP i = ΣP j (by the pigeonhole principle). Then Σ(P i ∪ P j ) = ΣP i + ΣP j = 0, so P i ∪ P j is not an edge. To see that this construction can achieve the stated Turán density, choose the partition so that |V w | = |V |/(r − 1). Then a random (average) 2k-tuple is an edge with probability r−2 r−1 + o(1), as can be seen by conditioning on the positions of all but one element. This construction depends essentially on an algebraic structure, which only exists for certain values of r. We will show that this is an intrinsic feature of the problem, by proving a stronger upper bound on the Turán density of C (4) r when r is not of the form 2 p + 1. Theorem 1.2 Suppose r ≥ 3, and let H be a 4-uniform hypergraph on n vertices with at least r−2 r−1 − 10 −33 r −70 n 4 edges. If H is C (4) r -free, then r = 2 p + 1 for some integer p. In contrast to Theorem 1.1 this is a result showing that certain constructions do not exist, so it is perhaps surprising that its proof also uses a stability argument. We study the properties of a C (4) r -free hypergraph with density close to r−2 r−1 and show that it give rise to the edge coloring of the complete graph K r−1 with special properties. Next we prove that for such edge-coloring there is a natural GF (2) vector space structure on the colors. Of course, such a space has cardinality 2 p , for some p, so we get a contradiction unless r = 2 p + 1. A complication arising in Theorem 1.1 is that the optimum construction is not achieved by a partition into two equal parts. Finding t to maximise the number of edges in B (2k) (n, t) is an interesting problem in enumerative combinatorics, equivalent to finding the minima of binary Krawtchouk polynomials. This is a family of polynomials orthogonal with respect to the uniform measure on a n-dimensional cube that play an important rôle in the analysis of binary Hamming association schemes (see, e.g., [5]). Despite some uncertainty in the location of their minima, the known bounds are sufficient for us to show that some B (2k) (n, t) must be optimal. In the case k = 2 one can compute the size of B (2k) (n) precisely, and there are considerable simplifications of the argument, so in the next section for illustrative purposes we start by giving a separate proof for this case. Section 3 contains a stability theorem for C (2k) 3 and the general case of Theorem 1.1. Then in Section 4 we prove a stability result for C (4) r for all r, and use it to establish Theorem 1.2. The final section of the paper contains some concluding remarks. We will assume throughout this paper that n is sufficiently large. 2 The Turán number of C (4) 3 We start by proving Frankl's conjecture for 4-uniform hypergraphs. This will serve to illustrate our method, as it has fewer complications than the general case. In addition, in this case it is easy to compute the Turán numbers of C 3 precisely. We recall that C 3 is the 4-uniform hypergraph with three edges {abcd, abef, cdef }. We can obtain a large C 3 -free graph on n vertices by partitioning an n-element set into 2 parts and taking those edges which have 1 point in either class and 3 points in the other. To see this, think of an edge as being the union of 2 different types of pairs of vertices: one type consisting of pairs with both vertices in one class, the other consisting of pairs that have one point of each class. Given any 3 pairs there are 2 of the same type, and these do not form an edge in the construction. To maximise the number of edges in this bipartite construction, it is not the case that the two parts have sizes as equal as possible, but we will see that the difference in the sizes should be at most of order √ n. Let B(n, t) denote the 4-uniform hypergraph obtained by partitioning an n-element set into 2 parts with sizes n 2 + t and n 2 − t, and taking those edges which have 1 point in either class and 3 points in the other. Let b(n, t) be the number of edges in B(n, t) and let d(n, t) be the degree of any vertex belonging to the side with size n 2 + t. Then the vertices on the side with size n 2 − t have degree d(n, −t). We will start with some estimates on these parameters. By definition, b(n, t) = n 2 + t n 2 − t 3 + n 2 − t n 2 + t 3 = n 4 − 6n 3 + 8n 2 − 16t 4 − 32t 2 + 24t 2 n 48 = 1 48 n 2 − 3n + 4 2 − 4t 2 − 3n + 4 2 .(1) Thus to maximise b(n, t) we should pick a value of t that minimises 4t 2 − 3n + 4, subject to the restriction that when n is even t has to be an integer, and when n is odd t + 1 2 has to be an integer. Let B n denote a hypergraph B(n, t * ), where t * is such a value of t. By symmetry we can take t * > 0. There is usually a unique best choice of t * , but for some n there are 2 equal choices of t * . Note that for any best choice we certainly have t * − 3n/4 − 1 ≤ 1/2. Let b(n) be the number of edges in B n . Then 48b(n) − (n 2 − 3n + 4) 2 = 4(t * ) 2 − 3n + 4 2 < 50n . It will be useful later to consider the following estimate which follows immediately from the last inequality for sufficiently large n b(n) − b(n − 1) > 1 12 n 3 − 1 2 n 2 .(2) Next we give an explicit formula for the degrees in B(n, t) d(n, t) = n 2 − t n/2 + t − 1 2 + n/2 − t 3 = n 3 − 6n 2 + 8n + 12t 2 12 + 6tn − 8t 3 − 16t 12 .(3) We finish these calculations with an upper bound on the maximum degree of B n ∆(n) = 1 12 n 3 − 6n 2 + 8n + 12(t * ) 2 + 1 12 6t * n − 8(t * ) 3 − 16t * < 1 12 n 3 − 1 2 n 2 + n 3/2 .(4) The first step in the proof is to show that any C 3 -free 4-uniform hypergraph H with density close to 1/2 has the correct approximate structure. To do so we need a few definitions. If we have a partition of the vertex set of H as V (H) = V 1 ∪ V 2 we call a 4-tuple of vertices good if it has either 1 point in V 1 and 3 points in V 2 or 1 point in V 2 and 3 points in V 1 ; otherwise we call it bad. With respect to H, we call a 4-tuple correct if it is either a good edge or a bad non-edge; otherwise we call it incorrect. We obtain the following stability result. Theorem 2.1 For every ǫ > 0 there is η > 0 so that if H is a C(4) 3 -free 4-uniform hypergraph with e(H) > b(n) − ηn 4 then there is a partition of the vertex set as V (H) = V 1 ∪ V 2 such that all but ǫn 4 4-tuples are correct. In the proof of this result we need a special case of the Simonovits stability theorem [10] for graphs, which we recall. It states that for every ǫ ′ > 0 there is η ′ > 0 such that if G is a triangle free graph on N vertices with at least (1−η ′ ) N 2 /2 edges then there is a partition of the vertex set as V (G) = U 1 ∪U 2 with e G (U 1 ) + e G (U 2 ) < ǫ ′ N 2 . Proof of Theorem 2.1. Define an auxiliary graph G whose vertices are all pairs of vertices of H, and where the pairs ab and cd are adjacent exactly when abcd is an edge of H. Since H is C 3 -free we see that G is triangle-free. Also, each edge of H creates exactly 3 edges in G (corresponding to the 3 ways of breaking a 4-tuple into pairs) so e(G) > 3 b(n) − ηn 4 > 1 − 50η 1 2 n 2 2 . Choose η so that Simonovits stability applies with η ′ = 50η, N = n 2 and ǫ ′ = ǫ 2 /500. We can also require that η < ǫ 2 /500. We get a partition of the pairs of vertices of H as U 1 ∪ U 2 , where all but ǫ ′ N 2 < ǫ 2 n 4 /2000 edges of H are formed by taking a pair from U 1 and a pair from U 2 . We will call the pairs in U 1 red, and the pairs in U 2 blue. A 4-tuple abcd will be called properly coloured if either (i) abcd is an edge of H and each of the 3 sets {ab, cd},{ac, bd},{ad, bc} has one red pair and one blue pair, or (ii) abcd is not an edge and each of the 3 sets {ab, cd},{ac, bd},{ad, bc} consists of two pairs with the same colour. An improperly coloured 4-tuple is either an edge that is the union of two pairs of the same colour or a non-edge which is the union of two pairs with different colours. There are at most ǫ 2 n 4 /2000 of the former 4-tuples, and the number of latter is at most |U 1 ||U 2 | − e(G) − ǫ ′ N 2 ≤ 50η 2 N 2 2 + ǫ ′ N 2 ≤ 50η 16 + ǫ ′ /4 n 4 < ǫ 2 n 4 /140 . Therefore all but ǫ 2 /140 + ǫ 2 /2000 n 4 < ǫ 2 n 4 /130 4-tuples are properly coloured. A simple counting argument shows that there is a pair ab so that for all but 4 2 ǫ 2 n 4 /130 / n 2 < ǫ 2 n 2 /10 other pairs cd the 4-tuple abcd is properly coloured. Without loss of generality ab is red. Partition the vertices of V − ab into 4 sets according to the colour of the edges they send to {a, b}. We label these sets RR,BB,RB,BR, where R means 'red', B means 'blue' and a vertex c belongs to the set that labels the colours of the edges ca, cb in this order. Note that if c is in RR and d is in RB then ca and db are coloured red and blue, whereas cb and da are are both red, so abcd is improperly coloured. We deduce that one of RR and RB has size at most ǫn/3, since otherwise we would have at least ǫ 2 n 2 /9 improperly colored 4-tuples containing ab. The same argument applies when take one point from each of BB and RB, or RR and BR, or BB and BR. Therefore, either RB and BR each have size at most ǫn/3, or RR and BB each have size at most ǫn/3. In the case when RB and BR each have size at most ǫn/3 we look at the pairs in RR ∪ BB. If c and d are both in RR then both of the opposite pairs {ac, bd} and {ad, bc} are coloured red. If cd is coloured blue then abcd is improperly coloured, so all but at most ǫ 2 n 2 /10 pairs in RR are coloured red. Similarly all but at most ǫ 2 n 2 /10 pairs in BB are coloured red, and all but at most ǫ 2 n 2 /10 pairs with one vertex in RR and one in BB are coloured blue. Define a partition V = V 1 ∪ V 2 , where V 1 contains RR, V 2 contains BB and the remaining vertices are distributed arbitrarily. Note that all the incorrect 4-tuples with respect to this partition belong to the one of the following three groups. (i) Improperly colored 4-tuples. There are at most ǫ 2 n 4 /130 of those. (ii) Properly colored 4-tuples which use at least one vertex in RB ∪ BR. There are at most 2ǫn/3 n 3 such 4-tuples. (iii) Properly colored 4-tuples which contain either a red pair of vertices with one vertex in RR and one in BB, or contain a blue pair of vertices from RR or from BB. There at most 3ǫ 2 n 2 /10 n 2 such 4-tuples. Therefore all but at most ǫ 2 n 4 130 + 2 ǫn The case when RR and BB each have size at most ǫn/3 can be treated similarly. Here the conclusion is that all but at most ǫ 2 n 2 /5 pairs within RB or BR are coloured blue, and all but at most ǫ 2 n 2 /10 pairs with one vertex in RB and one in BR are coloured red. Then, similarly as above one can show that with respect to a partition where V 1 contains RB, V 2 contains BR and the remaining vertices are distributed arbitrarily, all but at most ǫn 4 4-tuples are correct. Using the stability theorem we can now prove the following exact Turán result. Theorem 2.2 Let H be a 4-uniform hypergraph on n vertices that does not contain a copy of C 3 and let n be sufficiently large. Then the number of edges in H is at most b(n), with equality only when H is one of at most 2 hypergraphs B n . Proof. Let H be a 4-uniform hypergraph on n vertices, which has e(H) ≥ b(n) and contains no C First we claim that we can assume that H has minimum degree at least b(n) − b(n − 1). Indeed, suppose that we have proved the result under this assumption for all n ≥ n 0 . Construct a sequence of hypergraphs H = H n , H n−1 , · · · where H m−1 is obtained from H m by deleting a vertex of degree less than b(m) − b(m − 1). By setting f (m) = e(H m ) − b(m) we have f (n) ≥ 0 and f (m) ≥ f (m + 1) + 1. If we can continue this process to obtain a hypergraph H n 0 then n − n 0 ≤ n−1 m=n 0 f (m) − f (m + 1) ≤ f (n 0 ) ≤ n 0 4 , which is a contradiction for n sufficiently large. Otherwise we obtain a hypergraph H n ′ with n > n ′ > n 0 having minimal degree at least b(n ′ ) − b(n ′ − 1) and without a C (4) 3 . Then by the above assumption e(H n ′ ) ≤ b(n ′ ) and again we obtain a contradiction, since e(H) = e(H n ) ≤ b(n ′ ) + n ′ <m≤n b(m) − b(m − 1) − 1 < b(n) . Substituting from equation (2) we can assume H has minimum degree δ(H) ≥ b(n) − b(n − 1) > 1 12 n 3 − 1 2 n 2 .(5) Given a partition of V (H) = V 1 ∪ V 2 , we call an edge abcd of H good if abcd is a good 4-tuple (as defined before) with respect to this partition; otherwise we call it bad. By Theorem 2.1 there is a partition with all but at most 10 −25 n 4 edges of H being good. Let V (H) = V 1 ∪ V 2 be the partition which minimises the number of bad edges. With respect to this partition, every vertex belongs to at least as many good edges as bad edges, or we can move it to the other class of the partition. Also, by definition, there are at most b(n) good 4-tuples with respect to any partition. We must have |V 1 | − n/2 < 10 −6 n and |V 2 | − n/2 < 10 −6 n. Otherwise by equation (1) we get e(H) < 1 48 n 2 − 3n + 4 2 − 4 · 10 −12 n 2 − 3n + 4 2 + 10 −25 n 4 < b(n), which is a contradiction. Note that there is no pair of vertices ab for which there are both 10 −10 n 2 pairs cd such that abcd is a good edge and 10 −10 n 2 pairs ef such that abef is an bad edge. Indeed, each such cd and ef which are disjoint give a 4-tuple cdef which is good, but cannot be an edge as it would create a C 3 . Moreover, every 4-tuple can be obtained at most 3 times in this way, and every cd is disjoint from all but at most 2n pairs ef . Thus at least 10 −10 n 2 10 −10 n 2 − 2n /3 > 10 −21 n 4 good 4-tuples are not edges of H, and therefore e(H) < b(n) − 10 −21 n 4 + 10 −25 n 4 < b(n), which is a contradiction. The next step of the proof is the following claim. Proof. Suppose some vertex a belongs to 10 −5 n 3 bad edges. Call another vertex b good if there are at most 10 −10 n 2 pairs cd such that abcd is a bad edge, otherwise call b bad. By the above discussion, for every bad vertex b there are at most 10 −10 n 2 pairs ef such that abef is a good edge. Note that there are at least 10 −5 n bad vertices, otherwise we would only have at most 10 −5 n · n 2 + (1 − 10 −5 )n · 10 −10 n 2 < 10 −5 n 3 bad edges through a, which is contrary to our assumption. By choice of partition there are at least as many good edges containing a as bad. We know that a has degree at least 1 12 n 3 − 1 2 n 2 , at least half of which is good, so there are at least n/24 good vertices. Suppose that the number of good vertices is αn, and so there are (1 − α)n − 1 bad vertices. We can count the edges containing a as follows. By definition there are at most 10 −10 n 3 such good edges containing a bad vertex, and at most 10 −10 n 3 such bad edges containing a good vertex. Now we bound the number of remaining good edges. Note that these edges only contain good vertices. Looking at the vertices of such an edge in some order, we can select the first 2 vertices in αn(αn − 1) ways. Since the edge is good, the choice of 2 vertices together with a restricts the fourth vertex to lie in some particular class V i , so it can be chosen in at most 1 2 + 10 −6 n ways. Note that we have counted each edge 6 times, so we get at most αn(αn − 1) 1 2 + 10 −6 n/6 < α 2 + 1 2 · 10 −5 1 2 n 3 edges. Similarly there are at most (1 − α) 2 + 1 2 · 10 −5 1 2 n 3 remaining bad edges through a. Since 1/24 ≤ α ≤ 1 − 10 −5 , in total the number of edges containing a is bounded by α 2 + 1 2 · 10 −5 1 2 n 3 + (1 − α) 2 + 1 2 · 10 −5 1 2 n 3 + 2 · 10 −10 n 3 < 1 12 n 3 − 1 2 n 2 < δ(H) . This contradiction proves the claim. Now write |V 1 | = n/2 + t, |V 2 | = n/2 − t with −10 −6 n < t < 10 −6 n. By possibly renaming the classes (i.e. replacing t with −t) we can assume that d(n, t) < d(n, −t). Then any vertex of V 1 belongs to d(n, t) good 4-tuples. Now d(n, t) is the minimum degree of B(n, t), which is certainly at most the maximum degree of B n . Comparing with equation (4) we see that any vertex of V 1 belongs to at most 1 12 n 3 − 1 2 n 2 + n 3/2 good 4-tuples. From now on this will be the only property of V 1 we use that might possibly not be a property of V 2 . We will eventually end up showing the same bound on the number of good 4-tuples containing a vertex of V 2 . Then the whole argument will apply verbatim switching V 1 for V 2 . We will use this property in the following manner. Suppose a is a vertex of V 1 for which K of the good 4-tuples containing a are not edges of H. Then there are at most 1 12 n 3 − 1 2 n 2 + n 3/2 − K good edges containing a, so by (5) there must be at least δ(H) − 1 12 n 3 − 1 2 n 2 + n 3/2 − K ≥ 1 12 n 3 − 1 2 n 2 − 1 12 n 3 − 1 2 n 2 + n 3/2 − K = K − n 3/2 bad edges containing a. Similarly, if a ′ is a vertex in V 2 then it belongs to at most d(n, −t) = 1 12 (n 3 − 6n 2 + 8n + 12t 2 ) + 1 12 |6tn − 8t 3 − 16t| < 1 12 n 3 − 1 2 n 2 + 10 −6 n 3 good edges. Thus, if it belongs to L good 4-tuples which are not edges of H then it must belong to at least L − 10 −6 n 3 bad edges. Suppose for the sake of contradiction that there is some bad edge incident with V 1 . Denote the set of bad edges containing some vertex v by Z(v). Let a be a vertex in V 1 belonging to the maximum number of bad edges and let Z = |Z(a)|. Note that Z > 0. For every bad edge abcd containing a, consider a partition of its vertices into pairs, say ac and bd. Recall that there are 2 types of pairs, one type consisting of pairs with both vertices in one class, the other consisting of pairs that have one point of each class. By definition of a bad edge, ac and bd are pairs of the same type. If ef is any pair of the other type which is disjoint from both of them, then acef and bdef are good 4-tuples. One of them is not an edge of H, or we get a C 3 . The number of such pairs ef is clearly at least min |V 1 | − 4 |V 2 | − 4 , |V 1 | − 4 2 + |V 2 | − 4 2 ≥ 1 4 − 10 −12 n 2 − O(n) > n 2 /5 . Let Z 1 (a) be those bad edges for which there is some partition into pairs ac and bd, so that for at least n 2 /10 of the pairs ef defined above, the good 4-tuple acef is not an edge. Let Z 2 (a) = Z(a) − Z 1 (a), and write Z i = |Z i (a)| for i = 1, 2. Then one of Z 1 ,Z 2 is at least Z/2. Case 1: Suppose Z 1 ≥ Z/2. Let C be the (non-empty) set of vertices c such that there is some edge abcd in Z 1 (a), and acef is a good non-edge for at least n 2 /10 pairs ef . Then we have at least |C|n 2 /30 good non-edges containing a, as we count each acef at most 3 times. This implies that there are at least |C|n 2 /30 − n 3/2 ≥ |C|n 2 /31 bad edges containing a and therefore n 2 /31 ≤ Z/|C|. Since every edge in Z 1 (a) contains at most 3 vertices of C there exists c ∈ C which is contained in at least |Z 1 (a)|/(3|C|) = Z 1 /(3|C|) ≥ Z/(6|C|) > n 2 /200 bad edges. Fix one such c. Note that a graph with n vertices and m edges contains a matching of size at least m/2n, since otherwise there is a set of fewer than m/n vertices that cover all the edges of the graph, which is impossible by direct counting. Consider the set of pairs bd such that abcd is a bad edge. Then there exists a matching M of size at least n/400 so that for each bd in M we have that abcd is a bad edge of H. Partition such an edge into pairs ab and cd. Then, as we explained above, there are at least n 2 /5 pairs ef such that one of the 4-tuples abef and cdef is a good non-edge. Since M is a matching we count each such 4-tuple at most 3 times, so one of a or c belongs to at least 1 3 · 1 2 · n 2 5 · n 400 = n 3 /12000 good non-edges. Therefore it belongs to at least n 3 /12000 − 10 −6 n 3 > 10 −5 n 3 bad edges, which contradicts Claim 2.3. Case 2: Now suppose Z 2 ≥ Z/2. Note that every bad edge containing a contains at least one other point of V 1 , so there is some b ∈ V 1 belonging to at least Z 2 /n edges of Z 2 (a). Fix one such b. Suppose cd is a pair such that abcd is in Z 2 (a), and consider any partition of abcd into pairs p 1 , p 2 with a in p 1 and b in p 2 . Then, by definition of Z 2 (a), there are at least n 2 /10 pairs ef such that p 2 ∪ ef is a good non-edge. Let C be the set of vertices c for which there exists a vertex d such that abcd is an edge of Z 2 (a). Then there are at least |C|n 2 /30 good non-edges containing b, as we count each bcef at most 3 times. Thus, there are at least |C|n 2 /30 − n 3/2 > |C|n 2 /50 bad edges containing b. By maximality of Z we have |C|n 2 /50 ≤ |Z(b)| ≤ Z. Note that each edge in Z 2 (a) that contains b is obtained by picking a pair of vertices in C, so Z/(2n) ≤ Z 2 /n ≤ |C| 2 < 1250Z 2 /n 4 . Therefore Z ≥ n 3 /2500, which again contradicts Claim 2.3. We conclude that there are no bad edges incident to the vertices of V 1 , i.e. all bad edges have all 4 vertices in V 2 . We can use this information to give more precise bounds on the sizes of V 1 and V 2 . We recall that |V 1 | = n/2 + t, |V 2 | = n/2 − t and d(n, t) < d(n, −t). Suppose that |t| ≥ √ n, so that 6|t|n − 8|t| 3 − 16|t| < −2n 3/2 and by (3) d(n, t) = 1 12 n 3 − 6n 2 + 8n + 12t 2 + 1 12 6|t|n − 8|t| 3 − 16|t| < 1 12 n 3 − 1 2 n 2 − n 3/2 12 < δ(H) . This is a contradiction, since the vertices of V 1 only belong to good edges, of which there are at most d(n, t) < δ(H). Therefore |t| < √ n. Now we can bound the number of good 4-tuples containing a vertex of V 2 . By (3), this number is at most d(n, −t) = 1 12 n 3 − 6n 2 + 8n + 12t 2 + 1 12 8|t| 3 − 6|t|n + 16|t| < 1 12 n 3 − 1 2 n 2 + n 3/2 . Now the same argument as we used to show that no bad edges are incident with the vertices of V 1 shows that none are incident with V 2 either. We conclude that all edges are good. Then by definition of b(n) we have e(H) ≤ b(n), with equality only when H is a B n , so the theorem is proved. Proof of Frankl's conjecture In this section we will prove the general case of the Frankl conjecture. We recall that C (2k) 3 is the 2kuniform hypergraph with three edges {P 1 ∪ P 2 , P 2 ∪ P 3 , P 3 ∪ P 1 }, where P 1 , P 2 , P 3 are pairwise disjoint sets of k vertices. We can obtain a large C (2k) 3 -free graph on n vertices by partitioning an n-element set V into 2 parts V 1 , V 2 and taking those edges which intersect each part V i in an odd number of elements. To see this, consider any P 1 , P 2 , P 3 that are pairwise disjoint sets of k vertices. Then |V 1 ∩ P i | and |V 1 ∩ P j | have the same parity for some pair ij, so P i ∪ P j is not an edge. Note that this construction is the same as the one we described for C (4) 3 when k = 2. In the 4uniform case we were able to calculate the sizes of the parts that maximise the number of edges. For general k this is an interesting problem in enumerative combinatorics, that is equivalent to finding the minima of binary Krawtchouk polynomials. These polynomials play an important rôle in the analysis of binary Hamming association schemes and so many of their properties are well-known in this context (see, e.g., [5]). In particular, the location of their roots is an important problem, but we will need here only a crude estimate that follows easily from known results. In the first subsection of this section we will state this estimate and apply it to various parameters of our construction. The rest of the proof follows the same broad outline as that of the 4-uniform case, in that it falls naturally into two parts. We will prove the stability part in the second subsection, and the full result we defer to the final subsection. Binary Krawtchouk polynomials Let B (2k) (n, t) denote the 2k-uniform hypergraph obtained by partitioning an n-element set into two parts with sizes n 2 + t and n 2 − t, and taking as edges all 2k-tuples with odd intersection with each part. Let b 2k (n, t) be the number of edges in B (2k) (n, t) and let d 2k (n, t) be the degree of any vertex belonging to the side with size n 2 + t. Then the vertices on the side with size n 2 − t have degree d 2k (n, −t). The binary Krawtchouk polynomials K n m (x) can be defined by the generating function n m=0 K n m (x)z m = (1 − z) x (1 + z) n−x . From here we get the explicit expression K n m (x) = m i=0 (−1) i x i n−x m−i . Recall that b 2k (n, t) was the number of 2k-tuples with odd intersection with both parts in the above partition of an n-element set and so n 2k − b 2k (n, t) is the number of 2k-tuples with even intersection with these parts. This implies that n 2k − b 2k (n, t) − b 2k (n, t) = 2k i=0 (−1) i n/2+t i n/2−t 2k−i = K n 2k (n/2 + t), which gives b 2k (n, t) = 1 2 n 2k − K n 2k (n/2 + t) ,(6) so maximising b 2k (n, t) is equivalent to finding the minimum of K n 2k (x). Similarly, we can also express the degrees of B (2k) (n, t) in terms of Krawtchouk polynomials. Indeed, by definition, d 2k (n, t) is the number of (2k − 1)-tuples with even intersection with the first part in the partition of an (n − 1)element set in two parts with sizes n/2 + t − 1 and n/2 − t, and therefore n−1 2k−1 − d 2k (n, t) is the number of (2k − 1)-tuples with odd intersection with this part. Then, d 2k (n, t) − n−1 2k−1 − d 2k (n, t) = 2k−1 i=0 (−1) i n/2+t−1 i n/2−t 2k−1−i = K n−1 2k−1 (n/2 + t − 1), i.e. d 2k (n, t) = 1 2 n − 1 2k − 1 + K n−1 2k−1 (n/2 + t − 1)(7) Note that K n m (x) is a polynomial of degree m. It is known that it has m simple roots, symmetric with respect to n/2. The smallest root is given by the following formula obtained by Levenshtein [6]: r = n/2 − max m−2 i=0 x i x i+1 (i + 1)(n − i) , where the maximum is taken over x i with m−1 0 x 2 i = 1. From the Cauchy-Schwartz inequality we see that n/2 − r < √ mn. Note that K n 2k (0) = K n 2k (n) = n 2k > 0, so the minimum of K n 2k (x) occurs in the range n/2 ± √ 2kn. Let t * be chosen to maximise the number of edges in B (2k) (n, t), and denote any hypergraph obtained in this manner by B (2k) n . Note that t * may not be unique, but must satisfy |t * | < √ 2kn. Also, by symmetry we can assume that t * > 0. Write b 2k (n) for the number of edges in B (2k) n . Lemma 3.1 (i) K n m (n/2 + t) = m/2 i=0 (−1) i+m n/2−t i 2t m−2i . (ii) If c > 1 and 0 ≤ s ≤ c √ n then d 2k (n, ±s) − 1 2 n−1 2k−1 < (10c 2 ) k n k−1/2 . (iii) b 2k (n) − 1 2 n 2k < (20kn) k , d 2k (n, ±t * ) − 1 2 n−1 2k−1 < (20k) k n k−1/2 . (iv) If C > 20 k then d 2k (n, C √ n) < 1 2 n−1 2k−1 − 20 k n k−1/2 . (v) b 2k (n, ǫn) − 1 2 n 2k − 1 2 2ǫn 2k < (10ǫ) k n 2k−1 , d 2k (n, ǫn) − 1 2 n−1 2k−1 − 1 2 2ǫn−1 2k−1 < (10ǫ) k n 2k−2 . Proof. (i) Rewrite the generating function as n m=0 K n m (n/2 + t)z m = (1 − z 2 ) n/2−t (1 − z) 2t and expand. (ii) Using part (i) with t = s − 1/2, and applying (7), we get d 2k (n, s) − 1 2 n − 1 2k − 1 = 1 2 K n−1 2k−1 (n/2 + s − 1) = 1 2 k−1 i=0 (−1) i+1 n/2 − s i 2s − 1 2k − 1 − 2i < k · (2c √ n) 2k−1 < (10c 2 ) k n k−1/2 . The corresponding inequality for d 2k (n, −s) can be obtained similarly. (iii) The second statement follows from part (ii) with c = √ 2k ≥ t * / √ n. To prove the first statement, we use (6), part (i) and again the fact that 0 < t * < √ 2kn. Altogether they imply b 2k (n) − 1 2 n 2k = 1 2 K n 2k (n/2 + t * ) < k i=0 n/2 i 2 √ 2kn 2k − 2i < (k + 1) 2 √ 2kn 2k < (20kn) k . (iv) By (7) we have d 2k (n, C √ n) − 1 2 n − 1 2k − 1 = 1 2 K n−1 2k−1 n/2 + C √ n − 1 = 1 2 k−1 i=0 (−1) i+1 n 2 − C √ n i 2C √ n − 1 2k − 1 − 2i < − 1 2 2C √ n − 1 2k − 1 + 1 2 (k − 1)(n/2) 2C √ n − 1 2k − 3 = 1 + o(1) − (2C √ n) 2k−1 2(2k − 1)! + (k − 1)n 4 (2C √ n) 2k−3 (2k − 3)! < − C 2 (2k − 1) 2 − k − 1 2 (2C) 2k−3 (2k − 3)! n k−1/2 < −20 k n k−1/2 . (v) Using formula for K n 2k (n/2 + t) from part (i) together with (6) we obtain that b 2k (n, ǫn) − 1 2 n 2k − 1 2 2ǫn 2k = 1 2 K n 2k (n/2 + ǫn) − 1 2 2ǫn 2k = 1 2 k i=1 (−1) i n/2 − ǫn i 2ǫn 2k − 2i < n 2 · (2ǫn) 2k−2 + O n 2k−2 < (10ǫ) k n 2k−1 . The proof of the inequality for d 2k (n, ǫn) can be obtained similarly and we omit it here. We remark that these simple estimates are sufficient for our purposes, but the location of the roots and asymptotic values for Krawtchouk polynomials in the oscillatory region are known with more precision (see, e.g., [5]). With this information one could find better estimates for b 2k (n), and possibly how many different choices of t give the maximum number of edges. We conclude this section with an estimate on the difference of successive values of b 2k (n). Lemma 3.2 b 2k (n) − b 2k (n − 1) ≥ 1 2 n−1 2k−1 Proof. Suppose that H = B (2k) (n − 1) has b 2k (n − A stability result for C (2k) 3 In this subsection we prove a stability result for C = x(x − 1) · · · (x − k + 1)/k!. The following result appears in [7] (Exercise 13.31). Suppose we have a 2k-uniform hypergraph H and a partition of the vertex set V (H) = V 1 ∪ V 2 . Our terminology for 2k-tuples matches that of the 4-uniform case. We call a 2k-tuple of vertices good if it intersects each V i in an odd number of elements; otherwise we call it bad. We call a 2k-tuple correct if it is either a good edge or a bad non-edge; otherwise we call it incorrect. Theorem 3.4 For every ǫ > 0 there is η > 0 so that if H is a C (2k) 3 -free 2k-uniform hypergraph with e(H) > 1 2 n 2k − ηn 2k then there is a partition of the vertex set as V (H) = V 1 ∪ V 2 such that all but ǫn 2k 2k-tuples are correct. Proof. Define an auxiliary graph G whose vertices are all k-tuples of vertices of H, and where the k-tuples P 1 and P 2 are adjacent exactly when P 1 ∪ P 2 is an edge of H. Since H is C Choose η so that the Simonovits stability theorem (see Section 2) applies with η ′ = (k!) 2 2 2k η, N = n k and ǫ ′ = 10 −6k 2 ǫ k . We can also require that η < 10 −6k 2 ǫ k . We get a partition of the k-tuples of vertices of H as U 0 ∪ U 1 , where all but ǫ ′ N 2 = ǫ ′ n k 2 < 10 −6k 2 ǫ k n 2k edges of H are formed by taking a k-tuple from U 0 and a k-tuple from U 1 . We will think of the sets U i as determining a 2-coloring of all k-tuples, and say that the k-tuples in U i have colour i. A 2k-tuple I will be called properly coloured if, either it is an edge of H and however we partition I into k-tuples P 1 and P 2 they have different colours, or it is not an edge of H and for any partition of I into two k-tuples they have the same color. An improperly coloured 2k-tuple is either an edge that is the union of two k-tuples of the same colour or a non-edge which is the union of two k-tuples with different colours. There are at most 10 −6k 2 ǫ k n 2k of the former 2k-tuples, and the number of latter is at most |U 1 ||U 2 | − e(G) − ǫ ′ N 2 ≤ (k!) 2 2 2k η 2 N 2 2 + ǫ ′ N 2 ≤ (k!) 2 2 2k η 4(k!) 2 + ǫ ′ (k!) 2 n 2k ≤ 10 −5k 2 −1 ǫ k n 2k . Therefore all but 10 −6k 2 ǫ k + 10 −5k 2 −1 ǫ k n 2k < 10 −5k 2 ǫ k n 2k 2k-tuples are properly coloured. A simple counting argument shows that there is a k-tuple P so that for all but 2k k 10 −5k 2 ǫ k n 2k / n k < 10 −4k 2 ǫ k n k other k-tuples Q the 2k-tuple P ∪ Q is properly coloured. Without loss of generality P has colour 0. We will call a k-tuple Q proper if P ∪ Q is properly coloured; otherwise it is improper. Then by definition there are at most 10 −4k 2 ǫ k n k improper k-tuples. Call a (k − 1)-tuple X ⊂ V − P abnormal if there are at least 2 −3k ǫn vertices x ∈ V − (P ∪ X) for which X ∪ x is improper; otherwise call it normal. It is easy to see that there are at most k · 10 −4k 2 ǫ k n k /(2 −3k ǫn) < 10 −3k 2 ǫ k−1 n k−1 abnormal (k − 1)-tuples. We partition the vertices of V − P according to the colour of the k-tuples that they form when they replace an element of P . To be precise, we fix an order p 1 , · · · , p k of P and partition into 2 k parts V − P = V s , where s = (s 1 , . . . , s k ) ∈ {0, 1} k and a vertex x belongs to V s iff (P − p i ) ∪ x has colour s i for every 1 ≤ i ≤ k. Consider a (k − 1)-tuple X = x 1 · · · x k−1 and suppose a is a vertex such that X ∪ a is proper. Fix 1 ≤ i ≤ k and consider the partitions P ∪ X ∪ a = (P ) (X ∪ a) = (P − p i ) ∪ a (X ∪ p i ). Let V s be the class containing a, so that (P − p i ) ∪ a has colour s i . We recall that P has colour 0, so if also s i = 0 then to be properly coloured X ∪ a must have the same colour as X ∪ p i . On the other hand, if s i = 1 then X ∪ a and X ∪ p i must have different colours. If we write c X (v) for the colour of X ∪ v for any vertex v, then this can be summarised as If a ∈ V s and X ∪ a is proper, then c X (a) + s i = c X (p i ) (mod 2) . Suppose there are 2 classes V s and V s ′ both of size at least 2 −2k ǫn. Since 2 −2k ǫn k−1 > 10 −3k 2 ǫ k−1 n k−1 some (k−1)-tuple X ⊂ V s is normal. This means that there are at most 2 −3k ǫn vertices x ∈ V −(P ∪X) for which X ∪ x is improper, so there is a ∈ V s and b ∈ V s ′ such that X ∪ a and X ∪ b are proper. For any pair of indices i, j we have c X (a) + s i = c X (p i ), c X (a) + s j = c X (p j ), c X (b) + s ′ i = c X (p i ) and c X (b) + s ′ j = c X (p j ). Adding these equations gives s i + s j + s ′ i + s ′ j = 0. If s and s ′ differ in some co-ordinate i then this equation shows that they must also differ in any other co-ordinate j. In other words, if s ′ = s we must have s ′ = s, where s denotes the sequence whose ith entry is 1 − s i . Let V s be the largest class, and write m = |V s |. Clearly m ≥ 2 −k (n − k). Then all other classes, except possibly V s , have size at most 2 −2k ǫn. Let A i be the set of proper k-tuples contained in V s that have colour i. Then |A 0 | + |A 1 | > m k − 10 −4k 2 ǫ k n k > (1 − 10 −3k 2 ǫ) m k . Write |A i | = α i m k , so that α 0 + α 1 > 1 − 10 −3k 2 ǫ. Suppose both α i are at least 10 −2k 2 ǫ. Observe that |A i | = α 1/k i m k + O(m k−1 ), so by Proposition 3.3 we have |∂A i | ≥ α 1/k i m k − 1 + O(m k−2 ) = α (k−1)/k i m k − 1 + O(m k−2 ). Note that if z ≤ 2 −k we have that z −1/k ≥ 2 and therefore z (k−1)/k + (1 − 10 −3k 2 ǫ − z) (k−1)/k ≥ z (k−1)/k + 1 − 10 −3k 2 ǫ − z ≥ 2z + 1 − 10 −3k 2 ǫ − z = 1 + z − 10 −3k 2 ǫ. Since z (k−1)/k +(1−10 −3k 2 ǫ−z) (k−1)/k is concave we have α (k−1)/k 0 +α (k−1)/k 1 ≥ 1+10 −2k 2 ǫ−10 −3k 2 ǫ ≥ 1 + 10 −3k 2 ǫ. We deduce that |∂A 0 ∩ ∂A 1 | > 0, i.e. there is a (k − 1)-tuple X and points a 0 , a 1 such that X ∪ a i is proper, with c X (a i ) = i. But equation (8) gives i + s 1 = c X (a i ) + s 1 = c X (p 1 ), for i = 0, 1, which is a contradiction. We conclude that there is t ∈ {0, 1} for which α 1−t < 10 −2k 2 ǫ, and so all but at most 10 −2k 2 ǫ m k + 10 −4k 2 ǫ k n k < 10 −2k 2 ǫn k k-tuples inside V s have the same colour t. For 0 ≤ i ≤ k let D i be all k-tuples with i points in V s and k − i points in V s and let θ i = 10 −2k 2 2k2 2k i ǫ. We claim that for each i all but at most θ i n k k-tuples of D i have colour t + i (mod 2). Otherwise, choose the smallest i for which this is not true. By the above discussion i > 0, and there are at least θ i n k k-tuples in D i with color 1 − (t + i) = t + i − 1 (mod 2). Since i was the smallest such index all but at most θ i−1 n k k-tuples of D i−1 have colour t + i − 1 (mod 2). Let E i−1 be the (k − 1)-tuples Y with i − 1 points in V s and k − i points in V s for which there are at least 2 −2k n points y ∈ V s such that Y ∪ y does not have colour t + i − 1 (mod 2). Then |E i−1 | ≤ kθ i−1 n k /(2 −2k n) = 1 2 θ i n k−1 , so at most 1 2 θ i n k k-tuples contain an element of E i−1 . Recall that there are at most 10 −4k 2 ǫ k n k improper k-tuples and at most 10 −3k 2 ǫ k−1 n k−1 · n = 10 −3k 2 ǫ k−1 n k k-tuples that contain some abnormal (k − 1)-tuple. Since 10 −4k 2 ǫ k + 10 −3k 2 ǫ k−1 < 10 −2k 2 −1 ǫ < θ i /2 we can find a proper k-tuple K ∈ D i such that K has color t + i − 1 (mod 2) and for any (k − 1)-tuple Y ⊂ K we have Y normal and Y / ∈ E i−1 . Since i > 0, there is x ∈ K ∩ V s . Let Y = K − x. Since Y is normal there are at most 2 −3k ǫn vertices y such that Y ∪ y is improper, and by definition of E i−1 there are at most 2 −2k n points y ∈ V s such that Y ∪ y does not have colour t + i − 1 (mod 2). Since 2 −3k ǫn + 2 −2k n < 2 −k (n − k) there is y ∈ V s such that Y ∪ y is proper and has colour t + i − 1 (mod 2). Then c Y (x) = c Y (y) = t + i − 1 (mod 2). But x ∈ V s and y ∈ V s , so c Y (x) + 1 − s 1 = c Y (p 1 ) and c Y (y) + s 1 = c Y (p 1 ), both mod 2. This is a contradiction, so we conclude that all but at most θ i n k k-tuples of D i have colour t + i. Now partition V into 2 classes V 1 , V 2 so that V s ⊂ V 1 , V s ⊂ V 2 , and the other vertices are distributed arbitarily. Incorrect 2k-tuples with respect to this partition belong to the one of the following three groups. (i) Improperly colored 2k-tuples. There are at most 10 −5k 2 ǫ k n 2k of those. (ii) Properly colored 2k-tuples which use at least one vertex not in V s ∪ V s . There are at most 2 k 2 −2k ǫn n 2k−1 < 2 −k ǫn 2k such 2k-tuples. (iii) Properly colored 2k-tuples which contain a k-tuple of D i with colour t + i − 1 (mod 2). There are at most k i=0 θ i n k n k < θ k n 2k = 10 −2k 2 2k2 2k k ǫn 2k < 10 −k 2 ǫn 2k such 2k-tuples. Therefore all but at most 10 −5k 2 ǫ k + 2 −k ǫ + 10 −k 2 ǫ n 2k < ǫn 2k 2k-tuples are correct with respect to this partition. This completes the proof of the theorem. The Turán number of C (2k) 3 In this subsection we complete the proof of Frankl's conjecture. Proof of Theorem 1.1. Let H be a 2k-uniform hypergraph on n vertices, which has e(H) ≥ b 2k (n) and contains no C (2k) 3 . By the same argument given in the proof in the case k = 2 we can assume that H has minimum degree at least b 2k (n) − b 2k (n − 1). Applying Lemma 3.2 gives δ(H) ≥ 1 2 n − 1 2k − 1(9) For convenience of notation we set η = (100k) −10 k . By Theorem 3.4 there is a partition with all but at most (η/20k) 2k n 2k edges of H being good, i.e., they have odd intersection with both parts. Let V (H) = V 1 ∪ V 2 be the partition which minimises the number of bad edges. Then every vertex belongs to at least as many good edges as bad edges, or we can move it to the other class of the partition. Recall that, by definition, the number of good 2k-tuples with respect to this partition is at most b 2k (n). We must have |V 1 | − n/2 < 1 10 ηn and |V 2 | − n/2 < 1 10 ηn. Otherwise by Lemma 3.1, part (5) e(H) < 1 2 n 2k − 1 2 2 · 1 10 ηn 2k + (10 · 1 10 η) k n 2k−1 + (η/20k) 2k n 2k < b 2k (n), which is a contradiction. Note that there is no k-tuple of vertices P for which there are both (10k) −k ηn k k-tuples Q such that P ∪Q is a good edge and (10k) −k ηn k k-tuples R such that P ∪R is a bad edge. Indeed, each such Q and R which are disjoint give a 2k-tuple Q ∪ R which is good, but cannot be an edge as it would create a C (2k) 3 . Moreover, every 2k-tuple can be obtained at most 1 2 2k k times in this way, and every Q is disjoint from all but at most k n k−1 k-tuples R. Thus at least (10k) −k ηn k (10k) −k ηn k − k n k−1 / 1 2 2k k > 2(η/20k) 2k n 2k good 2k-tuples are not edges of H, and therefore e(H) < b 2k (n) − 2(η/20k) 2k n 2k + (η/20k) 2k n 2k < b 2k (n), which is a contradiction. Proof. Suppose some vertex a belongs to ηn 2k−1 bad edges. Call a (k − 1)-tuple X good if there are at most (10k) −k ηn k k-tuples Q such that a ∪ X ∪ Q is a bad edge, otherwise call X bad. By the above discussion, for every bad (k − 1)-tuple X there are at most (10k) −k ηn k k-tuples R such that a ∪ X ∪ R is a good edge. There are at least ηn k−1 bad (k − 1)-tuples or we would only have ηn k−1 · n k + n−1 k−1 − ηn k−1 · (10k) −k ηn k < ηn 2k−1 bad edges through a. By choice of partition there are at least as many good edges containing a as bad. From (9) we see that a is in at least Suppose there are α n k−1 good (k − 1)-tuples, where by the above we see that (2k) −k−1 ≤ α ≤ 1 − (k − 1)!η. We can count the good edges containing a as follows. By definition there are at most n k−1 · (10k) −k ηn k such good edges containing a bad (k − 1)-tuple. Note that in the remaining good edges every (k − 1)-tuple is good. Given any such edge W containing a we consider ordered triples (X, Y, b), where X and Y are (k − 1)-tuples, b is a vertex and X ∪ Y ∪ b ∪ a = W . Each edge gives rise to k 2k−1 k−1 such triples. To bound the number of triples recall that X and Y are good, so can be chosen in at most α n k−1 2 ways. Once X and Y have been chosen, to make E good b is constrained to lie in some particular class V i of the partition, so can be chosen in at most 1 2 + 1 10 η n ways. This shows that the number of good edges not containing bad (k − 1)-tuples is at most α n k − 1 2 1 2 + 1 10 η n k 2k − 1 k − 1 < α 2 + 3 · 1 10 η 1 2 n − 1 2k − 1 We can count the bad edges similarly, and deduce that the total number of edges containing a is at most α 2 + (1 − α) 2 + 6 · 1 10 η 1 2 n − 1 2k − 1 + 2 · n k − 1 · (10k) −k ηn k . From the bounds (2k) −k−1 ≤ α ≤ 1 − (k − 1)!η we see that this is at most 1 2 − η/2 n−1 2k−1 . This contradicts equation (9), so the claim is proved. Now write |V 1 | = n/2 + t, |V 2 | = n/2 − t with − 1 10 ηn < t < 1 10 ηn. By possibly renaming the classes (i.e. replacing t with −t) we can assume that d(n, t) < d(n, −t). Now any vertex of V 1 belongs to d(n, t) good 2k-tuples, and d(n, t) is the minimum degree of B (2k) (n, t), which is certainly at most the maximum degree of B 2k−1 + (20kn) k−1/2 but we will only use the weaker bound d(n, t) < 1 2 n−1 2k−1 + 10 4k 2 n k−1/2 . Later we will show that this weaker bound also holds for d(n, −t), and then the subsequent argument will apply switching V 1 and V 2 . Proof. 1. By the preceding remarks a belongs to at most 1 2 n−1 2k−1 + 10 4k 2 n k−1/2 good 2k-tuples and therefore it belongs to at most 1 2 n−1 2k−1 + 10 4k 2 n k−1/2 − K good edges. Then by equation (9) a belongs to at least K − 10 4k 2 n k−1/2 bad edges. Before proving the next claim we make a remark that will be used on several occasions without further comment. Suppose W is a bad edge, so that |W ∩ V i | is even for i = 1, 2. If we partition W = P ∪ Q with |P | = |Q| = k then |P ∩ V i | = |Q ∩ V i | (mod 2) for i = 1, 2. Then for any k-tuple R with |R ∩ V i | = |P ∩ V i | + 1 (mod 2) both 2k-tuples P ∪ R and Q ∪ R are good. We can obtain such a k-tuple R ⊂ V − (P ∪ Q) by picking any (k − 1)-tuple, and then another vertex which, because of parity, is constrained to lie in some particular V i . This counts each k-tuple k times, so the number of choices for R is at least k −1 n − 2k k − 1 1 2 − 1 10 η n − 3k > n k /(3 · k!). Claim 3.7 Suppose t ≤ k and T is a t-tuple of vertices belonging to θn 2k−t bad edges, for some θ > (20k) k η. Then any S ⊂ T with |S| = t − 1 belongs to at least (10k) −k θn 2k−t+1 good non-edges. Proof. Write T = S ∪ v. Consider a bad edge W containing T and a partition W = T ∪ X ∪ Y , where |X| = k − 1 and |Y | = k + 1 − t. By the above remark, there are at least n k /(3 · k!) k-tuples R for which v ∪ X ∪ R and S ∪ Y ∪ R are both good 2k-tuples. Note that they can't both be edges, or we would have a copy of C (2k) 3 . Suppose that for at least 1 2 θn 2k−t such W there is a partition W = T ∪ X ∪ Y for which there are at least n k /2(3 · k!) k-tuples R for which v ∪ X ∪ R is a good non-edge. This clearly gives at least 1 2 θn k−1 choices for X. Each such non-edge can be partitioned in at most 2k−1 k ways in the form v ∪ X ∪ R, so there are at least 2k − 1 k −1 1 2 θn k−1 n k 2(3 · k!) > (10k) −k θn 2k−1 good non-edges containing v. Now Claim 3.6 shows that there are at least (10k) −k θn 2k−1 − (ηn/5) 2k−1 > (20k) −k θn 2k−1 > ηn 2k−1 bad edges containing v, which contradicts Claim 3.5. It follows that for at least 1 2 θn 2k−t such W and any partition of W = T ∪ X ∪ Y we have a good non-edge S ∪ Y ∪ R for at least n k /2(3 · k!) k-tuples R. This gives at least 1 2 θn k+1−t choices for Y . Each such non-edge has at most 2k−t+1 k representations as S ∪ Y ∪ R, so there are at least 2k − t + 1 k −1 1 2 θn k+1−t n k 2(3 · k!) > (10k) −k θn 2k−t+1 good non-edges containing S. Suppose for the sake of contradiction that there is some bad edge incident with V 1 . Denote the set of bad edges containing some vertex v by Z(v). Let a be a vertex in V 1 belonging to the maximum number of bad edges and let Z = |Z(a)|. Note that Z > 0. Claim 3.8 Suppose t ≤ k, (20k) k η < φ < (100k) −k and F is a set of at least φZn −(2k−t) t-tuples containing a such that each F ∈ F is contained in at least φn 2k−t bad edges. Then there are at least φ 5 Zn −(2k−t+1) (t − 1)-tuples containing a each of which is contained in at least φ 5 n 2k−t+1 bad edges. Proof. Let G be the set of (t − 1)-tuples containing a that are contained in a member of F. By claim 3.7 each G ∈ G is contained in at least (10k) −k φn 2k−t+1 good non-edges. Each such good nonedge is counted by at most 2k−1 t−2 different G's, so there are at least 2k−1 t−2 −1 |G|(10k) −k φn 2k−t+1 > (40k) −k |G|φn 2k−t+1 good non-edges containing a. Since a ∈ V 1 Claim 3.6 gives at least (40k) −k |G|φn 2k−t+1 − 10 4k 2 n k−1/2 > (50k) −k |G|φn 2k−t+1 bad edges containing a, so by definition of Z we get |G| < (50k) k φ −1 Zn −(2k−t+1) . Let G 1 ⊂ G consist of those G that belong to at least φ 3 n members of F. Then φZn −(2k−t) ≤ |F| < |G 1 |n + |G|φ 3 n < |G 1 |n + (50k) k φ 2 Zn −(2k−t) so |G 1 | > φ 5 Zn −(2k−t−1) with room to spare. For each G ∈ G 1 there are at least φ 3 n sets of F each contributing φn 2k−t bad edges containing G. Each such bad edge is counted by at most 2k − t + 1 different F ∈ F, so G belongs to at least (2k − t + 1) −1 φ 3 n · φn 2k−t > φ 5 n 2k−t+1 bad edges. Let Z 1 (a) be those bad edges W containing a for which there is some partition into two k-tuples W = P ∪ Q with a ∈ P so that there are at least n k /2(3 · k!) k-tuples R for which P ∪ R is a good non-edge. Let Z 2 (a) = Z(a) − Z 1 (a), and write Z i = |Z i (a)| for i = 1, 2. Then one of Z 1 ,Z 2 is at least Z/2. Case 1: Suppose Z 1 ≥ Z/2. Let P be the (non-empty) set of k-tuples P containing a such that there is some edge P ∪ Q in Z 1 (a), and P ∪ R is a good non-edge for at least n k /2(3k!) ktuples R. Each such good non-edge is counted by at most 2k−1 k−1 different P 's, so there are at least 2k−1 k−1 −1 |P|n k /2(3 · k!) > (10k) −k |P|n k good non-edges containing a. Now Claim 3.6 gives at least (10k) −k |P|n k − 10 4k 2 n k−1/2 > (20k) −k |P|n k bad edges containing a, so by definition of Z, |P| < (20k) k Zn −k . On the other hand, let P 1 ⊂ P consist of those P that belong to at least 1 10 (20k) −k n k bad edges. Then Z/2 ≤ Z 1 < |P 1 |n k + |P| 1 10 (20k) −k n k < |P 1 |n k + Z/10 so |P 1 | > 0.4 Zn −k . Now apply Claim 3.8 k − 1 times, starting with t = k and φ = (100k) −k . We deduce that a belongs to at least φ 5 k−1 n 2k−1 > ηn 2k−1 bad edges, which contradicts Claim 3.5. Case 2: Now suppose Z 2 ≥ Z/2. Note that every bad edge containing a contains at least one other point of V 1 , so there is some b ∈ V 1 belonging to at least Z 2 /n edges of Z 2 (a). Fix one such b. Let X be the set of (k − 1)-tuples X for which there exists a (k − 1)-tuple Y such that W = a ∪ b ∪ X ∪ Y is an edge of Z 2 (a). By definition of Z 2 (a) for any such partition of W , there are at least n k /2(3·k!) k-tuples R such that b ∪ X ∪ R is a good non-edge. This gives at least n k 2(3·k!) |X | > (10k) −k |X |n k good non-edges containing b, and since b ∈ V 1 Claim 3.6 gives at least (10k) −k |X |n k − 10 4k 2 n k−1/2 > (20k) −k |X |n k bad edges containing b. Thus, by definition of Z, |X | < (20k) k Zn −k . Note that each edge in Z 2 (a) that contains b is obtained by picking a pair of (k − 1)-tuples in X , so Z/(2n) ≤ Z 2 /n ≤ |X | 2 < 1 2 (20k) 2k Z 2 n −2k . Therefore Z > (20k) −2k n 2k−1 > ηn 2k−1 , which contradicts Claim 3.5. We conclude that there are no bad edges incident to the vertices of V 1 , i.e. all bad edges are entirely contained in V 2 . As in the case k = 2 this gives a more precise bound on t, defined by |V 1 | = n/2 + t, |V 2 | = n/2 − t. If |t| ≥ 20 k √ n then Lemma 3.1, part (4) gives d 2k (n, t) < 1 2 n−1 2k−1 − 20 k n k−1/2 . This is a contradiction, since the vertices of V 1 only belong to good edges, of which there are at most d 2k (n, t) < δ(H). Therefore |t| < 20 k √ n. Now Lemma 3.1, part (2) gives d 2k (n, −t) < 1 2 n − 1 2k − 1 + 10(20 k ) 2 k n k−1/2 < 1 2 n − 1 2k − 1 + 10 4k 2 n k−1/2 . As we remarked earlier, this bound allows us to repeat the above argument interchanging V 1 and V 2 , so we deduce that there are no bad edges incident with V 2 either, i.e. all edges are good. Then by definition of b 2k (n) we have e(H) ≤ b 2k (n), with equality only when H is a B (2k) n , so the theorem is proved. Hypergraphs without C (4) r We recall that C (2k) r is the 2k-uniform hypergraph obtained by letting P 1 , · · · , P r be pairwise disjoint sets of size k and taking as edges all sets P i ∪ P j with i = j. In this section we will be concerned with the case k = 2 and general r. Sidorenko [8] showed that the Turán density of C (2k) r is at most r−2 r−1 . This is a consequence of Turán's theorem applied to an auxiliary graph G constructed from a 2k-uniform hypergraph H of order n. The vertices of G are the k-tuples of vertices of H, and two k-tuples P 1 ,P 2 are adjacent if P 1 ∪ P 2 is an edge of H. It is easy to see that the graph G has n k vertices, 1 2 2k k e(H) edges and contains no K r . Thus the upper bound on the number of edges of H follows immediately from Turán's theorem. The following construction from [8] gives a matching lower bound when r is of the form 2 p + 1. Let W be a vector space of dimension p over the field GF (2), i.e. the finite field with 2 elements {0, 1}. Partition a set of vertices V as w∈W V w , |V w | = |V |/(r − 1). Given t and a t-tuple of vertices X = x 1 · · · x t with x i ∈ V w i we define ΣX = t 1 w i . Define a 2k-uniform hypergraph H, where a 2ktuple X is an edge iff ΣX = 0. Observe that this doesn't contain a copy of C (2k) r . Indeed, if P 1 , · · · , P r are disjoint k-tuples then there is some i = j with ΣP i = ΣP j (by the pigeonhole principle). Then Σ(P i ∪ P j ) = ΣP i + ΣP j = 0, so P i ∪ P j is not an edge. This construction depends essentially on an algebraic structure, which only exists for certain values of r. Perhaps surprisingly, we will show that this is an intrinsic feature of the problem, by proving Theorem 1.2, which gives a stronger upper bound on the Turán density of C (4) r , when r is not of the form 2 p + 1. We make no attempt to optimize the constant in this bound. In addition, our proof of this theorem implies that, for r = 2 p +1, any C The rest of this section is organized as follows. In the first subsection we will prove a lemma showing that certain edge-colourings of the complete graph K s exist only if s is a power of 2. In the following subsection we will recall a proof of the Simonovits stability theorem so that we can calculate some explicit constants. The final subsection contains the proof of Theorem 1.2. Proof. Since the number of colours is s − 1, every colour is a matching and the total number of edges in K s is s(s − 1)/2 it is easy to see that every colour is a perfect matching. Also, if wx and yz are disjoint edges of the same colour, then by hypothesis only 3 different colours appear on wxyz, so wy and xz have the same colour, as do xy and wz. Denote the set of colours by C = {c 1 , · · · , c s−1 }. We define a binary operation + on C using the following rule. Pick a vertex x. Given c i and c j let e i = xy i and e j = xy j be the edges incident with x with these colours. These edges exist, as each colour is a perfect matching. Define c i + c j to be the colour of y i y j . A lemma on edge-colourings of a complete graph To see that this is well-defined, let x ′ be another vertex and suppose e ′ i = x ′ y ′ i has colour c i and e ′ j = x ′ y ′ j has colour c j . If y i = y ′ j then opposite edges of xy j y i x ′ have the same colours, so x ′ y j has colour c i , i.e. y j = y ′ i and there is nothing to prove. Therefore we can assume that all y i , y j , y ′ i , y ′ j are distinct. Consider the 4-tuple xx ′ y i y ′ i . Since xy i and xy ′ i have the same colour we deduce that xx ′ and y i y ′ i have the same colour. Similarly xx ′ and y j y ′ j have the same colour, from which we see that y i y ′ i and y j y ′ j have the same colour. Now looking at y i y ′ i y j y ′ j we see that y i y j and y ′ i y ′ j have the same colour, so c i + c j is well-defined. Let D be a set obtained by adjoining another element called 0 to C. Extend + to an operation on D by defining 0 + d = d + 0 = d and d + d = 0 for all d ∈ D. We claim that (D, +) is an abelian group. Note that + is commutative by definition, 0 is an identity and inverses exist. It remains to show associativity, i.e. for any d 1 , d 2 , d 3 we have (d 1 + d 2 ) + d 3 = d 1 + (d 2 + d 3 ) . This is immediate if any of the d i are 0 or if they are all equal. If d 1 = d 2 = d 3 then d 1 + d 2 = 0 and there is a triangle with colours d 1 , d 3 , d 1 + d 3 , so d 1 + (d 2 + d 3 ) = d 3 as required. The same argument applies when d 2 = d 3 = d 1 . If d 1 = d 3 then d 1 + d 2 = d 2 + d 3 by commutativity, and so (d 1 + d 2 ) + d 3 = d 1 + (d 2 + d 3 ) also by commutativity. So we can assume that the d i are pairwise distinct and non-zero. Pick a vertex x, let xy 1 be the edge of colour d 1 and xy 2 the edge of colour d 2 . Let y 2 z have colour d 3 . We can suppose z = y 1 , otherwise d 1 + d 2 = d 3 and d 2 + d 3 = d 1 and (d 1 + d 2 ) + d 3 = d 1 + (d 2 + d 3 ) = 0. Now y 1 y 2 has colour d 1 + d 2 and xz has colour d 2 + d 3 . Consider the edge y 1 z. From the triangle it forms with x we see that it has colour d 1 + (d 2 + d 3 ) and from the triangle with y 2 we see that it has colour (d 1 + d 2 ) + d 3 . This proves associativity, so D is an abelian group. Finally, note that every non-zero element has order 2, so D is in fact a vector space over the field with 2 elements. If p is its dimension then s = |D| = 2 p . The Simonovits stability theorem In this subsection we will recall a proof of the Simonovits stability theorem [10] so that we can calculate some explicit constants. Let T s (N ) be the s-partite Turán graph on N vertices, i.e. a complete s-partite graph with part sizes as equal as possible. Write t s (N ) for the number of edges in T s (N ). Then Turán's theorem states that any K s+1 -free graph on N vertices has at most t s (N ) edges, with equality only for T s (N ). It is easy to show that s−1 s N 2 /2 − s < t s (N ) ≤ s−1 s N 2 /2. Proof. By Turán's theorem G contains a copy of K s ; let A = {a 1 , · · · , a s } be its vertex set. Note that any vertex x not in A has at most s − 1 neighbours in A, or we get a K s+1 . Let B be those vertices with exactly s − 1 neighbours in A, and C = V (G) − A − B. Partition A ∪ B as U 1 ∪ · · · ∪ U s where U i consists of those vertices adjacent to A − a i . Then there are no edges inside any U i , as if xy is such an edge then xy + A − a i forms a K s+1 . Distribute the vertices of C arbitrarily among the U i . Counting edges between A and V − A gives sδ(G) ≤ e(A, V − A) ≤ (s − 1)|B| + (s − 2)|C| = (s − 1)(N − s) − |C| so |C| ≤ sαN − s(s − 1). Therefore e(U i ) < sαN 2 . Theorem 4.4 Suppose G is a K s+1 -free graph on N vertices with at least s−1 2s − c N 2 edges and c < 1/(4s 4 ). Then there is a partition of the vertex set of G as V (G) = U 1 ∪ · · · ∪ U s with e(U i ) < (2s + 1) √ c N 2 . Proof. Construct a sequence of graphs G = G N , G N −1 , · · · where if G m has a vertex of degree at most 1 − 1 s − 2 √ c m then we delete it to get G m−1 . Suppose can delete √ c N vertices by this process and reach a graph G (1− √ c)N . Then G (1− √ c)N is K s+1 -free and has at least s − 1 2s − c − √ c 1 − 1 s − 2 √ c N 2 > s − 1 2s (1 − √ c) 2 N 2 edges. This contradicts Turán's theorem, so the sequence terminates at some G m with m ≥ (1 − √ c)N and minimum degree at least 1 − 1 s − 2 √ c m. By Proposition 4.3 there is a partition V (G m ) = U 1 ∪ · · · ∪ U s with e(U i ) < 2s √ c N 2 . Proof of Theorem 1.2 Let V be the vertex set of H. Define a graph G whose vertices are all pairs in V , where the pairs ab and cd are adjacent exactly when abcd is an edge of H. Since H is C r -free we see that G is K r -free. Also, each edge of H creates exactly 3 edges in G (corresponding to the 3 ways of breaking a 4-tuple into pairs) so e(G) > 3 r − 2 r − 1 − 10 −33 r −70 n 4 > r − 2 2(r − 1) − 10 −33 r −70 N 2 , where N = n 2 . Applying Theorem 4.4 with s = r − 1 gives a partition of the pairs of vertices in V as r−1 1 P i with r−1 1 e(P i ) < 10 −16 r −34 N 2 . If there is some P i with |P i | < 1 r−1 − 10 −3 r −7 N then e(G) N 2 < r−2 2 (r − 2) 2 r − 2 r − 1 + 10 −3 r −7 2 + 1 r − 1 − 10 −3 r −7 r − 2 r − 1 + 10 −3 r −7 + 10 −16 r −34 < r − 2 2(r − 1) − 10 −6 r −14 /2 + 10 −16 r −34 . This is a contradiction so |P i | ≥ 1 r−1 − 10 −3 r −7 N for all i. Also if some |P i | > 1 r−1 + 10 −3 r −6 N , then there is j such that |P j | < 1 r−1 − 10 −3 r −7 N . Therefore for all i |P i | − 1 r − 1 N ≤ 10 −3 r −6 n 2 .(10) Note that all but at most 10 −16 r −34 n 4 edges of H are formed by taking a pair from P i and a pair from P j with i = j. We think of the P i as a colouring of pairs. A 4-tuple abcd will be called properly coloured if either (i) abcd is an edge and each of the 3 sets {ab, cd},{ac, bd},{ad, bc} contains two pairs with different colours, or (ii) abcd is not an edge and each of the 3 sets {ab, cd},{ac, bd},{ad, bc} consists of two pairs with the same colour. An improperly coloured 4-tuple is either an edge that is the union of two pairs of the same colour or a non-edge which is the union of two pairs with different colours. There are at most 10 −16 r −34 N 2 of the former 4-tuples, and the number of latter is at most r − 2 2(r − 1) N 2 − e(G) − 10 −16 r −34 N 2 ≤ 10 −16 r −34 + 10 −33 r −70 n 4 . Therefore all but 10 −15 r −34 n 4 4-tuples are properly coloured. Call a pair ab bad if there are at least 10 −12 r −32 n 2 pairs cd such that abcd is improperly coloured; otherwise call it good. Then there are at most 4 2 (10 −15 r −34 n 4 )/(10 −12 r −32 n 2 ) < 10 −2 r −2 n 2 bad pairs. Consider a graph on V whose edges are the pairs in P 1 . As noted in (10) it has at least 1 r−1 N − 10 −3 r −6 n 2 edges. For vertices a and b in V , let d(a) denote the degree of a and d(a, b) the codegree of a and b (i.e. the size of their common neighbourhood.) Then a,b∈V d(a, b) = c∈V d(c) 2 ≥ n d(c)/n 2 = n 2|P 1 |/n 2 > 1 5r 2 nN. Suppose there are at most m pairs (a, b) for which d(a, b) > n 10r 2 . Then 1 5r 2 nN < d(a, b) ≤ mn + N n 10r 2 , so 1 10r 2 N < m, i.e. there are at least 1 10r 2 n 2 pairs (a, b) for which d(a, b) > n 10r 2 . At least one such pair is good, as the number of bad pairs is at most 10 −2 r −2 n 2 < 1 20r 2 n 2 . Let (a, b) be such a pair and suppose it belongs to P t . Let B be the set of pairs cd for which abcd is improperly coloured. Since ab is good we have |B| ≤ 10 −12 r −32 n 2 . Therefore there are at most |B| n 2 < 10 −12 r −32 n 4 4-tuples of vertices that contain any pair of B. We will call a 4-tuple normal if it is properly coloured and does not contain a pair from B; otherwise we call it abnormal. Then all but at most 10 −12 r −32 n 4 + 10 −15 r −34 n 4 < 10 −11 r −32 n 4 4-tuples are normal. Partition the vertices of V − ab into (r − 1) 2 sets U ij , where c is in U ij iff ac ∈ P i and bc ∈ P j . Then by the above discussion |U 11 | ≥ n 10r 2 . Now we claim that for i = j we have |U ij | < 10 −3 r −11 n. For suppose that |U ij | ≥ 10 −3 r −11 n. Let P k be the colour that appears most frequently among pairs joining vertices of U 11 to U ij . Then there are at least 1 r−1 |U 11 ||U ij | pairs of color P k with one endpoint in U 11 and the other in U ij . Consider the 4-tuples of the form c 1 c 2 d 1 d 2 , with c 1 , c 2 ∈ U 11 , d 1 , d 2 ∈ U ij and c 1 d 1 , c 2 d 2 ∈ P k . There are at least 1 r − 1 |U 11 ||U ij | 1 r − 1 |U 11 ||U ij | − 2n /4 > r −2 10 −1 r −2 n · 10 −3 r −11 n 2 /4 > 10 −11 r −32 n 4 such 4-tuples, so some c 1 c 2 d 1 d 2 is normal. By definition of normality each of its pairs forms a properly coloured 4-tuple with ab. Since ac 1 and bc 2 are in P 1 and ab is in P t we deduce that c 1 c 2 is in P t as well. Also ad 1 ∈ P i , bd 2 ∈ P j and i = j, so d 1 d 2 cannot be in P t . But c 1 d 1 and c 2 d 2 both belong to P k so c 1 c 2 d 1 d 2 is improperly coloured. This contradicts the definition of normality, so we do have |U ij | < 10 −3 r −11 n. For convenience write U i = U ii . Then all but at most (r −1) 2 10 −3 r −11 n ≤ 10 −3 r −9 n vertices belong to one of the U i . Suppose cd is a pair such that abcd is properly coloured. Since ab is a good pair, this is the case for all but at most 10 −11 r −32 n 2 pairs cd. If c and d both belong to some U i then ac and bd both have colour i. Since ab has colour t we see that cd has colour t. Similarly, if c ∈ U i and d ∈ U j with i = j we see that cd cannot have colour t. Let E i denote the pairs with both endpoints in U i , so that |E i | = |U i | 2 . By the above discussion, all but at most 10 −12 r −32 n 2 pairs in ∪ i E i belong to P t . Suppose |U i | < 1 r−1 − 10 −1 r −3 n for some i, so that By (10), this gives the following contradiction. Therefore |U i | ≥ 1 r−1 − 10 −1 r −3 n for each i. Let E ij denote the edges with one endpoint in U i and the other in U j . We claim that one colour is dominant among these edges, i.e. there is some q such that all but 10 −2 r −4 n 2 edges of E ij belong to P q . Indeed, suppose that there are colours q 1 and q 2 for which there are at least 10 −2 r −4 n 2 edges in E ij of color q i for i = 1, 2. Then there are at least (10 −2 r −4 n 2 )(10 −2 r −4 n 2 − 2n) > 10 −11 r −32 n 4 4-tuples c 1 c 2 d 1 d 2 with c 1 , c 2 in U i , d 1 , d 2 in U j and c i d i of colour q i . At least one such 4-tuple c 1 c 2 d 1 d 2 is normal, since there at most 10 −11 r −32 n 4 abnormal 4-tuples. But then c 1 c 2 and d 1 d 2 both have colour t, so c 1 c 2 d 1 d 2 is improperly coloured, which is a contradiction. |E i | > 1 r − 1 − 10 − Consider the complete graph K r−1 on the vertex set {1, · · · , r − 1} and colour edge ij with the dominant colour of E ij . We show that this colouring satisfies the hypotheses of Lemma 4.2. First of all we show that colour t doesn't occur in this edge-coloring of K r−1 , i.e. there are only r − 2 colours. Suppose ij has colour t. Then is a contradiction. Now suppose that some colour ℓ is not a matching, i.e. there are edges ij and ik in K r−1 both of colour ℓ. Then all but at most 2 · 10 −2 r −4 n 2 pairs of E ij ∪ E ik have colour ℓ. Consider the 4-tuples of the form c 1 c 2 de, with c 1 , c 2 ∈ U i , d ∈ U j and e ∈ U k , such that c 1 d, c 2 d, c 1 e, c 2 e all have colour ℓ. There are at least such 4-tuples, so there is one such c 1 c 2 de which is normal. But then c 1 c 2 has colour t and de cannot have colour t, since by normality abde is properly colored. Therefore c 1 c 2 de is improperly coloured. This is a contradiction, so each colour forms a matching. |U i | 2 |U j ||U k | − 2 · 10 − It remains to show that if some 4 vertices x 1 x 2 x 3 x 4 in K r−1 do not span 6 different colours then they span only 3 colours. Suppose that x 1 x 2 and x 3 x 4 have colour α, x 1 x 3 has colour β and x 2 x 4 has colour γ. Recall that all but at most 10 −2 r −4 n 2 pairs in E x i x j have the corresponding color of x i x j . Consider the 4-tuples in H of the form c 1 c 2 c 3 c 4 with c i ∈ U x i such that c i c j has the same colour as x i x j . There are at least Finally it is not difficult to check that, when r = 2 p + 1, the above arguments together with the proof of Lemma 4.2 imply Corollary 4.1. Concluding remarks Among the various techniques that we used in this paper, the stability approach stands out as one that should be widely applicable in extremal combinatorics. The process of separating the argument into a stability stage and a refinement stage focuses attention on the particular difficulties of each, and often leads to progress where the raw problem has appeared intractable. For recent examples we refer to our proofs of the conjecture of Sós on the Turán number of the Fano plane [4], and a conjecture of Yuster on edge colorings with no monochromatic cliques [1]. Our methods probably apply to C (2k) r for general k when r is of the form 2 p + 1, although the reader who has grappled with the thornier aspects of this paper will note the formidable technical difficulties that would arise. It would be far more interesting to say more about the behaviour of the Turán density of C (2k) r for general r. Even C (4) 4 presents an enigma for which there is no obvious plausible conjecture. We find it remarkable that the seemingly similar hypergraphs C free hypergraph of density r−2 r−1 − o(1) looks approximately like Sidorenko's construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of C − c(r), where c(r) is a constant depending only on r. . Our first theorem proves this conjecture.Theorem 1.1 Let H be a 2k-uniform hypergraph on n vertices that does not contain a copy of C (2k) 3 and let n be sufficiently large. Then the number of edges in H is at most b 2k (n), with equality only when H is a hypergraph of the form B (2k) n . ǫn 4 4-tuples are correct with respect to this partition. Claim 2. 3 3Any vertex of H is contained in at most 10 −5 n 3 bad edges. 1) edges and has parts V (H) = A ∪ B. Let H 1 be obtained from H by adding a vertex v 1 to A, together with all the 2k-tuples containing v 1 and having odd intersections with A ∪ v 1 and B. Let H 2 be similarly obtained by adding a vertex v 2 to B, together with corresponding edges. By definition each H i has at most b 2k (n) edges, so the degree of each v i is a lower bound for b 2k (n) − b 2k (n − 1). On the other hand, for each (2k − 1)-tuple X of vertices in H there is exactly one i such that X ∪ v i is an edge of H i , so one of the v i has degree at least (2k) 3 . 3We start by recalling a version of the Kruskal-Katona theorem due to Lovasz. Write [m] = {1, · · · , m}, let [m] (k) denote the subsets of [m] of size k, and suppose A ⊂ [m] (k) . The shadow of A is ∂A ⊂ [m] (k−1) consisting of all sets of size k − 1 that are contained in some element of A. For any real x write x k Proposition 3. 3 3If A ⊂ [m] (k) and |A| = x k then |∂A| ≥ x k−1 . we see that G is triangle-free. Also, each edge of H creates exactly 1 2 2k k edges in G (corresponding to the ways of breaking a 2k-tuple into two k-tuples) − ηn 2k > 1 − (k!) 2 2 2k η Claim 3. 5 5Any vertex of H is contained in at most ηn 2k−1 bad edges. 1 − (10k) −k ηn 2k−1 / n−1 k ≥ n k−1 /(2k)! good (k − 1)-tuples. Claim 3 .6 1 . 31If a is a vertex of V 1 for which K of the good 2k-tuples containing a are not edges then there are at least K − 10 4k 2 n k−1/2 bad edges containing a. 2. If b is a vertex of V 2 for which L of the good 2k-tuples containing b are not edges then there are at least L − (ηn/5) 2k−1 bad edges containing b. -tuples, and the stated bound follows as in(1). − o(1) looks approximately like Sidorenko's construction. Corollary 4. 1 1Let r = 2 p + 1 be an integer and let W be a p-dimensional vector space over the field GF (2). For every ǫ > 0 there is η > 0 so that if H is a C ηn 4 then there is a partition of the vertex set as w∈W V w such that all but ǫn 4 edges X of H satisfy ΣX = 0. Lemma 4. 2 2Suppose that we have a colouring of the edges of the complete graph K s in s − 1 colours, so that every colour is a matching and each subset of 4 vertices spans edges of either 3 or 6 different colours. Then s = 2 p for some integer p. Proposition 4. 3 3Suppose G is a K s+1 -free graph on N vertices with minimum degree δ(G) ≥ 1 − 1 s − α N and α < 1/s 2 . Then there is a partition of the vertex set of G as V (G) = U 1 ∪ · · · U s with e(U i ) < sαN 2 . n 2 ≥ 2−3 r −6 n 2 ≥ |P t | ≥ i |E i | − 10 −12 r −32 n 2 + |U i ||U j | − 10 −2 r −4 ( − O(n 3 ) − 10 −2 r −4 n 4 > 10 −11 r −32 n 4 − 2 · 10 −2 r −4 n 4 > 10 −11 r −32 n 4 such 4-tuples. Since the the number of abnormal 4-tuples is at most 10 −11 r −32 n 4 , some such c 1 c 2 c 3 c 4 should be normal. Then β = γ, or c 1 c 2 c 3 c 4 would be improperly coloured. We see that opposite edges of x 1 x 2 x 3 x 4 have the same colour. Therefore we can apply Lemma 4.2 with s = r − 1 to deduce that r − 1 is of the form 2 p . Now distribute the √ cN deleted vertices arbitrarily among the U i . Then e(U i ) < (2s + 1) √ c N 2 . by a hidden algebraic feature, so are loathe even to speculate on the nature of the best construction for this case. The number of edge colorings with no monochromatic cliques. N Alon, J Balogh, P Keevash, B Sudakov, submittedN. Alon, J. Balogh, P. Keevash and B. Sudakov, The number of edge colorings with no monochro- matic cliques, submitted. Asymptotic solution of a Turán-type problem. P , Graphs and Combinatorics. 6P. Frankl, Asymptotic solution of a Turán-type problem. Graphs and Combinatorics 6 (1990), 223-227. Turán type problems, in: Surveys in combinatorics. Z Füredi, London Math. Soc. Lecture Note Ser. 166Cambridge Univ. PressZ. Füredi, Turán type problems, in: Surveys in combinatorics, London Math. Soc. Lecture Note Ser. 166, Cambridge Univ. Press, Cambridge, 1991, 253-300 The exact Turán number of the Fano plane. P Keevash, B Sudakov, submittedP. Keevash and B. Sudakov, The exact Turán number of the Fano plane, submitted. Survey of binary Krawtchouk polynomials, in: Codes and association schemes. I Krasikov, S Litsyn, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 56Amer. Math. SocI. Krasikov and S. Litsyn, Survey of binary Krawtchouk polynomials, in: Codes and association schemes, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 56, Amer. Math. Soc., Providence, RI, 2001, 199-211. Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces. V Levenshtein, IEEE Trans. Inform. Theory. 41V. Levenshtein, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Trans. Inform. Theory 41 (1995), 1303-1321. L Lovász, Combinatorial Problems and Exercises. North-Holland, AmsterdamL. Lovász, Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1993. An analytic approach to extremal problems for graphs and hypergraphs, in: Extremal problems for finite sets. A Sidorenko, Bolyai Soc. Math. Stud. 3A. Sidorenko, An analytic approach to extremal problems for graphs and hypergraphs, in: Ex- tremal problems for finite sets, Bolyai Soc. Math. Stud. 3, Budapest, 1994, 423-455. What we know and what we do not know about Turán numbers. A Sidorenko, Graphs and Combinatorics. 11A. Sidorenko, What we know and what we do not know about Turán numbers, Graphs and Combinatorics 11 (1995), 179-199. A method for solving extremal problems in graph theory, stability problems. M Simonovits, Theory of Graphs (Proc. Colloq. Tihany, 1966). New York, and Akad. Kiadó, BudapestAcademic PressM. Simonovits, A method for solving extremal problems in graph theory, stability problems, in: Theory of Graphs (Proc. Colloq. Tihany, 1966), Academic Press, New York, and Akad. Kiadó, Budapest, 1968, 279-319.
[]
[ "The Power Spectrum of Galaxies in the 2dF 100k Redshift Survey", "The Power Spectrum of Galaxies in the 2dF 100k Redshift Survey" ]
[ "Max Tegmark \nDept. of Physics\nUniv. of Pennsylvania\n19104PhiladelphiaPAUSA\n", "Andrew J S Hamilton \nJILA and Dept. Astrophysical & Planetary Sciences\nBox 440, U. Colorado80309BoulderCOUSA\n", "Yongzhong Xu \nDept. of Physics\nUniv. of Pennsylvania\n19104PhiladelphiaPAUSA\n" ]
[ "Dept. of Physics\nUniv. of Pennsylvania\n19104PhiladelphiaPAUSA", "JILA and Dept. Astrophysical & Planetary Sciences\nBox 440, U. Colorado80309BoulderCOUSA", "Dept. of Physics\nUniv. of Pennsylvania\n19104PhiladelphiaPAUSA" ]
[ "Mon. Not. R. Astron. Soc" ]
We compute the real-space power spectrum and the redshift-space distortions of galaxies in the 2dF 100k galaxy redshift survey using pseudo-Karhunen-Loève eigenmodes and the stochastic bias formalism. Our results agree well with those published by the 2dFGRS team, and have the added advantage of producing easy-tointerpret uncorrelated minimum-variance measurements of the galaxy-galaxy, galaxyvelocity and velocity-velocity power spectra in 27 k-bands, with narrow and wellbehaved window functions in the range 0.01 h/Mpc < k < 0.8 h/Mpc. We find no significant detection of baryonic wiggles, although our results are consistent with a standard flat Ω Λ = 0.7 "concordance" model and previous tantalizing hints of baryonic oscillations. We measure the galaxy-matter correlation coefficient r > 0.4 and the redshift-distortion parameter β = 0.49 ± 0.16 for r = 1 (β = 0.47 ± 0.16 without finger-of-god compression). Since this is an apparent-magnitude limited sample, luminosity-dependent bias may cause a slight red-tilt in the power spectum. A battery of systematic error tests indicate that the survey is not only impressive in size, but also unusually clean, free of systematic errors at the level to which our tests are sensitive. Our measurements and window functions are available at http : //www.hep.upenn.edu/ ∼max/2df.html together with the survey mask, radial selection function and uniform subsample of the survey that we have constructed.
10.1046/j.1365-8711.2002.05622.x
[ "https://arxiv.org/pdf/astro-ph/0111575v3.pdf" ]
16,378,362
astro-ph/0111575
b055e352119722b0b1acd53896b68e5980549f64
The Power Spectrum of Galaxies in the 2dF 100k Redshift Survey 27 October 2018 Max Tegmark Dept. of Physics Univ. of Pennsylvania 19104PhiladelphiaPAUSA Andrew J S Hamilton JILA and Dept. Astrophysical & Planetary Sciences Box 440, U. Colorado80309BoulderCOUSA Yongzhong Xu Dept. of Physics Univ. of Pennsylvania 19104PhiladelphiaPAUSA The Power Spectrum of Galaxies in the 2dF 100k Redshift Survey Mon. Not. R. Astron. Soc 000000027 October 2018Submitted to MNRAS December 2 2001, revised April 14 2002arXiv:astro-ph/0111575v3 7 (MN L A T E X style file v2.2)cosmology -large-scale structure of universe -galaxies: distances and redshifts -galaxies: statistics -methods: data analysis We compute the real-space power spectrum and the redshift-space distortions of galaxies in the 2dF 100k galaxy redshift survey using pseudo-Karhunen-Loève eigenmodes and the stochastic bias formalism. Our results agree well with those published by the 2dFGRS team, and have the added advantage of producing easy-tointerpret uncorrelated minimum-variance measurements of the galaxy-galaxy, galaxyvelocity and velocity-velocity power spectra in 27 k-bands, with narrow and wellbehaved window functions in the range 0.01 h/Mpc < k < 0.8 h/Mpc. We find no significant detection of baryonic wiggles, although our results are consistent with a standard flat Ω Λ = 0.7 "concordance" model and previous tantalizing hints of baryonic oscillations. We measure the galaxy-matter correlation coefficient r > 0.4 and the redshift-distortion parameter β = 0.49 ± 0.16 for r = 1 (β = 0.47 ± 0.16 without finger-of-god compression). Since this is an apparent-magnitude limited sample, luminosity-dependent bias may cause a slight red-tilt in the power spectum. A battery of systematic error tests indicate that the survey is not only impressive in size, but also unusually clean, free of systematic errors at the level to which our tests are sensitive. Our measurements and window functions are available at http : //www.hep.upenn.edu/ ∼max/2df.html together with the survey mask, radial selection function and uniform subsample of the survey that we have constructed. INTRODUCTION Three-dimensional maps of the Universe provided by galaxy redshift surveys place powerful constraints on cosmological models, which has motivated ever more ambitious observational efforts such as the the CfA/UZC (Huchra et al. 1990;Falco et al. 1999), LCRS (Shechtman et al. 1996) and PSCz (Saunders et al. 2000) surveys, each well in excess of 10 4 galaxies. This has been an exciting year in this regard, with early results released from two even more ambitions projects; the AAT two degree field galaxy redshift survey (2dFGRS; Colless et al. 2001) and the Sloan Digital Sky Survey (SDSS;York et al. 1999), which aim for 250,000 and 1 million galaxies, respectively. Analysis of the first 147,000 2dFGRS galaxies (Peacock et al. 2001;Percival et al. 2001;Norberg et al. 2001a;Madgwick et al. 2001) and the first 29,000 SDSS galaxies (Zehavi et al. 2002) have supported a flat dark-energy dominated cosmology, as have angular clustering analyses of the parent catalogs underlying the 2dFGRS (Efstathiou & Moody 2001) and SDSS (Scranton et al. 2002;Connolly et al. 2002;Tegmark et al. 2002;Szalay et al. 2002;Dodelson et al. 2002). Tantalizing evidence for baryonic wiggles in the galaxy power spectrum has been discussed (Percival et al. 2001;Miller et al. 2001), and cosmological models have been constrained in conjunction with cosmic microwave background (CMB) data (Efstathiou et al. 2002). The 2dFGRS team has kindly made their first 102,000 redshifts publicly available. Given the huge effort involved in creating this state-of-the-art sample, it is clearly worthwhile to subject it to an independent power spectrum analysis. This is the purpose of the present paper, focusing on large (k ∼ < 0.3 h/Mpc) scales. Since the cosmological constraints from galaxy surveys are only as accurate as our modeling of bias, extinction, integral constraints, geometry-induced power smearing and other real-world effects, we will employ a number of recently developed techniques for tackling these issues. Compared with the solid and thorough analysis by the 2dFGRS team in Peacock et al. (2001) and Percival et al. (2001), our main improvements will be in the following areas: • By using an approach based on information theory, involving pseudo-Karhunen-Loève eigenmodes, quadratic estimators and Fisher matrix decorrelation, we are able to produce uncorrelated measurements of the linear power spectrum with minimal error bars and quite narrow window functions. This allows the power spectrum to be plotted in an easy-to-interpret model-independent way and, because of the narrow windows, minimizes aliasing from non-linear scales when fitting to linear models. • Using the stochastic bias formalism, we measure independently not one power spectrum but three, encoding clustering anisotropy. On large scales where redshift distortions are linear (Kaiser 1987), these three curves are the realspace power spectra of the galaxies, their velocity divergence (related to the matter density) and the cross-correlation between the two. On smaller scales, the information they encode can be extracted using simulations. The rest of this paper is organized as follows. In Section 2, we describe the 2dFGRS data used and construct an easy-to-interpret subsample that is strictly magnitude limited after taking various real-world complications into account. We perform our basic analysis in Section 3 and report the results in Section 4. In Section 5, we test for a variety of systematic errors in Section 5 and quantify the effect of non-linearity and non-Gaussianity on our measurements. In Section 6, we discuss our results, fit to cosmological models and compare our results with those in the literature. DATA MODELING The 2dFGRS is described in detail in Colless et al. (2001, hereafter C01). The publicly released 2dFGRS sample consists of 102,426 unique objects (excluding duplicates), of which 93,843 have survey quality redshifts (quality factor 3). Of these 5,131 objects have heliocentric redshifts z 0.002 and are therefore probable stars, while a further 240 galaxies lie outside the defined angular boundaries of the survey (usually inside a hole in one of the parent UKST fields, occasionally marginally outside one of the 381 surveyed 2 • fields). This leaves a sample of 88,472 galaxies with survey quality redshifts. To do full justice to the quality of this data set in a power spectrum analysis, it is crucial to model accurately the three-dimensional selection functionn(r), which gives the expected (not observed) number density of galaxies as a function of 3D position. This is the goal of the present section. As will be described in Section 3, our method for measuring the power spectrum requires, in its current implementation, that the selection function be separable into the product of an angular part and a radial part: n(r) =n( r)n(r), where r = r r and r is a unit vector. The angular part n( r) may take any value between 0 and 1, and gives the completeness as a function of position, i.e., the fraction of all survey-selected galaxies for which survey quality redshifts are actually obtained, whilen(r) gives the radial selection function. Although it would be possible to generalize the method to a non-separable selection function (by breaking up the selection function into a sum of piece-wise separable parts), we have chosen to stick to the simple case of a separable selection function, for two reasons. First, although the selection function of the 2dF 100k release is not separable, it is nearly so (the survey was originally designed so that it would be), and the gain from allowing a nonseparable selection function has seemed insufficient to justify the extra complexity. Second, as described in Sections 5.3.1 and 5.3.2, we wish to be able to test for possible systematic effects arising from a misestimate of extinction, which would cause a purely angular modulation of density fluctuations, or from a misestimate of the radial selection function, which would cause a purely radial modulation of the density. Such tests are facilitated if the selection function is separable. There are two complications that cause slight departures from such separability (C01): (i) The magnitude limit varies slightly across the sky, because both the photometric calibration of the parent UKST fields, and the extinction correction at each angular position was improved after the survey had begun. (ii) Seeing issues lead to lower completeness for faint galaxies, and weather variations therefore cause the magnitude-dependent completeness fraction to vary in different 2 • fields. Below we will eliminate both of these complications with appropriate cuts on the data set, obtaining a uniform subsample with a separable selection function as in equation (1). The basic angular mask In this subsection, we describe our modeling of the angular maskn( r) for the full sample. In subsequent subsections, we will shrink and re-weight this mask slightly to eliminate the above-mentioned complications, obtaining the final result shown in Figure 1. Once the 2dFGRS is complete, it will contain a total of 1192 circular 2 • fields, including 450 fields in a 75 • × 10 • strip near the North Galactic Pole, 643 fields in an 85 • × 15 • strip near the South Galactic Pole, and a further 99 fields distributed randomly around the Southern strip. The various intersections of these fields with each other yield 7189 non-overlapping intersection regions, referred to as sectors. Parts of sectors are excluded if they fall outside the boundaries of the 314 rectangular UKST plates of the parent APM survey, or inside one of the holes excised from the plates in order to eliminate e.g. bright stars and satellite trails. The data release specifies 2024 holes, of which 1670 lie within, or overlap, those parts of the UKST plates designated as part of the 2dF survey. The 100k release is a subset of the survey, containing data from 381 circular 2 • fields, including 39 random fields. Eventually, when the survey is done, the observed region will be complete, but in the interim the released fields are variably incomplete, with a different completeness fraction n( r) in each sector, as described in C01. As part of the 2dFGRS data release, Peder Norberg and Shaun Cole provide software that evaluatesn( r) in each of approximately 2.5 million 3 ′ × 3 ′ pixels, taking all the various complications into account. However, we wish to adopt a different angular mask that admits a separable selection function, and we also wish to be able to compute Figure 1. The upper half shows the 59832 2dF galaxies in our baseline sample, in equatorial 1950 coordinates. The lower half shows the corresponding angular mask, the relative probabilities that galaxies in various directions get included. the spherical harmonics of the angular mask using the fast, analytic method described in Appendix A of Hamilton (1993). We therefore use a more explicit geometric (not pixellized) specification of the mask, described immediately below. All field, plate and hole boundaries are simple arcs on the celestial sphere, corresponding to the intersection of the sphere with some appropriate plane. This means that any spherical polygon (a field, plate, hole, sector, etc.) can be defined as the intersection of a set of caps, where a cap is the set of directions r satisfying a· r > b for some unit vector a and some constant b ∈ [−1,1]. For instance, a 2 • field is a single cap, and a rectangular plate is the intersection of four caps. We define masks such as that the one plotted in Figure 1 as a list of non-overlapping polygons such that n( r) is constant in each one. We construct the basic 2dFGRS mask as follows: (i) We generate a list of 8903 polygons comprised of 7189 sectors and 1670 holes, plus 44 polygons defining boundaries of UKST plates. (ii) Whenever two polygons intersect, we split them into non-intersecting parts, thereby obtaining a longer list of 12066 non-overlapping polygons. Although slightly tricky in practice, such an algorithm is easy to visualize: if you draw all boundary lines on a sphere and give it to your child as a coloring exercise, using four crayons and not allowing identically colored neighbors, you would soon be looking at such a list of non-overlapping polygons. (iii) We compute the completeness n( r) for each of these new polygons, originally using the Norberg-Cole software, but subsequently using our own computations, described in the following subsections. (iv) We simplify the result by omitting polygons with zero weight and merging adjacent polygons that have identical weight. With the original Norberg and Cole completenesses, the result is a list of 3765 polygons, with a total (unweighted) area of 983 square degrees, and an effective (weighted) area n( r)dΩ of of 537 square degrees. With the revised completenesses described in Section 2.2, there are 3614 polygons, with an (unweighted) area of 711 square degrees, and an effective (weighted) area n( r)dΩ of 431 square degrees. This angular mask, and the polygons into which it resolves, are illustrated in Figure 1. Section 2.2 explains how we eliminate the two abovementioned complications, the variations in the magnitude limit, and the variations in the weather, so as to produce an angular mask with the same radial selection function at all points. The reader uninterested in such details can safely skip all this, jumping straight to Section 2.3, remembering only the simple bottom line: we create a uniform sample with 64,844 galaxies over 711 square degrees that is complete down to bJ magnitude 19.27. Cutting to a uniform magnitude limit The 2dFGRS aimed to be complete to a limiting bJ magnitude m = 19.45 after correction for extinction. However, the actual limiting magnitude varies slightly across the sky as described in C01. This is because after the survey began, there have been improvements in both the photometric calibrations of the underlying parent catalog (Maddox et al. 2001, in preparation) and in the extinction corrections (Schlegel, Finkbeiner & Davis 1998). We eliminate this complication by creating a subsample that is complete down to a slightly brighter limiting magnitude m * , applying the following two cuts: (i) Reject all galaxies whose extinction-corrected magnitude bJ is fainter than m * . (ii) Reject all sectors whose extinction-corrected magnitude is brighter than m * . The magnitude limit of a sector is defined in the most conservative possible fashion: it is the brightest among all the magnitude limits at the position of each galaxy and of each Norberg-Cole pixel within the sector. The extinction at each position is evaluated using the extinction map of Schlegel, Finkbeiner & Davis (1988). Figure 2 shows the number of surviving galaxies as a function of m * . As we increase m * , the first cut eliminates fewer galaxies whereas the second cut eliminates more galaxies. The result is seen to be a rather sharply peaked curve, taking its maximum for m * = 19.27, for which 66,050, or 75 percent, of the 88,472 galaxies survive. The choice m * = 19.27 turns out to maximize not only the number of galaxies, but the effective survey volume as well. As the flux cut F * is made fainter, the depth of the survey (∝ F −1/2 * in the Euclidean limit) increases, but the area decreases because there are fewer sectors complete down to F * . Therefore the survey volume ∝ ∼ (area)×F , and this also happens to peak for F * corresponding to magnitude 19.27. Angular selection function Modeling the weather One of the more time-consuming aspects of our analysis was modeling another departure from uniformity in the 2dFGRS: spatial dependence of the magnitude-dependent incompleteness. As described by C01, although the successrate P for measuring reliable redshifts (quality 3) for targeted galaxies is in general quite high, it depends on weather. The poorer the seeing is when a given field is observed, the lower the success rate. Moreover, this weather modulation affects fainter galaxies more than bright ones. C01 found the success rate to be well fit by an expression of the form P (F ) = γ[1 − (Φ f /F ) a ],(2) where F is the observed flux from the galaxy, γ = 0.99, a = 2.5/ ln(10) ≈ 1.086 and Φ f is a parameter that is fit for separately for each observed field f , interpretable as the faintest observable flux. Note that since this observational selection effect depends only on magnitude and weather, this issue can be analyzed and resolved in terms of apparent magnitudes alone, without explicitly involving redshifts. The Φ f -values computed by the 2dFGRS team were not part of the public release, but it is straightforward to generate values from the data provided. Whereas C01 estimated Φ f from the observed completeness fraction for each field, we performed a maximum-likelihood fit over the fluxes of all objects (galaxies and stars) targeted for observation in each of the 381 field-nights, the likelihood being a product of terms P (Fi) for all successful observations (those yielding a survey-quality redshift), and terms [1 − P (Fi)] for all unsuccessful observations. Maximizing over 382 parameters (Φ f for each of 381 distinct field-nights f , and a global value of a, with γ fixed equal to 1), we obtain a best fit exponent a = 0.96 ± 0.04. Since the exponent is consistent with unity, we set a = 1 for simplicity. We repeated the analysis with a permitted to vary separately in each field, but the likelihood is consistent with constant values. As a cross-check, we repeated the entire analysis sectorby-sector instead of field-by-field, obtaining reassuringly similar results. Random sampling to a sharp magnitude limit As mentioned, our power spectrum analysis requires a selection function of the separable form of equation (1). Yet the discussion above shows that the shape of the radial selection functionn(r) varies across the sky, since the success rate P f (F ) is different for each of the 381 field-nights f , as given by equation (2). We remedy this problem by sparse-sampling the galaxies in such a way that the shape of the success rate P (F ) (as opposed to its amplitude) becomes the same for all fields. The amplitude variations can then be absorbed into the angular maskn( r), restoring separability. There are clearly infinitely many ways of doing this -we wish to find the way that maximizes the effective volume of the survey for measuring large scale power. If we throw away galaxies at random, keeping galaxies in a given field f with a probability p f that depends on their observed flux F , then the original success rate P f (F ) for the field from equation (2) gets replaced by P f (F )p f (F ). Our goal then becomes to choose these probabilities p f (F ) such that P f (F )p f (F ) = w f P * (F ),(3) where P * (F ) is the desired uniform, global success rate and the weights w f are are scaling factors that will be absorbed into the angular mask. Since the functions P f (F ) are known, equation (3) immediately specifies how we should choose the probabilities once the function P * and the weights have been fixed. To maximize the number of surviving galaxies, we want to make p f and hence w f as large as possible. Since probabilities cannot exceed unity, this implies that the best weights are w f = min F P f (F ) P * (F ) .(4) It remains to choose the target success rate P * (F ). Since we are interested in large scale power, our aim is to maximize not so much the number of galaxies, but rather the effective volume of the survey, and we must accomplish this goal by adjusting a function P * (F ) of apparent flux F . The way to do this is to choose P * (F ) so as to retain all galaxies at the faint limit of the survey, and then to make P * (F ) as large as possible at all other fluxes. Given that the original P (F ) decreases monotonically to fainter fluxes for all values of the weather parameter Φ f , and that Φ f includes cases of perfectly observed fields (Φ f = ∞), the solution is simply to choose P * (F ) to be constant, which can be taken to equal 1 without loss of generality, at all values brighter than the flux limit. This is delightfully simple and convenient: it means that the best choice is a pure magnitude-limited sample with no magnitude incompleteness to keep track of! The corresponding weights are w f = min F P f (F ) = P f (F * ) = 1 − Φ f /F * (5) where F * is the flux limit. The scheme thus keeps all galaxies at the flux limit F * , and discards a progressively larger fraction of the brighter galaxies in each sector so as to cancel exactly the magnitude-dependence of the incompleteness. The magnitude limit 19.27 arrived at in the previous subsection turns out to maximize the number of galaxies not only before sparse-sampling, but also after sparse-sampling. The final result is a list of 3614 polygons with associated weights. available at http://www.hep.upenn.edu/∼max/2df .html together with the uniform galaxy sample and our power spectrum measurements. The total area is 711 square degrees, and the effective area n( r)dΩ is 432 square degrees. The radial selection function After the modeling of angular effects above, it remains to measure the radial selection functionn(r) for the uniform sample. It is important to do this as accurately as possible, since errors in the selection function translate into spurious large scale power. The radial selection functionn(r) that results from the analysis described immediately below is shown in Figure 3. In addition to imposing a faint magnitude limit of bJ = 19.27, we follow the advice of the 2dFGRS team (Matthew Colless 2001, private communication) in cutting the survey to a bright limit of bJ = 15. We use a maximum likelihood method based on the C − method of Lynden-Bell (1971), which assumes that luminosity is uncorrelated with position. We generate an initial approximation to the selection function using a continuous version of the Turner (1979) method, which yields the exact maximum likelihood solution for the case of a survey with a sharp faint flux limit. The Turner method has the merit of being exceedingly fast (less than one CPU second), but it works only if the survey is flux-limited at one end (e.g. the faint end). Starting from the Turner solution, we use an iterative method designed to converge towards the exact maximum likelihood solution for the selection function, which can be shown (Hamilton Figure 3. The redshift distribution of the galaxies in our sample is shown both as a histogram (top) and relative to the expected distribution (bottom), in comoving coordinates assuming a flat Ωm = 0.3 cosmology. The curves correspond to the the radial selection functionn(r) employed in our analysis (solid) and by C01 (dotted). The four vertical lines indicate the redshift limits employed in our analysis (10 h −1 Mpc < r < 650 h −1 Mpc) and where spectral type subsamples are available (33 h −1 Mpc < r < 538 h −1 Mpc). & Tegmark 2002) to be a step function with steps at the limiting distance of each of the ∼ 60,000 galaxies in the sample. To implement the Bayesian prejudice that the selection function should be smooth, we interpolate the resulting 60,000-point function at ∼ 500 points, through which we pass a cubic spline. We follow the 2dFGRS team in assuming a flat ΩΛ = 0.7 cosmology when converting redshifts to comoving distances r. We transform the galaxy positions into the Local Group frame assuming that the solar motion relative to the Local Group is 306 km/s toward l = 99 • , b = −4 • (Courteau & van den Bergh 1999). We model k-corrections and luminosity evolution (ε-corrections) together as a power law luminosity evolution ∝ (1+z) κ with exponent κ = −0.7. This exponent was chosen so as to make the comoving density shown in the lower panel in Figure 3 as flat as possible, i.e., by assuming minimal evolution in the comoving number density of galaxies. Similar results have been reported by Cole et al. (2001), C01, Cross et al. (2001) and Madgewick et al. (2001) and Norberg et al. (2001b). The slight differences between ourn(r) and that of C01 seen Figure 3 are due to our different methods for estimating this function from the data, and below we find that they do not have a major impact on the final power spectrum. We truncate the sample radially by eliminating objects with r < 10 h −1 Mpc (to eliminate stellar contamination) and r > 650 h −1 Mpc (where Figure 3 shows evidence of incompleteness). This leaves 59832 galaxies in the sample. METHOD AND BASIC ANALYSIS In this section, we analyze the uniform galaxy sample described in the previous section, measuring the power spectrum and redshift space distortions of the galaxy density field. We adopt the matrix-based approach described in Tegmark et al. (1998, hereafter THSVS98), using the mode expansion of Hamilton & Culhane (1996) and including the stochastic bias formalism. Our analysis consists of the following five steps: (i) Finger-of-god compression (ii) Pseudo-Karhunen-Loève compression (iii) True Karhunen-Loève expansion (iv) Quadratic band-power estimation (v) Fisher decorrelation and flavor disentanglement We will now describe these steps in more detail. We will see that step (iii) is not required in practice, and we use it only for systematics tests. Step 1: Finger-of-god compression Since our analysis uses the linear Kaiser approximation for redshift space distortions, it is crucial that we are able to empirically quantify our sensitivity to the so-called fingerof-god (FOG) effect whereby radial velocities in virialized clusters make them appear elongated along the line of sight. We therefore start our analysis by compressing (isotropizing) FOGs, as illustrated in Figure 4. The FOG compression involves a tunable threshold density, and in Section 5.4 below we will study how the final results change as we gradually change this threshold to include lesser or greater numbers of FOGs. We use a standard friends-of-friends algorithm, in which two galaxies are considered friends, therefore in the same cluster, if the density windowed through an ellipse 10 times longer in the radial than transverse directions, centered on the pair, exceeds a certain overdensity threshold. To avoid linking well-separated galaxies in deep, sparsely sampled parts of the survey, we impose the additional constraint that friends should be closer than r ⊥max = 5 h −1 Mpc in the transverse direction. The two conditions are combined into the following single criterion: two galaxies separated by r in the radial direction and by r ⊥ in the transverse direction are considered friends if (r /10) 2 + r 2 wheren is the selection function (geometrically averaged) at the position of the pair, and δc is an overdensity threshold. Note that δc represents not the overdensity of the pair as seen in redshift space, but rather the overdensity of the pair after their radial separation has been reduced by a factor of 10. In other words, δc is intended to approximate the threshold overdensity of a cluster in real space, not the overdensity of the elongated FOG seen in redshift space. Having identified a cluster by friends-of-friends in this fashion, we measure the dispersion of galaxy positions about the center of the cluster in both radial and transverse The slice has thickness 2 • and has been rotated to lie in the plane of the page. From left to right, the panels show all 15,055 galaxies in the slice, the 6,211 that are identified as belonging to FOGs (with density threshold 100) and the same galaxies after FOG compression, respectively. directions. If the 1-dimensional radial dispersion exceeds the transverse dispersion, then the cluster is deemed a FOG, and the FOG is then compressed radially so that the cluster becomes round, that is, the transverse dispersion equals the radial dispersion. We perform the entire analysis five times, using progressively more aggressive compression with density cutoffs 1+δc = ∞, 200, 100, 50 and 25, respectively. The infinite threshold 1+δc = ∞ corresponds to no compression at all. Figure 4 illustrates FOG compression with threshold density 1+δc = 100, which is the baseline case adopted in this paper. It corresponds to fairly aggressive FOG removal since the overdensity of a cluster is around 200 at virialization and rises as the Universe expands and the background density continues to drop. Step 2: Pseudo-KL pixelization The raw data consists of N gal = 59,832 three-dimensional vectors rα, α = 1, ..., N gal , giving the measured positions of each galaxy in redshift space. As in THSVS98, we define the density in Nx "pixels" xi, i = 1, ..., Nx by xi ≡ n(r) n(r) ψi(r)d 3 r(7) for some set of functions ψi and work with the Nxdimensional data vector x instead of the the 3 × N gal numbers rα. Although these are perhaps more aptly termed "modes" since we will choose quite non-local functions ψi, we will keep referring to them as pixels to highlight the useful analogy with CMB map analysis. Galaxies are (from a cosmological perspective) deltafunctions in space, so the integral in equation (7) reduces to a discrete sum over galaxies. We do not rebin the galaxies spatially, which would artificially degrade the resolution. It is convenient to isolate the mean density into a single mode ψ1(r) =n(r), with amplitude x1 = n(r)d 3 r = N gal ,(8) and to arrange all other modes to have zero mean xi = ψi(r)d 3 r = 0 (i = 0).(9) The covariance matrix of the vector x of amplitudes is a sum of noise and signal terms ∆x∆x t = C ≡ N + S,(10) where the shot noise covariance matrix is given by Nij = ψi(r)ψj (r) n(r) d 3 r(11) and the signal covariance matrix is Sij = ψi(k) ψj(k) * P (k) d 3 k (2π) 3(12) in the absence of redshift-space distortions. Here hats denote Fourier transforms andn is the three-dimensional selection function described in Section 2, i.e.,n(r)dV is the expected (not the observed) number of galaxies in a volume dV about r. P (k) is the power spectrum, which for a random field of density fluctuations δ(r) is defined by δ(k) * δ(k ′ ) = (2π) 3 δDirac(k − k ′ ). As our functions ψi(r), we use the pseudo-Karhunen-Loève (PKL) eigenmodes defined in Hamilton, Tegmark & Padmanabhan (2000; hereafter "HTP00"). To provide an intuitive feel for the nature of these modes, a sample is plotted in Figure 5 and Figure 6. We use these modes because they have the following desirable properties: (i) They form a complete set of basis functions probing successively smaller scales, so that a finite number of them (we use the first 4,000) allow essentially all information about the density field on large scales to be distilled into the vector x. (ii) They allow the covariance matrices N and S to be fairly rapidly computed. (iii) They are the product of an angular and a radial part, i.e., take the separable form ψi(r) = ψi( r)ψi(r), which accelerates numerical computations. (iv) A range of potential sources of systematic problems are isolated into special modes that are orthogonal to all other modes. This means that we can test for the presence of such problems by looking for excess power in these modes, and immunize against their effects by discarding these modes. We have four types of such special modes: (i) The very first mode is the mean density, ψ1(r) = n(r). The mean mode is used in determining the maximum likelihood normalization of the selection function, but is then discarded from the analysis, since it is impossible to measure the fluctuation of the mean mode. The idea of solving the so-called integral constraint problem by making all modes orthogonal to the mean goes back to Fisher et al. (1993). . The triangles show the 4,000 elements x i of the data vector x (the pseudo-KL expansion coefficients) for the baseline galaxy sample. If there were no clustering in the survey, merely shot noise, they would have unit variance, and about 68% of them would be expected to lie within the blue/dark grey band. If our prior power spectrum were correct, then the standard deviation would be larger, as indicated by the shaded yellow/light grey band. while in the radial direction they take the form specified by equation (4.42) of Hamilton (1997c). Mode 2 is a pure monopole mode (multiplied by the angular mask), and is present because the survey is not all-sky. The other three Local Group modes are dipole modes with admixtures of the Local Group monopole mode 2, such as to make them orthogonal to the mean mode 1. (iii) Purely radial modes (for example mode 104 in Figure 6) are to first order the only ones affected by misestimates of the radial selection functionn(r). (iv) Purely angular modes (for example mode 148 in Figure 6) are to first order the only ones affected by misestimates of the angular selection functionn( r), as may result from inadequate corrections for, e.g., extinction, the variable magnitude limit, the variable magnitude completeness or photometric zero-point offsets. As described in HTP00, the modes ψi are computed in the logarithmic spherical wave basis (Hamilton & Culhane 1996), which are orthonormal eigenfunctions Z ωℓm (r) = (2π) −1/2 e −(3/2+iω) ln r Y ℓm (r) of the complete set of commuting Hermitian operators i ∂ ∂ ln r + 3 2 = −i ∂ ∂ ln k + 3 2 , L 2 , Lz .(13) Slightly better numerical behavior is obtained by expanding not ψi(r) itself but rather ψi(r)/n(r) 1/2 (the denominator is the square root of the radial part of the selection function only, not the angular part) in logarithmic spherical waves, since this mitigates some difficulties that arise from the fact that the radial selection functionn(r) varies by orders of magnitude. The merits of working in a basis of spherical harmonics were first emphasized by Fisher, Scharf & Lahav (1994) and by Heavens & Taylor (1995). The advantages of working with logarithmic radial waves e −(3/2+iω) ln r , compared for example to spherical Bessel functions, are both numerical and physical: (i) Numerically, the logarithmic radial wave basis permits rapid transformation between real, ω, and Fourier space using Fast Fourier Transforms. The transformation is mathematically equivalent to the Fast Fourier-Hankel-Bessel Transform FFTLog described in Appendix B of Hamilton (2000). (ii) Physically, logarithmic radial waves are well matched to real galaxy surveys like the 2dFGRS, which are finely sampled nearby, and coarsely sampled far away. (iii) The linear redshift distortion operator is diagonal in this basis (Hamilton & Culhane 1996). The logarithmic radial wave basis discretizes naturally on to a logarithmically equispaced grid (in both real and Fourier space), and is periodic over a logarithmic interval. To avoid potential problems of aliasing between small and large scales, we embed the survey inside a suitably large logarithmic interval of depths, extending in real space from 10 −2 h −1 Mpc to 10 4 h −1 Mpc. As remarked in Section 2.3, we truncate the survey to radial depths 10-650 h −1 Mpc within this interval. The dimensionless log-frequency ω in the radial eigenmode e −(3/2+iω) ln r is a radial analogue (in a precise mathematical sense) of the dimensionless angular harmonic number ℓ. Similar resolution in the radial and angular directions is secured by choosing the maximum logfrequency to be about equal to the the maximum harmonic number, ωmax ≈ ℓmax. The maximum log-frequency is related to the radial resolution ∆ ln r by ωmax = π/∆ ln r. We adopt a maximum harmonic number of ℓmax = 40, and a radial resolution of 32 points per decade, so ∆ ln r = (ln 10)/32, giving ωmax = 43.7 (the same as in HTP00). These choices ensure comparable effective resolutions in radial and angular directions. A maximum angular harmonic number of ℓmax = 40 gives (ℓmax+1) 2 = 1681 spherical harmonics, while 32 points per radial decade over 6 decades gives 192 radial modes. Thus there is a potential pool of 41 2 × 192 ≈ 320,000 modes from which we would like to construct Karhunen-Loève (KL) modes. The usual way to construct such modes would be to diagonalize a 320,000 × 320,000 matrix, but this is evidently utterly intractable numerically. How do we build the PKL modes in practice? To make the problem tractable, we instead proceed hierarchically, first constructing angular PKL modes, and then constructing a set of radial PKL modes associated with each angular KL mode. The procedure is possible because we have required the selection function to be separable into angular and radial parts, equation (1). We refer to the resulting modes as pseudo Karhunen-Loève (PKL) modes. The PKL basis contains almost as much information as a true KL basis, but it circumvents the need to diagonalize an impossibly huge matrix. Our procedure is the same as that of HTP00. A different, but similar in spirit, hierarchical approach to the KL problem has been proposed by Taylor et al. (2001). As we proceed from angular PKL mode to angular PKL mode, extending each into 3D PKL modes by computing associated radial functions, we retain only the Nx = 4000 PKL modes with the highest expected signal-to-noise. As detailed below, we make this truncation both to render the various Nx × Nx matrices numerically tractable and to limit sensitivity to small, nonlinear scales. As the signal-to-noise of the angular PKL mode decreases, fewer and fewer of the associated radial PKL modes make the cut into the pool of PKL modes. We stop when 10 successive angular PKL modes have contributed no new PKL mode. In practice only 140 of the angular PKL modes actually contribute to the PKL modes. The reduction from 1681 to 140 angular modes with little information loss is possible because the spherical harmonics are overcomplete and redundant on the modest fraction of the sky actually covered by the 2dFGRS. The orthogonality of the PKL modes to the mean and the properties of the "special" modes are enforced in the construction of the modes. We perform the PKL decomposition after selecting out the special modes (rather than doing the KL decomposition and then making them orthogonal to the special modes), since we find that this makes better PKL modes. We do this as decribed in Appendix B of THSVS98, with the complication that we make the non-special modes exactly orthogonal to the masked mean and the masked LG modes, not merely orthogonal up to the finite order of the discrete matrices. The pixelized data vector x is shown in Figure 7. This data compression step has thus distilled the largescale information about the galaxy density field from N gal = 59,832 galaxy position vectors into 4,000 PKL-coefficients. The functions ψi are normalized so that Nii = 1, i.e., so that the shot noise contribution to their variance is unity. If there were no cosmological density fluctuations in the survey, merely Poisson fluctuations, the PKL-coefficients xi would thus have unit variance, and about 68% of them would be expected to lie within the blue/dark grey band. Figure 7 shows that the fluctuations are considerably larger than Poisson, especially for the largest-scale modes (to the left), demonstrating that cosmological density fluctuations are present, as expected. Step 3: Expansion into true KL modes Karhunen-Loève (KL) expansion (Karhunen 1947) was first introduced into large-scale structure analysis by Vogeley & Szalay (1996). It has since been applied to the Las Campanas redshift survey (Matsubara et al. 1999), the UZC survey (PTH01) and the SDSS (Szalay et al. 2002;Tegmark et al. 2002) and has been successfully applied to Cosmic Microwave Background data as well, first by Bond (1995) and Bunn (1995). Given x, N and S from the previous section, it is straightforward to compute the true Karhunen-Loève (KL) coefficients. They are defined by y ≡ B t x,(14) where b, the columns of the matrix B, are the Nx eigenvectors of the generalized eigenvalue problem Sb = λNb,(15) sorted from highest to lowest eigenvalue λ and normalized so that b † Nb = I. This implies that yiyj = δij(1 + λi),(16) which means that the transformed data values y have the desirable property of being uncorrelated. In the approximation that the distribution function of x is a multivariate Gaussian, this also implies that they are statistically independent -then y is merely a vector of independent Gaussian random variables. Moreover, equation (15) shows that the eigenvalues λi can be interpreted as a signalto-noise ratio S/N . Since the matrix B is invertible, the final data set y clearly retains all the information that was present in x. In summary, the KL transformation partitions the information content of the original data set x into Nx chunks that are mutually exclusive (independent), collectively exhaustive (jointly retaining all the information), and sorted from best to worst in terms of their information content. In most applications, the chief use of KL-coefficients is for data compression, discarding modes containing almost no information and thereby accelerating subsequent calculations. The KL-coefficients for our dataset are plotted in Figure 8, and it is seen that even the worst coefficients still have non-negligible signal-to-noise, bearing numerical testimony to the quality of the PKL-modes we have used. This means that KL-compression would not accelerate our particular analysis, and we will indeed work directly with the uncompressed data x in the following subsections. Rather, the reason we have computed KL-coefficients is as an additional check against systematic errors and incorrect assumptions, to verify that we modeled not only the diagonal terms in C correctly (as seen in Figure 7), but the off-diagonal correlations as well. As discussed in many of the above-mentioned KL-papers, inspection of the KLcoefficients as in Figure 8 provides yet another opportunity to detect suspicious outliers and to check whether the variance predicted by the prior power spectrum is consistent with the data. We will provide a detailed test based on the KL-coefficients in Section 5.1. 3.4 What we wish to measure: three power spectra, not one Before analyzing the x-vector in the following subsections, let us first discuss precisely what we want to measure. Cosmological constraints based on galaxy power spectrum measurements are only as accurate as our understanding of biasing. We will therefore perform our analysis in a way that avoids making any assumptions about the relation between galaxies and matter, as described in and HTP00. Unfortunately, bias is complicated. The commonly used assumption that the matter density fluctuations δ(r) and the galaxy number density fluctuations g(r) obey g(r) = b δ(r)(17) for some constant b (the bias factor) appears to be violated in a number of ways. It has been long known (Davis & Geller 1976;Dressler 1980) (17), but If there were no clustering in the survey, merely shot noise, they would have unit variance, and about 68% of them would be expected to lie within the blue/dark grey band. If our prior power spectrum were correct, then the standard deviation would be larger, as indicated by the shaded yellow/light grey band. The green/grey curve is the rms of the data points x i , averaged in bands of width 25, and is seen to agree better with the yellow/light grey band than the blue/dark grey band. that bias is inherently somewhat stochastic (Dekel & Lahav 1999) -this has been demonstrated in both simulations and real data (Tegmark & Bromley 1999). The term stochastic does of course not imply any randomness in the galaxy formation process, merely that additional factors besides density may be important (gas temperature, say). The good news for our present analysis is that, restricting attention to second moments, all the information about stochasticity is contained in a single new function r(k) (Pen 1998;Tegmark & Peebles 1998). Grouping the fluctuations into a two-dimensional vector x ≡ δ g(18) and assuming nothing except translational invariance, its Fourier transform x(k) ≡ e −ik·r x(r)d 3 r obeys x(k) x(k ′ ) † = (2π) 3 δ D (k − k ′ ) P (k) P×(k) P×(k) Pgg(k)(19) for some 2 × 2 power spectrum matrix that we will denote P(k). Here P is the conventional power spectrum of the mass distribution, Pgg is the power spectrum of the galaxies, and P× is the cross spectrum. It is convenient to rewrite this covariance matrix as P(k) = P (k) 1 b(k)r(k) b(k)r(k) b(k) 2(20) where b ≡ (Pgg/P ) 1/2 is the bias factor (the ratio of galaxy and total fluctuations) and the new function r ≡ P×/(P Pgg) 1/2 is the dimensionless correlation coefficient between galaxies and matter. Note that both b and r generally depend on scale k. The Schwarz inequality shows that the special case r = 1 implies the simple deterministic equation (17), and the converse is of course true as well. On large scales where linear perturbation theory is valid, redshift distortions (Kaiser 1987;Hamilton 1998) conveniently allow all three of these functions to be measured. Specifically, the correlation between the observed densities at any two points depends linearly on these three power spectra: galaxy-galaxy power : Pgg(k) = b(k) 2 P (k) galaxy-velocity power : Pgv(k) = r(k)b(k)f P (k) velocity-velocity power : Pvv(k) = f 2 P (k) Here f ≈ Ω 0.6 m is the dimensionless growth rate for linear density perturbations (see Hamilton 2001). More correctly, the 'velocity' here refers to minus the velocity divergence, which in linear theory is related to the mass (not galaxy) overdensity δ by f δ + ∇ · v = 0, where ∇ denotes the comoving gradient in velocity units. Note that Pgv(k) = f P×(k) and that the parameter f is conveniently eliminated by defining β(k) ≡ f /b(k), which gives Pgv(k) = β(k)r(k)Pgg(k), Pvv(k) = β(k) 2 Pgg(k).(22) Step 4: Quadratic compression into band powers In this step, we perform a much more radical data compression by taking certain quadratic combinations of the data vector that can easily be converted into power spectrum measurements. We parametrize the three power spectra Pgg(k), Pgv(k) and Pvv(k) as piecewise constant functions, each with 49 "steps" of height pi, which we term the band powers. To avoid unnecessarily jagged spectra, we take k 1.5 P rather than P to be constant within each band. We group these 3 × 49 numbers into a 147-dimensional vector p. We choose our 49 k-bands to be centered on logarithmically equispaced k-values ki = 10 i−41 16 h/Mpc, i = 1, ..., 49, i.e., ranging from 0.00316 h/Mpc to 3.16 h/Mpc. For instance, Pgg(k) = (k/ki) −1.5 pi for | lg k − lg ki| < 1/32. This should provide fine enough k-resolution to resolve any baryonic wiggles and other spectral features that may be present in the power spectrum. For instance, baryon wiggles have a characteristic scale of order ∆k ∼ 0.1, so we oversample the first one around k ∼ 0.1 by a factor ∆k/(k26 − k25) ∼ 16/ ln 10 ∼ 7. This parametrization means that we can write the pixel covariance matrix of equation (10) as C = 147 i=0 piC, i,(23) where the derivative matrix C, i ≡ ∂C/∂pi is the contribution from the i th band. For notational convenience, we have included the noise term in equation (23) by defining C,0 ≡ N, corresponding to an extra dummy parameter p0 = 1 giving the shot noise normalization. As in and HTP00, we in practice redefine the parameters pi to be ratio of the actual band power to the Figure 9. The 147 quadratic estimators q i , normalized so that their window functions equal unity and with the shot noise contribution f i (dashed curve) subtracted out. They cannot be directly interpreted as power spectrum measurements, since each point probes a linear combination of all three power spectra over a broad range of scales, typically centered at a k-value different than the nominal k where it is plotted. Moreover, nearby points are strongly correlated, causing this plot to overrepresent the amount of information present in the data. The solid curves show the prior power spectrum used to compute the error bars. prior band power. As long as the prior agrees fairly well with the measured result, this has the advantage of giving better behaved window functions, as described in . Our quadratic band power estimates are defined by qi ≡ 1 2 x t C −1 C,iC −1 x,(24) i = 0, ..., 147. These numbers are shown in Figure 9, and we group them together in a 148-dimensional vector q. Note that whereas x (and therefore C) is dimensionless, p has units of power, i.e., volume. Equation (24) therefore shows that q has units of inverse power, i.e., inverse volume. It is not immediately obvious that the vector q is a useful quantity. It is certainly not the final result (the power spectrum estimates) that we want, since it does not even have the right units. Rather, it is a useful intermediate step. In the approximation that the pixelized data has a Gaussian probability distribution (a good approximation in our case because of the central limit theorem, since N gal is large) q has been shown to retain all the power spectrum information from the original data set (Tegmark 1997, hereafter "T97"). The numbers qi have the additional advantage (as compared with, e.g., maximum-likelihood estimators) that their properties are easy to compute: their mean and covariance are given by q = Fp,(25)qq t − q q t = F,(26) where F is the Fisher information matrix (Tegmark et al. 1997) Fij = 1 2 tr C −1 C,iC −1 C,j .(27) Quadratic estimators were first derived for galaxy survey applications (Hamilton 1997ab). They were accelerated and first applied to CMB analysis (T97; Bond, Jaffe & Knox 2000). In conclusion, this step takes the vector x and its covariance matrix C from Figure 7 and compresses it into the smaller vector q and its covariance matrix F, illustrated in Figure 9 and Figure 10. Although equation (25) shows that we can obtain unbiased estimates of the true powers p by computing F −1 q, there are better options, as will be described in the next subsection. 3.6 Step 5: Fisher decorrelation and flavor disentanglement Let us first eliminate the shot-noise dummy parameter p0, since we know its value. We define f to be the 0 th column of the Fisher matrix defined above (fi ≡ Fi0) and restrict the indices i and j to run from 1 to 147 from now on, so f , q and p are 147-dimensional vectors and F is a 147 × 147 matrix. Since p0 = 1, equation (25) then becomes q = Fp + f . We now define a vector of shot noise corrected band power estimates p ≡ M(q − f ),(28) where M is some matrix whose rows are normalized so that the rows of MF sum to unity. Using equations (25) and (26), this gives the mean and covariance p = Wp,(29)Σ ≡ p p t − p p t = MFM t ,(30) where W ≡ MF. We will refer to the rows of W as window functions, since they sum to unity and equation (29) shows that pi probes a weighted average of the true band powers pj, the i th row of W giving the weights. Correlated, anticorrelated and uncorrelated band powers For the purpose of fitting models p to our measurements p, we are already done -the last two equations tell us how to compute χ 2 for any given p, and the result χ2 = ( p − p ) t Σ −1 ( p − p ) t(31) is independent of the choice of M. However, since one of the key goals of our analysis is to provide model-independent measurement of the three power spectra, the choice of M is crucial. Ideally, we would like both uncorrelated error bars (diagonal Σ) and well-behaved (narrow, unimodal and nonnegative) window functions W that do not mix the three power spectra. There are a number of interesting choices of M that each have their pros and cons (Tegmark & Hamilton 1998;. The simple choice where M is diagonal gives the "best guess" estimates in the sense of having minimum variance (Hamilton 1997a;T97;Bond, Jaffe & Knox 2000), and also has the advantage of being independent of the number of bands used in the limit of high spectral resolution. It was used for Figure 9 and Figure 10. Here the window functions are simply the rows of the Fisher matrix, and are seen to be rather broad. All entries of F are guaranteed to be positive as proven in PTH01, which means not only that all windows are positive (which is good) but also that all measurements are positively correlated (which is bad). Another interesting choice is (T97) M = F −1 , which gives W = I. In other words, all window functions are Kronecker delta functions, and p gives completely unbiased estimates of the band powers, with pi = pi regardless of what values the other band powers take. This gives an answer similar to the maximum-likelihood method (THSVS98), and the covariance matrix of equation (30) reduces to F −1 . A serious drawback of this choice is that that if we have sampled the power spectrum on a scale finer than the inverse survey size in an attempt to retain all information about wiggles etc., this covariance matrix tends to give substantially larger error bars (∆pi ≡ M 1/2 ii = [(F −1 )ii] 1/2 ) than the first method, anti-correlated between neighboring bands. The two above-mentioned choices for M both tend to produce correlations between the band power error bars. The minimum-variance choice generally gives positive correlations, since the Fisher matrix cannot have negative elements, whereas the unbiased choice tends to give anticorrelation between neighboring bands. The choice (Tegmark & Hamilton 1998;) M = F −1/2 with the rows renormalized has the attractive property of making the errors uncorrelated, with the covariance matrix of equation (30) diagonal. The corresponding window functions W are plotted in Figure 11, and are seen to be quite well-behaved, even narrower than those in Figure 10 while remaining positive. 1 This choice, which is the one we make in this paper, is a compromise between the two first ones: it narrows the minimum variance window functions at the cost of only a small noise increase, with uncorrelated noise as an extra bonus. The minimum-variance band power estimators are essentially a smoothed version of the uncorrelated ones, and their lower variance was paid for by correlations which reduced the effective number of independent measurements. Disentangling the three power spectra The fact that we are measuring three power spectra rather than one introduces an additional complication. As illustrated by Figure 12, an estimate of the power in one of the three spectra generally picks up unwanted contributions ("leakage") from the other two, making it complicated to interpret. Although the above-mentioned F −1 -method in principle eliminates leakage completely, the cost in terms of increased error bars is found to be prohibitive. We therefore follow HTP00 in adopting the following procedure for disentangling this three power spectra: For each of the 49 k-bands, we take linear combinations of the gg, gv and vv measurements such that the unwanted parts of the window functions average to zero. This procedure is mathematically identical to that used in Tegmark & de Oliveira-Costa (2001) for separating different types of CMB polarization, so the interested reader is referred there for the explicit equations. The procedure is illustrated in Figure 12, and by construction has the property that leakage is completely eliminated if the true power has the same 1 The reader interested in mathematical challenges will be interested to know that it remains a mystery to the authors why this F 1/2 method works so well. We have been unable to prove that F 1/2 has no negative elements (indeed, counterexamples can be contrived), yet the method works like magic in practice in all LSS and CMB applications we have tried. Figure 11. The window functions (rows of the gg-portion of W) are shown using decorrelated estimations. The i th row of W typically peaks at the i th band, the scale k that the band power estimator p i was designed to probe. Comparison with Figure 10 shows that decorrelation makes all windows substantially narrower. shape (not necessarily the same amplitude) as the prior. We find that this method works quite well (in the sense that the unwanted windows do not merely average to zero) for the most accurately measured band powers, in particular the central parts of the gg-spectrum, with slightly poorer leakage elimination for bands with larger error bars. The window functions plotted in Figure 11 are the gg-windows after disentanglement. Note that although our disentanglement introduces correlations between the gg, gv and vv measurements for a given k-band, different k-bands remain uncorrelated. RESULTS The three power spectra Our basic results are shown in Figure 13. The single most striking feature of this plot is clearly that the 2dFGRS is an amazing data set with unprecedented constraining power. The window functions in Figure 11 are seen to be quite narrow despite the complicated survey geometry. The galaxy-galaxy power is constrained to 20% or better over an order of magnitude in length scale, in about a dozen uncorrelated bands centered around k ∼ 0.1 h/Mpc. Whereas the increase in error bars on large scales reflects the finite survey volume, the lack of information on small scales is caused by our analysis being limited to the first 4000 PKL-modes. Comparing Figure 13 with Figure 9 serves as a sobering reminder of the importance of decorrelating and disentangling the measurements to avoid a misleadingly rosy picture of how well one can do. Whereas Pgg(k) is well measured, Figure 13 shows that the information about Pgv(k) is quite limited and that on Pvv(k) almost nonexistent. To avoid excessive Figure 12. The window function for our measurement of the 25th band of the galaxy-galaxy power is shown before (left) and after (right) disentanglement. Whereas unwanted leakage of gv and vv power is present initially, these unwanted window functions both average to zero afterward. The success of this method hinges on the fact that since the three initial functions (left) have similar shape, it is possible to take linear combinations of them that almost vanish (right). cluttering in Figure 13, band-power measurements with very low information content have been binned into fewer (still uncorrelated) bands. The main cause of these large error bars is that the information on Pvv and Pgg comes from the quadrupole and hexadecapole moments of the clustering anisotropy, which are intrinsically small and hence poorly constrained quantities. However, the problem may be exacerbated by the lack of large contiguous angular regions in the current data, impeding accurate comparisons of angular and radial clustering (the situation is similar for the SDSS; Zehavi et al. 2002), and should improve as the survey nears completion and gets more filled in. This effect is evident from a comparison with the results from the much more contiguous PSCz survey (HTP00): the error bars on Pgg are appreciably larger for PSCz than 2dFGRS, but those on redshift distortions (say β) are comparable. In the remainder of this paper, we will address two separate issues in turn: redshift-space distortions/biasing (β,r) and the detailed shape of the galaxy-galaxy power spectrum (model fits, evidence for baryonic wiggles, etc.). Constraints on redshift space distortions As seen from Figure 13, the constraints on Pgv(k) and Pvv(k) from 2dFGRS are too weak to allow β(k) and r(k) to be measured reliably as a function of scale. As data on Large Scale Structure improve, it should become possible to accomplish such a measurement, and thereby to establish quantitatively the scale dependence of biasing at linear scales. In the meantime we limit ourselves to the less ambitious goal of measuring overall parameters β and r, simply treating them as scale-independent constants. This has not been previously done for the case of r. Such scaleindependence of bias on linear scales is a feature of local bias models (Coles 1993;Fry and Gaztañaga 1993;Scherrer & Weinberg 1998;Coles, Melott & Munshi 1999;Heavens, Matarrese & Verde 1999). For our redshift-distortion analysis, we employ a simple scale-invariant power spectrum Pgg(k) of the BBKS form (Bardeen et al. 1986), parametrized by an amplitude σ8 and a "shape parameter" Γ that on a log plot shifts the curve vertically and horizontally, respectively. We will use more physically motivated power spectra with baryon wiggles etc. in Section 6.2 -we tried various alternative parametrizations, and found that the detailed form had essentially no effect on the (r, β)-constraints, since they come from the ratios of the three spectra, not from their shapes. Our model for the underlying band power vector p thus depends on four parameters (Γ, σ8, β, r). We map out the likelihood function L = e −χ 2 /2 using equation (31) on a fine grid in this parameter space, and compute constraints on individual parameters by marginalizing over the other parameters as described in Tegmark & Zaldarriaga (2000), maximizing rather than integrating over them. The results are plotted in figures 14, 15 and 16. Figure 13. Decorrelated and disentangled measurements of the galaxy-galaxy power spectrum (top), the galaxy-velocity power spectrum (middle) and the velocity-velocity) power spectrum (bottom) for the baseline galaxy sample. Red points represent negative values -since the points are differences between two positive quantities (total power minus expected shot noise power), they can be negative when the signal-to-noise is poor. Each points is plotted at the k-value that is the median of its window function, and 68% of this function is contained within the range of the horizontal bars. The curves shows our prior power spectrum. Note that most of the information in the survey is on the galaxy-galaxy spectrum. Band-power measurements with very low information content have been binned into fewer (still uncorrelated) bands. Figure 14. The blue/grey band shows the 1σ allowed range for β, assuming r = 1 and the shape of the prior Pgg(k) but marginalizing over the power spectrum normalization, using FOG compression with density threshold 1+δc = 100. These fits are performed cumulatively, using all measurements for all wavenumbers k. From bottom to top, the five curves show the best fit β for FOG thresholds 1+δc = ∞ (no FOG compression), 200, 100 (heavy), 50 and 25. Figure 15. 1-dimensional likelihood curves for Γ, β and r are shown after marginalizing over the power spectrum normalization and the other parameters using our baseline (1+δc = 100) fingerof-god compression. The 68% and 95% constraints are where the curves intersect the dashed horizontal lines. The dashed curve in the middle panel shows how the β-constraints tighten up when assuming r = 1. Figure 16. Constraints in the (β, r) plane are shown for our baseline (1+δc = 100) finger-of-god compression, using all measurements with k < 0.3h/Mpc and marginalizing over the power spectrum normalization for fixed spectral shape. The four contours correspond to ∆χ 2 = 1, 2.29, 6.18 and 11.83, and would enclose 39%, 68%, 95% and 99.8% of the probability, respectively, if the likelihood function were Gaussian. Figure 14 assumes Γ = 0.14, r = 1 (the best fit values) and explores how the results change as we include information from smaller and smaller scales. As will be discussed in more detail in Section 6, non-linear effects invalidate the Kaiser approximation for redshift space distortions on small scales. A smoking gun signature of such nonlinearities is r and hence the best-fit β dropping and ultimately going negative, as nonlinear "fingers of god" (FOGs) reverse the effect of linear redshift distortions. The fact that Figure 14 does not show this effect is reassuring evidence that little small-scale information is present in our data. This is of course by design, since our PKL-modes contain contributions only from ℓ 40, corresponding to a comoving distance around 20 h −1 Mpc at the characteristic survey depth of 400 h −1 Mpc. This lack of small-scale information in our PKL-modes is also reflected in the error bars on β, which are seen to stop decreasing around k ∼ 0.2 h/Mpc. Figure 14 also shows how the results depend on the FOG removal described in Section 3.1. The curves are seen to diverge markedly around k ∼ 0.2 h/Mpc, with the FOGrelated uncertainty becoming as large as the statistical error bars for k ∼ 1 h/Mpc. We will return to these nonlinearity issues in Section 5.4 below. Figure 15 shows the constraints on Γ, β and r after marginalizing over the other parameters. The best fit model is Γ = 0.14, β = 0.50, r = 1, σ8 = 0.99. The reason that the constraints on β are so weak is illustrated in Figure 16: there is a degeneracy with r. Figure 13 shows that our information about redshift distortions is coming predominantly from Pgv(k), not from the poorly constrained Pvv(k), so we are to first order measuring the combination βr rather than β and r individually. Imposing the prior r = 1, as was implicitly done in Peacock et al. and almost all prior work, therefore tightens the upper limit on β substantially, as shown by the dashed curve in Figure 15. The galaxy-galaxy power spectrum alone The previous subsection discussed the 2dFGRS constraints on redshift space distortions, essentially the ratios of the power spectra Pgg(k), Pgv(k) and Pvv(k), without regard to their shape. Let us now do the opposite, and focus on the shape of the galaxy power spectrum Pgg(k). The success of the disentanglement scheme illustrated in Figure 12 implies that the galaxy power spectrum plotted in Figure 13 is robust, essentially independent of what the power spectra Pgv(k) and Pvv(k) are doing. However, this robustness came at a price in terms of increased error bars. Assuming that all three power spectra have essentially the same shape, but not the same amplitudes, we compute a more accurate estimate of Pgg(k) as follows. We first assume some fixed values for β and r. This allows us to eliminate Pgv(k) and Pvv(k) using equation (22), reducing the size of our parameter vector p from 3×49 = 147 to 49 and our Fisher matrix to size 49 × 49, and gives 49 decorrelated estimators of Pgg(k). The result assuming β = 0.5, r = 1 (our best fit values) is shown in Figure 17. We perform no binning here except averaging the noisy bands with k < 0.02 and k > 0.8 into single bins to reduce clutter. We then repeat this exercise for a range of values of β and r consistent with our analysis in the previous subsection Figure 17. The decorrelated galaxy-galaxy power spectrum is shown for the baseline galaxy sample assuming β = 0.5 and r = 1. As discussed in the text, uncertainty in β and r contribute to an overall calibration uncertainty of order 12% which is not included in these error bars. to quantify the uncertainty these parameters introduce. We find these uncertainties to be quite small, as expected considering the small initial leakage of gv and vv power (see Figure 12), and can therefore quantify the added uncertainty δPgg to first order as δ ln Pgg(k) = ∂ ln Pgg(k) ∂(βr) δ(βr) + ∂ ln Pgg(k) ∂(β 2 ) δ(β 2 ). (32) Numerically, we find these two derivatives to be approximately −0.2 and −0.04, respectively, essentially independent of k. This scale-independence is not surprising in the smallangle limit, where these derivatives would involve simply various average moments of µ, the angle between the kvector and the line of sight. Assuming uncertainties δβ = 0.15 and and δr = 0.5, equation (32) thus gives δ ln Pgg(k) ≈ 0.12, the second term being negligible relative to the first. In conclusion, the uncertainties in Figure 17 induced by uncertainties about β and r can be summarized as simply an overall multiplicative calibration error of order 12% for the measured power spectrum. HOW RELIABLE ARE OUR RESULTS? How reliable are the results presented in the previous section? In this section, we perform a series of tests, both of our software and algorithms and of potential systematic errors. We also discuss the underlying assumptions that are likely to be most important for interpreting the results. Validation of method and software Since our analysis consists of a number of numerically nontrivial steps, it is important to test both the software and the underlying methods. We do this by generating Nmonte = 100 Monte Carlo simulations of the 2dFGRS catalog with a known power spectrum, processing them through our analysis pipeline and checking whether they give the correct answer on average and with a scatter corresponding to the predicted error bars. We found this end-to-end testing to be quite useful in all phases of this project -indeed, we had to run the pipeline 43 times until everything finally worked... The mock survey generator Standard N-body simulations would not suffice for our precision test, because of a slight catch-22 situation: the true non-linear power spectrum of which an N-body simulation is a realization (with shot noise added) is not known analytically, and is usually estimated by measuring it from the simulation -but this is precisely the step that we wish to test. We therefore generate realizations that are firmly in the linear regime, returning to nonlinearity issues below. We do this as described in PTH01, with a test power spectrum of the simple Gaussian form P (k) ∝ e −(Rk) 2 /2 with R = 32 h −1 Mpc, normalized so that the rms fluctuations δ 2 1/2 = 0.2. Figure 18 shows the result of processing the Monte Carlo simulations through the first step of the analysis pipeline, i.e., computing the corresponding Pseudo-KL expansion coefficients xi. This is a sensitive test of the mean correction given by equation (9), which can be a couple of orders of magnitude larger than the scatter in Figure 18 for some modes. A number of problems with the radial selection function integration and the spherical harmonic expansion of the angular mask in our code were discovered in this way. After fixing these problems, the coefficients xi became consistent with having zero mean as seen in the figure. Figure 18 also shows that the scatter in the modes is consistent with the predicted standard deviation σi = (Cii/Nmonte) 1/2 (shaded region), with most of the the fluctuations being localized to modes probing large scales (with i being small). A more sensitive test of this scatter is shown in Figure 19, which shows that the theoretically predicted variance for each mode agrees with what is observed in the 100 Monte Carlo realizations. Since crowding makes it hard to verify all modes in this plot, an alternative representation of this test is shown in Figure 20. Testing the PKL pixelization Although these tests verify that the mean and variance of each mode come out as they should, they are not sensitive to errors in the off-diagonal elements of the covariance matrix C, i.e., to incorrect correlations between the mode coefficients. To close this loophole, Figure 21 shows the scatter in the true KL-modes (y = Bx), illustrating agreement with the theoretical variance prediction even in this alternative basis where all coefficients yi should be uncorrelated. Note that the expected variance decreases monotonically here, as opposed to in Figure 19, since the true KL-modes are strictly sorted by decreasing variance. convolved with the window functions, and the observed scatter is seen to be consistent with the predicted error bars (Figure 23). These two figures therefore constitute an end-to-end test of our data analysis pipeline, since errors in any of the many intermediate steps would have shown up here at some level. Since information from large numbers of modes contributes to each pi, the scatter is seen to be small. Therefore, even quite subtle bugs and inaccuracies can be (and were!) discovered and remedied as a result of this test. Robustness to method details Our analysis pipeline has a few "knobs" that can be set in more than one way. This section discusses the sensitivity to such settings. Effect of changing the prior The analysis method employed assumes a "prior" power spectrum via equation (23), both to compute band power error bars and to find the galaxy pair weighting that minimizes them. As mentioned, an iterative approach was adopted starting with a simple BBKS model, then shifting it vertically and horizontally to better fit the resulting measurements and recomputing the measurements a second time. To what extent does this choice of prior affect the results? On purely theoretical grounds (e.g., Tegmark, Taylor & Heavens 1997), one expects a grossly incorrect prior to give unbiased results but with unnecessarily large variance. If the prior is too high, the sample-variance contribution to error bars will be overestimated and vice versa. This hypothesis has been extensively tested and Figure 23. Same as the previous figure, but testing the error bars ∆p i rather than the power itself. The triangles show the observed rms of the power spectrum estimates from 100 simulations and the solid blue curve shows the predicted curve around which they should scatter. confirmed in the context of power spectrum measurements from both the Cosmic Microwave Background (e.g., Bunn 1995) and galaxy redshift surveys (PTH01), confirming that the correct result is recovered on average even when using a grossly incorrect prior. In our case, the prior by construction agrees quite well with the actual measurements (see Figure 13), so the quoted error bars should be reliable as well. Effect of changing the number of PKL modes We have limited our analysis to the first N = 4000 PKL modes whose angular part is spanned by spherical harmonics with ℓ 40. This choice was a tradeoff between the desire to capture as much information as possible about the galaxy survey and the need to stay away from small scales where non-linear effects invalidate the Kaiser approximation to redshift distortions. To quantify our sensitivity to these choices, we repeated the entire analysis using 500, 1000, 2000 and 4000 modes. Our power spectrum measurements on the very largest scales were recovered even with merely 500 modes. As we added more and more modes (more and mode small-scale information), the power measurements converged to those in Figure 13 for larger and larger k. The rising part of the envelope in Figure 10 remained essentially unchanged, merely continuing to grow further as more modes were added, so the turnover of this envelope directly shows the k-scale beyond which we start running out of information. The version of Figure 10 shown in this paper indicates that our 4000 PKL modes have captured Figure 24. Numerical convergence. The figure shows for how many of our 4000 PKL modes the numerical calculations are converged to accurately measure the power up to a given wavenumber k. From left to right, the 12 curves correspond to truncation at ℓcut =20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220 and 240. essentially all cosmological information from the 2dFGRS for k ∼ < 0.1. Numerical issues The computation of the matrices Pi involves a summation over multipoles ℓ that should, strictly speaking, run from ℓ = 0 to ℓ = ∞, since the angular mask itself has sharp edges involving harmonics to ℓ = ∞. In practice, this summation must of course be truncated at some finite multipole ℓcut. To quantity the effect of this truncation, we plot the diagonal elements of the P-matrices as a function of ℓcut and study how they converge as ℓcut increases. We define a given PKLmode as having converged by some multipole if subsequent ℓ-values contribute less than 1% of its variance. Figure 24 plots the number of usable PKL-modes as a function of wavenumber k, defining a mode to be usable for our analysis only if it is converged for all smaller wavenumbers k ′ < k for all three power flavors (Pgg, Pgv and Pvv). We use ℓcut = 260 in our final analysis, since this guarantees that all 4000 modes are usable for wavenumbers k in the range 0 − 0.5 h/Mpc, i.e., comfortably beyond the large scales 0 − 0.3 h/Mpc that are the focus of this paper. With this cutoff, the computation of the P -matrices (which scales as ℓ 2 cut asymptotically), took about a week on a SunBlade1000 workstation. Our power spectrum estimates are likely to remain fairly accurate as far out as we plot them, i.e., to k ∼ 1 h/Mpc, since Figure 24 shows most modes remaining usable out to this scale, and since we find that even the ones that do not meet our strict 1% convergence criterion at every single band are generally fairly accurately treated. Indeed, we repeated our entire analysis with ℓcut = 120 and obtained almost indistinguishable power spectra. Figure 25. Constraints on excess power in special modes. Our 2dF power spectrum measurements from Figure 17 are averaged into fewer bands and compared with measurements using only special (radial, angular and local group) modes and only generic (the remaining) modes (dashed). Tests for problems with data modeling In Section 2, we performed detailed modeling of the way in which the 2dFGRS data was selected, and produced a uniform galaxy sample fully characterized by a selection functionn(r) of the separable form of equation (1). Let us now assess how sensitive our results are to potential misestimates ofn, both angularly and radially, by discarding purely angular and radial modes from our analysis. Robustness to angular problems Angular modulations caused by dust extinction tend to have a power spectrum rising sharply toward the largest scales (Vogeley 1998), and is therefore of particular concern for the interpretation of our leftmost bandpower estimates. The galaxy magnitudes are extinction corrected by the 2dFGRS team, using extinction map produced by Schlegel, Finkbeiner & Davis (1998), so any inaccuracies in this extinction model would masquerade as excess large-scale power. Inaccuracies in zero-point offsets or in the magnitude dependent completeness correction that we applied in Section 2.2.2 could also introduce spurious angular power. Of our 4000 modes, 147 are purely angular (see Figure 6 for an example), and as described in Section 3.2, the remaining 3853 are orthogonal to them. This means that to first order, angular problems affect only these 147 PKLcoefficients xi. We repeated our entire analysis with these coefficients discarded, and found that the error bars became so large for k ∼ < 0.03 h/Mpc that no signal could be detected there. In other words, the information on the power spectrum on the very largest scales comes mainly from the purely angular modes. On smaller scales, the measured power spectrum remained essentially unchanged. Although we have no indication that angular problems are actually present, it may be prudent to follow the 2dFGRS team and discard the information on the very largest scales -to be conservative, we therefore use only the measurements for k 0.01 h/Mpc to be conservative in our likelihood analyses (for β, r and cosmological parameters). Robustness to problems with the radial selection function 45 of our 4000 modes are purely radial (see Figure 6 for an example), and are to first order the only ones affected by mis-estimates of the radial selection function n(r). Since accurate k-corrections and evolution modeling are notoriously challenging to perform, we repeated our entire analysis with these 45 modes omitted as a precaution. This resulted in a slight increase in error bars on the largest scales, but much less noticeable than when we removed the angular modes as described above. This can be readily understood geometrically. If we count the number of modes that probe mainly scales k < k * , then the number of purely radial, purely angular and arbitrary modes will grow as k * , k 2 * and k 3 * , respectively. This means that "special" modes (radial and angular) will make up a larger fraction of the total pool on large scales (at small k), and that the purely radial ones will be outnumbered by the purely angular ones. Percival et al. (2001) report that slight changes inn(r) did not have a strong effect on the recovered 2dFGRS power spectrum, and we confirm this. We repeated our analysis with a number of different radial selection functionsn(r), including the one from Colless et al. (2001) (the dashed curve in Figure 3), finding only changes smaller than the error bars for P (k) on the largest scales and no noticeable changes for larger k. A final end-to-end test for problems with any special (angular, radial, or local group) modes is shown in Figure 25. Here we have repeated the entire analysis twice, once excluding all the special modes and once using only the special modes (except the monopole). The latter is seen to give quite large error bars since only 196 modes are used (4 local group, 147 angular and 45 radial), but all three are seen to be reassuringly consistent. In contrast, systematic problems with any special modes would tend to add power to the special modes. This shows that any misestimates of special modes is having a negligible impact on our final results. Non-linearity issues A key assumption (essentially the only one) underlying our analysis is that the Kaiser (1987) linear perturbation theory approach to redshift space distortions is valid. This approximation is known to break down on small scales where nonlinear effects become important, which is why we have limited our analysis to large scales. To be more precise, our basic measurement of Pgg(k), Pgv(k) and Pvv(k) assumes nothing at all, and measures the quantities that reduce to the monopole, quadrupole and hexadecapole of power in the in small-angle approximation (Hamilton 1998). However, relating these three measured functions to β(k) and r(k) via equation (22) does require the Kaiser approximation to be valid. Substantial progress has recently been made in quantifying nonlinear effects on redshift distortions, using both perturbation theory, gravitational N -body simulations and semianalytic galaxy formation theory (Hatton & Cole 1997Scoccimarro et al. 1999;Heavens, Matarrese & Verde 1999;Scoccimarro, Zaldarriaga & Hui 1999;Hamilton 2000;Seljak 2001;Scoccimarro & Sheth in preparation). The consensus is that nonlinear effects may be important even on scales as large as k ∼ 0.1 − 0.3 h/Mpc), although the critical scale is sensitive to the type of galaxies involved via their bias properties (Seljak 2001). Moreover, a generic smokinggun signature of nonlinear effects is found to be that the ratio Pgv(k)/Pgg(k) starts dropping and eventually becomes negative, as nonlinear fingers-of-god reverse the signature of linear infall. The ratio Pvv(k)/Pgg(k) increases sharply in this regime. Ideally, to do full justice to the 2dFGRS data set, one would like to perform a suite of nonlinear simulations until a realistic biasing scheme is found that reproduces all observed characteristics of the data. The fast PTHalos approach suggests that such an ambitious approach may ultimately be feasible. In the interim, the results obtained with analytic approximations must be interpreted with great caution. Peacock et al. (2001) use the widespread approach of adding a nuisance parameter to the Kaiser formula, interpreted as a small-scale velocity dispersion (cite), and marginalizing over it. This gives β = 0.43 ± 0.07 from 141,000 2dFGRS galaxies. Hatton & Cole (1999) and Scoccimarro & Sheth (in preparation) argue that this is approximation is inaccurate, underestimating the nonlinear corrections (hence underestimating β) on large scales, and that the approximation of Hatton & Cole (1999) is preferable. Given these important uncertainties, we adopt a more empirical approach, using the above-mentioned Pgv-drop in the data as an indicator of where to stop trusting the results. This was also done in the PSCz analysis of Hamilton et al. (2001), where β was found to start dropping for k ∼ > 0.3 h/Mpc. Figure 13 shows no indication of Pgv(k)/Pgg(k) (basically the quadrupole-to-monopole ratio) dropping, suggesting that our linear approximation is not seriously biasing our results on the large scales probed by our PKL modes (which recover information fully down to k ∼ 0.1 as described above). To quantify further the effect of non-linearities empirically, we performed our entire analysis five times with different levels of finger-of-god (FOG) compression as described in Section 3.1. The five curves in Figure 14 correspond to progressively more aggressive compression with overdensity cutoffs 1+δc = ∞, 200, 100, 50 and 25, respectively. This corresponds to 6677, 7820, 8643 and 9124 FOG's compressed, involving 18544, 24031, 29807 and 36098 galaxies, respectively. Figure 14 shows that more aggressive FOG-compression has an effect with the expected sign, increasing the best-fit β-value for k ∼ > 0.1, and that the effect is reassuringly small compared to the statistical error bars. Since a cluster is expected to have an overdensity around 200 when it virializes, more later since the background density drops, thresholds 1+δc < 100 are likely to be overkill -we included the cases 1+δc = 50 and 25 in the figure merely to explore an extreme range of remedies. By removing essentially all structures that are elongated along the line of sight, one of course creates an artificial excess of flattened structures, leading to an overestimate of β. In conclusion, we believe that our estimate β = 0.49 ± 0.16 is not severely affected by nonlinearities. A conservative approach would be to take our measurement without FOG compression and use it merely as a lower limit, giving β > 0.26 at 90% confidence. Non-linearities affect our analysis in a different way as well, leading to slight underestimates of error bars. Our power spectrum measurements are simply certain second moments of the data, and remain valid regardless of whether the underlying density field is Gaussian or not. The power spectrum variance, however, involves fourth moments, and we have computed our error bars by making the Gaussian approximation to calculate these moments. The standard rule of thumb is that this approximation underestimates the error bars on the correlation function ξ(r) by a factor [1+ξ(r)] 1/2 . Norberg et al. (2001) fit the 2dFGRS correlation function to a power law ξ(r) = (r/r * ) −γ with correlation length r * = 4.9 h −1 Mpc and slope γ = 1.71. Taking k ∼ π/r, this gives error bar correction factors [1 + (r * k/π) γ )] 1/2 ≈ 2%, 7% and 13% at 0.1, 0.2 and 0.3 h/Mpc, respectively. Here ξ(r) should refer to the correlation function of the matter, not of the galaxies, so if the 2dFGRS galaxies are biased with b > 1, the correction factors will be smaller. In conclusion, although nonlinear error bar corrections certainly become important on very small scales, they are likely to be of only minor importance on the large scales k < 0.3 h/Mpc that are the focus of this paper. Bias issues Although our basic measurement of Pgg(k), Pgv(k) and Pvv(k) assumes nothing about biasing, a bias model is obviously necessary before the results can be used to constrain cosmological models. We therefore comment briefly on the bias issue here. Substantially larger data sets such as the complete SDSS catalog hold the promise of measuring β(k) and r(k) with sufficient accuracy to quantity their scale-dependence, if any. Figure 13 shows that our present sample is still not quite large enough to place strong constraints of this type. An alternate route to constraining b(k) involves comparing the clustering amplitudes of various subsamples, selected by, say, luminosity or spectral type. Such comparisons can also constrain r directly (Tegmark & Bromley 1999;Blanton 2000). It has been long known that bright elliptical galaxies are more clustered than spirals, presumably because the former are more likely to reside in clusters. Recent subsample analysis of the 2dFGRS (Norberg et al. 2001a) and SDSS (Zehavi et al. 2002) have confirmed and further quantified this effect. Since recent cosmological parameter analyses using P (k)-measurements (most recently Wang et al. 2002 andEfstathiou et al. 2002) have assumed that the bias factor b is scale-independent on linear scales, it is important to note that slight scale-dependence of bias is likely to be present in Pgg(k)-measurements from a heterogeneous galaxy sample such as the 2dFGRS. Most of the information about Pgg(k) on large scales comes from distant parts of the survey, where bright ellipticals are over-represented since dimmer galaxies get excluded by the faint magnitude limit. This could cause b(k) to rise as k → 0. If uncorrected, this effect could masquerade as evidence for a redder power spectrum, i.e., one with a smaller spectral index n. Figure 17 indeed suggests slightly more 2dFGRS power on the largest scales than currently favored cosmological models with constant bias would suggest, although this excess may also be caused by the angular or radial issues mentioned above. Detailed power spectrum analysis of subsamples should settle this issue. DISCUSSION AND CONCLUSIONS To place our results in context, we will now briefly discuss how they compare with other recent power spectrum measurements and with cosmological models. Figure 26 compares our 2dFGRS power spectrum measurements from Figure 17 (averaged into fewer bands to reduce clutter) with measurements from other recent surveys. The PSCz and UCZ redshift surveys were analyzed with the same basic method that we have employed here 2 , so a direct comparison involves no method-related interpretational issues. The 2dFGRS sample is seen to be slightly more biased than PSCz, but slightly less biased than UZC. Figure 26 also suggests that 2dFGRS may have a slightly redder power spectrum than PSCz. This would also be consistent with the scale-dependent bias scenario mentioned above -the PSCz survey would probably be less afflicted than 2dFGRS, since the IRAS-selected galaxies in PSCz tend to avoid clusters. Comparison with other surveys Although the 2dFGRS error bars are seen to be small compared the PSCz and UZC ones, due to the larger sample size and survey volume, the horizontal bars show that the 2dFGRS window functions are somewhat broader. This is easy to understand: whereas PSCz and UZC cover large contiguous sky regions, the 2dFGRS sky coverage is currently fragmented into a multitude of regions of small angular extent, exacerbating aliasing problems. Indeed, since the characteristic width of 2dFGRS patches in the narrowest direction is more than an order of magnitude smaller than for PSCz or UZC (of order 2 • rather than ∼ 60 • ), the fact that the windows are only 2-3 times wider reflects the quality of the 2dFGRS survey design and the power of the quadratic estimator method. The remaining two power spectra are interesting since they were measured without use of redshift information and thus without the additional complications introduced by redshift space distortions. The APM points are from the likelihood analysis of Efstathiou & Moody (2001), using a few million galaxies, and reflect the full uncertainty even on the largest scales. Here the vertical bands have Figure 26. Comparison with other power spectrum measurements. Our 2dF power spectrum measurements from Figure 17 are averaged into fewer bands and compared with measurements from the PSCz (HTP00) and UZC (this work) redshift surveys as well as angular clustering in the APM survey (Efstathiou & Moody 2001) and the SDSS (the points are from Tegmark et al. 2002 for galaxies in the magnitude range 21 < r ′ < 22 -see also Dodelson et al. 2002). a different interpretation, indicating the bands used in the likelihood analysis. Note that although the 2dFGRS galaxies are a subset of the APM galaxies, they need not have the exact same bias. Since the 2dFGRS subset involves on average brighter and more luminous galaxies, one might expect them to be slightly more clustered. The SDSS points (from Tegmark et al. 2002) are for about a million galaxies in the magnitude range 21 < r ′ < 22, and the vertical bars have the same interpretation as for the 2dFGRS points (redshift information obviously helps tighten up the windows). In contrast, the parameterized SDSS power spectrum in Dodelson et al. (2002) can be interpreted like the APM one. A direct comparison of our power spectrum results with those reported by the 2dFGRS team (Percival 2001) is unfortunately not possible at this time, since their window functions are of crucial importance and have not yet been made publicly available. However, an indirect comparison is possible as described in the next section, indicating good agreement. Our β-constraints are consistent with those reported in Peacock et al. (2001). Figure 17 are averaged into fewer bands and compared with theoretical models. The BBKS model is the wiggle-free prior used for our calculation. The flat ΛCDM "concordance" models from Wang et al. (2002) and Efstathiou et al. (2002), both renormalized to our 2dF measurements, are seen to be quite similar. The wigglier curve corresponds to the best-fit high baryon model in the upper right corner of Figure 28. Only data to the left of the dashed vertical line are included in our fits. dependent bias. The measurements are seen to be in good agreement with both our simple BBKS prior and the recent concordance model from Efstathiou et al. (2002) -specifically, this is fit B from their paper, a flat scaleinvariant scalar model with ΩΛ = 0.71, h = 0.69, baryon density ω b = 0.021 and dark matter density ω d = 0.12. Cosmological constraints (ω b ≡ h 2 Ω b , ω d ≡ h 2 Ω d .) Both of these are of course good fits by construction: we iterated our analysis until we found a prior that was consistent with the data, and Efstathiou et al. (2002) searched for models fitting both the 2dFGRS power spectrum and CMB data. However, the fact that the Efstathiou et al. (2002) model fits our data so well provides an important cross-check between the 2dFGRS team power spectrum measurement (Percival et al. 2001) and ours, indicating good agreement. Figure 27 also shows the concordance model from Wang et al. (2002), resulting from a fit to all CMB data and the PSCz galaxy power spectrum. It is a flat scalar model with ΩΛ = 0.66, h = 0.64, baryon density ω b = 0.020, dark matter density ω d = 0.12 and a slight red-tilt, ns = 0.91, here renormalized to the PSCz data. The fact that these pre-2dF and post-2dF concordance models agree so well is a reassuring indication that such multi-parameter analyses are converging to the correct answer, and that the final numbers are not overly sensitive to bias issues or methodological technicalities. A full multiparameter analysis of our results along the lines of Wang et al. (2002) and Efstathiou et al. (2002) is clearly beyond the scope of the present paper. However, since evidence for baryonic wiggles in the galaxy power spectrum has generated strong recent interest, first from the PSCz data (HTP00) and then more strikingly from the 2dF data (Percival et al. 2001;Miller et al. 2001), we perform a limited analysis to address the baryon issue. We consider flat scale-invariant scalar models parametrized by the total matter content Ωm, the baryon fraction Ω b /Ωm, the hubble parameter h and the spectral index ns. We map out the likelihood function L = e −χ 2 /2 using equation (31) on a fine grid in this parameter space, and compute constraints on individual parameters by marginalizing over the other parameters. Figure 28 shows the result of fixing ns = 1 and h = 0.72, the best-fit value from Freedman et al. (2001). Here the axes have been chosen to facilitate comparison with Figure 5 from Percival et al. (2001) 3 . The general agreement between the two figures is seen to be good, both in terms of the shape and location of the banana-shaped degeneracy track, and in that there are two distinct favored regions -a low-baryon solution like the concordance models in Figure 27 and a high-baryon solution that is inconsistent with both Big Bang Nucleosynthesis (Burles et al. 2001) and CMB constraints. To illustrate the nature of the banana degeneracy in Figure 28 , we have plotted the best fit high-baryon model in Figure 27. It has Ωm = 0.75 and ω b = 0.18, and is seen to provide a slightly better fit to the data around k = 0.04 h/Mpc at the expense of slight difficulties on smaller scales. There is, however, one notable difference between Figure 28 and its twin in Percival et al. (2001). Whereas the latter excluded Ω b /Ωm = 0, we find no significant detection of baryons. This is of course not an indication of problems with either analysis, since the Percival et al. figure excludes zero baryons only at modest significance. Most importantly, as emphasized by Efstathiou et al. (2002), the constraints get much weaker when allowing small variations in other parameters, most strikingly the spectral index ns. We confirm this effect by marginalizing over ns and h with various priors. This means that the full statistical power of the complete 2dF and SDSS data sets will be needed to provide unequivocal evidence for baryonic signatures in the galaxy distribution. Outlook We have computed the real-space power spectrum and the redshift-space distortions of the first 10 5 galaxies in the 2dFGRS using pseudo-Karhunen-Loève eigenmodes and the stochastic bias formalism, providing easy-to-interpret uncorrelated power measurements with narrow and wellbehaved window functions in the range 0.01 h/Mpc < k < 1 h/Mpc. A battery of systematic error tests indicate that the survey is not only impressive in size, but also unusually clean. Figure 28. Constraints in on the matter density Ωm and the baryon fraction Ω b /Ωm from the linear power spectrum over the range 0.01 h/Mpc < k < 0.3 h/Mpc, after marginalizing over the power spectrum amplitude. These constraints assumes a flat, scale-invariant cosmological model with h = 0.72. For comparison with Percival et al (2001), contours have been plotted at the level for one-parameter confidence of 68% and two-parameter confidence of 68%, 95% and 99% (i.e., χ2−χ2 min = 1, 2.3, 6.0, 9.2. Marginalizing over the Hubble parameter h and limiting the analysis to scales k < 0.15h/Mpc as in Percival et al (2001) further weakens the constraints. Galaxy redshift surveys are living up to expectations. The striking early successes of the 2dFGRS and SDSS projects have firmly established galaxy redshift surveys as a precision tool for constraining cosmological models. However, it is important to bear in mind that this is only the beginning, and that many of the most exciting cosmological applications of these surveys still lie ahead. As discussed above, detailed comparisons with grids of fast simulations are likely to place information extracted from redshift distortions on a firmer footing and allow substantially more velocity information to be extracted from translinear scales. A bivariate analysis of how clustering depends jointly on both spectral type and luminosity should improve our quantitative understanding of biasing and allow possibilities such as the above-mentioned artificial red-tilt to be quantified and eliminated. With such progress combined with an order-of-magnitude increase in sample size, to more than 10 6 galaxies from 2dFGRS and SDSS combined, exciting opportunities will abound over the next few years, from definitive constraints on baryons and neutrinos to things that have not even been thought of yet. Figure 2 . 2Number of galaxies surviving as a function of uniform magnitude cut. Figure 4 . 4The effect of our Fingers-of-god (FOG) removal is shown in the southern slice δ = −27.7 • , −35 • < RA < 53 • . Figure 5 . 5A sample of four angular pseudo-KL (PKL) modes are shown in Hammer-Aitoff projection in equatorial coordinates, with grey representing zero weight, and lighter/darker shades indicating positive/negative weight, respectively. From top to bottom, they are angular modes 1 (the mean mode), 3, 20 and 106, and are seen to probe successively smaller angular scales. Figure 6 . 6A sample of six pseudo-KL modes are shown in the plane of the southern 2dF slice with δ = −27.7 • , −35 • < RA < 53 • . Grey represents zero weight, and lighter/darker shades indicate positive/negative weight, respectively. From left to right, top to bottom, these are modes 1 (the mean mode), 14, 104, 148, 58 and 178, and are seen to probe successively smaller scales. Those in the middle panel are examples of purely radial (left) and purely angular (right) modes. (ii) Modes 2-5 are associated with the motion of the Local Group through the Cosmic Microwave Background at 622 km/s towards (B1950 FK4) RA = 162 • , Dec = −27 • (Lineweaver et al. 1996; Courteau & van den Bergh 1999). In the angular direction, these Local Group modes are monopole and dipole modes multiplied by the angular mask, Figure 7 7Figure 7. The triangles show the 4,000 elements x i of the data vector x (the pseudo-KL expansion coefficients) for the baseline galaxy sample. If there were no clustering in the survey, merely shot noise, they would have unit variance, and about 68% of them would be expected to lie within the blue/dark grey band. If our prior power spectrum were correct, then the standard deviation would be larger, as indicated by the shaded yellow/light grey band. Figure 8 . 8The triangles show the 3999 uncorrelated elements y i of the transformed data vector y = Bx (the true KL expansion coefficients) for the baseline galaxy sample. Figure 10 . 10The rows of the gg-portion of the Fisher matrix F. The i th row typically peaks at the i th band, the scale k that the band power estimator q i was designed to probe. All curves have been renormalized to unit area, so the highest peaks indicate the scales the the window functions obtained are narrowest. The turnover in the envelope at k ∼ 0.1 h/Mpc reflects our running out of information due to omission of modes probing smaller scales. For comparison with the next figure, these are the rows of W when M is diagonal. Figure 18 . 18The triangles show the elements x i of the data vector x (the pseudo-KL expansion coefficients) averaged over 100 Monte-Carlo simulations of the baseline galaxy sample. If the algorithms and software are correct, then their mean should be zero and about 68% of them should lie within the shaded yellow/grey region giving their standard deviation. Figure 19 .Figure 22 Figure 20 . 192220The triangles show the rms fluctuations of the elements x i from 100 Monte-Carlo simulations. If the algorithms and software are correct, then the expectation value of this rms is given by the thin blue curve, and most of them should scatter in the yellow/grey region.5.1.3 Testing the quadratic compression, Fisher decorrelation and disentanglementFigures 22 and 23 show the result of processing the Monte Carlo simulations through the remaining steps of the analysis pipeline, i.e., computing the raw quadratic estimator vector q and, from it, the decorrelated and disentangled band-power vector p. The mean recovered power spectra are seen to be in excellent agreement with the Gaussian prior used in the simulations (In this alternative representation of the test fromFigure 19, most of the vertical lines should intersect the 45 • line if the algorithms and software are correct. Figure 21 . 21The triangles show the rms fluctuations of the elements (Bx) i from 100 Monte-Carlo simulations. If the algorithms and software are correct, then the expectation value of this rms is given by the thin blue curve, and most of them should scatter in the yellow/grey banana-shaped region. Figure 22 . 22The triangles show the decorrelated and disentangled band-power estimates p i , averaged over 100 Monte-Carlo simulations of the baseline galaxy sample. If the algorithms and software are correct, then this should recover the window-convolved input power spectrum Wp, plotted as a thin blue line. The thin shaded yellow/grey band indicates the expected scatter. The harmless discontinuity in the middle panel is an artifact of the disentangled galaxy-velocity windows having negative area on the largest scales where there is essentially no information available. Figure 27 Figure 27 . 2727compares our 2dFGRS measurements with theoretical predictions from a series of models. No corrections have been made for non-linear evolution or scale-Our 2dF power spectrum measurements from that b must depend on galaxy type. However, there is also evidence that it depends on scale (see e.g. Mann et al. 1997; Blanton et al. 1999; Hamilton & Tegmark 2002 and references therein) and on time (Fry 1996; Tegmark & Peebles 1998; Giavalisco et al. 1998). Finally, there are good reasons to believe that there is no deterministic relation that can replace equation Since the UZC analysis in PTH01 did not include redshift space distortions, we performed a complete reanalysis of that data set for this figure, expanding the 13342 galaxies surviving the cuts described in PTH01 in 1000 PKL modes. As a technical point, Percival et al. included band powers up to a nominal wavenumber k = 0.15 in their figure. Since our window functions are narrower, we have included band powers up to k = 0.3 inFigure 28to ensure that we do not use less small-scale information. & NAG5-10763, the University of Pennsylvania Research Foundation, the Zaccheus Daniel Foundation and the David and Lucile Packard Foundation. ACKNOWLEDGEMENTSThe authors wish to thank the 2dFGRS team for kindly making the data from this superb survey public and Shaun Cole, Mathew Colless and Karl Glazebrook in particular for helpful information about technical survey details. Thanks to Martin Kunz and Michael Vogeley for helpful comments. Support for this work was provided by NSF grants AST-0071213 & AST-0134999, NASA grants NAG5-9194, NAG5- BBKS. J M Bardeen, J R Bond, N Kaiser, A S Szalay, ApJ. 30415Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, ApJ, 304, 15 ("BBKS") . M Blanton, ApJ. 54463Blanton, M. 2000, ApJ, 544, 63 . M Blanton, R Cen, J P Ostriker, M A Strauss, M Tegmark, ApJ. 5311Blanton, M., Cen, R., Ostriker, J. P., Strauss, M. A., & Tegmark, M. 2000, ApJ, 531, 1 . J R Bond, Phys. Rev. Lett. 744369Bond, J. R. 1995, Phys. Rev. Lett., 74, 4369 . J R Bond, A H Jaffe, L E Knox, ApJ. 53319Bond, J. R., Jaffe, A. H., & Knox, L. E. 2000, ApJ, 533, 19 . E F. ; U C Bunn, S Berkeley Burles, K M Nollett, M S Turner, ApJ. 5521Ph.D. Thesis,Bunn, E. F. 1995, Ph.D. Thesis, U.C. Berkeley Burles, S., Nollett, K. M., & Turner, M. S. 2001, ApJ, 552, L1 . S Cole, MNRAS. 326255Cole, S. et al. 2001, MNRAS, 326, 255 . P Coles, MNRAS. 2621065Coles, P. 1993, MNRAS, 262, 1065 . P Coles, A Melott, D Munshi, ApJ. 5215Coles, P., Melott, A., & Munshi, D. 1999, ApJ, 521, 5 . N Cross, MNRAS. 324825Cross, N. et al. 2001, MNRAS, 324, 825 . M Colless, MNRAS. 3281039Colless, M. et al. 2001, MNRAS, 328, 1039 ("C01") . A Connolly, astro-ph/0107417ApJ. 118337AJConnolly, A. et al. 2001, astro-ph/0107417, ApJ, in press Courteau, S., & van den Bergh, S. 1999, AJ, 118, 337 . M Davis, M J Geller, ApJ. 20813Davis, M., & Geller, M. J. 1976, ApJ, 208, 13 . A Dekel, O Lahav, ApJ. 52024Dekel, A., & Lahav, O. 1999, ApJ, 520, 24 . S Dodelson, astro-ph/0109152MNRAS. 3251603MNRASDodelson, S. et al. 2002, astro-ph/0107421, ApJ, in press Efstathiou, G. et al. 2002, astro-ph/0109152, MNRAS, in press Efstathiou, G., & Moody, S. J. 2001, MNRAS, 325, 1603 . E E Falco, PASP. 111438Falco, E. E. et al. 1999, PASP, 111, 438 . K B Fisher, M Davis, M A Strauss, A Yahil, J P Huchra, ApJ. 40242Fisher, K. B., Davis, M., Strauss, M. A., Yahil, A., & Huchra, J. P. 1993, ApJ, 402, 42 . K B Fisher, C A Scharf, O Lahav, MNRAS. 266219Fisher, K. B., Scharf, C. A., & Lahav, O. 1994, MNRAS, 266, 219 . W L Freedman, ApJ. 55347Freedman, W. L. 2001, ApJ, 553, 47 . J N Fry, ApJ. 46165Fry, J. N. 1996, ApJ, 461, L65 . J N Fry, E Gaztañaga, ApJ. 413447Fry, J. N., & Gaztañaga, E. 1993, ApJ, 413, 447 . M Giavalisco, ApJ. 503543Giavalisco, M. et al. 1998, ApJ, 503, 543 . A J Hamilton, MNRAS. 289285Hamilton, A. J. S. 1997a, MNRAS, 289, 285 . A J Hamilton, MNRAS. 289295Hamilton, A. J. S. 1997b, MNRAS, 289, 295 . A J S Hamilton, astro-ph/9708102Hamilton, A. J. S. 1997c, astro-ph/9708102 . A J S Hamilton, MNRAS. 312257Hamilton, A. J. S. 2000, MNRAS, 312, 257 . A J Hamilton, MNRAS. 322419Hamilton, A. J. S. 2001, MNRAS, 322, 419 . A J S Hamilton, M Culhane, MNRAS. 27873Hamilton, A. J. S. & Culhane, M. 1996, MNRAS, 278, 73 . A J S Hamilton, M Tegmark, MNRAS. 312285Hamilton, A. J. S., & Tegmark, M. 2000, MNRAS, 312, 285 . A J S Hamilton, M Tegmark, MNRAS. 330506Hamilton, A. J. S., & Tegmark, M. 2002, MNRAS, 330, 506 HTP00. A J S Hamilton, M Tegmark, N Padmanabhan, MNRAS. 23Hamilton, A. J. S., Tegmark, M., & Padmanabhan, N. 2000, MNRAS, 317, L23 ("HTP00") . S J Hatton, S Cole, MNRAS. 29610Hatton, S. J., & Cole, S. 1998, MNRAS, 296, 10 . S J Hatton, S Cole, MNRAS. 3101137Hatton, S. J., & Cole, S. 1999, MNRAS, 310, 1137 . A . F Heavens, S Matarrese, L Verde, MNRAS. 301797Heavens, A.. F., Matarrese, S., & Verde, L. 1999, MNRAS, 301, 797 . A F Heavens, A N Taylor, MNRAS. 483497Heavens, A. F., & Taylor, A. N. 1995, MNRAS, 483, 497 . J P Huchra, M J Geller, V De Lapparent, Ii G CorwinJr, ApJS. 72433Huchra, J. P., Geller, M. J., de Lapparent, V., & Corwin, II. G. Jr. 1990, ApJS, 72, 433 . N Kaiser, MNRAS. 2271Kaiser, N. 1987, MNRAS, 227, 1 K Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung (Kirjapaino oy. sana: Helsinki). Karhunen, K. 1947,Über lineare Methoden in der Wahrscheinlichkeitsrechnung (Kirjapaino oy. sana: Helsinki) . C H Lineweaver, L Tenorio, G F Smoot, P Keegstra, A J Banday, P Lubin, ApJ. 47038Lineweaver, C. H., Tenorio, L., Smoot, G. F., Keegstra, P., Banday, A. J., & Lubin, P. 1996, ApJ, 470, 38 . D Lynden-Bell, MNRAS. 15595Lynden-Bell, D. 1.971, MNRAS, 155, 95 . R G Mann, J A Peacock, A F Heavens, MNRAS. 293209Mann, R. G., Peacock, J. A., & Heavens, A. F. 1998, MNRAS, 293, 209 . T Matsubara, A S Szalay, S D Landy, ApJ. 5351Matsubara, T., Szalay, A. S., & Landy, S. D. 2000, ApJ, 535, 1 . C J Miller, R C Nichol, D J Batuski, Science. 2922302Miller, C. J., Nichol,, R. C., & Batuski, D. J. 2001, Science, 292, 2302 . P Norberg, MNRAS. 32864Norberg, P. et al. 2001a, MNRAS, 328, 64 . P Norberg, astro-ph/0111011Norberg, P. et al. 2001b, astro-ph/0111011 . N Padmanabhan, M Tegmark, Hamilton A J S, ApJ. 55052Padmanabhan, N., Tegmark, M., & Hamilton A J S 2001, ApJ, 550, 52 . Peacock, Nature. 410169Peacock, et al.. 2001, Nature, 410, 169 . U Pen, ApJ. 504601Pen, U. 1998, ApJ, 504, 601 . W J Percival, MNRAS. 3271297Percival, W. J. et al. 2001, MNRAS, 327, 1297 . W Saunders, MNRAS. 31755Saunders, W. et al. 2000, MNRAS, 317, 55 . R J Scherrer, D H Weinberg, ApJ. 504607Scherrer, R. J., & Weinberg, D. H. 1998, ApJ, 504, 607 . D J Schlegel, D P Finkbeiner, M Davis, ApJ. 500525Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 . S A Schectman, ApJ. 470172Schectman, S. A. et al. 1996, ApJ, 470, 172 . R Scoccimarro, H M P Couchman, J A Frieman, ApJ. 517531Scoccimarro, R., Couchman, H. M. P., & Frieman, J. A. 1999, ApJ, 517, 531 . R Scoccimarro, R K Sheth, MNRAS. 329629Scoccimarro, R., & Sheth, R. K. 2002, MNRAS, 329, 629 . R Scoccimarro, R Sheth, R Scranton, astro-ph/0107416MNRAS. 3251359ApJScoccimarro, R., & Sheth, R. 2001b, in preparation Scranton, R. et al. 2002, astro-ph/0107416, ApJ, in press Seljak, U. 2001, MNRAS, 325, 1359 . A Szalay, astro-ph/0107419Phys. Rev. D. 555895ApJSzalay, A. et al. 2002, astro-ph/0107419, ApJ, in press Tegmark, M. 1997, Phys. Rev. D, 55, 5895 M Tegmark, astro-ph/9809185Wide Field Surveys in Cosmology. Colombi, S., & Mellier, Y.ParisEditions Frontières43Tegmark, M. 1998, astro-ph/9809185, in Wide Field Surveys in Cosmology, ed. Colombi, S., & Mellier, Y. (Editions Frontières: Paris), p43 . M Tegmark, B C Bromley, ApJL. 51869Tegmark, M., & Bromley, B. C. 1999, ApJL, 518, L69 . M Tegmark, A Oliveira-Costa, PRD. 64Tegmark, M., & de Oliveira-Costa, A. 2001, PRD, 64, 063001- 063015 M Tegmark, astro-ph/9702019Relativistic Astrophysics & Cosmology. SingaporeWorld Scientific270Tegmark, M. et al. 2002, astro-ph/0107418, ApJ, in press Tegmark, M., & Hamilton, A. J. S. 1998, astro-ph/9702019, in Relativistic Astrophysics & Cosmology, ed. Olinto, A. V., Frieman, J. A., & Schramm, D. (World Scientific: Singapore), p270 THSVS98. M Tegmark, A J S Hamilton, M A Strauss, M S Vogeley, A S Szalay, ApJ. 499555Tegmark, M., Hamilton, A. J. S., Strauss, M. A., Vogeley, M. S., & Szalay, A. S. 1998, ApJ, 499, 555 ("THSVS98") . M Tegmark, P J E Peebles, ApJL. 50079Tegmark, M., & Peebles, P. J. E. 1998, ApJL, 500, 79 . M Tegmark, A N Taylor, A F Heavens, ApJ. 48022Tegmark, M., Taylor, A. N., & Heavens, A. F. 1997, ApJ, 480, 22 . M Tegmark, M Zaldarriaga, A J S Hamilton, Phys. Rev. D. 6343007Tegmark, M., Zaldarriaga, M., & Hamilton, A. J. S. 2001, Phys. Rev. D, 63, 43007 M S Vogeley, astro-ph/9805160Ringberg Workshop on Large-Scale Structure. Hamilton, D.Kluwer: AmsterdamVogeley, M. S. 1998, astro-ph/9805160, in Ringberg Workshop on Large-Scale Structure, ed. Hamilton, D. (Kluwer: Amster- dam) . M S Vogeley, A S Szalay, ApJ. 46534Vogeley, M. S., & Szalay, A. S. 1996, ApJ, 465, 34 . X Wang, M Tegmark, M ; D Zaldarriaga, astro- ph/0105091AJ. 1201579PRD, in press YorkWang, X., Tegmark, M., & Zaldarriaga, M. 2002, astro- ph/0105091, PRD, in press York, D. et al. 2000, AJ, 120, 1579 . I Zehavi, astro-ph/0106476ApJ. in pressZehavi, I. et al. 2002, astro-ph/0106476, ApJ, in press
[]
[ "Characterization of Magnetic Components in the Diluted Magnetic Semiconductor Zn 1−x Co x O by X-ray Magnetic Circular Dichroism", "Characterization of Magnetic Components in the Diluted Magnetic Semiconductor Zn 1−x Co x O by X-ray Magnetic Circular Dichroism" ]
[ "M Kobayashi \nDepartment of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan\n", "Y Ishida \nDepartment of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan\n", "J L Hwang \nDepartment of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan\n", "T Mizokawa \nDepartment of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan\n", "A Fujimori \nDepartment of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan\n", "K Mamiya \nSynchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan\n", "J Okamoto \nSynchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan\n", "Y Takeda \nSynchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan\n", "T Okane \nSynchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan\n", "Y Saitoh \nSynchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan\n", "Y Muramatsu \nSynchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan\n", "A Tanaka \nDepatment of Quantum Matter\nADSM\nHiroshima University\nHigashi-Hiroshima739-8530Japan\n", "H Saeki \nInstitute of Scientific and Industrial Research\nOsaka University\n567-0047IbarakiOsakaJapan\n", "H Tabata \nInstitute of Scientific and Industrial Research\nOsaka University\n567-0047IbarakiOsakaJapan\n", "T Kawai \nInstitute of Scientific and Industrial Research\nOsaka University\n567-0047IbarakiOsakaJapan\n" ]
[ "Department of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan", "Department of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan", "Department of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan", "Department of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan", "Department of Physics\nDepartment of Complexity Science and Engineering\nUniversity of Tokyo\n277-8561KashiwaChibaJapan", "Synchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan", "Synchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan", "Synchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan", "Synchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan", "Synchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan", "Synchrotron Radiation Research Center\nJapan Atomic Energy Research Institute\n679-5148MikazukiHyogoJapan", "Depatment of Quantum Matter\nADSM\nHiroshima University\nHigashi-Hiroshima739-8530Japan", "Institute of Scientific and Industrial Research\nOsaka University\n567-0047IbarakiOsakaJapan", "Institute of Scientific and Industrial Research\nOsaka University\n567-0047IbarakiOsakaJapan", "Institute of Scientific and Industrial Research\nOsaka University\n567-0047IbarakiOsakaJapan" ]
[]
We report on the results of x-ray absorption (XAS), x-ray magnetic circular dichroism (XMCD), and photoemission experiments on n-type Zn1−xCoxO (x = 0.05) thin film, which shows ferromagnetism at room temperature. The XMCD spectra show a multiplet structure, characteristic of the Co 2+ ion tetrahedrally coordinated by oxygen, suggesting that the ferromagnetism comes from Co ions substituting the Zn site in ZnO. The magnetic field and temperature dependences of the XMCD spectra imply that the non-ferromagnetic Co ions are strongly coupled antiferromagnetically with each other.
10.1103/physrevb.72.201201
[ "https://arxiv.org/pdf/cond-mat/0505387v1.pdf" ]
119,387,751
cond-mat/0505387
db02dc3a09ee31e77f180d4a26a06f7a9c3874da
Characterization of Magnetic Components in the Diluted Magnetic Semiconductor Zn 1−x Co x O by X-ray Magnetic Circular Dichroism 16 May 2005 M Kobayashi Department of Physics Department of Complexity Science and Engineering University of Tokyo 277-8561KashiwaChibaJapan Y Ishida Department of Physics Department of Complexity Science and Engineering University of Tokyo 277-8561KashiwaChibaJapan J L Hwang Department of Physics Department of Complexity Science and Engineering University of Tokyo 277-8561KashiwaChibaJapan T Mizokawa Department of Physics Department of Complexity Science and Engineering University of Tokyo 277-8561KashiwaChibaJapan A Fujimori Department of Physics Department of Complexity Science and Engineering University of Tokyo 277-8561KashiwaChibaJapan K Mamiya Synchrotron Radiation Research Center Japan Atomic Energy Research Institute 679-5148MikazukiHyogoJapan J Okamoto Synchrotron Radiation Research Center Japan Atomic Energy Research Institute 679-5148MikazukiHyogoJapan Y Takeda Synchrotron Radiation Research Center Japan Atomic Energy Research Institute 679-5148MikazukiHyogoJapan T Okane Synchrotron Radiation Research Center Japan Atomic Energy Research Institute 679-5148MikazukiHyogoJapan Y Saitoh Synchrotron Radiation Research Center Japan Atomic Energy Research Institute 679-5148MikazukiHyogoJapan Y Muramatsu Synchrotron Radiation Research Center Japan Atomic Energy Research Institute 679-5148MikazukiHyogoJapan A Tanaka Depatment of Quantum Matter ADSM Hiroshima University Higashi-Hiroshima739-8530Japan H Saeki Institute of Scientific and Industrial Research Osaka University 567-0047IbarakiOsakaJapan H Tabata Institute of Scientific and Industrial Research Osaka University 567-0047IbarakiOsakaJapan T Kawai Institute of Scientific and Industrial Research Osaka University 567-0047IbarakiOsakaJapan Characterization of Magnetic Components in the Diluted Magnetic Semiconductor Zn 1−x Co x O by X-ray Magnetic Circular Dichroism 16 May 2005(Dated: January 31, 2018)numbers: 7550Pp7960Dp7870Dm7820Ls We report on the results of x-ray absorption (XAS), x-ray magnetic circular dichroism (XMCD), and photoemission experiments on n-type Zn1−xCoxO (x = 0.05) thin film, which shows ferromagnetism at room temperature. The XMCD spectra show a multiplet structure, characteristic of the Co 2+ ion tetrahedrally coordinated by oxygen, suggesting that the ferromagnetism comes from Co ions substituting the Zn site in ZnO. The magnetic field and temperature dependences of the XMCD spectra imply that the non-ferromagnetic Co ions are strongly coupled antiferromagnetically with each other. Diluted magnetic semiconductors (DMS's), in which a portion of atoms of the non-magnetic semiconductor hosts are replaced by magnetic ions, are key materials for "spintronics" (spin electronics), which is intended to manipulate both the spin and charge degrees of freedom by use of coupling between the spins of the magnetic ions and the charge carriers of the host semiconductors [1]. Indeed, using ferromagnetic DMS's, it has been successful to realize spin-related new techniques such as spin injection [2], electrical manipulation of magnetization reversal [3], and current-induced domain-wall switching [4]. However, because the Curie temperature (T C ) of the prototypical ferromagnetic DMS Ga 1−x Mn x As is below the room temperature (T C <200 K), it is still difficult to utilize DMS's in practical applications. Recently, oxidebased DMS's [5], especially ZnO-based DMS's [6,7,8], have attracted much attention as candidates for room temperature ferromagnetic DMS's. The wide band gap of ZnO is also expected to expand the range of applications. Theoretical studies have predicted that intrinsic ferromagnetism of Co-doped ZnO can be stabilized by electron doping [9,10]. However, possible extrinsic origins of the ferromagnetism such as precipitated Co metal clusters [11] have not been excluded and the ferromagnetism of Zn 1−x Co x O is still in strong dispute. Although magnetization and anomalous Hall effect measurements are suitable to investigate magnetic properties, it is not straight forward to judge from these measurements whether the ferromagnetism is intrinsic or extrinsic [12]. X-ray magnetic circular dichroism (XMCD), which is the difference in core-level absorption spectra between right-and left-handed circularly polarized x-rays, is an element specific probe sensitive to the magnetic polarization of each element, and therefore enables us to directly extract the local electronic structure related to particular magnetic properties of the substituted transitionmetal ions [13]. In this work, we have performed com-bined x-ray absorption (XAS), XMCD and photoemission spectroscopy (PES) studies of Zn 1−x Co x O to determine the electronic structure and the magnetic properties associated with the Co ions. In particular, the XMCD line shape and the intensity under varying magnetic field and temperature have implied that the ferromagnetism of Zn 1−x Co x O is indeed caused by the Co 2+ ions substituting the Zn site. A Zn 1−x Co x O (x = 0.05) thin film was epitaxially grown on a α-Al 2 O 3 (0001) substrate by the pulsed laser deposition technique using an ArF excimer laser with energy density 1.0 J/cm 2 . During the deposition, the substrate was kept at a temperature of ∼300 • C in an oxygen pressure of 1.0×10 −5 mbar. The total thickness of the Zn 1−x Co x O layer was ∼ 2000Å on a 500Å ZnO buffer layer. X-ray diffraction confirmed that the thin film had the wurtzite structure and no secondary phase was observed. Details of the sample fabrication are given in Ref. [14]. Ferromagnetism with T C above the room temperature was confirmed by magnetization measurements using a SQUID magnetometer (Quantum Design, Co. Ltd.). XAS and XMCD measurements at the Co 2p→3d (Co L 2,3 ) edge were performed at beam line BL23SU [15] of SPring-8 in the total-electron yield mode. The monochromator resolution was E/∆E > 10000. Righthanded (µ + ) and left-handed (µ − ) circularly polarized xray absorption spectra were obtained by reversing photon helicity at each photon energy. External magnetic field was applied perpendicular to the sample surface. In the XMCD experiment, the magnetic field (H) is changed from 2.0 to 7.0 T at 20 K and the temperature (T ) from 20 to 220 K at 7.0 T. The background of the XAS spectra was assumed to be a hyperbolic tangent function as usual. Circularly polarized x-ray absorption spectra under each experimental condition have been normalized to the maximum height of the Co L 2,3 edge XAS [(µ + + µ − )/2] spectra as 100. Ultraviolet photoemission (UPS) measurements were performed at BL-18A of Photon Factory (PF), High Energy Accelerator Research Organization (KEK). Spectra were taken at room temperature in a vacuum below 7.5×10 −10 Torr. The total resolution of the spectrometer (VG CLAM hemispherical analyzer) including temperature broadening was ∼ 200 meV. X-ray photoemission (XPS) measurements were performed using a Gammadata Scienta SES-100 hemispherical analyzer and an AlKα source (hν = 1486.6 eV) in a vacuum below 1.0×10 −9 Torr. In both UPS and XPS measurements, photoelectrons were collected in the angle integrated mode. Sample surface was cleaned by cycles of Ar + -ion sputtering at 1.5 kV and annealing at 250 • C. Cleanliness of the sample surface was checked by the absence of a high binding-energy shoulder in the O 1s spectrum and C 1s contamination by XPS. Figure 1(a) shows the Co L 2,3 XAS spectrum compared with spectra calculated using atomic multiplet theory. The calculation was carried out for Co 2+ and Co 3+ with the positive and negative crystal-field parameter 10Dq representing the octahedral and tetrahedral coordinations of oxygen atoms for Co, respectively. The calculated multiplet splitting for Co 3+ is more spread than experiment both for the octahedral and tetrahedral crystal fields, and the calculated spectra for Co 2+ better reproduce the experiment. Furthermore, the negative 10Dq better reproduces the measured XAS spectrum. Hence, we conclude that the Co ion is divalent and is tetrahedrally coordinated by four oxygen atoms. Further information about the local electronic structure of the Co 2+ ion, namely, hybridization of the Co 3d orbital with the host oxygen 2p orbital as well as the d-d Coulomb interaction can be studied by photoemission spectroscopy [16,17]. The Co core-level 2p XPS spectrum of Zn 0.95 Co 0.05 O shown in Fig. 1(b) is similar to that of CoO [18]. We have made a configurationinteraction (CI) cluster-model analysis of the Co 2p XPS spectrum using a [Co 2+ (O 2− ) 4 ] 6− cluster and estimated the electronic structure parameters: the ligandto-3d charge-transfer energy ∆ = 5.0±0.5 eV, the d-d Coulomb interaction energy U = 6.0±0.5 eV, and the Slater-Koster parameter (pdσ) = −1.6±0.1 eV. These parameters are in good agreement with those obtained from the previous XAS study [19] and are consistent with the chemical trend in transition-metal-doped II-VI DMS's [20]. Here, Racah parameters have been fixed at the values of the free ion: B = 0.138 eV and C = 0.54 eV. The Co 3d-2p core hole Coulomb attraction energy Q is related to U through U = βQ, where at β = 0.7. The ratio between (pdσ) and (pdπ) has been fixed (pdσ)/(pdπ) = −2.16. Figure 1(b) shows that the main peak of the spectrum dominantly consists of chargetransferred states, i.e. cd 8 L, where c and L denote a hole in the Co 2p and oxygen 2p orbitals, respectively. cal contact with the sample. The absorption spectrum shows that Co 3p→3d absorption occurs at hν∼61 eV. Constant-initial-state (CIS) spectra at various E B 's (not shown) indicate that the Co 3d partial density of states (PDOS) is primarily located at E B ∼3.0 and ∼ 7.0 eV. Figure 2(b) shows the Co 3d PDOS of Zn 0.95 Co 0.05 O, which has been obtained by subtracting the off-resonance (hν = 60 eV) spectrum from the on-resonance (hν = 61.5 eV) one. Here, the off-resonance spectrum was multiplied by the integrated (0<E B <9 eV) intensity ratio between hν = 61.5 and 60 eV of pure ZnO. The Co 3d PDOS shows a peak at E B ∼3.0 eV, which is similar to that of the polycrystalline Zn 0.9 Co 0.1 O [21], and a satellite at E B ∼7.0 eV. Because the energy difference between the top of the O 2p band and E F is nearly equal to the band gap of ZnO, E F is supposed to be located near the conduction band minimum (composed of Zn 4s and possibly of Co d states), meaning that the sample is n-type. Although local-density approximation (LDA) calculations have predicted that ferromagnetism is mediated by carriers and therefore needs a high density of states (DOS) at E F [9], we could not clearly observe a finite DOS at E F , consistent with the low carrier density. The energy difference between the main structure and the satellite of the Co 3d PDOS was as large ∼9 eV for CoO [18] while it is ∼4 eV for Zn 1−x Co x O, probably because of the different co-ordinations of oxygen atoms between CoO and Zn 1−x Co x O. This can be well explained by the CI clustermodel calculation using the same ∆, U , (pdσ) as shown in Fig. 2(b). Although the XAS spectra are independent of the magnetic field and µ + and µ − are nearly identical on the scale of Fig. 1(a), there were weak but reprodusible XMCD signals (µ + − µ − ) as shown in Fig. 3(a). The intensity of XMCD spectra increases with increasing magnetic field as shown in Fig. 3(b), while it is rather independent of temperature as shown in Fig. 3(c). Note that, in Fig. 3(b) and (c), the spin (M spin ), orbital (M orb ), and total (M tot ) magnetic moments of Co estimated using XMCD sum rules [22,23] are plotted rather than the raw XMCD intensities. Part of the XMCD signals which linearly increases with H represents the paramagnetic component, while XMCD signals which persist at H∼0 T represent the ferromagnetic component [24]. The difference between XMCD spectra under H = 2.0 and 4.5 T, which reflects the paramagnetic component, shows nearly the same line shape as the XMCD spectrum taken at the lowest magnetic field of 2.0 T, as shown in Fig. 3(a). Therefore, it seems that the Co ions have similar electronic structures in the paramagnetic and ferromagnetic components. It should be emphasized that the XMCD spectra also show a multiplet structure, unlike those of Co metal [25], indicating that the magnetism in the present sample is not due to metallic Co clusters but due to Co ions with localized 3d electrons. As in the case of the XAS spectrum, the XMCD spectra are also compared with theoretical XMCD spectra calculated using atomic multiplet theory as shown in Fig. 4(a). The calculated ratio M orb /M spin is compared with experiment. The line shape of the calculated spectra for Co 3+ are different from the experimental one and the line shape for Co 2+ with tetrahedral oxygen coordination best agrees with experiment. Comparison of the ratio M orb /M spin between the calculated spectra and experiment indicates that the Co 2+ ion with tetrahedral oxygen co-ordination and not octahedral one is consistent with experiment. Therefore, the calculated spectrum for Co 2+ with 10Dq = −0.7 eV best reproduces the experimental XMCD. To examine more details of the XAS and XMCD line shapes, CI cluster-model calculations were performed as shown in Fig. 4(b). We have used electronic structure parameters ∆ = 5.0 eV, U = 5.0 eV, and (pdσ) = −1.6 eV, nearly the same as those estimated from the PES experiments. The calculated M orb /M spin is closer to the experimental value than that of atomic multiplet theory. This gives further support that the substituted Co ions under tetrahedral crystal field are responsible for the ferromagnetism in Zn 1−x Co x O. Finally we comment on possible origins of the temperature-independent paramagnetic component of the XMCD signals. This signal cannot be due to Pauli paramagnetism of conduction electrons because their XMCD spectra show characteristic of the Co 2+ ion and also that its susceptibility χ exp ∼ 1.43 × 10 −2 (µ B /T per Co) is several orders of magnitude larger than the Pauli paramagnetism expected for the conduction electron concentration n ∼ 1.0×10 17 cm −3 of the present sample. If the temperature-independent paramagnetism is due to the Co 2+ ions, there should be strong antiferromagnetic interaction between the Co ions because, if the Co ions do not interact with each other, they would show the Curie behavior. The temperature-independent paramagnetic component may arise from the susceptibility of the antiferromagnetic Co ions having a Néel temperature above the room temperature. Such magnetic susceptibility is estimated to be g 2 S(S + 1)/T N ∼ 8.4 × 10 −3 (µ B /T per Co) for g = 2, S = 3/2, and T N = 400 K, in reasonable agreement with slope of the XMCD intensity [ Fig. 3(b)] of χ exp ∼ 1.43 × 10 −2 (µ B /T per Co). Although both the ferromagnetic and paramagnetic/antiferromagnetic Co ions has the same 2+ valence and the tetrahedral crystal field, subtle differences such as neighboring defects and local lattice distortion may have lead to the different magnetic behaviors. In order to confirm the above conjectures, more precise and systematic XMCD measurements on samples with varying carrier concentrations are necessary. In summary, we have performed XAS, XMCD, and PES experiments on the diluted ferromagnetic semiconductor n-type Zn 1−x Co x O (x = 0.05). The XMCD spectra show a multiplet structure, characteristic of the Co 2+ ion tetrahedrally coordinated by oxygen. This implies that the ferromagnetism in Zn 1−x Co x O is caused by the substituted Co 2+ ions at the Zn site. The magnetic field and temperature dependences of the XMCD intensity suggest that the non-ferromagnetic Co ions are strongly coupled antiferromagnetically with each other. We thank T. Okuda, A. Harasawa, and T. Kinoshita for technical help at PF. We also thank the Materials Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo, for the use of the SQUID magnetometer. This work was supported by a Grant-in-Aid for Scientific Research in Priority Area "Semiconductor Nanospintronics" (14076209) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The experiment at PF was approved by the Photon Factory Program Advisory Committee (Proposal No. 2002G027). MK acknowledges support from the Japan Society for the Promotion of Science for Young Scientists. Figure 2 ( 2a) shows the valence-band UPS spectra of Zn 0.95 Co 0.05 O taken at various photon energies in the Co 3p→3d core-excitation region. The absorption spectrum in the same energy region is shown in the inset. Binding energies (E B 's) are referenced to the Fermi level (E F ) of a metallic sample holder which is in electri- FIG. 3 : 3Magnetic field and temperature dependences of the Co L2,3 XMCD spectra of Zn0.95Co0.05O. (a) XMCD spectra under different magnetic fields at 20 K. Open circle shows the difference of the spectra between H = 4.5 and 2.0 T. (b) Average magnetic moments Mspin, M orb , and Mtot as functions of magnetic field, estimated using the XMCD sum rules. (c) The same as (b) as function of temperature. Co L 3 Co L 3 FIG. 4 : 334Co L2,3 XMCD spectra of Zn0.95Co0.05O. (a) Comparison with atomic multiplet calculation, in which the valence of Co and the sign and magnitude of the crystal-field splitting are varied. (b) CI cluster-model analysis for the XAS and XMCD spectra. Intensity (arb. units)FIG. 2: Valence-band UPS spectra of Zn0.95Co0.05O. (a) A series of photoemission spectra for photon energies in the Co 3p→3d core-excitation region. Inset: Absorption spectrum recorded in the total electron yield mode. (b) Top: Onresonance (hν = 61.5 eV) and off-resonance (hν = 60.0 eV) spectra. The difference between these spectra represents the Co 3d PDOS. Bottom: CI cluster-model analysis for the Co 3d PDOS.12 8 4 0 Binding Energy (eV) 60 61 62 hν = 65 eV 55 57 58 59 63 Zn 3d O 2p (a) Total Yield (a.u.) 80 70 60 50 40 Photon Energy (eV) Intensity (arb. units) 8 6 4 2 0 Binding Energy (eV) Co PDOS CI Calc. Back Ground On-Reso. Off-Reso. (b) sat. . H Ohno, F Matsukura, Y Ohno, JSAP International. 54H. Ohno, F. Matsukura, and Y. Ohno, JSAP Interna- tional 5, 4 (2002). . Y Ohno, D K Young, B Beschoten, F Matsukura, H Ohno, D D Awschalom, Nature. 402790Y. Ohno, D. K. Young, B. Beschoten, F. Matsukura, H. Ohno, and D. D. Awschalom, Nature 402, 790 (1999). . D Chiba, M Yamanouchi, F Matsukura, H Ohno, Science. 301943D. Chiba, M. Yamanouchi, F. Matsukura, and H. Ohno, Science 301, 943 (2003). . M Yamanouchi, D Chiba, F Matsukura, H Ohno, Nature. 428539M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature 428, 539 (2004). . S J Pearton, W H Heo, M Ivill, D P Norton, T Steiner, Semicond. Sci. Technol. 1959S. J. Pearton, W. H. Heo, M. Ivill, D. P. Norton, and T. Steiner, Semicond. Sci. Technol. 19, R59 (2004). . K Ueda, H Tabata, T Kawai, Appl. Phys. Lett. 79988K. Ueda, H. Tabata, and T. Kawai, Appl. Phys. Lett. 79, 988 (2001). . H Saeki, H Tabata, T Kawai, Solid State Commun. 120439H. Saeki, H. Tabata, and T. Kawai, Solid State Commun. 120, 439 (2001). . P Sharma, A Gupta, K V Rao, F J Owens, R Sharma, R Ahuja, J M O Guillen, B Johansson, G A Gehring, Nat. Mater. 2673P. Sharma, A. Gupta, K. V. Rao, F. J. Owens, R. Sharma, R. Ahuja, J. M. O. Guillen, B. Johansson, and G. A. Gehring, Nat. Mater. 2, 673 (2003). . K Sato, H Katayama-Yoshida, Jpn. J. Appl. Phys. 40334K. Sato and H. Katayama-Yoshida, Jpn. J. Appl. Phys. 40, L334 (2001). . E.-C Lee, K J Chang, Phys. Rev. B. 6985205E.-C. Lee and K. J. Chang, Phys. Rev. B 69, 085205 (2004). . J H Kim, H Kim, D Kim, Y.-E Ihm, W K Choo, J. Euro. Ceram. Soc. 241847J. H. Kim, H. Kim, D. Kim, Y.-E. Ihm, and W. K. Choo, J. Euro. Ceram. Soc. 24, 1847 (2004). . S R Shinde, S B Ogale, J S Higgins, H Zheng, A J , S. R. Shinde, S. B. Ogale, J. S. Higgins, H. Zheng, A. J. . V N Millis, R Kulkarni, R L Ramesh, T Greene, Venkatesan, Phys. Rev. Lett. 92166601Millis, V. N. Kulkarni, R. Ramesh, R. L. Greene, and T. Venkatesan, Phys. Rev. Lett. 92, 166601 (2004). . D J Keavney, D Wu, J W Freeland, E Johnston-Halperin, D D Awschalom, J Shi, Phys. Rev. Lett. 91187203D. J. Keavney, D. Wu, J. W. Freeland, E. Johnston- Halperin, D. D. Awschalom, and J. Shi, Phys. Rev. Lett. 91, 187203 (2003). . H Saeki, H Matsui, T Kawai, H Tabata, J. Phys.: Condens. Mat. 165533H. Saeki, H. Matsui, T. Kawai, and H. Tabata, J. Phys.: Condens. Mat. 16, S5533 (2004). . J Okamoto, K Mamiya, S.-I Fujimori, T Okane, Y Saitoh, Y Muramatsu, A Fujimori, S Ishiwata, M Takano, AIP Conf. Proc. 7051110J. Okamoto, K. Mamiya, S.-I. Fujimori, T. Okane, Y. Saitoh, Y. Muramatsu, A. Fujimori, S. Ishiwata, and M. Takano, AIP Conf. Proc. 705, 1110 (2004). . J Okabayashi, A Kimura, O Rader, T Mizokawa, A Fujimori, T Hayashi, M Tanaka, Phys. Rev. B. 584211J. Okabayashi, A. Kimura, O. Rader, T. Mizokawa, A. Fujimori, T. Hayashi, and M. Tanaka, Phys. Rev. B 58, R4211 (1998). . J Okabayashi, A Kimura, T Mizokawa, A Fujimori, T Hayashi, M Tanaka, Phys. Rev. B. 592486J. Okabayashi, A. Kimura, T. Mizokawa, A. Fujimori, T. Hayashi, and M. Tanaka, Phys. Rev. B 59, R2486 (1999). . Z.-X Shen, J W Allen, P A P Lindberg, D S Dessau, B O Wells, A Borg, W Ellis, J S Kang, S.-J Oh, I Lindau, W E Spicer, Phys. Rev. B. 421817Z.-X. Shen, J. W. Allen, P. A. P. Lindberg, D. S. Dessau, B. O. Wells, A. Borg, W. Ellis, J. S. Kang, S.-J. Oh, I. Lindau, and W. E. Spicer, Phys. Rev. B 42, 1817 (1990). . J Okabayashi, K Ono, M Mizuguchi, M Oshima, S S Gupta, D D Sarma, T Mizokawa, A Fujimori, M Yuri, C T Chen, T Fukumura, M Kawasaki, H Koinuma, J. Appl. Phys. 953573J. Okabayashi, K. Ono, M. Mizuguchi, M. Oshima, S. S. Gupta, D. D. Sarma, T. Mizokawa, A. Fujimori, M. Yuri, C. T. Chen, T. Fukumura, M. Kawasaki, and H. Koinuma, J. Appl. Phys. 95, 3573 (2004). . T Mizokawa, A Fujimori, Phys. Rev. B. 566669T. Mizokawa and A. Fujimori, Phys. Rev. B 56, 6669 (1997). . S C Wi, J.-S Kang, J H Kim, S.-B Cho, B J Kim, S Yoon, B J Suh, S W Han, K H Kim, K J Kim, B S Kim, H J Song, H J Shin, J H Shim, B I Min, Appl. Phys. Lett. 844233S. C. Wi, J.-S. Kang, J. H. Kim, S.-B. Cho, B. J. Kim, S. Yoon, B. J. Suh, S. W. Han, K. H. Kim, K. J. Kim, B. S. Kim, H. J. Song, H. J. Shin, J. H. Shim, and B. I. Min, Appl. Phys. Lett. 84, 4233 (2004). . B T Thole, P Carra, F Sette, G Van Der Laan, Phys. Rev. Lett. 681943B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992). . P Carra, B T Thole, M Altarelli, X Wang, Phys. Rev. Lett. 70694P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys. Rev. Lett. 70, 694 (1993). . A Fujimori, J Okabayashi, Y Takeda, T Mizokawa, J Okamoto, K Mamiya, Y Saitoh, Y Muramatsu, M Oshima, S Ohya, M Tanaka, J. Electron Spectrosc. Relat. Phenom. 701A. Fujimori, J. Okabayashi, Y. Takeda, T. Mizokawa, J. Okamoto, K. Mamiya, Y. Saitoh, Y. Muramatsu, M. Oshima, S. Ohya, and M. Tanaka, J. Electron Spectrosc. Relat. Phenom. 144-147, 701 (2005). . C T Chen, Y U Idzerda, H.-J Lin, N V Smith, G Meigs, E Chaban, G H Ho, E Pellegrin, F Sette, Phys. Rev. Lett. 75152C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and F. Sette, Phys. Rev. Lett. 75, 152 (1995).
[]
[ "Back to Boundaries in Billiards", "Back to Boundaries in Billiards" ]
[ "Leonid Bunimovich ", "Yaofeng Su " ]
[]
[]
We prove Poisson limit laws for open billiards where the holes are on the boundaries of billiard tables (rather than some abstract holes in the phase space of a billiard). Such holes are of the main interest for billiard systems, especially for applications. Sinai billiards with or without a finite horizon, diamond billiards, and semi-dispersing billiards, as well as focusing billiards with slow decay of correlations, are considered.Contents
null
[ "https://arxiv.org/pdf/2203.00785v1.pdf" ]
247,218,532
2203.00785
9c26cca3dc65fe3b5e72ebba0811b1100d38285e
Back to Boundaries in Billiards March 3, 2022 Leonid Bunimovich Yaofeng Su Back to Boundaries in Billiards March 3, 2022 We prove Poisson limit laws for open billiards where the holes are on the boundaries of billiard tables (rather than some abstract holes in the phase space of a billiard). Such holes are of the main interest for billiard systems, especially for applications. Sinai billiards with or without a finite horizon, diamond billiards, and semi-dispersing billiards, as well as focusing billiards with slow decay of correlations, are considered.Contents The studies of Poisson approximations for recurrences to small subsets in the phase spaces of chaotic dynamical systems are developed now into a large active area. Another view at this type of problems is a subject of the theory of open dynamical systems, where some positive measure subset of the phase space is named a hole, and the process of escape through the hole is studied. In a general set up, one picks a small measure subset (a hole) in the phase space of a hyperbolic (chaotic) ergodic dynamical system and attempts to prove that in the limit, when measure of the hole approaches zero, the corresponding process of recurrences to the hole converges to the Poisson process. This area received an essential boost after L-S.Young papers [22,23], where a new general framework was introduced for analysis of statistical properties of hyperbolic dynamical systems. This approach employs representation of the phase space of a dynamical system as a tower (later called Young tower, Gibbs-Markov-Young tower, etc), which allow to study dynamics by analysing recurrences to the base of the tower. Particularly, in the papers [3,19,20] the holes, which are balls, shrinking a point in the phase space of billiard systems, were studied. The paper [4] deals with the holes (shrinking to a curve) in the phase space of Sinai billiards with a finite horizon. For positive measure ("physical") holes the paper [18] studied holes inside the Sinai's billiard tables, and [20] considered holes near corners of a diamond-shaped billiard tables. Such holes correspond to strip-shaped holes in the phase space which are shrinking to broken line segments, while the projection of the hole to configuration space shrinks to a point. Finally, the paper [11] should be mentioned, which is dealing with various holes shrinking to null sets in the phase space, rather than to a point. Some of the systems, which were considered in this paper, can not be modelled by Young towers, but they have milder singularities than the ones of the billiard systems. Here we present a new development of this area, which, particularly, allows to prove Poisson approximations for various billiard systems with arbitrarily slow decay of correlations. Moreover, we also consider holes located on the boundaries of billiard tables. Such holes are natural to consider for the process of escape in billiard systems. Moreover, exactly such holes are studied in real systems, e.g. in physics [9,15,16,17,21]. In the phase space such holes tend to straight segments, when a hole at the boundary shrinks to a point. In comparison to previous papers we obtain here new results, some of which are a kind of unexpected ones. 1. The technique used in [4,18] works only for Sinai billiards with a finite horizon. Our Theorem 2.13 is applicable to a larger class of billiard systems. The technique used here is also new for general open dynamical systems. 2. The approach employed in the papers [4,18,20] requires to verify the socalled short return conditions specifically for each billiard systems, while our main Theorem 2.13 assumes only some natural general, and easy to verify, conditions. 3. Unlike [11], Theorem 2.13 shows surprisingly that the validity of the Poisson approximation for billiard systems does not depend on how slow correlations decay, i.e., it holds for any rate of decay of correlations. 4. The papers [3,19] require that the contraction rates on (un)stable manifolds must be sufficiently large. Our results show that this condition can be weakened. The structure of the paper is the following. The section 2 includes some notations, definitions, and formulation of the main Theorem 2.13. The sections 3-8 contain a proof of this Theorem 2.13. We start by presenting a general result on a Poisson approximation for general point processes. Then we simplify it, step by step, from the section 3 to the section 8. The section 9 presents applications to various billiard systems, especially to slow mixing billiard systems, which are the main focus in this paper. Definitions and Main Results We start by introducing some notations 1. C z denotes a constant depending on z. 2. The notation "a n z b n " ("a n = O z (b n )") means that there is a constant C z ≥ 1 such that (s.t.) a n ≤ C z b n for all n ≥ 1, whereas the notation "a n b n " (or "a n = O(b n )") means that there is a constant C ≥ 1 such that a n ≤ Cb n for all n ≥ 1. Next, "a n ≈ z b n " and a n = C ±1 z b n mean that there is a constant C z ≥ 1 such that C −1 z b n ≤ a n ≤ C z b n for all n ≥ 1. Further, the notations "a n = C ±1 b n " and "a n ≈ b n " means that there is a constant C ≥ 1 such that C −1 b n ≤ a n ≤ Cb n for all n ≥ 1. Finally, "a n = o(b n )" means that lim n→∞ |a n /b n | = 0. 3. The notation P refers to a probability distribution on the probability space, where a random variable lives, and E denotes the expectation of a random variable. 4. µ A denotes a probability (normalized) measure on a set A. T (A) denotes a tangent bundle of a (sub)manifold A. 6. N = {1, 2, 3, · · · }, N 0 = {0, 1, 2, 3, · · · }. 7. S d ⊆ R d+1 is a d-dimensional unit half-sphere. Definition 2.1 (Billiard tables and phase spaces). Let Q be a compact Riemannian manifold. Its boundary ∂Q is a sub-Riemannian manifold (connected or non-connected), consisting of finitely many at least C 3smooth manifolds with uniformly bounded sectional curvatures, endowed with an induced Riemannian metric d ∂Q on ∂Q. Further, B r (q) denotes a geodesic ball in ∂Q with a radius r and a center q ∈ M. In what follows we say that Q is a billiard table, ∂Q is the boundary of a billiard table, and B r (q) is a hole in the boundary ∂Q. Let S d be a d-dimensional unit half-sphere, and d = dim ∂Q. Denote a phase space by M := {(q, v) ∈ ∂Q × S d : n(q), v ≥ 0}, where n(q) is an inward unit normal vector at q ∈ ∂Q. The space M is endowed with a natural Riemannian metric d M and Riemannian volume Leb M . A dynamics on M is a map f : M → M. For a subset X ⊆ M with Leb M (X) > 0 its first return time to X is R : X → N and the first return map is f R : X → X. Definition 2.2 (Singularities and (un)stable manifolds). Denote by S ⊆ X a singularity set for f R . We assume that S has zero Lebesgue measure and S c ⊆ X consists of countably many open connected components. Unstable (resp. stable) manifolds are the connected components of ( i≥0 (f R ) i S) c (resp. ( i≥0 (f R ) −i S) c ). A closed and connected part of the unstable (resp. stable) manifold will be called an unstable (resp. stable ) disk. We denote each unstable (resp. stable) manifold/disk by γ u (resp. γ s ) and its tangent vectors by v u (resp. v s ) . Remark 2.3. The singularity set S consists of the points in X which are not "wellbehaved". So it includes the discontinuities and the points where the map f R is not differentiable. It may also include other points in X with some "bad" properties. We say that (X, f R ) is a CMZ structure of (M, f ) if there are constants C > 0 and β ∈ (0, 1) such that the following conditions hold 1. Hyperbolicity. For any n ∈ N, v u and v s , |D(f R ) n v u | ≥ Cβ −n |v u |, |D(f R ) n v s | ≤ Cβ n |v s |, where | · | is the Riemannian metric induced from M to (un)stable manifolds. SRB measures and u-SRB measures. There is a probability measure µ X Leb X with dµ X d Leb X ∈ L ∞ and dµ X d Leb X > 0 Leb X -a.s. on X such that (X, f R , µ X ) is a K-system. A corresponding measurable K-partition consists of smooth pieces of stable manifolds. Moreover, conditional distributions on γ u (say µ γ u ) are absolutely continuous w.r.t. Lebesgue measure Leb γ u on γ u . 3. Distortion bounds. Let d γ u (·, ·) be the distance measured along γ u . By det D u f R we denote the Jacobian of Df R along the unstable manifolds. Then, if x, y ∈ X belong to a γ u such that (f R ) n is smooth on γ u , the following relation holds log det D u (f R ) n (x) det D u (f R ) n (y) ≤ ψ d γ u (f R ) n x, (f R ) n y , where ψ(·) is some function, which does not depend on γ u , and lim s→0 + ψ(s) = 0. 4. Bounded curvatures. The curvatures of all γ u are uniformly bounded by C. 5. Absolute continuity. Consider a holonomy map h : γ u 1 → γ u 2 , which maps a point x ∈ γ u 1 the point h(x) ∈ γ u 2 , such that both x and h(x) belong to the same γ s . We assume that the holonomy map satisfies the following relation det D u (f R ) n (x) det D u (f R ) n h(x) = C ±1 for all n ≥ 1 and x ∈ γ u 1 , 6. Growth lemmas. There exist N ∈ N, sufficiently small δ 0 > 0 and constants κ, σ > 0 which satisfy the following condition. For any sufficiently small δ > 0 and for any disk on a smooth stable manifold γ u with diam γ u ≤ δ 0 , denote by U δ ⊆ γ u a δ-neighborhood of the subset γ u 0≤i≤N (f R ) −i S within the set γ u . 3. It follows from [5,7] that (X, f R , µ X ) can be modelled by a hyperbolic Young tower [22]. 4. It follows from [7,8] that the mixing rates for the dynamical system (M, f, µ M ) are determined by the decay rate of µ X (R > n). However, we do not use this fact in the present paper. 5. (M, f, µ M ) is a K-system (and therefore mixing) because of the condition that f R is K-mixing and gcd{R} = 1. Definition 2.7 (Dynamical point processes). We define now a dynamical point process N r,q on R + . For any measurable A ⊆ R + N r,q (A) := i·µ M (Br(q)×S d )∈A 1 Br(q)×S d • f i , which is equivalent to N r,q (A) = i·µ ∆ (Ar)∈A 1 Ar • F i , where A r := π −1 (B r (q) × S d ). Definition 2.8 (Poisson point processes). We say that P is a Poisson point process on R + if 1. P is a random counting measure on R + . P(A) is a Poisson-distributed random variable for any Borel set A ⊆ R + . 3. If A 1 , A 2 , · · · , A n ⊆ R + are pairwise disjoint, then P(A 1 ), · · · , P(A n ) are independent. EP(A) = Leb(A) for any Borel set A ⊆ R + . Definition 2.9 (Poisson approximations). We say that N r,q → d P if N r,q f → d Pf for any f ∈ C + c (R + ), i.e., lim r→0 exp (−t · N r,q f )dµ M = exp (−t · Pf )dP for all t > 0. This is equivalent to the relation (N r,q I 1 , N r,q I 2 , · · · , N r,q I k ) → d (PI 1 , PI 2 , · · · , PI k ) for any k ∈ N where I 1 , · · · I k ⊆ R + are intervals. See, e.g., Theorem 16.16 in [13]. Definition 2.10 (Sections and quasi-sections). Recall that π X : ∆ → X is defined by π X (x, n) = x. We say that S r ⊆ M is a section if π X : π −1 S r → X is injective for any small r > 0, and that Q r ⊆ M is a quasi-section if there is a section S r ⊆ Q r , such that µ M (Q r \ S r ) = o µ M (Q r ) for sufficiently small r > 0. In what follows, Q r is chosen as B r (q) × S d . Assumption 2.11 (Geometric assumptions). 1. For a.e. q ∈ ∂Q the set B r (q) × S d is a quasi-section for sufficiently small r > 0. 2. i≥1 ∂X i ⊆ S (recall that R is constant on each X i ). 3. There are constants C > 0 and α ∈ (0, 1] such that for any γ k , k = u or s (which implies that γ k ⊆ {x ∈ X : R(x) = i} for some i ≥ 1), and for any x, y ∈ γ k , d f j γ k (f j x, f j y) ≤ Cd γ k (x, y) α for all j < R(x). 4. There exist two cones C u , C s ⊆ T (M) such that dim(int C u int C s ) < d, (Df )C u ⊆ C u , (Df ) −1 C s ⊆ C s , and for all n ≥ 1 and Leb-a.e. q ∈ ∂Q dim T ({q} × S d ) int C u < d, dim T ({q} × S d ) int C s < d, (Df ) n T ({q} × S d ) ⊆ int C u , (Df ) −n T ({q} × S d ) ⊆ int C s , where int C u (resp. int C s ) is the interior of C u (resp. C s ). Remark 2.12. 1. We will present later an easy to implement scheme to verify existence of C u , C s for billiard systems. The cone condition dim(int C u int C s ) < d means that these two open sets are transversal (but not necessarily uniformly transversal). This condition in Assumption 2.11 is called an aperiodic condition. 2. The Hölder condition is natural, and it is traditionally used for hyperbolic systems, and, particularly, for billiards. Theorem 2.13 (Poisson limit laws). Suppose that dynamical system (M, f ) has a CMZ structure (X, f R ) (see Definition 2.4) and the Assumption 2.11 holds. Then N r,q → d P (see Definition 2.9) for Leb-a.s. q ∈ ∂Q. Remark 2.14. 1. Unlike [11,18], Theorem 2.13 claims that Poisson limit laws do not depend on the rate of correlations decay. In other words, decay of correlations can be arbitrarily slow as long as the first return time R ∈ L 1 , i.e., R must be just integrable. 2. The technique used in [4,18] works only for Sinai billiards with finite horizons, while our approach works for Sinai billiards with unbounded horizons as well as for slowly mixing billiards. 3. The papers [3,19] established the Poisson limit under condition that contraction rates α along (un)stable manifolds and dimensions of holes are sufficiently large, i.e. α · dim (hole) > 1. These conditions fail for the billiards with focusing components of the boundary and for holes in the boundary of billiard tables (e.g. if α = 1 and dim (hole) = 1 for stadium billiards). For instance, in Sinai billiards even making holes in the boundary does not make sense. Indeed, in Sinai billiards the boundary is formed by convex scatterers with smooth boundaries. Therefore it is not clear what happens to the particle when it enters to a scatterer through its boundary. 4. One of the main difficulties in proving Poisson limit laws, related to short returns, was outlined in [19]. The papers [4,11,18,19,20] handled it by inspecting the original dynamics f , which leads to the requirement of a fast mixing rate. Our approach via an inducing method allows to restrict a hole to a good set, and it works for systems with an arbitrarily slow mixing rates. Another challenge for proving Poisson limit laws, called a corona, comes from the condition α · dim (hole) > 1 (see [19]). The failure of α · dim (hole) > 1 in this paper causes essential difficulties in proving Poisson limit laws. Our techniques, the inducing method together with an approximation method, allow to overcome this challenge. [4,20] studied general holes in M for Sinai billiards with bounded horizons and diamond billiards. Although in the present paper we consider a special type of holes, i.e., the ones in B r (q) × S d , our technique can be adapted for more general holes. The holes we consider here are the most natural for billiard systems, and consideration of a general type holes would make the paper much longer and even more technical. The papers 6. Unlike [3], our result does not provide convergence rates to Poisson limit laws. We believe though that it is possible to get convergence rates by placing more restricting conditions on the first return time R and expect to deal with convergence rates in another paper. Corollary 2.15 (First hitting). Under the same conditions as in Theorem 2.13 consider a moment of time when the first hitting (passage) of the hole occur, i.e., τ r,q (x) := inf n ≥ 1 : f n (x) ∈ B r (q) × S d for any x ∈ M. Then for any t > 0 and almost every q ∈ ∂Q, the following relation holds for the first hitting probability lim r→0 µ M τ r,q > t µ(B r (q) × S d ) = e −t . Proof. Clearly µ M τ r,q > t/µ(B r (q) × S d ) = µ M N r,q 0, t/µ(B r (q) × S d ) = 0 . Apply now a relevant one of Theorems 2.13, and the corollary holds. 3 Poisson limit laws: from quasi-sections to sections To prove Poisson limit laws, (see Definition 2.9), we will need one result from [13]. For any q ∈ ∂Q a convergence N r,q → d P holds if 1. For any compact set K ⊆ [0, ∞), lim sup r→0 N r,q (K)dµ M ≤ Leb(K). For any disjoint bounded intervals J 1 , J 2 , · · · , J n ⊆ [0, ∞), lim r→0 µ M N r,q ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 . (3.1) Now we will verify these conditions. Lemma 3.2. For any compact set K ⊆ [0, ∞), lim sup r→0 N r,q (K)dµ M ≤ Leb(K). Proof. Assume that K ⊆ [0, T ) for sufficiently large T > 0. Then [0, T ) \ K = i≥1 (a i , b i ) for some disjoint open intervals (a i , b i ). Therefore, K = N ≥1 i≤N [0, T ) (a i , b i ) c . Let K N = i≤N [0, T ) (a i , b i ) c , which is a disjoint union of finitely many intervals J 1 , J 2 , · · · , J i N . Hence N r,q (J i )dµ M = j·µ M (Br(q)×S d )∈J i 1 Br(q)×S d • f j dµ M ≤ 1 + Leb(J i )/µ M (B r (q) × S d ) µ M (B r (q) × S d ). Therefore N r,q (K)dµ M ≤ N r,q (K N )dµ M = j·µ M (Br(q)×S d )∈K N 1 Br(q)×S d • f j dµ M ≤ i N · µ M B r (q) × S d + Leb(K N ). Thus lim sup r→0 N r,q (K)dµ M ≤ Leb(K N ). We conclude the proof by letting N → ∞. We will now further modify the relation (3.1) as follows. Suppose that B r (q) × S d is a quasi-section for some small r > 0. Hence there is a section S r ⊆ B r (q) × S d such that µ M B r (q) × S d \ S r = o µ M (B r (q) × S d ) for any sufficiently small r > 0. Define N r,q s (A) := i·µ M (Sr)∈A 1 Sr • f i , (3.2) where A is a measurable set in [0, ∞). Then for any disjoint bounded intervals J 1 , J 2 , · · · , J n ⊆ [0, ∞), lim r→0 µ M N r,q s ( + µ M j∈ J i f j / ∈ S r − µ M j∈ J i f j / ∈ B r (q) × S d ≤ µ M j∈ J i f j / ∈ S r − µ M j∈ (J i J i ) f j / ∈ S r + µ M j∈ J i f j / ∈ S r − µ M j∈ (J i J i ) f j / ∈ S r + µ M j∈ J i f j / ∈ S r j∈ J i f j ∈ B r (q) × S d ≤ µ M j∈ J i f j / ∈ S r j∈ (J i J i ) f j ∈ S r + µ M j∈ J i f j / ∈ S r j∈ (J i J i ) f j ∈ S r + µ M j∈ J i f j ∈ B r (q) × S d \ S r . (3.3) There is a small r J > 0 depending on J 1 , J 2 , · · · J n such that, for any r ∈ (0, r J ) and all i ≤ n, the interval J i is the only possible one that intersect J i and Leb J i ≥ 1, Leb J i ≥ 1. Therefore j∈ J i f j / ∈ S r j∈ (J i J i ) f j ∈ S r ⊆ j∈ (J i \J i ) f j ∈ S r , j∈ J i f j / ∈ S r j∈ (J i J i ) f j ∈ S r ⊆ j∈ (J i \J i ) f j ∈ S r , and J i \ J i , J i \ J i contain at most Leb(J i \ J i ) + 1, Leb(J i \ J i ) + 1 positive integers respectively. For any r ∈ (0, r J ) by making use of f * µ M = µ M we can continue the estimate (3.3) as ≤ Leb( J i \ J i ) + Leb( J i \ J i ) + 2n µ M (S r ) + Leb( J i )µ M B r (q) × S d \ S r ≤ 2n max i J i 1/µ M (S r ) − 1/µ M (B r (q) × S d ) + 2n µ M (S r ) + n max i J i µ M B r (q) × S d \ S r µ M B r (q) × S d , where max i J i is the maximal positive number in i J i . By letting r → 0 we conclude a proof of this lemma. It follows from Lemma 3.3 and (3.1) that, in order to prove Poisson limit laws for quasi-sections B r (q) × S d , we just need to prove Poisson limit laws for sections S r ⊆ B r (q) × S d , i.e., lim r→0 µ M N r,q s ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 ,(3.4) where J 1 , · · · , J n ⊆ [0, ∞) are disjoint bounded intervals. The rest of the paper deals with a proof of the relation (3.4). Inducing and approximations We begin with some notions used in this section for the CMZ structure (see Definition 2.4). Definition 4.1 (Truncated towers). For each m ≥ 0 let a truncated sub-tower of ∆ be ∆ m := ∆ (X × {0, 1, 2, · · · , m}). Define now projections π X : ∆ m → X and π N : ∆ m → N 0 (the same notations as that in (2.1)) so that for any (x, n) ∈ ∆ m π X (x, n) = x, π N (x, n) = n. The first return time R m : ∆ m → N is R m (x) = inf{n ≥ 1 : F n (x) ∈ ∆ m } for any x ∈ ∆ m . Thus we also have a first return map F Rm : ∆ m → ∆ m . Define the i-th return times R i m recursively as R 1 m := R m , R i m := R i−1 m + R m • F Rm for any i ≥ 2. A probability distribution on ∆ m is defined by µ ∆m := ( min{R, m + 1}dµ X ) −1 j≤m (F j ) * µ X | {R>j} . Now, a map π ∆m : ∆ → ∆ m is defined so that for any (x, n) ∈ ∆, π m (x, n) = (x, n), n ≤ m (x, m), n > m , i.e., π ∆m pulls every element of ∆ m back to the roof of ∆ m and keeps others elements unchanged. Note that the relation R m dµ ∆m = µ ∆ (∆ m ) −1 follows from the Kac's lemma (see [12]). Observe also that, if R(x) > m for x ∈ X, then F R m+1 m (x) = F R (x) = f R (x). Now we introduce some point processes for the section S r , which is contained in the quasi-section B r (q) × S d ⊆ M. 1 π −1 Sr • F i , N r i (A) := i·µ ∆m (π ∆m π −1 Sr)∈A 1 π ∆m π −1 Sr • (F Rm ) i . Remark 4.3. Note that N r s can be viewed as a point process on (∆, µ ∆ ). Hence N r s = d N r,q s , where the last one was defined in (3.2). Remark 4.4. Since S r is a section, then π X : π −1 S r → X is injective, and thus π X : π ∆m π −1 S r → X is injective also. Lemma 4.5 (Inducing). Suppose that for any disjoint bounded intervals J 1 , · · · , J n ⊆ [0, ∞), lim r→0 µ ∆m N r i ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 , Then for any disjoint bounded intervals J 1 , · · · , J n ⊆ [0, ∞), lim r→0 µ ∆m N r s ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 . Proof. For any x ∈ ∆ m define τ s (x) := inf{n ≥ 1 : F n (x) ∈ π −1 S r }, τ p (x) := inf{n ≥ 1 : F n (x) ∈ π ∆m π −1 S r }, τ i (x) := inf{n ≥ 1 : (F Rm ) n (x) ∈ π ∆m π −1 S r }. The corresponding j-th return times τ j s , τ j p , τ j i are defined for any j ≥ 1 so that τ s := τ 1 s < τ 2 s < · · · < τ j s < τ j+1 s < · · · , τ p := τ 1 p < τ 2 p < · · · < τ j p < τ j+1 p < · · · , τ i := τ 1 i < τ 2 i < · · · < τ j i < τ j+1 i < · · · . By making use of the fact that S r is a section and Remark 4.4 , we get that for any j ≥ 1, τ j p = R m + R m • F Rm + · · · + R m • (F Rm ) τ j i , |τ j p − τ j s | ≤ R m • (F Rm ) τ j i . From the Birkhoff's ergodic theorem we obtain lim n→∞ i≤n R i m n = R m dµ ∆m = µ ∆ (∆ m ) −1 µ ∆m -a.s. lim n→∞ R m • (F Rm ) n n = 0 µ ∆m -a.s. Therefore, for any small > 0 and a.e. x ∈ ∆ m , there is N ,x > 0, such that for any n > N ,x , i≤n R i m n − µ ∆ (∆ m ) −1 ≤ , R m • (F Rm ) n n ≤ . Let G n, := {x ∈ ∆ m : n ≥ N ,x }. Then G n, ⊆ G n+1, for any n ≥ 1. Hence lim n→∞ µ ∆m (G n, ) = 1. Therefore, there is N > 0, such that for any n ≥ N , µ ∆m (G c n, ) ≤ . Let x ∈ G N , . We have that x ∈ G n, for any n ≥ N and i≤n R i m n − µ ∆ (∆ m ) −1 ≤ , R m • (F Rm ) n n ≤ . In particular, if τ i ≥ N , (which implies τ j i ≥ N for all j ≥ 1), then for all j ≥ 1, τ j p τ j i ∈ [µ ∆ (∆ m ) −1 − , µ ∆ (∆ m ) −1 + ], τ j s τ j i − τ j p τ j i ≤ R m • (F Rm ) τ j i τ j i ≤ . Therefore, for any j ≥ 1 τ j s τ j i ∈ [µ ∆ (∆ m ) −1 − 2 , µ ∆ (∆ m ) −1 + 2 ]. Now let H := {x ∈ ∆ m : τ i (x) > N }. Then µ ∆m (H c ) = µ ∆m {x ∈ ∆ m : τ i (x) ≤ N } ≤ N µ ∆m (π ∆m π −1 S r ) = N Rdµ X min{R, m + 1}dµ X µ ∆ (π −1 S r ) = N Rdµ X min{R, m + 1}dµ X µ M (S r ), where the second equality holds because S r is a section. Therefore, there is r > 0, such that for any r ∈ (0, r ), µ ∆m (H c ) ≤ , µ ∆m (H c G c N , ) ≤ 2 . Besides for any x ∈ H G N , and any j ≥ 1, τ j s (x) τ j i (x) ∈ [µ ∆ (∆ m ) −1 − 2 , µ ∆ (∆ m ) −1 + 2 ]. Let aJ i := {aj : j ∈ J i } for any a > 0, and J i := µ ∆ (π −1 S r ) −1 J i , J i : = (µ ∆ (∆ m ) −1 − 2 ) −1 µ ∆ (π −1 S r ) −1 J i (µ ∆ (∆ m ) −1 + 2 ) −1 µ ∆ (π −1 S r ) −1 J i = (µ ∆ (∆ m ) −1 − 2 ) −1 µ ∆ (π ∆m π −1 S r ) −1 J i (µ ∆ (∆ m ) −1 + 2 ) −1 µ ∆ (π ∆m π −1 S r ) −1 J i = (1 − 2µ ∆ (∆ m ) ) −1 µ ∆m (π ∆m π −1 S r ) −1 J i (1 + 2µ ∆ (∆ m ) ) −1 µ ∆m (π ∆m π −1 S r ) −1 J i , where the second equality holds because S r is a section. NextĴ i := (1 − 2µ ∆ (∆ m ) ) −1 J i (1 + 2µ ∆ (∆ m ) ) −1 J i , which implies that µ ∆m (π ∆m π −1 S r ) −1Ĵ i = J i . Also notice that {x ∈ ∆ m \ π −1 S r : N r s ( i≤n J i )(x) = 0} = {x ∈ ∆ m \ π −1 S r : τ j s (x) / ∈ i≤n J i for all j ≥ 1}, {x ∈ ∆ m \ π ∆m π −1 S r : N r i ( i≤nĴ i )(x) = 0} = {x ∈ ∆ m \ π ∆m π −1 S r : τ j i (x) / ∈ i≤n J i for all j ≥ 1}. Now we can estimate N r s via N r i : µ ∆m {x ∈ ∆ m : N r s ( i≤n J i )(x) = 0} ≤ µ ∆m {x ∈ ∆ m \ π −1 S r : τ j s (x) / ∈ i≤n J i for all j ≥ 1} + µ ∆m (π −1 S r ) ≤ µ ∆m {x ∈ H G N , : τ j s (x) / ∈ i≤n J i for all j ≥ 1} + 2 + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ H G N , : τ j i (x) / ∈ k≤n J k for all j ≥ 1} + 2 + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ ∆ m : τ j i (x) / ∈ k≤n µ ∆m (π ∆m π −1 S r ) −1Ĵ k for all j ≥ 1} + 2 + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ ∆ m \ π ∆m π −1 S r : τ j i (x) / ∈ k≤n µ ∆m (π ∆m π −1 S r ) −1Ĵ k for all j ≥ 1} + 2 + 2µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ ∆ m : N r i ( k≤nĴ k )(x) = 0} + 2 + 2 Rdµ X min{R, m + 1}dµ X µ M (S r ). From the conditions of this lemma we have lim sup r→0 µ ∆m {x ∈ ∆ m : N r s ( i≤n J i )(x) = 0} ≤ e − Leb( k≤nĴ k ) + 2 . Let → 0. Then Leb( k≤nĴ k ) → Leb( k≤n J k ) and lim sup r→0 µ ∆m {x ∈ ∆ m : N r s ( i≤n J i )(x) = 0} ≤ e − Leb( k≤n J k ) . Similarly, letĴ i : = (1−2µ ∆ (∆ m ) ) −1 J i (1+2µ ∆ (∆ m ) ) −1 J i , J i := µ ∆m (π ∆m π −1 S r ) −1Ĵ i , J i : = (µ ∆ (∆ m ) −1 − 2 )µ ∆m (π ∆m π −1 S r ) −1Ĵ i (µ ∆ (∆ m ) −1 + 2 )µ ∆m (π ∆m π −1 S r ) −1Ĵ i = (1 − 2µ ∆ (∆ m ) )µ ∆ (π ∆m π −1 S r ) −1Ĵ i (1 + 2µ ∆ (∆ m ) )µ ∆ (π ∆m π −1 S r ) −1Ĵ i = (1 − 2µ ∆ (∆ m ) )µ ∆ (π −1 S r ) −1Ĵ i (1 + 2µ ∆ (∆ m ) )µ ∆ (π −1 S r ) −1Ĵ i , where the third equality holds because S r is a section. Clearly J i = (1 − 2µ ∆ (∆ m ) )Ĵ i (1 + 2µ ∆ (∆ m ) )Ĵ i , µ ∆ (π −1 S r ) −1 J i = J i . Also note that {x ∈ ∆ m \ π ∆m π −1 S r : N r i ( k≤nĴ k )(x) = 0} = {x ∈ ∆ m \ π ∆m π −1 S r : τ j i (x) / ∈ k≤n J k for all j ≥ 1}, {x ∈ ∆ m \ π −1 S r : N r s ( k≤n J k )(x) = 0} = {x ∈ ∆ m \ π −1 S r : τ j s (x) / ∈ k≤n J k for all j ≥ 1}. Now we can estimate N r i via N r s as µ ∆m {x ∈ ∆ m : N r i ( i≤nĴ i )(x) = 0} ≤ µ ∆m {x ∈ ∆ m \ π ∆m π −1 S r : τ j i (x) / ∈ k≤n J k for all j ≥ 1} + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ H G N , : τ j i (x) / ∈ k≤n J k for all j ≥ 1} + 2 + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ H G N , : τ j s (x) / ∈ k≤n J k for all j ≥ 1} + 2 + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ ∆ m : τ j s (x) / ∈ k≤n µ ∆ (π −1 S r ) −1 J k for all j ≥ 1} + 2 + µ ∆m (π ∆m π −1 S r ) ≤ µ ∆m {x ∈ ∆ m \ π −1 S r : τ j s (x) / ∈ k≤n µ ∆ (π −1 S r ) −1 J k for all j ≥ 1} + 2 + 2µ ∆m (π ∆m π −1 S r ) = µ ∆m {x ∈ ∆ m : N r s ( k≤n J k )(x) = 0} + 2 + 2 Rdµ X min{R, m + 1}dµ X µ M (S r ). From the condition of this lemma we have lim inf r→0 µ ∆m {x ∈ ∆ m : N r s ( i≤n J i )(x) = 0} ≥ e − Leb( k≤nĴ k ) − 2 . Let → 0. Then Leb( k≤nĴ k ) → Leb( k≤n J k ) and lim inf r→0 µ ∆m {x ∈ ∆ m : N r s ( i≤n J i )(x) = 0} ≥ e − Leb( k≤n J k ) . Therefore lim r→0 µ ∆m {x ∈ ∆ m : N r s ( i≤n J i )(x) = 0} = e − Leb( k≤n J k ) = P P( i≤n J i ) = 0 for any disjoint bounded intervals J 1 , · · · , J n ⊆ [0, ∞), which concludes a proof of this lemma. Remark 4.6. In [2] a similar result was obtained for return statistics to balls. We are dealing here with a more general hitting statistics for sections. Lemma 4.7 (Approximations). Suppose that for any large m ≥ 1, and any disjoint bounded intervals J 1 , · · · , J n ⊆ [0, ∞), lim r→0 µ ∆m N r i ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 . (4.1) Then for any disjoint bounded intervals J 1 , · · · , J n ⊆ [0, ∞) we have lim r→0 µ ∆ N r s ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 . From Remark 4.3 we also get the required relation (3.4) lim r→0 µ M N r,q s ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 , which concludes a proof for Poisson limit laws. Proof. It follows from Lemma 4.5 that for any m 1 lim r→0 µ ∆m N r s ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 , Therefore, lim r→0 µ ∆ N r s ( i≤n J i ) = 0 = µ ∆ (∆ m ) lim r→0 µ ∆m N r s ( i≤n J i ) = 0 + O µ ∆ (∆ c m ) = µ ∆ (∆ m )P P( i≤n J i ) = 0 + O µ ∆ (∆ c m ) . Let m → ∞. Then lim r→0 µ ∆ N r s ( i≤n J i ) = 0 = P P( i≤n J i ) = 0 . Remark 4.8. In the remaining sections we will show that (∆ m , F Rm , µ ∆m ) is uniformly hyperbolic dynamical system with exponential decay of correlations and arbitrarily large contraction rates α along (un)stable manifolds. It allows us to skip verification of the condition α · dim (hole) > 1 (see [19]), which fails for many slowly mixing billiard systems. Thanks to Lemmas 4.7, 3.3 and 3.2, in order to prove that N r,q → d P we just need to verify (4.1), which is the focus of the remaining sections. Thicker hyperbolic and expanding Young towers In order to prove (4.1), we will model dynamical systems (∆ m , F Rm , µ ∆m ) by hyperbolic (although non-mixing) Young towers (see [22,23]), but with exponential expansion and contraction. Hyperbolic Young towers for (X, f R , µ X ) In this subsection we will consider a dynamical system (X, f R , µ X ). To simplify notations denote f R or F R by g. According to [5,7,14] and Definition 2.4 (CMZ structures), there exists a compact set Λ ⊆ i≤N g i S c i≤N g −i S c ⊆ X with a hyperbolic product structure. Besides, there are such families of C 1 -stable disks (i.e., closed connected parts of stable manifolds) Γ s := {γ s } and families of C 1unstable disks (i.e., closed connected parts of unstable manifolds) Γ u := {γ u } such that the following conditions hold. 1. Λ = ( γ s ) ( γ u ), 2. dim γ s + dim γ u = dim M, 3. each γ s intersects every γ u at exactly one point, 4. stable and unstable manifolds are transversal, and the angles between them are uniformly bounded away from 0, 5. Γ u := {γ u } is a continuous family, i.e., there is a compact set K s , a unit disk O u in some Euclidean space and a map φ u : K s × O u → M such that (a) γ u = φ u ({x} × O u ) is an unstable manifold, (b) φ u maps K s × O u homeomorphically onto its image, (c) x → φ u | {x}×O u defines a continuous map from K s to Emb 1 (O u , M), where Emb 1 (O u , M) is the space of C 1 -embeddings of O u into M. Γ s := {γ s } is also a continuous family in the same sense. 6. Lebesgue detectability: there exists γ ∈ Γ u such that Leb γ (Λ γ) > 0, where Leb γ is the Lebesgue measure on γ. 7. Markov property: there exist pairwise disjoint s-subsets Λ 1 , Λ 2 , · · · ⊆ Λ, that is, each Λ i = γ s ∈Γ s i γ s γ u ∈Γ u γ u for some Γ s i ⊆ Γ s , such that (a) Leb γ Λ \ ( i≥1 Λ i ) = 0 on each γ ∈ Γ u , (b) there is a return time function R e : Λ → N and a return map g R e : Λ → Λ, such that for each i ≥ 1 R e Λ i = R e i ≡ 0 (mod N ), g R e Λ i = g R e i Λ i , g R e i (Λ i ) is a u-subset (i.e., each g R e i (Λ i ) = γ s ∈Γ s γ s γ u ∈Γ u i γ u for some Γ u i ⊆ Γ u ), and for all x ∈ Λ i , g R e i γ s (x) ⊆ γ s g R e i (x) , g R e i γ u (x) ⊇ γ u g R e i (x) , where γ u (y) (resp. γ s (y)) is an element of Γ u (resp. Γ s ) which contains y ∈ Λ. Moreover, there exist constants C ≥ 1 and 0 < β < 1, such that the following conditions hold. 8. Exponential contraction of stable disks: for any γ s ∈ Γ s , x, y ∈ γ s , n ≥ 1, d g n (x), g n (y) ≤ Cβ n . 9. Backward exponential contraction of unstable disks: for any γ u ∈ Γ u , x, y ∈ γ u , n ≥ 1, d g −n (x), g −n (y) ≤ Cβ n . 10. Bounded distortion: for any γ ∈ Γ u and x, y ∈ γ Λ i for some Λ i , log det(D u g R e )(x) det(D u g R e )(y) ≤ Cβ s(g R e (x),g R e (y)) , where s(x, y) is the separation time, i.e., for any x, y ∈ Λ, s(x, y) := min{n ≥ 0 : (g R e ) n (x) and (g R e ) n (y) belong to different sets Λ i }. 11. Regularity of the stable foliations: for each γ, γ ∈ Γ u , define Θ γ,γ : γ Λ → γ Λ by Θ γ,γ (x) = γ s (x) γ. Then the following properties hold (a) Θ γ,γ is absolutely continuous, and for any x ∈ γ Λ d (Θ γ,γ ) * Leb γ d Leb γ (x) = n≥0 det(D u g) (g n (x)) det(D u g) g n (Θ −1 γ,γ x) = C ±1 , (b) for any x, y ∈ γ Λ, log d(Θ γ,γ ) * Leb γ d Lebγ (x) d(Θ γ,γ ) * Leb γ d Lebγ (y) ≤ Cβ s(x,y) . 12. Decay rate of the return times R e : for any γ ∈ Γ u Leb γ (R e > n) ≤ Cβ n . (5.1) Remark 5.1. Note that gcd {R e i } ≥ N . Indeed, it follows from the fact that (f R ) N satisfies the growth lemma for a CMZ structure. Now we can construct a hyperbolic Young tower ∆ e with dynamics F e : ∆ e → ∆ e , where ∆ e := {(x, l) ∈ Λ × {0, 1, 2, · · · } : 0 ≤ l < R e (x)}, F e (x, l) := (x, l + 1), l < R e (x) − 1 g R e (x), 0 , l = R e (x) − 1 . The equivalence relation ∼ on Λ is then x ∼ y if and only if x, y ∈ γ s for some γ s ∈ Γ s . Another equivalence relation ∼ on ∆ e is (x, n) ∼ (y, m) if and only if x, y ∈ γ s for some γ s ∈ Γ s , n = m. By making use of these equivalence relations we can define a quotient tower ∆ e := ∆ e / ∼, which has a quotient product structure Λ := Λ/ ∼ with canonical projections π ∆e : ∆ e → ∆ e and π Λ : Λ → Λ. We identify Λ, Λ with ∆ e (Λ×{0}), ∆ e ( Λ×{0}), respectively. The quotient maps F e : ∆ e → ∆ e , g R e : Λ → Λ, a quotient return time R e : ∆ e → N, and a quotient separation time s on Λ× Λ are defined via the following relations F e = F e • π ∆e , R e = R e • π Λ , g R e = F e R e , s = s • ( π Λ , π Λ ). It follows from [22] that there exists a measure m on Λ such that log det D F e R e (x) det D F e R e (y) ≤ Cβ s( Fe R e x, Fe R e y) , (5.2) where det D F e R e is the Radon-Nikodym derivative of F e R e with respect to m. The set ( ∆ e , F e ), together with (5.2), is called an expanding quotient Young tower. [22,23]). There are constants C ≥ 1 and 0 < β < 1, such that the following holds. Lemma 5.2 (See 1. There exists a probability distribution µ Λ on Λ, which is constructed only from m and F e R e , such that ( F e R e ) * µ Λ = µ Λ , dµ Λ dm = C ±1 . (5.3) A probability distribution on ∆ e , defined by µ ∆e := ( R e dµ Λ ) −1 j ( F e j ) * µ Λ | { R e >j} , is an invariant measure, i.e., F e * µ ∆e = µ ∆e . 2. Further, there exists a probability measure µ Λ on Λ such that ( π Λ ) * µ Λ = µ Λ , (g R e ) * µ Λ = µ Λ , (µ Λ ) γ u Leb γ u , d(µ Λ ) γ u d Leb γ u = C ±1 , (5.4) where (µ Λ ) γ u is the conditional probability of µ Λ on γ u ∈ Γ u . A probability measure on ∆ e defined by µ ∆e := ( R e dµ Λ ) −1 j (F j e ) * µ Λ | {R e >j} satisfies the following condition ( π ∆e ) * µ ∆e = µ ∆e , π e * µ ∆e = µ X , F e * µ ∆e = µ ∆e , where π e : ∆ e → X is the projection, defined as π e (x, l) := g l (x). Suppose that gcd {R e i } = N e ≥ N . Now we define new towers ∆ e := {(x, lN e ) ∈ Λ × {0, 1, 2, · · · } : 0 ≤ l < R e (x)/N e }, ∆ e := {(x, lN e ) ∈ Λ × {0, 1, 2, · · · } : 0 ≤ l < R e (x)/N e }, which are sub-towers of ∆ e and ∆ e , respectively. Then the maps F Ne e : ∆ e → ∆ e and F e Ne : ∆ e → ∆ e preserve probability measures µ ∆ e := ( R e /N e dµ Λ ) −1 j (F jNe e ) * µ Λ | {R e /Ne>j} , µ ∆ e := ( R e /N e dµ Λ ) −1 j ( F e jNe ) * µ Λ | { R e /Ne>j} , respectively. Further, π e * µ ∆ e is exactly µ X , since (X, g, µ X ) is mixing (see Definition 2.4). A family of partitions (Q k ) k≥0 of ∆ e , ( Q k ) k≥0 of ∆ e , defined as Q 0 := {Λ i × {lN e }, i ≥ 1, 0 ≤ l < R e i /N e }, Q k := 0≤i≤k (F Ne e ) −i Q 0 , Q 0 := { Λ i × {lN e }, i ≥ 1, 0 ≤ l < R e i /N e }, Q k := 0≤i≤k ( F e Ne ) −i Q 0 , satisfies the relations diam π e • F Nek e (Q) ≤ Cβ kNe , π ∆e Q ∈ Q 2k for any Q ∈ Q 2k . 5. Finally, for any n > 2k ≥ 2,any ( Q i ) i≥1 ⊆ Q k , and any bounded function h : ∆ e → R, we have the following estimate for decay of correlations 1 i≥1 Q i h•( F e Ne ) n dµ ∆ e −µ ∆ e i≥1 Q i hdµ ∆ e ≤ Cβ n−2k µ ∆ e i≥1 Q i || h|| ∞ . (5.5) From (5.5) and ( π ∆e ) * µ ∆e = µ ∆e , immediately follows an estimate of a rate of decay of correlations. Namely, for any n > 2k ≥ 2, any (Q i ) i≥1 ⊆ Q k and any σ( k≥0 Q k )-measurable function h : ∆ e → R, 1 i≥1 Q i h•(F Ne e ) n dµ ∆ e −µ ∆ e i≥1 Q i hdµ ∆ e ≤ Cβ n−2k µ ∆ e i≥1 Q i ||h|| ∞ . (5.6) Remark 5.3. It is easy to prove that for any unstable manifold/disk γ u we have (µ X ) γ u Leb γ u , where (µ X ) γ u is the conditional measure of µ X on an unstable manifold/disk γ u . This µ X is called a Sinai-Ruelle-Bowen measure (SRB measure). Thanks to all preparations above, we can construct now thicker hyperbolic and expanding Young towers. Thicker hyperbolic Young towers for (∆ m , F R m , µ ∆ m ) Lemma 5.4. (∆ m , F Rm , µ ∆m ) is K-mixing for sufficiently large m ≥ 1. (Observe, that it is not true for a small m). Proof. By Definition 2.4, gcd{R} = 1. Suppose that gcd{i 1 , i 2 , · · · , i k } = 1 so that gcd{R} = gcd{R| {R=i 1 } , R| {R=i 2 } , · · · , R| {R=i k } } = 1. Then choose m ≥ max{i 1 , i 2 , · · · , i k }, we have gcd{R m } = gcd{R} = 1. Besides, F Rm = g : X → X is K-mixing. Therefore, (∆ m , F Rm , µ ∆m ) is K-mixing, and thus mixing. From now on we will consider only large enough m. Define a new tower as ∆ e,m = i≥1 0≤j≤R e i −1 Λ i × {j} × {0, 1, 2, · · · , m i,j }, where m i,j = min{m, R g j (x) − 1} for 0 ≤ j ≤ R e i − 1 and any x ∈ Λ i . Lemma 5.5. m i,j is well-defined. Proof. To prove this we just need to show that R g j (x) does not depend on any x ∈ Λ i . It follows from j ≤ R e i − 1 that g R e i −j is smooth on g j (Λ i ). Therefore g j (Λ i ) ⊆ S c . By Assumption 2.11, i≥1 ∂X i ⊆ S. Thus g j (Λ i ) X k for some k. Since R is constant on X k , then R(y) = R| X k for any y ∈ g j (Λ i ), and m i,j is well-defined. We identify Λ i × {j} × {0} with Λ i × {j}, and Λ × {0} × {0} with Λ. Then ∆ e is a sub-tower of ∆ e,m . Define now a map F e,m : ∆ e,m → ∆ e,m , so that for any x ∈ Λ i and for some i ≥ 1 F e,m (x, j, k) :=      (x, j, k + 1), j ≤ R e i − 1, k < m i,j (x, j + 1, 0), j < R e i − 1, k = m i,j g R e i (x), 0, 0 , j = R e i − 1, k = m i,j . The set Λ, as the base of ∆ e,m , has a return time R e,m | Λ i := j<R e i 1 + m i,j , such that F Re,m e,m = g R e : Λ → Λ is the induced map for the tower (∆ e,m , F e,m ). Define a probability measure µ ∆e,m on ∆ e,m as µ ∆e,m := ( R e,m dµ Λ ) −1 j (F j e,m ) * µ Λ | {Re,m>j} . A projection π e,m : ∆ e,m → ∆ m is defined by π e,m (x, j, k) := F k • g j (x). Lemma 5.6. For any γ ∈ Γ u we have R e ≤ R e,m ≤ 2mR e , Leb γ (R e,m > n) ≤ Cβ n/(2m) . Proof. The first inequality is obvious, since m i,j ≤ m for any i, j. By (5.1), Leb γ (R e,m > n) ≤ Leb γ (R e > n/2m) ≤ Cβ n/(2m) . Lemma 5.7. (π e,m ) * µ ∆e,m = µ ∆m and π e,m is a semi-conjugacy, that is, π e,m •F e,m = F Rm • π e,m . Proof. First we prove a semi-conjugacy. For any ( x, j, k) ∈ ∆ e,m , where x ∈ Λ i for some i ≥ 1, suppose that k < m i,j , π e,m • F e,m (x, j, k) = π e,m (x, j, k + 1) = F k+1 • g j (x), F Rm • π e,m (x, j, k) = F Rm • F k • g j (x) = F k+1 • g j (x), where the last equality holds because π N F k • g i (x) = k < m i,j . Suppose that k = m i,j , j ≤ R e i − 1, π e,m • F e,m (x, j, k) = π e,m (x, j + 1, 0) = g j+1 (x), F Rm • π e,m (x, j, k) = F Rm • F k • g j (x) = g • g j (x) = g j+1 (x), where F Rm • F k = g follows from the fact that F k • g j (x) is already on the roof of ∆ m . Therefore π e,m is a semi-conjugacy. Next we prove that (π e,m ) * µ ∆e,m = µ ∆m . Denote j (F j e ) * (µ Λ | R e >j ) by ν. Then for any A × {k} ⊆ ∆ m , where A ⊆ {R = i} for some i ≥ 1, µ ∆m (A × {k}) = ( min{R, m + 1}dµ X ) −1 µ X (A) = ( min{R, m + 1}dµ X ) −1 µ ∆e (π −1 e A) = ( min{R, m + 1}dµ X ) −1 µ ∆e {(x, n) ∈ Λ × {0, 1, 2, · · · } : g n (x) ∈ A} = ( min{R, m + 1}dµ X ) −1 ( R e dµ Λ ) −1 ν(π −1 e A). (π e,m ) * µ ∆e,m (A × {k}) = µ ∆e,m {π −1 e,m (A × {k})} = µ ∆e,m {(x, j, i) ∈ ∆ e,m : F i g j (x) ∈ A × {k}} = µ ∆e,m {(x, j, k) ∈ ∆ e,m : g j (x) ∈ A} = ( R e,m dµ Λ ) −1 ν(π −1 e A). Note that min{R, m + 1}dµ X = ( R e dµ Λ ) −1 min{R, m + 1} • π e dν = ( R e dµ Λ ) −1 i≥1 j<R e i min{R, m + 1}| g j Λ i µ Λ (Λ i ) = ( R e dµ Λ ) −1 i≥1 j<R e i (1 + m i,j )µ Λ (Λ i ) = ( R e dµ Λ ) −1 R e,m dµ Λ . Then ( min{R, m + 1}dµ X ) −1 ( R e dµ Λ ) −1 = ( R e,m dµ Λ ) −1 . Therefore (π e,m ) * µ ∆e,m (A × {k}) = µ ∆m (A × {k}) for any measurable set A × {k} ⊆ ∆ m . Since such set generates the σ-algebra of ∆ m , then (π e,m ) * µ ∆e,m = µ ∆m . Thicker expanding quotient Young tower for (∆ m , F R m , µ ∆ m ) Introduce an equivalence relation ∼ on ∆ e,m as (x, j, k) ∼ (y, i, l) if and only if x, y ∈ γ s for some γ s ∈ Γ s , j = i, k = l. Using this equivalence relation we can define a quotient tower ∆ e,m := ∆ e,m / ∼, with canonical projections π ∆e,m : ∆ e,m → ∆ e,m . We identify Λ with ∆ e,m ( Λ × {0} × {0}). A quotient map F e,m : ∆ e,m → ∆ e,m , and a quotient return time R e : ∆ e → N are defined via the following relations F e,m = F e,m • π ∆e,m , R e,m = R e,m • π Λ . They satisfy g R e = F e,m Re,m , which is easy to prove from the construction of ∆ e,m and F e,m . Define a probability measure µ ∆e,m on ∆ e,m as µ ∆e,m := ( R e,m dµ Λ ) −1 j ( F e,m j ) * µ Λ | { Re,m>j} . Since g R e = F e R e , it is easy to see that ( π ∆e,m ) * µ ∆e,m = µ ∆e,m . By ( with respect to m. Therefore ( ∆ e,m , F e,m ), together with (5.7), is the thicker expanding quotient Young tower for (∆ m , F Rm , µ ∆m ). Decay of correlations Let R e,m,i := R e,m | Λ i . Suppose that gcd {R e,m,i } = N e,m . Now we can define new towers ∆ e,m := {(x, lN e,m ) ∈ Λ × {0, 1, 2, · · · } : 0 ≤ l < R e,m (x)/N e,m }, ∆ e,m := {(x, lN e,m ) ∈ Λ × {0, 1, 2, · · · } : 0 ≤ l < R e,m (x)/N e,m },µ ∆ e,m := ( R e,m /N e,m dµ Λ ) −1 j (F jNe,m e,m ) * µ Λ | {Re,m/Ne,m>j} , µ ∆ e,m := ( R e,m /N e,m dµ Λ ) −1 j ( F e,m jNe,m ) * µ Λ | { Re,m/Ne,m>j} , respectively. Since (∆ m , F Rm , µ ∆m ) is mixing (see Lemma 5.4), then by using the same argument as that on the page 607 of [22] we have (π e,m ) * µ ∆ e,m = µ ∆m . A family of partitions (Q m k ) k≥0 of ∆ e,m , ( Q m k ) k≥0 of ∆ e,m is defined by Q m 0 := {Λ i × {lN e,m }, i ≥ 1, 0 ≤ l < R e,m,i /N e,m }, Q m k := 0≤i≤k (F Ne,m e,m ) −i Q m 0 , Q m 0 := { Λ i × {lN e,m }, i ≥ 1, 0 ≤ l < R e,m,i /N e,m }, Q m k := 0≤i≤k ( F e,m Ne,m ) −i Q m 0 . Lemma 5.8. There is a constant C α > 0 such that diam π • π e,m • F Ne,mk e,m (Q) ≤ C α β αNe,mk 2m and π ∆e,m Q ∈ Q m 2k for any Q ∈ Q m 2k and any k > m + 1. Moreover, there are constants β m ∈ (0, 1) and C m > 0, such that for any n > 2k ≥ 2, any ( Q i ) i≥1 ⊆ Q m k , and any bounded function h : ∆ e,m → R, we have the following estimate for decay of correlations 1 i≥1 Q i h•( F e,m Ne,m ) n dµ ∆ e,m −µ ∆ e,m i≥1 Q i hdµ ∆ e,m ≤ C m β n−2k m µ ∆ e,m i≥1 Q i || h|| ∞ . (5.8) From (5.8) and π ∆e,m * µ ∆e,m = µ ∆e,m , we get the following estimate for decay of correlations. Namely, for any n > 2k ≥ 2, any (Q i ) i≥1 ⊆ Q m k , and for any σ( k≥0 Q m k )-measurable function h : ∆ e,m → R 1 i≥1 Q i h • (F Ne,m e,m ) n dµ ∆ e,m − µ ∆ e,m i≥1 Q i hdµ ∆ e,m ≤ C m β n−2k m µ ∆ e,m i≥1 Q i ||h|| ∞ . (5.9) Proof. Since the return time for ∆ e,m is R e,m /N e,m , we have gcd{ R e,m /N e,m } = 1. m{x ∈ Λ : R e,m (x)/N e,m > n} ≤ Cµ Λ {x ∈ Λ : R e,m (x)/N e,m > n} = C ( π Λ ) * µ Λ {x ∈ Λ : R e,m (x)/N e,m > n} = Cµ Λ {x ∈ Λ : R e,m • π Λ (x)/N e,m > n} = Cµ Λ {x ∈ Λ : R e,m (x)/N e,m > n} = C µ γ u {x ∈ γ u : R e,m (x)/N e,m > n}dµ Λ ≤ C 2 Leb γ u {x ∈ γ u : R e,m (x)/N e,m > n}dµ Λ ≤ C 3 β Ne,m 2m n , where we applied Lemma 5.6 to the last inequality. By making use of Theorem 2 in [23] we get (5.8) and (5.9). Next we estimate diam π • π e,m • F Ne,mk e,m (Q) . For any γ s ⊆ Q, π • π e,m • F Ne,mk e,m (γ s ) returns to X at least N e,m k/(1 + m) times. Therefore, it is contracted by g at least N e,m k/(2m) times. Assume that q ∈ [ Ne,mk 2m , N e,m k] is the last return time to X. Then π e,m • F q e,m (γ s ) is a smooth stable disk in X and π • π e,m • F j e,m (γ s ) ⊆ X c for any q < j ≤ N e,m k. Therefore diam π • π e,m • F q e,m (γ s ) ≤ Cβ q diam γ s ≤ Cβ Ne,mk 2m diam γ s . Then by Assumption 2.11 and by Definition 2.4, R is a constant on π e,m •F q e,m (γ s ), and there is a constant C α > 0, such that diam π • π e,m • F Ne,mk e,m (γ s ) ≤ C diam π • π e,m • F q e,m (γ s ) α ≤ C α β Ne,mkα 2m /2. On the other hand, for any γ u ⊆ Q consider a smooth unstable disk γ u 0 = π•π e,m • F Ne,m2k e,m (γ u ). Assume that π N π e,m • F Ne,mk e,m (γ u ) = j ≤ m, π N π e,m • F Ne,m2k e,m (γ u ) = j ≤ m. Then π • π e,m • F Ne,mk−j e,m (γ u ) ∈ X, π • π e,m • F Ne,mk−j e,m (γ u ) = g −q f −j γ u 0 , where q ∈ [ Ne,mk+j m+1 , N e,m k + j ]. By Assumption 2.11 and by Definition 2 .4, R is constant on π • π e,m • F Ne,mk−j e,m (γ u ), and there is a constant C α > 0, such that diam π • π e,m • F Ne,mk e,m (γ u ) ≤ C diam π • π e,m • F Ne,mk−j e,m (γ u ) α ≤ C 1+α β αq diam π • π e,m • F 2Ne,mk−j e,m (γ u ) α ≤ C α β Ne,mkα 2m /2. Finally, since any γ u , γ s ⊆ Q intersect exactly at one point, then for any x, y ∈ Q there is o ∈ Λ, such that x ∈ γ u (o), y ∈ γ s (o), and d π • π e,m • F Ne,mk e,m (x), π • π e,m • F Ne,mk e,m (y) ≤ diam π • π e,m • F Ne,mk e,m γ u (o) + diam π • π e,m • F Ne,mk e,m γ s (o) ≤ C α β Ne,mkα 2m . Therefore, diam π • π e,m • F Ne,mk e,m (Q) ≤ C α β Ne,mkα 2m . Poisson limit laws for non-mixing hyperbolic Young towers Now we are ready to prove (4.1). The approach for Poisson approximations, developed in [3,19,20], works for a mixing hyperbolic Young tower. However, from the previous section, we know that our Young tower for (∆ e,m , F e,m , µ ∆e,m ) is generally non-mixing. In this section we will prove Poisson limit laws for the dynamics F Rm , which can be described by the hyperbolic Young tower (∆ e,m , F e,m , µ ∆e,m ). For any i ≥ 0, we let X i := 1 π ∆m π −1 Sr • (F Rm ) i , X X X i := 1 π ∆m π −1 Sr • (F Rm ) Ne,mi , 1 π ∆m π −1 Sr • (F Rm ) Ne,mi+1 , · · · , 1 π ∆m π −1 Sr • (F Rm ) Ne,m(i+1)−1 . Observe that (X X X i ) i≥0 is stationary since µ ∆m is (F Rm ) Ne,m -invariant. Denote by {X X X i } i≥0 i.i.d. random vectors defined on a probability space (Ω,P), such that for each i ≥ 0 X X X i = dX X X i , i.e., they have the same distribution. Throughout this section the notation h( • , • , · · · , • k ) means that a function h is defined on R k for some k ≥ 1, and h ∈ [0, 1] means that a function h takes values in [0, 1]. Further, E is the expectation of µ ∆m ⊗P. For any vector a a a ∈ {0, 1} Ne,m denote by 0 0 0 the zero vector in {0, 1} Ne,m . Then a a a ≥ 1 means that at least one of the coordinates of a a a is not zero, and a a a = 0 0 0 means that all coordinates of a a a are zero. Lemma 6.1. For any n ≥ 1 and any integer p ∈ (0, n), sup h∈[0,1] Eh(X X X 0 , X X X 1 , · · · , X X X n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) m R 1 + R 2 + R 3 , where R 1 := 0≤l≤n−p sup a a a≥1 sup h∈[0,1] E 1 X X X 0 =a a a h(X X X p , · · · , X X X n−l ) − E1 X X X 0 =a a a Eh(X X X p , · · · , X X X n−l ) R 2 := sup a a a≥1 nE 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 R 3 := pnµ ∆m (π ∆m π −1 S r ) 2 + pµ ∆m (π ∆m π −1 S r ), where a constant in " m " depends only on m. Proof. sup h∈[0,1] Eh(X X X 0 , X X X 1 , · · · , X X X n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) = sup h∈[0,1] 0≤l≤n Eh X X X 1 , · · · ,X X X l−1 , X X X l , · · · , X X X n − Eh X X X 1 , · · · ,X X X l−1 ,X X X l , X X X l+1 , · · · , X X X n ≤ sup h∈[0,1] 0≤l≤n Eh l X X X l , X X X l+1 , · · · , X X X n − Eh l X X X l , X X X l+1 , · · · , X X X n , where h l ( • ) := h(X X X 1 , · · · ,X X X l−1 , • ) . SinceX X X 1 , · · · ,X X X l−1 are independent of other random variables, then, without loss of generality, we can assume that the function h l does not depend onX X X 1 , · · · ,X X X l−1 . Note that X X X l = dX X X l are {0, 1} Ne,m -valued random vectors. Thus Eh l X X X l , X X X l+1 , · · · , X X X n − Eh l X X X l , X X X l+1 , · · · , X X X n = E 1 X X X l =0 0 0 h l (0 0 0, X X X l+1 , · · · , X X X n ) + a a a≥1 E 1 X X X l =a a a h l (a a a, X X X l+1 , · · · , X X X n ) − E1X X X l =0 0 0 Eh l (0 0 0, X X X l+1 , · · · , X X X n ) − a a a≥1 E1X X X l =a a a Eh l (a a a, X X X l+1 , · · · , X X X n ) = a a a≥1 E 1 X X X l =a a a h l (0 0 0, X X X l+1 , · · · , X X X n ) + a a a≥1 E 1 X X X l =a a a h l (a a a, X X X l+1 , · · · , X X X n ) − a a a≥1 E1X X X l =a a a Eh l (0 0 0, X X X l+1 , · · · , X X X n ) − a a a≥1 E1X X X l =a a a Eh l (a a a, X X X l+1 , · · · , X X X n ) ≤ 2 a a a≥1 sup h∈[0,1] E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) . Therefore, sup h∈[0,1] Eh(X X X 0 , X X X 1 , · · · , X X X n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) ≤ 2 0≤l≤n a a a≥1 sup h∈[0,1] E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) . (6.1) We will start with estimating the terms with l ≤ n − p in (6.1). E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) = E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) + E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) + E1 X X X l =a a a Eh(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) = E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) + E1 X X X l =a a a E h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − h(X X X l+1 , · · · , X X X n ) + E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) . Observe that |h(X X X l+1 , · · · , X X X n ) − h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n )| ≤ 21 l+1≤j≤l+p−1 X X X j ≥1 . Now, in view of stationarity of (X X X i ) i≥0 , we can continue the estimates as ≤ E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh (0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) + 2E 1 X X X l =a a a 1 l+1≤j≤l+p−1 X X X j ≥1 + 2E1 X X X l =a a a E1 l+1≤j≤l+p−1 X X X j ≥1 ≤ E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) + 2E 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 + 2E1 X X X 0 =a a a E1 1≤j≤p−1 X X X j ≥1 . Observe that { 1≤j≤p−1 X X X j ≥ 1} = Ne,m≤j≤Ne,mp−1 (F Rm ) −j (π ∆m π −1 S r ), a a a ≥ 1, {X X X 0 = a a a} ⊆ 0≤j≤Ne,m−1 (F Rm ) −j (π ∆m π −1 S r ). Hence, we can continue the sequence of inequalities above as m E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) + E 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 + pµ ∆m (π ∆m π −1 S r ) 2 . Therefore for the terms with l ≤ n − p in (6.1) we have E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) m E 1 X X X l =a a a h(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(0 0 0, · · · , 0 0 0, X X X l+p , · · · , X X X n ) + E 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 + pµ ∆m (π ∆m π −1 S r ) 2 . Consider now the terms with l > n − p in (6.1). Since a a a ≥ 1 and {X X X l = a a a} ⊆ lNe,m≤j≤(l+1)Ne,m−1 (F Rm ) −j (π ∆m π −1 S r ), ||h|| ∞ ≤ 1, then E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) m µ ∆m (π ∆m π −1 S r ). Therefore (6.1) = 2 0≤l≤n sup a a a≥1 sup h∈[0,1] E 1 X X X l =a a a h(X X X l+1 , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+1 , · · · , X X X n ) E 1 X X X l =a a a h(X X X l+p , · · · , X X X n ) − E1 X X X l =a a a Eh(X X X l+p , · · · , X X X n ) + sup a a a≥1 nE 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 + pnµ ∆m (π ∆m π −1 S r ) 2 + pµ ∆m (π ∆m π −1 S r ). By making use of stationarity of (X X X i ) i≥0 , the last expression above can be estimated as E 1 X X X 0 =a a a h(X X X p , · · · , X X X n−l ) − E1 X X X 0 =a a a Eh(X X X p , · · · , X X X n−l ) + sup a a a≥1 nE 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 + pnµ ∆m (π ∆m π −1 S r ) 2 + pµ ∆m (π ∆m π −1 S r ), which concludes a proof. Denote by {X i } i≥0 i.i.d. random variables, which do not depend on (X i ) i≥0 and (X X X i ) i≥0 , and which are defined on a probability space (Ω,P), such that for each i ≥ 0, X i = dXi . Define now random vectors Y Y Y i := (X iNe,m ,X iNe,m+1 , · · · ,X (i+1)Ne,m−1 ). As the next step we will approximate (X X X i ) i≥0 by (Y Y Y i ) i≥0 . Lemma 6.2. For any n ≥ 1, sup h∈[0,1] Eh(Y Y Y 0 , Y Y Y 1 , · · · , Y Y Y n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) m nE1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + nµ ∆m (π ∆m π −1 S r ) 2 , where a constant in " m " depends only on m. Proof. Assume thatX X X 0 := (Z 0 , Z 1 , · · · , Z Ne,m−1 ). Note that all Z i are not independent, and for all 0 ≤ i ≤ N e,m − 1, we have Z i = d X i = dXi . We can start now required estimate. sup h∈[0,1] Eh(Y Y Y 0 , Y Y Y 1 , · · · , Y Y Y n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) = sup h∈[0,1] 0≤l≤n Eh X X X 1 , · · · ,X X X l−1 , Y Y Y l , · · · , Y Y Y n − Eh X X X 1 , · · · ,X X X l−1 ,X X X l , Y Y Y l+1 , · · · , Y Y Y n ≤ sup h∈[0,1] 0≤l≤n Eh l (Y Y Y l ) − Eh l (X X X l ) ≤ 0≤l≤n sup h∈[0,1] Eh l (Y Y Y l ) − Eh l (X X X l ) , where h l ( • ) := h(X X X 1 , · · · ,X X X l−1 , • , Y Y Y l+1 , · · · , Y Y Y n ). SinceX X X 1 , · · · ,X X X l−1 , Y Y Y l+1 , · · · , Y Y Y n do not depend on Y Y Y l andX X X l , then, without loss of generality, h l can be regarded as a function which does not depend onX X X 1 , · · · ,X X X l−1 , Y Y Y l+1 , · · · , Y Y Y n . By stationarity of (Y Y Y i ) i≥0 and (X X X i ) i≥0 , we have sup h∈[0,1] Eh l (Y Y Y l ) − Eh l (X X X l ) ≤ sup h∈[0,1] Eh(Y Y Y l ) − Eh(X X X l ) = sup h∈[0,1] Eh(Y Y Y 0 ) − Eh(X X X 0 ) = sup h∈[0,1] 0≤l≤Ne,m−1 Eh X 0 , · · · ,X l−1 , Z l , · · · , Z Ne,m−1 − Eh X 0 , · · · ,X l−1 ,X l , Z l+1 , · · · , Z Ne,m−1 ≤ sup h∈[0,1] 0≤l≤Ne,m−1 Eh l (Z l , · · · , Z Ne,m−1 ) − Eh l (X l , Z l+1 , · · · , Z Ne,m−1 ) ≤ 0≤l≤Ne,m−1 sup h∈[0,1] Eh l (Z l , · · · , Z Ne,m−1 ) − Eh l (X l , Z l+1 , · · · , Z Ne,m−1 ) , where h l ( • ) := h(X 1 , · · · ,X l−1 , • ). As before, h l can be regarded as a function which does not depend onX 1 , · · · ,X l−1 . Note that X l = dXl = d Z l are {0, 1}-valued random variables. Thus Eh l Z l , Z l+1 , · · · , Z Ne,m−1 − Eh l X l , Z l+1 , · · · , Z Ne,m−1 = E 1 Z l =0 h l (0, Z l+1 , · · · , Z Ne,m−1 ) + E 1 Z l =1 h l (1, Z l+1 , · · · , Z Ne,m−1 ) − E1X l =0 Eh l (0, Z l+1 , · · · , Z Ne,m−1 ) − E1X l =1 Eh l (1, Z l+1 , · · · , Z Ne,m−1 ) = E 1 Z l =1 h l (0, Z l+1 , · · · , Z Ne,m−1 ) + E 1 Z l =1 h l (1, Z l+1 , · · · , Z Ne,m−1 ) − E1X l =1 Eh l (0, Z l+1 , · · · , Z Ne,m−1 ) − E1X l =1 Eh l (1, Z l+1 , · · · , Z Ne,m−1 ) ≤ 2 sup h∈[0,1] E 1 Z l =1 h(Z l+1 , · · · , Z Ne,m−1 ) − E1 Z l =1 Eh(Z l+1 , · · · , Z Ne,m−1 ) = 2 sup h∈[0,1] E 1 Z l =1 h(Z l+1 , · · · , Z Ne,m−1 ) − 1 Z l =1 h(0, · · · , 0) − E1 Z l =1 Eh(Z l+1 , · · · , Z Ne,m−1 ) − h(0, · · · , 0) . Using that |h(Z l+1 , · · · , Z Ne,m−1 ) − h(0, · · · , 0)| ≤ 21 l+1≤j≤Ne,m−1 Z j ≥1 , and stationarity of (X i ) Ne,m−1 i=0 and of (Z i ) Ne,m−1 i=0 , and the relation (X 0 , X 1 , · · · , X Ne,m−1 ) = d (Z 0 , Z 1 , · · · , Z Ne,m−1 ), we can continue the estimate above as E1 Z l =1 1 l+1≤j≤Ne,m−1 Z j ≥1 + E1 Z l =1 E1 l+1≤j≤Ne,m−1 Z j ≥1 ≤ E1 Z 0 =1 1 1≤j≤Ne,m−1 Z j ≥1 + E1 Z 0 =1 E1 1≤j≤Ne,m−1 Z j ≥1 = E1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + E1 X 0 =1 E1 1≤j≤Ne,m−1 X j ≥1 ≤ E1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + µ ∆m (π ∆m π −1 S r )E1 1≤j≤Ne,m−1 X j ≥1 ≤ E1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + N e,m µ ∆m (π ∆m π −1 S r ) 2 , where the last inequality holds because { 1≤j≤Ne,m−1 X j ≥ 1} ⊆ 0≤j≤Ne,m−1 (F Rm ) −j (π ∆m π −1 S r ). By combining all estimates above we get sup h∈[0,1] Eh(Y Y Y 0 , Y Y Y 1 , · · · , Y Y Y n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) ≤ 0≤l≤n sup h∈[0,1] Eh l (Y Y Y l ) − Eh l (X X X l ) ≤ 0≤l≤n 0≤l ≤Ne,m−1 sup h∈[0,1] Eh l (Z l , · · · , Z Ne,m−1 ) − Eh l (X l , Z l +1 , · · · , Z Ne,m−1 ) ≤ 0≤l≤n 0≤l ≤Ne,m−1 E1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + N e,m µ ∆m (π ∆m π −1 S r ) 2 m nE1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + nµ ∆m (π ∆m π −1 S r ) 2 , which concludes a proof. By making use of Lemmas 6.1 and 6.2 we can simplify now (4.1). Lemma 6.3. For any ∈ (0, 1) and for any disjoint bounded intervals J 1 , J 2 , · · · , J k ⊆ [0, ∞), let n := max{i : i · µ ∆m (π ∆m π −1 S r ) ∈ J j , j = 1, 2, · · · , k}, p := n 1− . Then E 1 X X X 0 =a a a h(X X X p , · · · , X X X n−l ) − E1 X X X 0 =a a a Eh(X X X p , · · · , X X X n−l ) (6.2) µ ∆m N r i ( i≤k J i ) = 0 − P P( i≤k J i ) = 0 m T 1 + T 2 + T 3 ,T 2 := µ ∆m (π ∆m π −1 S r ) −1 E 1 X 0 =1 1 1≤j≤pNe,m−1 X j ≥1 (6.3) T 3 := µ ∆m (π ∆m π −1 S r ) , where a constant in " m " depends only on m. Since T 3 m µ M (S r ) → 0 when r → 0, then in order to prove (4.1), we just need to show that (6.2) and (6.3) converge to zero when r → 0. We will say in what follows that (6.3) is a short return. Proof. Let J i := {j : jµ ∆m (π ∆m π −1 S r ) ∈ J i }, X J i := j∈J i 1 π ∆m π −1 Sr • (F Rm ) j , and X J i := j∈J iX j , where {X i } i≥0 are i.i.d. random variables such thatX i = d X i = 1 π ∆m π −1 Sr • (F Rm ) i .µ ∆m N r i ( i≤k J i ) = 0 − P P( i≤k J i ) = 0 ≤ sup h∈[0,1] Eh(X J 1 , · · · , X J k ) − h P(J 1 ), · · · , P(J k ) ≤ sup h∈[0,1] Eh(X J 1 , · · · , X J k ) − h(X J 1 , · · · ,X J k ) + sup h∈[0,1] Eh P(J 1 ), · · · , P(J k ) − h(X J 1 , · · · ,X J k ) ≤ sup h∈[0,1] Eh(X 0 , X 1 , · · · , X n ) − h(X 0 ,X 1 , · · · ,X n ) + sup h∈[0,1] Eh P(J 1 ), · · · , P(J k ) − h(X J 1 , · · · ,X J k ) . By applying Theorem 2 from [1] to (X i ) i≥0 we get sup h∈[0,1] Eh P (J 1 ), · · · , P (J k ) − h X J 1 , · · · ,X J k nµ ∆m (π ∆m π −1 S r ) 2 , Using now Lemmas 6.1 and 6.2 we can continue the estimate above as m sup h∈[0,1] Eh(X 0 , X 1 , · · · , X n ) − h(X 0 ,X 1 , · · · ,X n ) + nµ ∆m (π ∆m π −1 S r ) 2 ≤ sup h∈[0,1] Eh(X X X 0 , X X X 1 , · · · , X X X n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) + nµ ∆m (π ∆m π −1 S r ) 2 + sup h∈[0,1] Eh(Y Y Y 0 , Y Y Y 1 , · · · , Y Y Y n ) − h(X X X 0 ,X X X 1 , · · · ,X X X n ) m R 1 + R 2 + R 3 + nE1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 + nµ ∆m (π ∆m π −1 S r ) 2 . Note that n µ ∆m (π ∆m π −1 S r ) −1 , where a constant in " " depends on max{n : n ∈ J i , i = 1, 2, · · · , k} < ∞, and R 2 = sup a a a≥1 nE 1 X X X 0 =a a a 1 1≤j≤p−1 X X X j ≥1 µ ∆m (π ∆m π −1 S r ) −1 E 1 X 0 =1 1 1≤j≤pNe,m−1 X j ≥1 , nE1 X 0 =1 1 1≤j≤Ne,m−1 X j ≥1 µ ∆m (π ∆m π −1 S r ) −1 E 1 X 0 =1 1 1≤j≤pNe,m−1 X j ≥1 . Next we can continue our estimate above as E 1 X X X 0 =a a a h(X X X p , · · · , X X X n−l ) − E1 X X X 0 =a a a Eh(X X X p , · · · , X X X n−l ) + µ ∆m (π ∆m π −1 S r ) −1 E 1 X 0 =1 1 1≤j≤pNe,m−1 X j ≥1 + µ ∆m (π ∆m π −1 S r ) , which concludes a proof. Short returns In this section we will estimate short returns (6.3). The papers [4,18,20] provided effective methods to estimate (6.3) for Sinai billiards with bounded horizons and for diamond billiards. Some specific properties (e.g. bounded free paths and complexity of singularities for these two billiards) were used there. In contrast, we are using here only the hyperbolic product structure Λ in hyperbolic Young towers. At first we will show that the short return (6.3) on ∆ m can be reduced to the short return on X. Lemma 7.1 (Reduce (6.3)). E 1 X 0 =1 1 1≤j≤pNe,m−1 X j ≥1 m 1 π X π −1 Sr 1 1≤j≤pNe,m (f R ) −j (π X π −1 Sr) dµ X , µ ∆m (π ∆m π −1 S r ) ≈ m µ X (π X π −1 S r ). Proof. Since S r is a section, then for any x ∈ ∆ m , 1 π ∆m π −1 Sr (x)1 pNe,m−1 j=1 (F Rm ) −j (π ∆m π −1 Sr) (x) ≤ 1 π X π −1 Sr (π X x)1 pNe,m j=1 (f R ) −j (π X π −1 Sr) (π X x), (π X ) * (µ ∆m | π ∆m π −1 Sr ) = ( min{R, m + 1}dµ X ) −1 µ X | π X π −1 Sr Therefore, E 1 X 0 =1 1 1≤j≤pNe,m−1 X j ≥1 ≤ π ∆m π −1 Sr 1 π X π −1 Sr • π X 1 pNe,m j=1 (f R ) −j (π X π −1 Sr) • π X dµ ∆m m 1 π X π −1 Sr 1 pNe,m j=1 (f R ) −j (π X π −1 Sr) dµ X . The relation µ ∆m (π ∆m π −1 S r ) ≈ m µ X (π X π −1 S r ) holds since S r is a section, which concludes a proof of lemma. Therefore, in order to prove that (6.3) converges to zero, the relation lim r→0 1 µ X (π X π −1 S r ) 1 π X π −1 Sr 1 1≤j≤pNe,m−1 (f R ) −j (π X π −1 Sr) dµ X = 0 will be proved in the following subsections. We will use for that decay of correlations in the dynamical system (X, (f R ) Ne , µ X ) and mixing hyperbolic Young towers (∆ e , F Ne e , µ ∆ e ), where N e is the one from Lemma 5.2. Throughout this section, in order to simplify notations, we let g = f R , and still use π e to denote the semi-conjugacy from ∆ e to X, i.e., π e : ∆ e → X satisfies g Ne • π e = π e • F Ne e . This is possible since (π e ) * µ ∆ e = µ X by Lemma 5.2. Besides, define π ∂ : M = ∂Q × S d → ∂Q by π ∂ (q, v) = q, and suppose that R| Xn = R n for some R n ∈ N. This is the case because X is divided by a part of S into countably many pieces X = i X i , so that R is constant on each X i . Return statistics In this subsection, we will prove that for a.e. m m m ∈ M, lim inf r→0 log Z 2r (m m m) − log µ ∆m (π ∆m π −1 S r ) ≥ 1, where for any m m m ∈ M, Z r (m m m) := min{n ≥ 1 : g n (π X π −1 m m m) ∈ π X π −1 B r (π ∂ m m m) × S d }, S r is the section contained in the quasi-section B r (π ∂ m m m) × S d . Remark 7.2. It is worthwhile to mention that an estimate of Z 2r (m m m) does not require that B r (π ∂ m m m) × S d are quasi-sections for some m m m ∈ M. The fact that S r is a section, is used only for the estimate µ ∆m (π ∆m π −1 S r ) ≈ r d . Define for any n ≥ 1 G n,r := {m m m ∈ M : (g Ne ) n (π X π −1 m m m) ∈ A r (π ∂ m m m)}, where A r (π ∂ m m m) := 0≤i≤Ne−1 g i π X π −1 B r (π ∂ m m m) × S d . Fix p ∈ N, which will be exactly determined later. From Lemma 5.2 almost surely For each Z ∈ Q 2p , we know that π e F Nep e Z is the intersection of a family of stable disks and a family of unstable disks. Thus by Assumption 2.11 π e F Nep e Z is completely contained in some X i . Therefore for each X i we define a family of sets (Z ji ) j ⊆ Q 2p , such that each Z ji satisfies the relation π e F Nep e Z ji ⊆ X i . Therefore j π e F Nep e Z ji ⊆ X i . In fact we have a stronger result. Lemma 7.3. j Z ji = F −Nep e π −1 e X i almost surely, which implies that j π e F Nep e Z ji = X i . Proof. For a.e. x ∈ X i there is z ∈ Z ⊆ ∆ e for some Z ∈ Q 2p , such that π e F Nep e z = x. By Assumption 2.11 we know that π e F Nep e Z is completely contained in some X j , which must be X i , since π e F Nep e Z also contains x ∈ X i . From the way how Z ji was chosen, we know that Z is one of the Z ji . Therefore j Z ji = F −Nep e π −1 e X i almost surely. Since M = i k<R i f k X i , in order to estimate µ M (G n,r ), we will estimate a measure of f k X i G n,r = f k ( j π e F Nep e Z ji ) G n,r , where k < R i . And finally we will sum up these estimates. For each Z ji , we choose and fix a m jik ∈ f k (π e F Nep e Z ji ) G n,r . Then the following statement holds. Lemma 7.4. There is C α > 0 such that for each k < R i , µ M (f k X i G n,r ) ≤ j µ ∆ e Z ji F −Nen e F −Nep e π −1 e A r+Cαβ αpNe (π ∂ m jik ) . Proof. For any m m m ∈ f k (π e F Nep e Z ji ) G n,r we have (g Ne ) n (π X π −1 m m m) ∈ A r (π ∂ m m m), (g Ne ) n (π X π −1 m jik ) ∈ A r (π ∂ m jik ). Since Z ji has a product structure, then f k (π e F Nep e Z ji ) is close to a product structure in M. So it is an intersection of families of stable and unstable disks, where each stable disk γ s and each unstable disk γ u intersect exactly at one point in f k π e F Nep e Z ji , but the angle between unstable and stable disks is not uniformly bounded away from 0. Then there is z ∈ Z ji , such that m m m ∈ f k π e F Nep e γ u (z) and m jik ∈ f k π e F Nep e γ s (z), and by Lemma 5. Therefore, d(m m m, m jik ) ≤ C α β αpNe for some C α > 0. Having this estimate we can compare A r (π ∂ m m m) and A r (π ∂ m jik ). Claim: A r (π ∂ m m m) ⊆ A r+Cαβ αpNe (π ∂ m jik ). For each x ∈ A r (π ∂ m m m) there exists 0 ≤ i ≤ N e − 1 such that x ∈ g i π X π −1 B r (π ∂ m m m) × S d ⊆ g i π X π −1 B r+Cαβ αpNe (π ∂ m jik ) × S d , which means that x ∈ A r+Cαβ αpNe (π ∂ m m m). So this claim holds. Claim: Z ji F −Nep e π −1 e f −k G n,r ⊆ Z ji F −Nep e π −1 e f −k ππ −1 X (g Ne ) −n A r+Cαβ αpNe (π ∂ m jik ). For any z ∈ Z ji F −Nep e π −1 e f −k G n,r we have f k π e F Nep e z ∈ f k (π e F Nep e Z ji ) G n,r , and (g Ne ) n (π X π −1 f k π e F Nep e z) ∈ A r (π ∂ f k π e F Nep e z) ⊆ A r+Cαβ αpNe (π ∂ m jik ), i.e., z ∈ Z ji F −Nep e π −1 e f −k ππ −1 X (g Ne ) −n A r+Cαβ αpNe (π ∂ m jik ). So this claim holds. Using the claims above, Lemma 7.3, the relations f * µ M = µ M and (F Ne e ) * µ ∆ e = µ ∆ e we can estimate µ M (f k X i G n,r ) = µ M (X i f −k G n,r ) = µ ∆ e (π −1 e X i π −1 e f −k G n,r ) = µ ∆ e (F −Nep e π −1 e X i F −Nep e π −1 e f −k G n,r ) = µ ∆ e ( j Z ji F −Nep e π −1 e f −k G n,r ) ≤ j µ ∆ e Z ji F −Nep e π −1 e f −k ππ −1 X (g Ne ) −n A r+Cαβ αpNe (π ∂ m jik ) . Since π e F pNe e Z ji ⊆ X i and k < R i , then π X π −1 f k = π X F k π −1 is an identity map on π e F pNe e Z ji . Using this we can continue our estimate above as ≤ j µ ∆ e Z ji F −Nep e π −1 e (g Ne ) −n A r+Cαβ αpNe (π ∂ m jik ) ≤ j µ ∆ e Z ji F −Nen e F −Nep e π −1 e A r+Cαβ αpNe (π ∂ m jik ) . Thus the lemma is proved. In order to proceed with further estimates we need to study F −Nep e π −1 e A r+Cαβ αpNe (π ∂ m jik ). Define a family of sets (Z ) ⊆ Q 2p , such that Z ⊇ F −Nep e π −1 e A r+Cαβ αpNe (π ∂ m jik ),(7.1) and let each Z in this family satisfies Z F −Nep e π −1 e A r+Cαβ αpNe (π ∂ m jik ) = ∅. Lemma 7.5. µ ∆ e ( Z ) (r + C α β αpNe ) d for some C α > 0. Proof. By Lemma 5.2 and by the definition of Z we have that diam π e F Nep e Z ≤ Cβ pNe , Then there exists the smallest integer k ∈ [0, N e ), which depends on Z , such that g −k π e F Nep e Z π X π −1 B r+Cαβ αpNe (π ∂ m jik ) × S d = ∅, that is, π e F Nep−k e Z π X π −1 B r+Cαβ αpNe (π ∂ m jik ) × S d = ∅. For each X i , we collect all Z , e.g., j Z j i , such that j π e F Nep e Z j i ⊆ X i . For each Z j i , k = k Z j i ∈ [0, N e ), and denote Z j i by Z j i k . Then π e F Nep−k e Z j i k π X π −1 B r+Cαβ αpNe (π ∂ m jik ) × S d = ∅. Therefore, j π e F Nep e Z j i = Ne−1 k =0 j π e F Nep e Z j i k , and µ ∆ e ( Z ) ≤ µ ∆ e (F −Nep e π −1 e π e F Nep e Z ) = µ X (π e F Nep e Z ) ≤ i µ X (π e F Nep e j Z j i ) ≤ i k µ X (π e F Nep e j Z j i k ) = i k µ X (g −k π e F Nep e j Z j i k ) = i k µ X (π e F Nep−k e j Z j i k ) i k µ ∆ (π e F Nep−k e j Z j i k ) = i k µ ∆ F K i π e F Nep−k e j Z j i k ,(7.2) where K i := min π N X i ×N 0 π −1 B r+Cαβ αpNe (π ∂ m jik )×S d on X i , which is well-defined and is a constant on π e F Nep−k e j Z j i k since π e F Nep−k e j Z j i k ⊆ X i . Claim: diam(π e F Nep−k e Z j i k ) ≤ 2Cβ Nep−Ne . For any x, y ∈ Z j i k , there is o ∈ Z j i k , such that x ∈ γ u (o) and y ∈ γ s (o). By Therefore diam(π e F Nep−k e Z j i k ) ≤ 2Cβ Nep−Ne , and the claim holds. Claim: F K i π e F Nep−k e j Z j i k ⊆ X i × N 0 π −1 B r+C α β αpNe (π ∂ m jik ) × S d for some C α > 0. We just need to show that F K i π e F Nep−k e Z j i k ⊆ X i ×N 0 π −1 B r+C α β αpNe (π ∂ m jik )× S d for each j . Clearly F K i π e F Nep−k e Z j i k ⊆ X i × N 0 , since π e F Nep−k e Z j i k ⊆ X i . By definition of K i and Assumption 2.11 we have then diam πF K i π e F Nep−k e Z j i k = diam f K i π e F Nep−k e Z j i k ≤ 2 α C 1+α β αNep−αNe , πF K i π e F Nep−k e Z j i k ⊆ B r+C α β αpNe (π ∂ m jik ) × S d , where C α := 2 α C 1+α β −αNe + C α . Hence the claim holds. Now, using the claims above and the relation ∆ i (X i × N 0 ) = ∆, we can continue an estimate of (7.2) as ≤ i k µ ∆ X i × N 0 π −1 B r+C α β αpNe (π ∂ m jik ) × S d ≤ N e i µ ∆ X i × N 0 π −1 B r+C α β αpNe (π ∂ m jik ) × S d µ ∆ π −1 B r+C α β αpNe (π ∂ m jik ) × S d = µ M B r+C α β αpNe (π ∂ m jik ) × S d (r + C α β αpNe ) d , where the last " " holds because dµ M d Leb M ∈ L ∞ , and because ∂Q has uniformly bounded sectional curvature. We choose p = n/4 and estimate now µ M (G n,r ). Lemma 7.6. µ M (G n,r ) ≤ C β n/2 + C (r + C α β αNen/4 ) d for some C > 0. Proof. By (7.1) and Lemma 7.4 we have µ M (G n,r ) = i k<R i µ M (f k X i G n,r ) ≤ i k<R i j µ ∆ e Z ji F −Nen e F −Nep e π −1 e A r+Cαβ αpNe (π ∂ m jik ) ≤ i k<R i j µ ∆ e Z ji F −Nen e Z − µ ∆ e (Z ji )µ ∆ e ( Z ) + µ ∆ e (Z ji )µ ∆ e ( Z ) i k<R i j β n−2p µ ∆ e (Z ji ) + µ ∆ e (Z ji )(r + C α β αpNe ) d , where the last " " is due to (5.6) and Lemma 7.5. By making use of Lemma 7.3, Rdµ X < ∞ and (F Ne e ) * µ ∆ e = µ ∆ e , we can now continue the estimate above as = i k<R i β n−2p µ ∆ e (F −Nep e π −1 e X i ) + µ ∆ e (F −Nep e π −1 e X i )(r + C α β αpNe ) d = i k<R i β n−2p µ ∆ e (π −1 e X i ) + µ ∆ e (π −1 e X i )(r + C α β αpNe ) d = i k<R i β n−2p µ X (X i ) + µ X (X i )(r + C α β αpNe ) d = i R i β n−2p µ X (X i ) + R i µ X (X i )(r + C α β αpNe ) d = Rdµ X β n−2p + Rdµ X (r + C α β αpNe ) d β n/2 + (r + C α β αNen/4 ) d , which concludes a proof of lemma. Now we can estimate the range of Z r (m m m) for a particular r. Proof. Let r = n −(1+δ)/d . Then µ M (G n,n −(1+δ)/d ) ) β n/2 + (n −(1+δ)/d + C α β αNen/4 ) d n −1−δ . By Borel-Cantelli lemma for a.e. m m m ∈ M there exists N m m m > 1, such that for any n > N m m m , (g Ne ) n (π X π −1 m m m) / ∈ 0≤i≤Ne−1 g i π X π −1 B n −(1+δ)/d (π ∂ m m m) × S d , which means that for any n > N e N m m m , g n−Ne (π X π −1 m m m) / ∈ π X π −1 B n/Ne −(1+δ)/d (π ∂ m m m) × S d . Furthermore, for any n ≥ m > N e N m m m , g m−Ne (π X π −1 m m m) / ∈ π X π −1 B n/Ne −(1+δ)/d (π ∂ m m m) × S d . Therefore, Z n/Ne −(1+δ)/d (m m m) = min{m ≥ 1 : g m (π X π −1 m m m) ∈ π X π −1 B n/Ne −(1+δ)/d (π ∂ m m m) × S d } ∈ {1, 2, · · · , N e N m m m − N e } (n − N e , ∞). So the lemma is proved. To rule out the set {1, 2, · · · , N e N m m m − N e }, we need the following lemma. Lemma 7.8 (Aperiodicity). For a.e. m m m ∈ M and for any k ∈ N, g k (π X π −1 m m m) / ∈ π X π −1 ({π ∂ m m m} × S d ). Proof. For any q ∈ ∂Q, let A 0 (q) := π X π −1 ({q} × S d ). If q is a periodic point, then there is x ∈ A 0 (q) such that there is k ≥ 1 satisfying g k (x) ∈ A 0 (q). Therefore g(x) ∈ g(A 0 (q)), g(x) ∈ g −(k−1) (A 0 (q)). If k > 1, then by Assumption 2.11, T g(x) g A 0 (q) ⊆ (Df i )T ({q} × S d ) ⊆ int C u for some i ≥ 1, T g(x) g −(k−1) A 0 (q) ⊆ (Df −j )T ({q} × S d ) ⊆ int C s for some j ≥ 1. If k = 1, then by Assumption 2.11, T g(x) g A 0 (q) ⊆ (Df i )T ({q} × S d ) ⊆ int C u for some i ≥ 1, T g(x) g −(k−1) A 0 (q) ⊆ (Df −j )T ({q} × S d ) ⊆ int C s or T ({q} × S d ) for some j ≥ 0. Thus dim T g(x) g A 0 (q) T g(x) g −(k−1) A 0 (q) < d. On the other hand, since A 0 (q) is a countable union of d-dimensional connected submanifolds, so are as well g A 0 (q) and g −(k−1) A 0 (q) . Their intersection is a countable union of d − 1dimensional connected submanifolds. Thus x ∈ g −1 g A 0 (q) g −(k−1) A 0 (q) belongs to a d − 1-dimensional submanifold contained in A 0 (q). Therefore, for any q ∈ ∂Q Leb A 0 (q) {x ∈ X : g k x ∈ A 0 (q) for some k ≥ 1} = 0, where Leb A 0 (q) means the Lebesgue measure conditioned on the submanifold A 0 (q). The same notations will be used below. By lifting it to ∆ we have Leb π −1 ({q}×S d ) {x ∈ ∆ : g k (π X x) ∈ π X π −1 ({q} × S d ) for some k ≥ 1} = 0. Since π is an isomorphism, then Leb {q}×S d {m m m ∈ M : g k (π X π −1 m m m) ∈ π X π −1 ({q} × S d ) for some k ≥ 1} = 0. By Fubini's theorem, µ M {m m m ∈ M : g k (π X π −1 m m m) ∈ π X π −1 ({π ∂ m m m} × S d ) for some k ≥ 1} Leb{m m m ∈ M : g k (π X π −1 m m m) ∈ π X π −1 ({π ∂ m m m} × S d ) for some k ≥ 1} = ∂Q Leb {q}×S d {m m m ∈ M : g k (π X π −1 m m m) ∈ π X π −1 ({π ∂ m m m} × S d ) for some k ≥ 1}dq = ∂Q Leb {q}×S d {m m m ∈ M : g k (π X π −1 m m m) ∈ π X π −1 ({q} × S d ) for some k ≥ 1}dq = 0, which concludes a proof of this lemma. Now we can proceed to proving the main result of this subsection. Lemma 7.9. For a.e. m m m ∈ M, lim inf r→0 log Z 2r (m m m) − log µ ∆m (π ∆m π −1 S r ) ≥ 1, where S r is the section contained in the quasi-section B r (π ∂ m m m) × S d . Proof. From Lemma 7.8 for a.e. m m m ∈ M we have that for any k = 1, 2, · · · , N e N m m m − N e , and any j < R(g k π X π −1 m m m), πF j (g k π X π −1 m m m) / ∈ {∂m m m} × S d . If it is not the case, then there is j < R(g k π X π −1 m m m), such that F j (g k π X π −1 m m m) ∈ π −1 ({∂m m m} × S d ), which implies that g k π X π −1 m m m ∈ π X π −1 ({∂m m m} × S d ). But this is in contradiction with Lemma 7.8. Choose now a small r m m m > 0, such that for any r ∈ (0, r m m m ), any k = 1, 2, · · · , N e N m m m − N e , and any j < R(g k π X π −1 m m m), πF j (g k π X π −1 m m m) / ∈ B r (∂m m m) × S d . This implies that for any k = 1, 2, · · · , N e N m m m − N e and for any r ∈ (0, r m m m ) g k π X π −1 m m m / ∈ π X π −1 B r (∂m m m) × S d . Furthermore, for any δ > 0, and any k = 1, 2, · · · , N e N m m m −N e , if n > N e r −(1+δ)/d m m m , then g k π X π −1 m m m / ∈ π X π −1 B n/Ne −(1+δ)/d ({∂m m m}) × S d . It follows from Lemma 7.7 that for any n > max{N e N m m m , N e r Note that B r (π ∂ m m m) × S d is a quasi-section, and if r > 0 is sufficiently small, then, because ∂Q has uniformly bounded sectional curvature and dµ M d Leb M ∈ L ∞ , we obtain −(1+δ)/d m m m }, Z n/Ne −(1+δ)/d (m m m) = min{m ≥ 1 : g m (π X π −1 m m m) ∈ π X π −1 B n/Ne −(1+δ)/d (π ∂ m m m) × S d } > n − N e .µ ∆m (π ∆m π −1 S r ) ≈ m µ ∆ (π −1 S r ) ≈ (π * µ ∆ ) B r (π ∂ m m m) × S d ≈ r d . Then lim inf r→0 log Z 2r (m m m) − log µ ∆m (π ∆m π −1 S r ) = lim inf r→0 log Z 2r (m m m) − log µ ∆m (π ∆m π −1 S 2r ) log µ ∆m (π ∆m π −1 S 2r ) log µ ∆m (π ∆m π −1 S r ) = lim inf r→0 log Z r (m m m) − log r d ≥ 1/(1 + δ). By letting now δ → 0 we conclude a proof of this lemma. Short returns on X In this subsection, we will prove short returns on X, i.e., lim r→0 1 µ X (π X π −1 S r ) 1 π X π −1 Sr 1 1≤j≤pNe,m−1 (f R ) −j (π X π −1 Sr) dµ X = 0. To simplify notations, we set Q r (q) = π X π −1 B r (q) × S d . Let us, at first, estimate µ X Q r (q) 1≤j≤pNe,m g −j Q r (q) .Q r (π ∂ m m m) 1≤j≤pNe,m g −j Q r (π ∂ m m m) ⊆ Q r (π ∂ m m m) π X π −1 {y ∈ M : Z 2r (y) ≤ pN e,m }. Proof. Let x ∈ Q r (π ∂ m m m) 1≤j≤pNe,m g −j Q r (π ∂ m m m). Then there is k 1 < R(x), 1 ≤ j ≤ pN e , k 2 < R(g j x), such that f k 1 (x) ∈ B r (π ∂ m m m) × S d and f k 2 g j (x) ∈ B r (π ∂ m m m) × S d . Therefore f k 2 g j (x) ∈ B 2r π ∂ f k 1 (x) × S d . It implies that π −1 f k 2 g j (x) ∈ π −1 B 2r π ∂ f k 1 (x) × S d , π X π −1 f k 2 g j (x) ∈ π X π −1 B 2r π ∂ f k 1 (x) × S d . Note that π X π −1 f k 2 g j (x) = π X F k 2 π −1 g j (x) = g j (x), π X π −1 f k 1 (x) = π X F k 1 π −1 (x) = x. Thus g j π X π −1 f k 1 (x) ∈ π X π −1 B 2r π ∂ f k 1 (x) × S d , which means that f k 1 (x) ∈ {y ∈ M : Z 2r ≤ pN e,m }, x ∈ π X π −1 {y ∈ M : Z 2r (y) ≤ pN e,m }, and this lemma holds. Lemma 7.11. For Leb-a.e. q ∈ ∂Q, lim r→0 1 µ X Q r (q) Qr(q) 1 pNe,m j=1 g −j Qr(q) dµ X = 0. Proof. Recall that S r is a section, B r (π ∂ m m m) is a quasi-section, and there is C > 0, such that p = n 1− = C ±1 µ ∆m (π ∆m π −1 S r ) −1+ , where ∈ (0, 1) is small enough. Using this and Lemma 7.10 we have for some constant C ,m ∈ R and any r 0 > r µ X Q r (π ∂ m m m) 1≤j≤pNe,m g −j Q r (π ∂ m m m) ≤ µ X Q r (π ∂ m m m) π X π −1 {y ∈ M : Z 2r (y) ≤ pN e,m } = µ X π X π −1 {B r (π ∂ m m m) × S d } π X π −1 {y ∈ M : Z 2r (y) ≤ pN e,m } µ ∆ π X π −1 {B r (π ∂ m m m) × S d } π X π −1 {y ∈ M : Z 2r (y) ≤ pN e,m } ≤ µ ∆ π −1 {B r (π ∂ m m m) × S d } π −1 {y ∈ M : Z 2r (y) ≤ pN e,m } = µ M {B r (π ∂ m m m) × S d } {y ∈ M : Z 2r (y) ≤ pN e,m } = µ M {B r (π ∂ m m m) × S d } {y ∈ M : Z 2r (y) ≤ C µ ∆m (π ∆m π −1 S r ) −1+ N e,m } = µ M {B r (π ∂ m m m) × S d } {y ∈ M : −C ,m + log Z 2r (y) − log µ ∆m (π ∆m π −1 S r ) ≤ 1 − } ≤ µ M {B r (π ∂ m m m) × S d } {y ∈ M : inf r <r 0 −C ,m + log Z 2r (y) − log µ ∆m (π ∆m π −1 S r ) ≤ 1 − } (7.3) By Lemma 7.9, we get lim r 0 →0 µ M y ∈ M : inf r <r 0 −C ,m + log Z 2r (y) − log µ ∆m (π ∆m π −1 S r ) ≤ 1 − = 0. (7.4) Let h := dµ M d Leb be the density of µ M , and Leb S d and Leb ∂Q are the Lebesgue measures on S d and ∂Q, respectively. For any q ∈ ∂Q, define now a measure µ q µ q (A) := A h(q, ·)d Leb S d (7.5) for any measurable A ⊆ S d . Let also U q,r 0 := v ∈ S d : inf r <r 0 −C ,m + log Z 2r (q, v) − log µ ∆m (π ∆m π −1 S r ) ≤ 1 − . Then (7.4) is equivalent to lim r 0 →0 ∂Q µ q (U q,r 0 )d Leb ∂Q = 0. For any δ > 0, let T r 0 ,δ := {q ∈ ∂Q : µ q (U q,r 0 ) > δ}. Then lim r 0 →0 Leb ∂Q (T r 0 ,δ ) = 0. Now using the Lebesgue differentiation theorem (which holds also on a Riemannian manifold ∂Q), for any r 0 , δ > 0, there is a full measure set Q r 0 ,δ in ∂Q, such that for a.e. q ∈ Q r 0 ,δ , lim r→0 1 Leb ∂Q B r (q) Br(q) 1 T r 0 ,δ d Leb ∂Q = 1 T r 0 ,δ (q). Hence, if q / ∈ T r 0 ,δ , then µ q (U q,r 0 ) < δ, Br(q) 1 T r 0 ,δ d Leb ∂Q = o Leb ∂Q B r (q) . Let r 0 = 1/n, δ = 1/k. Then Leb ∂Q ( n T 1/n,1/k ) = 0, Leb ∂Q ( k n T 1/n,1/k ) = 0. Choose π ∂Q m m m ∈ n,k Q 1/n,1/k ( k n T c 1/n,1/k ), which has a full measure. Then for any k 1 there is n k,π ∂Q m m m , such that for any n ≥ n k,π ∂Q m m m , π ∂Q m m m / ∈ T 1/n,1/k (here we used that T 1/t,1/k ⊇ T 1/(1+t),1/k for any t ≥ 1). We will use now these k, n for the estimate (7.3). = µ M {(q, v) ∈ M : q ∈ B r (π ∂ m m m), v ∈ U q,1/n } = Br(π ∂ m m m) µ q (U q,1/n )d Leb ∂Q = Br(π ∂ m m m) T 1/n,1/k µ q (U q,1/n )d Leb ∂Q + Br(π ∂ m m m) T c 1/n,1/k µ q (U q,1/n )d Leb ∂Q Br(π ∂ m m m) 1 T 1/n,1/k d Leb ∂Q + Br(π ∂ m m m) T c 1/n,1/k k −1 d Leb ∂Q = o Leb ∂Q B r (π ∂ m m m) + O Leb ∂Q B r (π ∂ m m m) k −1 , which means that lim r→0 µ X Q r (π ∂ m m m) 1≤j≤pNe,m g −j Q r (π ∂ m m m) Leb ∂Q B r (π ∂ m m m) = O(1/k) does not depend on n anymore. Let k → ∞. Then for any q ∈ n,k Q 1/n,1/k ( k n T c 1/n,1/k ), lim r→0 µ X Q r (q) 1≤j≤pNe,m g −j Q r (q) Leb ∂Q B r (q) = 0. Finally, if r > 0 is small enough, then µ X Q r (q) ≈ µ M (B r (q) × S d ) ≈ r d ≈ Leb ∂Q B r (q) because B r (q) × S d is a quasi-section, which concludes a proof of lemma. Now we can prove the main result of this subsection. Lemma 7.12. For Leb-a.e. q ∈ ∂Q lim r→0 1 µ X (π X π −1 S r ) 1 π X π −1 Sr 1 1≤j≤pNe,m−1 (f R ) −j (π X π −1 Sr) dµ X = 0. This implies that (6.3) converges to zero. Proof. Since B q × S d is a quasi-section, we can conclude a proof of lemma by noting that µ X (π X π −1 S r ) ≈ µ X Q r (q) and 1 π X π −1 Sr 1 1≤j≤pNe,m−1 (f R ) −j (π X π −1 Sr) dµ X ≤ Qr(q) 1 pNe,m j=1 g −j Qr(q) dµ X , if r > 0 is small enough. Conclusion of a proof of Theorem 2.13 Recall that (6.3) is already proved. Then it follows from lemma 6.3, that the last statement, which is required to prove, is an estimation of (6.2). Consider E 1 X X X 0 =a a a h(X X X p , · · · , X X X n−l ) − E1 X X X 0 =a a a Eh(X X X p , · · · , X X X n−l ) . Since there are finitely many a a a ≥ 1, we just need to estimate n · sup h∈[0,1] E 1 X X X 0 ≥1 h(X X X p , · · · , X X X n−l ) − E1 X X X 0 ≥1 Eh(X X X p , · · · , X X X n−l ) To simplify notations we set throughout this section T = F Rm , U = F Ne,m e,m , H = π ∆m π −1 S r . Let an integer k > m + 1, for any i = 0, 1, · · · N e,m − 1, will be determined later B i = U −k π −1 e,m T −i H, B B B = (1 B 0 , · · · , 1 B Ne,m−1 ). Lift now the dynamical system (∆ m , (F Rm ) Ne,m , µ ∆m ) to the mixing hyperbolic Young tower (∆ e,m , F Ne,m e,m , µ ∆ e,m ), as it is shown in the next lemma. E 1 X X X 0 ≥1 h(X X X p , · · · , X X X n−l ) − E1 X X X 0 ≥1 Eh(X X X p , · · · , X X X n−l ) = n · sup h∈[0,1] 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m − 1 B B B≥1 dµ ∆ e,m h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m . Proof. Note that X X X 0 = (1 H , 1 H • T · · · , 1 H • T Ne,m−1 ), X X X i = X X X 0 • T Ne,mi , B B B = X X X 0 • π e,m • U k . We have n · sup h∈[0,1] E 1 X X X 0 ≥1 h(X X X p , · · · , X X X n−l ) − E1 X X X 0 ≥1 Eh(X X X p , · · · , X X X n−l ) = n · sup h∈[0,1] E 1 X X X 0 ≥1 h(X X X 0 • T Ne,mp , · · · , X X X 0 • T Ne,m(n−l) ) − E1 X X X 0 ≥1 Eh(X X X 0 • T Ne,mp , · · · , X X X 0 • T Ne,m(n−l) ) = n · sup h∈[0,1] 1 X X X 0 •πe,m≥1 h(X X X 0 • π e,m • U p , · · · , X X X 0 • π e,m • U n−l ) dµ ∆ e,m − 1 X X X 0 •πe,m≥1 dµ ∆ e,m h(X X X 0 • π e,m • U p , · · · , X X X 0 • π e,m • U n−l )dµ ∆ e,m = n · sup h∈[0,1] 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m − 1 B B B≥1 dµ ∆ e,m h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m , where the last equality holds because U k * µ ∆ e,m = µ ∆ e,m . Now we will cover B i by elements Q of Q m 2k . Define B i := Q B i =∅ Q, ∂B i := Q B i \B i =∅ Q, A := {B B B ≥ 1}, A := Q A =∅ Q, ∂A := Q A\A =∅ Q, B B B := (1 B 0 , · · · , 1 B Ne,m−1 ), ∂B ∂B ∂B := (1 ∂B 0 , · · · , 1 ∂B Ne,m−1 ). Then B i = B i ∂B i , A = {B B B ≥ 1} = i B i , A = A ∂A, A = {B B B ≥ 1} = i B i , A \ A ⊆ i ∂B i = {∂B ∂B ∂B ≥ 1}. Lemma 8.2. 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m − µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m m [1 + nµ ∆ e,m (B B B ≥ 1)]µ ∆ e,m (∂B ∂B ∂B ≥ 1) + β p−2k m µ ∆ e,m (B B B ≥ 1), where a constant in " m " does not depend on h. Proof. 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m − µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m ≤ 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m + 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m − µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m + µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m ≤ 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m + 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m − µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m + (1 B B B≥1 − 1 B B B≥1 )h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m + µ ∆ e,m (B B B ≥ 1) − µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m + µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m ≤ 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l ) dµ ∆ e,m + 2µ ∆ e,m (∂B ∂B ∂B ≥ 1) + 1 B B B≥1 h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m − µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m + µ ∆ e,m (B B B ≥ 1) h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l )dµ ∆ e,m . Claim: h(B B B • U p , · · · , B B B • U n−l ) − h(B B B • U p , · · · , B B B • U n−l ) ≤ 21 p≤i≤n−l ∂B ∂B ∂B•U i ≥1 . For any x / ∈ p≤i≤n−l {∂B ∂B ∂B • U i ≥ 1} we have ∂B B B • U i (x) = 0 for any i ∈ [p, n − l]. Then B B B • U i (x) = B B B • U i (x)+ µ ∆ e,m (B B B ≥ 1)µ ∆ e,m ( p≤i≤n−l ∂B ∂B ∂B • U i ≥ 1) m 1 B B B≥1 1 p≤i≤n−l ∂B ∂B ∂B•U i ≥1 dµ ∆ e,m + µ ∆ e,m (∂B ∂B ∂B ≥ 1) + β p−2k m µ ∆ e,m (B B B ≥ 1) + µ ∆ e,m (B B B ≥ 1)µ ∆ e,m ( p≤i≤n−l ∂B ∂B ∂B • U i ≥ 1) m 1 B B B≥1 1 p≤i≤n−l ∂B ∂B ∂B•U i ≥1 dµ ∆ e,m − 1 B B B≥1 dµ ∆ e,m 1 p≤i≤n−l ∂B ∂B ∂B•U i ≥1 dµ ∆ e,m + µ ∆ e,m (B B B ≥ 1)µ ∆ e,m ( p≤i≤n−l ∂B ∂B ∂B • U i ≥ 1) + µ ∆ e,m (∂B ∂B ∂B ≥ 1) + β p−2k m µ ∆ e,m (B B B ≥ 1) + µ ∆ e,m (B B B ≥ 1)µ ∆ e,m ( p≤i≤n−l ∂B ∂B ∂B • U i ≥ 1) m µ ∆ e,m (B B B ≥ 1)µ ∆ e,m ( p≤i≤n−l ∂B ∂B ∂B • U i ≥ 1) + µ ∆ e,m (∂B ∂B ∂B ≥ 1) + β p−2k m µ ∆ e,m (B B B ≥ 1) + µ ∆ e,m (B B B ≥ 1)µ ∆ e,m ( p≤i≤n−l ∂B ∂B ∂B • U i ≥ 1) m [1 + nµ ∆ e,m (B B B ≥ 1)]µ ∆ e,m (∂B ∂B ∂B ≥ 1) + β p−2k m µ ∆ e,m (B B B ≥ 1), which concludes a proof of this lemma. To proceed with further estimates we will make some preparations. We need now to consider only Q ∈ Q m 2k , which is contained in ∂B i for any 0 ≤ i ≤ N e,m − 1. By Lemma 5.7, and because i + kN e,m < 2kN e,m , we have that T i π e,m U k Q = π e,m F i e,m U k Q = π e,m F i+kNe,m e,m Q does not contain any singularities of S. Therefore π X T i π e,m U k Q belongs to some X t with R| Xt = R t . This means, according to the definitions of X t and R t in section 7, that T i π e,m U k Q ⊆ X t × N 0 . Lemma 8.3. For any t ≥ 1, and any Q ⊆ ∂B i , which satisfies π X T i π e,m U k Q ⊆ X t , there is u t ∈ [0, R t ), which does not depend on Q, such that F ut T i π e,m U k Q π −1 S r = ∅. This lemma claims that DF ut is an identity on T i π e,m U k Q, i.e., F ut pushes T i π e,m U k Q upward until it hits π −1 S r (X t × N 0 ). Proof. For any Q ⊆ ∂B i we know that T i π e,m U k Q π ∆m π −1 S r = ∅. Besides, there is u t ∈ [0, R t ), such that F ut T i π e,m U k Q π −1 S r = ∅. We will show now that u t does not depend on Q. Since π X : π −1 S r → X and π X : π ∆m π −1 S r → ∆ m are injective, then π −1 S r (X t × N 0 ) ⊆ X t × {a} and π ∆m π −1 S r (X t × N 0 ) ⊆ X t × {b} for some a, b ∈ [0, R t ). Hence u t = a − b does not depend on Q, and thus this lemma holds. For any 0 ≤ i ≤ N e,m − 1, we collect all Q t,j ⊆ ∂B i , such that Q t,j ∈ Q m 2k , T i π e,m U k ( j Q t,j ) ⊆ X t × N 0 . Lemma 8.4. There exists a constant C α,m > 0, which does not depend on Q t,j , i, t, k, such that diam(π • F ut • T i • π e,m • U k Q t,j ) ≤ C α,m β Ne,mkα 2m . Proof. Since π X T i π e,m U k Q t,j does not intersect the singularity set S, then π X F ut T i π e,m U k Q t,j does not intersect S either. For any γ s ⊆ Q t,j , F ut π e,m •F i+Ne,mk e,m (γ s ) returns to X at least (i+N e,m k)/(1+m) times. Therefore, it is contracted by g at least N e,m k/(2m) times. Assume that q ∈ [ Ne,mk 2m , N e,m k + i] is the last hitting time to X, i.e., π e,m • F q e,m (γ s ) is a smooth stable disk in X, and π • π e,m • F j e,m (γ s ) ⊆ X c for any q < j ≤ N e,m k + i and F j π e,m • F i+Ne,mk e,m (γ s ) ⊆ X c for any 0 < j ≤ u t . Therefore, by Assumption 2.11 and by Definition 2.4, R is constant on π e,m • F q e,m Q t,j , and there is C α > 0, such that diam π • F ut • π e,m • F i+Ne,mk e,m (γ s ) ≤ C diam π e,m • F q e,m (γ s ) α ≤ C 1+α β αq diam γ s ≤ C α β αNe,mk 2m . On the other hand, for any γ u ⊆ Q t,j consider a smooth unstable disk γ u 0 = π•π e,m •F C α,m > 0, such that diam π • F ut • π e,m • F Ne,mk+i e,m (γ u ) ≤ C diam π • π e,m • F Ne,mk+i−j e,m (γ u ) α ≤ C 1+α β αq diam π • π e,m • F 2Ne,mk−j e,m (γ u ) α ≤ C 1+α β α Ne,mk+j −i m+1 diam π • π e,m • F 2Ne,mk−j e,m (γ u ) α ≤ C 1+α β αNe,m(k−1) m+1 diam π • π e,m • F 2Ne,mk−j e,m (γ u ) α ≤ C α,m β Ne,mkα 2m . Similar to the argument in the proof of Lemma 5.8, we have that there is a constant C α,m > 0, such that diam π • F ut • T i • π e,m • U k Q t,j = diam π • F ut • π e,m • F Ne,mk+i e,m Q t,j ≤ C α,m β Ne,mkα 2m . Hence this lemma holds. Lemma 8.5. For any 0 ≤ i ≤ N e,m − 1 we have µ ∆ e,m (∂B i ) ≤ C α,m β Ne,mkαd 2m , where a constant C α,m does not depend on i, k. Proof. Consider first any Q t,j ∈ Q m 2k , which is contained in ∂B i . Claim: F ut T i π e,m U k Q t,j π −1 S r = ∅ =⇒ πF ut T i π e,m U k Q t,j ⊆ B r+Cα,mβ Ne,mkα 2m (q) B r−Cα,mβ Ne,mkα 2m (q) × S d . By definition of ∂B i there are x, y ∈ Q t,j , such that T i π e,m U k x ∈ H = π ∆m π −1 S r and T i π e,m U k y / ∈ H. Then F ut T i π e,m U k x ∈ π −1 S r , and F ut T i π e,m U k y / ∈ π −1 S r . By Lemma 8.4 we conclude that πF ut T i π e,m U k Q t,j belongs to C α,m β Ne,mkα 2m −neighborhood of the boundary of B r (q) × S d . So, the lemma holds. From this claim and from the relations U * µ ∆ e,m = µ ∆ e,m , T * µ ∆m = µ ∆m we have µ ∆ e,m (∂B i ) ≤ µ ∆ e,m (U −k π −1 e,m π e,m U k Q) = µ ∆m (π e,m U k Q) ≤ µ ∆m (T −i T i π e,m U k Q) m µ ∆ (T i π e,m U k t j Q t,j ) m t µ ∆ (T i π e,m U k j Q t,j ) m t µ ∆ (Xt×N 0 ) (F ut T i π e,m U k j Q t,j ) m t µ ∆ (Xt×N 0 ) π −1 B r+Cα,mβ Ne,mkα 2m (q) B r−Cα,mβ Ne,mkα 2m (q) × S d m µ ∆ π −1 B r+Cα,mβ Ne,mkα 2m (q) B r−Cα,mβ Ne,mkα 2m (q) × S d m µ M B r+Cα,mβ Ne,mkα 2m (q) B r−Cα,mβ Ne,mkα 2m (q) × S d ≤ C α,m β Ne,mkαd 2m for some constant C α,m > 0, which is due to dµ M d Leb M ∈ L ∞ . Finally, we can prove now the main result of this section. E 1 X X X 0 ≥1 h(X X X p , · · · , X X X n−l ) − E1 X X X 0 ≥1 Eh(X X X p , · · · , X X X n−l ) → 0. Proof. At first we estimate µ ∆ e,m (B B B ≥ 1) and µ ∆ e,m (∂B ∂B ∂B ≥ 1). By making use of Lemma 8.5, U * µ ∆ e,m = µ ∆ e,m and of the section S r ⊆ B r (q) × S d we get x n = (q n , φ n ) = f n (x 0 ) for all n ∈ Z, where x 0 ∈ M. We list now several basic formulas (see e.g. [6]) for two-dimensional billiards µ ∆ e,m (B B B ≥ 1) ≤ 0≤i≤Ne,m−1 µ ∆ e,m (B i ) ≤ 0≤i≤Ne,m−1 µ ∆ e,m (B i ) + 0≤i≤Ne,m−1 µ ∆ e,m (∂B i ) = 0≤i≤Ne,m−1 µ ∆ e,m (U −k π −1 e,m T −i H) + 0≤i≤Ne,m−1 µ ∆ e,v := dφ/dq = B − cos φ + K = B + cos φ − K, 1/B − (x n+1 ) = τ n + 1/B + (x n ), Df (x n ) = −1 cos φ n+1 τ n K(q n ) + cos φ n τ n τ n K(q n )K(q n+1 ) + K(q n ) cos φ n+1 + K(q n+1 ) cos φ n τ n K(q n+1 ) + cos φ n+1 , (9.1) ||dx n+1 || p ||dx n || p = ||Df (dx n )|| p ||dx n || p = |1 + τ n B + (x n )|, (9.2) ||dx n || = ||dx n || p cos φ n 1 + ( dφ n dq n ) 2 (9.3) where τ n is the length of the free path from x n to x n+1 , ||dx n || p := cos φ n |dq n |, ||dx n || : = (dφ n ) 2 + (dq n ) 2 . In what follows we always assume that all boundary components of a billiard table are at least C 3 . Then there are cone fields C u , C s ⊆ T (M), which satisfy Assumption 2.11. Proof. We construct C u , C s fiber-wisely as follows. For any x = (q, φ) C u x := {(dq, dφ) ∈ T x M : K ≤ dφ/dq ≤ ∞}, int C u x := {(dq, dφ) ∈ T x M \ {0} : K < dφ/dq < ∞} for dispersing and flat boundary components, and C u x := {(dq, dφ) ∈ T x M : K ≤ dφ/dq ≤ 0}, int C u x := {(dq, dφ) ∈ T x M \ {0} : K < dφ/dq < 0} for focusing arcs. C s x := {(dq, dφ) ∈ T x M : 0 ≤ dφ/dq ≤ −K} int C s x := {(dq, dφ) ∈ T x M \ {0} : 0 < dφ/dq < −K} for focusing arcs. Define C u := x∈M C u x , int C u := x∈M int C u x , C s := x∈M C s x and int C s := x∈M int C s x in T M. It was shown in Theorem 8.9 of [6] that Df (C u x ) ⊆ C u f (x) , Df −1 (C s x ) ⊆ C s f −1 (x) . Clearly int C u int C s = ∅, dim(int C u int C s ) = 0 < d = 1. Now let x 0 = (q 0 , φ 0 ) ∈ {q} × S 1 . Then the following claims hold. Claim: (Df )T ({q} × S 1 ) ⊆ C u , (Df )T ({q} × S 1 ) ⊆ int C u if φ 1 = ±π/2. Suppose that dx 0 ∈ T ({q} × S 1 ). Then B + (x 0 ) cos φ 0 − K(q 0 ) = dφ 0 /dq 0 = ∞, i.e., B + (x 0 ) = ∞. It implies that dφ 1 dq 1 = K(q 1 ) + cos φ 1 τ 0 + 1/B + (x 0 ) = K(q 1 ) + cos φ 1 τ 0 ∈ [K(q 1 ), ∞). If q 1 belongs to a focusing arc, then by the SFC-condition τ 0 ≥ −2K(q 1 ) −1 cos φ 1 , which implies that dφ 1 dq 1 = K(q 1 ) + cos φ 1 τ 0 ≤ K(q 1 ) + cos φ 1 −2K(q 1 ) −1 cos φ 1 = K(q 1 )/2 < 0. Then dx 1 ∈ C u x 1 for any q ∈ ∂Q. Particularly, if φ 1 = ±π/2, then dφ 1 dq 1 > K(q 1 ). Thus dx 1 ∈ int C u x 1 , and the claim holds. Claim: (Df ) −1 T ({q} × S 1 ) ⊆ C s , (Df ) −1 T ({q} × S 1 ) ⊆ int C s if φ −1 = ±π/2. Suppose that dx 0 ∈ T ({q} × S 1 ). Then B − (x 0 ) cos φ 0 + K(q 0 ) = dφ 0 /dq 0 = ∞, i.e., B − (x 0 ) = ∞. Therefore 0 = 1/B − (x 0 ) = τ −1 + 1/B + (x −1 ), dφ −1 dq −1 = −K(q −1 ) + B + (x −1 ) cos φ −1 = −K(q −1 ) − cos φ −1 τ −1 ∈ (−∞, −K(q −1 )]. If q −1 belongs to a focusing arc, then by the SFC-condition we have τ −1 ≥ −2K(q −1 ) −1 cos φ −1 , which implies dφ −1 dq −1 = −K(q −1 ) − cos φ −1 τ −1 ≥ −K(q −1 ) − cos φ −1 −2K(q −1 ) −1 cos φ −1 = −K(q −1 )/2 > 0. Hence dx −1 ∈ C u x −1 for any q ∈ ∂Q. In particular, if φ −1 = ±π/2, then dφ −1 dq −1 < −K(q −1 ). Thus dx −1 ∈ int C u x −1 , which proves the claim. Claim: For the set Φ := ( i∈Z f −i {φ = ±π/2}) c ⊆ M we have µ M (Φ) = 1. This claim follows from the facts that µ M is f -invariant and µ{φ = ±π/2} = 0. Claim: For any x 0 ∈ Φ we have (Df ) int C u x 0 ⊆ int C u f (x 0 ) and (Df ) −1 int C s x 0 ⊆ int C s f −1 (x 0 ) . In view of the involution property of a billiard map, we just need to show that (Df ) int C u x 0 ⊆ int C u f (x 0 ) . Suppose that dx 0 ∈ int C u x 0 . Then dφ 0 dq 0 = −K(q 0 ) + B + (x 0 ) cos φ 0 ∈ (K(q 0 ), ∞) if K(q 0 ) ≥ 0, dφ 0 dq 0 = −K(q 0 ) + B + (x 0 ) cos φ 0 ∈ (K(q 0 ), 0) if K(q 0 ) < 0. To prove the relation dx 1 ∈ int C u x 1 , we will show that dφ 1 dq 1 = K(q 1 ) + cos φ 1 τ 0 + 1/B + (x 0 ) ∈ (K(q 1 ), ∞) if K(q 1 ) ≥ 0, dφ 1 dq 1 = K(q 1 ) + cos φ 1 τ 0 + 1/B + (x 0 ) ∈ (K(q 1 ), 0) if K(q 1 ) < 0, for case by case, depending on the positions of q 0 , q 1 , where x 0 ∈ Φ. If K(q 0 ) ≥ 0 and K(q 1 ) ≥ 0, then B + (x 0 ) > 0, dφ 1 /dq 1 ∈ (0, ∞). If now K(q 0 ) ≥ 0 and K(q 1 ) < 0, then B + (x 0 ) > 0, dφ 1 /dq 1 > K(q 1 ), and by SFC-condition we get dφ 1 dq 1 < K(q 1 ) + cos φ 1 τ 0 ≤ K(q 1 ) + cos φ 1 −2K(q 1 ) −1 cos φ 1 ≤ K(q 1 )/2 < 0. If K(q 0 ) < 0 and K(q 1 ) ≥ 0, then B + (x 0 ) ∈ 2K(q 0 ) cos φ 0 , K(q 0 ) cos φ 0 ⊆ (−∞, 0), and by the SFC-condition τ 0 + 1/B + (x 0 ) > τ 0 + K(q 0 ) −1 cos φ 0 ≥ −2K(q 0 ) −1 cos φ 0 + K(q 0 ) −1 cos φ 0 > 0 =⇒ dφ 1 dq 1 = K(q 1 ) + cos φ 1 τ 0 + 1/B + (x 0 ) ∈ (K(q 1 ), ∞). If K(q 0 ) < 0 and K(q 1 ) < 0, then B + (x 0 ) ∈ 2K(q 0 ) cos φ 0 , K(q 0 ) cos φ 0 , τ 0 + 1/B + (x 0 ) > 0 and dφ 1 dq 1 = K(q 1 ) + cos φ 1 τ 0 + 1/B + (x 0 ) > K(q 1 ). According to the SFC-condition (i.e., τ 0 ≥ −K(q 0 ) −1 cos φ 0 − K(q 1 ) −1 cos φ 1 ), dφ 1 dq 1 < K(q 1 ) + cos φ 1 τ 0 + K(q 0 ) −1 cos φ 0 ≤ K(q 1 ) + cos φ 1 −K(q 1 ) −1 cos φ 1 = 0. Therefore dx 1 ∈ int C u x 1 , and the claim holds. Combining all the claims above, we obtain for any x 0 = (q, φ 0 ) ∈ Φ, n ≥ 1, (Df ) n T ({q} × S 1 ) ⊆ int C u , (Df ) −n T ({q} × S 1 ) ⊆ int C s . Claim: For a.e. q ∈ ∂Q, we have (Df ) n T ({q} × S 1 ) ⊆ int C u , (Df ) −n T ({q} × S 1 ) ⊆ int C s . If it is not the case, then there exists a subset O ⊆ ∂Q with Leb ∂Q O > 0, so that for any q ∈ O, (Df ) n T ({q} × S 1 ) int C u or (Df ) −n T ({q} × S 1 ) int C s . Then µ M (O×S 1 ) > 0, (O×S 1 ) Φ = ∅, and for any x 0 = (q, φ 0 ) ∈ (O×S 1 ) Φ, (Df ) n T ({q} × S 1 ) int C u or (Df ) −n T ({q} × S 1 ) int C s . Thus we came to a contradiction. Hence the claim holds, which concludes a proof of this lemma. All billiard systems, which will be considered below, satisfy the conditions of Lemma 9.1. Therefore the condition on existence of the cone fields in Assumption 2.11 holds automatically. Sinai and diamond billiards Pictures of billiard tables of Sinai billiards and of diamond billiards are given in Figures 1 and 2. Choose the first return time R = 1. The Assumption 2.11 in this case holds automatically. Corollary 9.2. Theorem 2.13 holds for Sinai billiards with bounded or unbounded horizon (see e.g. [6]), and for diamond billiards (see e.g. [20]). Remark 9.3. In fact, if a billiard map of a two-dimensional billiard has a CMZ structure, i.e., R = 1, and if the boundary of a billiard table satisfies the conditions of Lemma 9.1, then Theorem 2.13 holds for such billiard. Moreover, such billiards have exponentially mixing rates (or exponential decay of correlations), i.e., of order O(ρ n ) for some ρ ∈ (0, 1). In the following subsections, we consider two-dimensional slowly mixing billiards, which were studied in [7,8]. The rates of mixing (decay of correlations) of these billiards are either of order O(n −1 ), or O(n −2 )). Squashes or Stadium-type billiards A billiard table Q of a squash billiard is a convex domain bounded by two circular arcs and two straight (flat) segments tangent to the arcs at their common endpoints. A squash billiard is called a stadium if flat sides are parallel, see Figure 3. (Initially called) squash billiards were later called sometimes "skewed" stadia, drive-belt billiards, etc, Figure 4. Note that squashes contain a boundary arc, which is longer than a half circle. We will verify now the Assumption 2.11 for this class of billiards. Let for a ("straight") stadium X ⊆ M be the region where the first collisions (in a series of consecutive collisions with one and the same circular arc) with circular arcs occur. Denote by R the first return time to X for the billiard map f . Let l be the length of each straight segment. Without loss of generality, we may assume that the radius of circular arcs equals 1. that f n 1 x = x 1 , f n 2 x = x 2 , and d(q, π ∂Q x 1 ) < r, d(q, π ∂Q x 2 ) < r, where x is a point of the first collision (in a series) with one of the circular arcs. Suppose that q belongs to a circular arc, and r is sufficiently small. Then π ∂ x 1 and π ∂ x 2 belong to a r-neighborhood of q. So the orbits of x 1 and of x 2 are sliding along the same circular arc, and f n 2 −n 1 x 1 = x 2 . Hence the angle |φ| for x 1 (and x 2 ) is greater or equal to π/2 − 2r/(2n 2 − 2n 1 ) ≥ π/2 − r. Therefore, a "non-injective" configuration is {(q , φ) ∈ M : d(q, q ) < r, |φ| > π/2 − r}. Clearly, it has a measure of order O(r 3 ). Suppose now that q belongs to the flat part of the boundary, and that r is sufficiently small. Since the collisions at x 1 , x 2 occur after the first collision at x on a circular arc and there were no reflection off another circular arc yet, then x 1 , x 2 must be bouncing on the boundary component, which contains q. Since the radius of the circular arc is 1, and at these points the angle of reflection |φ| is the same, then |φ| does not exceed arctan[2r/(2n 2 − 2n 1 )] ≤ arctan(r). Therefore, the set of "non-injective" configurations in this case is {(q , φ) ∈ M : d(q, q ) < r, |φ| < arctan(r)}, which has a measure of order O(r 2 ). Summarizing the arguments above, we have that the set of "non-injective" configurations has a measure of order O(r 2 ). Therefore B r (q) × S 1 is a quasisection. 3. Hölder continuity along small (un)stable manifolds. The reason here for using "small" manifolds is that in the Definition 2.2 (un)stable manifolds are supposed to be maximal, and may not be Hölder continuous. Therefore, besides the singularities S 1 of f R , other points in X have to be added into S (see Remark 2.3). We define S by (f R ) −1 S 1 S 1 f R S 1 , and define the "small" (un)stable manifolds in the same way as that in Definition 2.2. First we consider a stable manifold with the stable cone C s (x) := {dx = (dq, dφ) ∈ T x M :B + (x) ∈ [− 1 cos φ , 0] if K(q) = −1, B + (x) ∈ [−∞, 0] if K(q) = 0}. Suppose that the first collision on a circular arc is at a point x 0 ∈ X, and its stable manifold is γ s (x 0 ). Several cases will be considered. (a) Sliding along a circular arc, i.e., the points x 0 , x 1 , · · · , x k (k ≥ 0) belong to the same circular arc, and x k+1 does not. Then dx k ∈ C s (x k ), i.e., B + (x k ) = −2 cos φ k + 1 τ k−1 + 1 B + (x k−1 ) ∈ [− 1 cos φ k , 0] =⇒ B + (x k−1 ) ∈ [ −1 τ k−1 − cos φ k , −1 τ k−1 − cos φ k 2 ] = [ −1 cos φ 0 , −1 3 cos φ 0 2 ]. Inductively, we have for any i ∈ [0, k], B + (x i ) ∈ −1 cos φ 0 , −2(k − i) [2(k − i) + 1] cos φ 0 , and for any i ∈ [0, k), |1 + τ i B + (x i )| ≤ 1. By (9.2) and (9.3), for any i ∈ [0, k], ||dx i || p ||dx 0 || p = j≤i−1 |1 + τ j B + (x j )| ≤ 1, ||dx i || ||dx 0 || = ||dx i || p ||dx 0 || p cos φ 0 cos φ i 1 + ( dφ i dq i ) 2 1 + ( dφ 0 dq 0 ) 2 1, where the last " " holds thanks to the fact that on a circular arc cos φ 0 = cos φ i , and a slope of the stable manifold is uniformly bounded. Therefore, f i | γ s (x 0 ) is Lipschitz in this case. (b) Suppose that x k+1 , · · · , x k+n , (n > 1) are on the flat sides, and x k+n+1 is on another circular arc. Then B + (x k+n+1 ) = −2 cos φ k+n+1 + 1 τ k+n + 1 B + (x k+n ) ∈ [− 1 cos φ k+n+1 , 0] =⇒ B + (x n+k ) ∈ [ −1 τ n+k − cos φ n+k+1 , −1 τ n+k − cos φ n+k+1 2 ]. To continue computations, note that K(q n+k ) = 0. Then B + (x k+n ) = 0 cos φ k+n + 1 τ k+n−1 + 1 B + (x k+n−1 ) ∈ [ −1 τ n+k − cos φ n+k+1 , −1 τ n+k − cos φ n+k+1 2 ] =⇒ B + (x n+k−1 ) ∈ [ −1 τ n+k + τ n+k−1 − cos φ n+k+1 , −1 τ n+k + τ n+k−1 − cos φ n+k+1 2 ]. Inductively, we have for any 0 ≤ i < n B + (x n+k−i ) ∈ [ −1 j≤i τ n+k−j − cos φ n+k+1 , −1 j≤i τ n+k−j − cos φ n+k+1 2 ], |1 + τ n+k−i B + (x n+k−i )| ≤ 1, where the last inequality is due to the SFC-condition, i.e., τ n+k ≥ 2 cos φ n+k+1 . By (9.2), for any i ∈ [k + 1, k + n], ||dx i || p ||dx 0 || p = j≤i−1 |1 + τ j B + (x j )| ≤ 1. Since n > 1, then |φ i | is uniformly bounded away from π/2. Thus ||dx i || ||dx 0 || = ||dx i || p ||dx 0 || p cos φ 0 cos φ i 1 + ( dφ i dq i ) 2 1 + ( dφ 0 dq 0 ) 2 1, where the last " " holds due to the fact that a slope of a stable manifold is uniformly bounded. Therefore, f i | γ s (x 0 ) in this case is Lipschitz too. (c) Consider now transitions between circular arcs through a flat side, i.e., x k is the last in a series collision on a circular arc, and x k+1 is on a flat side while x k+2 is on another circular arc. We have ||dx k+1 || p ||dx 0 || p = j≤k |1 + τ j B + (x j )| ≤ 1, ||dx k+1 || ||dx 0 || = ||dx k+1 || p ||dx 0 || p cos φ 0 cos φ k+1 1 + ( dφ k+1 dq k+1 ) 2 1 + ( dφ 0 dq 0 ) 2 cos φ 0 cos φ k+1 = cos φ k cos φ k+1 . For each φ k a possible minimum of cos φ k+1 (i.e., the maximum of φ k+1 ) satisfies the relation cos 2 φ k = (1 − l 2 ) cos 2 max φ k+1 + 2l cos max φ k+1 sin max φ k+1 , which holds if q k+1 is at the end point of this flat side (or at the end point of another circular arc). Thus ||dx k+1 || ||dx 0 || l 1 cos φ k k. Since diam γ s (x 0 ) = O(1/k 2 ), we have diam f k+1 γ s (x 0 ) k diam γ s (x 0 ) [diam γ s (x 0 )] 1/2 . The arguments above show that there exists C > 0, such that for any γ s ⊆ i≥−1 (f R ) −i S 1 c , d f j γ s (f j x, f j y) ≤ Cd γ s (x, y) 1/2 for all j < R(x). Next we turn to unstable manifolds, with the unstable cone field C s (x) := {dx = (dq, dφ) ∈ T x M :B + (x) ∈ [− 2 cos φ , − 1 cos φ ] if K(q) = −1, B + (x) ∈ [0, ∞] if K(q) = 0}. Suppose that the first collision in a series on a circular arc is at x 0 ∈ X. The unstable manifold at this point is γ u (x 0 ). Again we will consider several cases. (a) Sliding along a circular arc, i.e., x 0 , x 1 , · · · , x k (k ≥ 0) belong to one and the same circular arc while x k+1 does not. Then dx 0 ∈ C s (x 0 ), i.e., B + (x 1 ) = −2 cos φ 1 + 1 τ 0 + 1 B + (x 0 ) , B + (x 0 ) ∈ [− 2 cos φ 0 , − 1 cos φ 0 ], =⇒ B + (x 1 ) ∈ [− 4 3 cos φ 1 , − 1 cos φ 1 ]. Inductively, we have for any i ∈ [0, k], B + (x i ) ∈ − 2i + 2 (2i + 1) cos φ i , − 1 cos φ i , and for any i ∈ [0, k), |1 + τ i B + (x i )| ≤ 2i + 3 2i + 1 . By (9.2) and (9.3), for any i ∈ [0, k), ||dx i || p ||dx 0 || p = j≤i−1 |1 + τ j B + (x j )| i ≤ k, ||dx i || ||dx 0 || = ||dx i || p ||dx 0 || p cos φ 0 cos φ i 1 + ( dφ i dq i ) 2 1 + ( dφ 0 dq 0 ) 2 k, where the last " " holds because cos φ 0 = cos φ i on a circular arc, and the slope of a stable manifold is uniformly bounded. Our small unstable manifold γ u (x 0 ) is contained in a connected component of the set i≥−1 (f R ) i S 1 c . Thus the length of γ u (x 0 ) is O(k −2 ) . Hence diam f i γ u (x 0 ) k diam γ u (x 0 ) [diam γ u (x 0 )] 1/2 . Therefore, f i | γ s (x 0 ) is Hölder continuous. (b) Bouncing on flat sides. Here the points x k+1 , · · · , x k+n , (n > 1) are on flat sides, and x k+n+1 belongs to another circular arc. B + (x k+1 ) = 0 cos φ k+1 + 1 τ k + 1 B + (x k ) , B + (x k ) ∈ [− 2(k + 2) (2k + 1) cos φ k , − 1 cos φ k ], B + (x k+2 ) = 0 cos φ k+2 + 1 τ k+1 + 1 B + (x k+1 ) = 1 τ k+1 + τ k + 1 B + (x k ) . Inductively, we have for any 0 ≤ i ≤ n B + (x k+i ) = 1 0≤j≤i−1 τ k+j + 1 B + (x k ) , |1 + τ k+i B + (x k+i )| = 0≤j≤i τ k+j + 1 B + (x k ) 0≤j≤i−1 τ k+j + 1 B + (x k ) . Since n > 1, then k may assume only a finite number of values. Then, it is bounded by a constant depending on Q only, while |φ k |, · · · , |φ k+n | are bounded away from π/2 by a positive constant, which also depends only on Q. By (9.2) for any i ∈ [k + 1, k + n], ||dx i || p ||dx 0 || p = j≤i−1 |1 + τ j B + (x j )| 0≤j≤i τ k+j + 1 B + (x k ) τ k + 1 B + (x k ) n, ||dx i || ||dx 0 || = ||dx i || p ||dx 0 || p cos φ 0 cos φ i 1 + ( dφ i dq i ) 2 1 + ( dφ 0 dq 0 ) 2 n, where the last " " holds thanks to the fact that the slope of a stable manifold is uniformly bounded. A small unstable manifold γ u (x 0 ) is contained in a connected component of i≥−1 (f R ) i S 1 c . Thus the length of γ u (x 0 ) is O(n −2 ) . Hence diam f i γ u (x 0 ) n diam γ u (x 0 ) [diam γ u (x 0 )] 1/2 . Therefore f i | γ s (x 0 ) is Hölder continuous. (c) Sliding on a flat side, i.e, x k corresponds to a point of the last collision with a circular arc, x k+1 is on a flat side, and x k+2 is on another circular arc. We have B + (x k ) ∈ [− 2(k + 2) (2k + 1) cos φ k , − 1 cos φ k ], =⇒ ||dx k+1 || p ||dx k || p = |1 + τ k B + (x k )| 1 cos φ k ≈ k, =⇒ ||dx k+2 || p ||dx k+1 || p = |1 + τ k+1 B + (x k+1 )| = |1 + τ k+1 τ k + 1/B + (x k ) | ≥ τ k + τ k+1 − 2k+1 2k+4 cos φ k τ k − 2k+1 2k+4 cos φ k ≥ 1, where the last " ≥ " is due to the SFC-condition. Therefore ||dx k+2 || ||dx k+1 || ≈ ||dx k+2 || p ||dx k+1 || p cos φ k+1 cos φ k+2 cos φ k+1 cos φ k+2 cos φ k+2 , where the argument for the last " " is the same as that for the case of "sliding on the flat sides" for stable manifolds. Since γ u (x 0 ) ⊆ i≥−1 (f R ) i S 1 c , then f k+2 γ u (x 0 ) belongs to a k -sliding cell, diam f k+2 γ u (x 0 ) = O(1/k 2 ) and cos φ k+2 ≈ 1/k , so ||dx k+2 || ||dx k+1 || cos φ k+2 ≈ 1/k . From Lemma 8.45 of [6] we have diam f k+1 γ u (x 0 ) k diam f k+2 γ u (x 0 ) [diam f k+2 γ u (x 0 ) ] 1/2 [diam γ u (x 0 )] 1/8 . The arguments above show that there is C > 0, such that for any γ u ⊆ i≥−1 (f R ) i S 1 c d f j γ u (f j x, f j y) ≤ Cd γ u (x, y) 1/8 for all j < R(x). Now we turn to a squash (or a"skewed" stadium). Since two flat sides are not parallel, we can suppose that the angle between these two flat sides is γ > 0. Following [7], define X to be the same set as that of "straight" stadiums. Verification of Assumption 2.11 is the same as for a "straight" stadium. Therefore we will skip the details and outline only the differences. 1. B r (q) × S 1 is a quasi-section. If q is on the flat sides, then the angle of reflection increases or decreases by γ between two consecutive bounces on flat sides. Then B r (q) × S 1 is a section, provided that r is much smaller than γ. Let now a point q belongs to a circular arc. The analysis in the case for orbits, sliding on these arcs, is the same as the one for a straight stadium. Another case is a bouncing on a circular arc. It is enough to consider this case for the bigger arc only. Let us estimate a measure of the set of "noninjective" configurations. Suppose that (p 1 , φ 1 ) = f n (x) and (p 2 , φ 2 ) = f m (x), where n < m, d(q, p 1 ) < r, d(q, p 2 ) < r, x = (p 0 , θ) ∈ X, n, m < R(x) and θ is sufficiently small. Assume that r is so small that all points q, p 0 , p 1 , p 2 are on the bigger arc. From an elementary geometry we have φ 1 = φ 2 = θ, d(p 0 , p 1 ) = 4nθ, d(p 0 , p 2 ) = 4mθ, and d(p 1 , p 2 ) = 4(m − n)θ. Therefore d(p 1 , p 2 ) = 4(m − n)θ < 2r, which implies that φ 1 = φ 2 = θ < r/2. Thus the set of "non-injective" configurations for bouncing on the circular arc is contained in {(p, φ) ∈ M : p ∈ B r (q), |φ| < r/2}, which has measure of order O(r 2 ). 2. Another difference is in verification of Hölder continuity. Besides the three cases studied for the (straight) stadiums, we also need to consider here a bouncing on the bigger circular arc. The argument there is exactly the same as that for "sliding on a circular arc". We can conclude now this subsection by stating the following Corollary 9.4. Theorem 2.13 holds for squash billiards (stadiums with unequal or equal focusing arcs). More general class of billiards with focusing components In this section we consider billiard tables Q for which each smooth component Γ i ⊆ ∂Q of the boundary is either dispersing, i.e., convex inwards, or focusing, i.e., convex outwards. A curvature of every dispersing component is bounded away from zero and infinity. We assume that every focusing component is an arc of a circle, and that there are no points of ∂Q on that circle or inside it, other than the arc itself (that is, the SFC-condition). We assume also that two dispersing components intersect (if they do) transversally (i.e., there are no cusps) and, besides, every focusing arc is not longer than a half of the corresponding circle, e.g. see the Figure 5. Denote the union of dispersing components by ∂Q + , and the union of focusing components by ∂Q − . Let X ⊆ M be X := (∂Q + × S 1 ) {x ∈ M : π ∂Q x ∈ ∂Q − , π ∂Q x and π ∂Q (f −1 x) belong to a different Γ i }, i.e., only the first entries into circular arcs and any collisions on the dispersing components are included in X. Therefore, the case with R > 1 may occur only in the series of reflections off a circular arc. The verification of Assumption 2.11 is similar (actually it is easier) to the one for stadium billiards. Therefore we just outline below the differences. 1. B r (q) × S 1 is a quasi-section. If a point q belongs to dispersing components, then clearly B r (q)×S 1 is a section. If q belongs to circular arc, then B r (q)×S 1 is a quasi-section. The argument in this case is exactly the same as for stadium billiards. 2. Hölder continuity. Note that R(x) > 1 only occurs if π ∂Q x ∈ ∂Q − . In this case, {x, f (x), f 2 x · · · f R−1 (x)} is a series of consecutive reflections off a circular arc. Thus the argument for Hölder continuity is exactly the same as that for stadium-type billiards. Now we can conclude this subsection by stating the following Corollary 9.5. Theorem 2.13 holds for the class of billiards analyzed in this subsection. Semi-dispersing billiards In this subsection we consider billiard tables of the following type. Let R 0 ⊆ R 2 be a rectangle, and scatterers B 1 , · · · , B r ⊆ int R 0 are open strictly convex sub-domains with smooth, (at least C 3 ), or piece-wise smooth boundaries whose curvature is bounded away from zero, and such that B i B j = ∅ for i = j. The boundary of a billiard table Q = R 0 \ i B i is partially dispersing (convex inwards) and partially neutral (flat), e.g. see Figure 6. The flat part is ∂R 0 . Denote by ∂Q + the union of dispersing components, and the union of four flat sides by ∂R 0 . Let X := {x ∈ M : π ∂Q x ∈ ∂Q + }, where R is the first return time to X. If sup R < ∞, then this billiard system has exponentially decay of correlations, i.e., it is not slowly mixing. So we assume that sup R = ∞. Clearly f R is a billiard map of a Sinai billiard with an infinite horizon. Verification of the Assumption 2.11 for such billiards has a few differences with the one for stadium billiards. Namely must intersect i B i . Therefore max{p, k} must not exceed M for such billiard orbit, in order it not to intersect i B i . It implies that there are finitely many pairs (p, k) with gcd(p, k) = 1, such that tan φ 1 = k/p + O(r). Then φ 1 has a measure of order O(r). So a measure of 'non-injective" configuration (q 1 , φ 1 ) ∈ B r (q) × S 1 is of order O(r 2 ). Hence B r (q) × S 1 is a quasi-section. 2. Hölder continuity. Consider first unstable manifolds in X (which are, actually, homogeneous unstable manifolds, defined, e.g., in section 5.4 of [6], see also Remark 2.3). Suppose that x 0 ∈ X, and that n := R(x 0 ) is sufficiently large. Then the orbit of x 0 hits ∂R 0 many times before getting back to ∂Q + . Clearly for billiards of this type all possible angles φ 1 are bounded by π/2. Consider now the worse case when φ 1 ≈ 0, i.e., |φ 1 | < φ Q for some small φ Q , (which depends only on exact shape of a billiard table under consideration), such that φ 1 = φ 2 = · · · = φ j−1 = φ j+1 = · · · = φ n−1 , and φ j = π/2 − φ 1 ≈ π/2 for some 2 ≤ j < n. Then for any i = j we have cos φ j ≈ 1/n, and cos φ i ≈ 1. Assume now that f n γ u (x 0 ) belongs to a homogeneity strip, for instance to cos φ n ≈ 1/k 2 . Observe also that k n 1/4 (see page 544 of [5]). It is known, see e.g. [6], that B + (x 0 ) ∈ [ 2K(q 0 ) cos φ 0 , 2K(q 0 ) cos φ 0 + O(1)]. Thus we have 1 B + (x j ) = 1 B − (x i ) = 0≤i≤j−1 τ (f i x 0 ) + 1 B + (x 0 ) =⇒ ||dx n || p ||dx j || p = |1 + j≤i<n τ (f i x)B + (x j )| = 0≤i≤n−1 τ (f i x 0 ) + 1 B + (x 0 ) 0≤i≤j−1 τ (f i x 0 ) + 1 B + (x 0 ) ≥ 1 =⇒ ||dx n || ||dx j || cos φ j cos φ n ||dx n || p ||dx j || p cos φ j cos φ n ≈ 1/n 1/k 2 1 k 2 =⇒ diam f j γ u (x 0 ) k 2 diam f n γ u (x 0 ) [diam f n γ u (x 0 ) ] 1/3 , where the last " " is due to O diam f n γ u (x 0 ) = 1/k 3 . We also have that ||dx n || p ||dx 0 || p = |1 + j<n τ (f j x)B + (x 0 )| ≈ n cos φ 0 =⇒ ||dx n || ||dx 0 || ≈ cos φ 0 cos φ n |1 + j<n τ (f j x)B + (x 0 )| ≈ n cos φ n . Let now s = cos φ n , then |ds| ≈ |dφ n | ≈ |dx n |. So we get diam γ u (x 0 ) = ||dx i || p ||dx 0 || p = |1 + j<i τ (f j x)B + (x 0 )| n cos φ 0 =⇒ ||dx i || ||dx 0 || ≈ cos φ 0 cos φ i |1 + j<n τ (f j x)B + (x 0 )| n cos φ i n =⇒ diam f i γ u (x 0 ) n diam γ u (x 0 ) [diam γ u (x 0 )] 1/2 . So far we proved Hölder continuity for the case when φ 1 < φ Q . Actually this argument is analogous to the one for a series of reflections off the flat sides in stadium billiards. If φ 1 > φ Q , then all φ i i ∈ [0, R(x 0 )) are uniformly bounded away from 0 and π/2. Then the argument is the same as for bouncing on the flat sides in stadium billiards. Actually, this case is much easier, and we skip a proof. Therefore we obtain Hölder continuity of unstable manifolds. For stable manifolds the argument is similar and, basically, the same as the one for stadium billiards. Thus we do not repeat it here. So we obtain the following result. Corollary 9.6. Theorem 2.13 holds for the class of semi-dispersing billiards considered in this subsection. Remark 9.7. From the proof of this corollary, it could be seen that the singularities for this class of semi-dispersing billiards have, in a sense, a similar structure as the ones in the stadium-type billiards. This is a reason why these two classes of billiards have the same rate of decay of correlations (see [7,8]). Young towers for (X, f R , µ X ) . . . . . . . . . . . . . . . . 19 5.2 Thicker hyperbolic Young towers for (∆ m , F Rm , µ ∆m ) . . . . . . . . . 24 5.3 Thicker expanding quotient Young tower for (∆ m , F Rm , µ ∆m ) . . . . . 27 5.4 Decay of correlations . . . . . . . . . . . . . . . . . . . . . . . . . . Definition 2. 4 ( 4Chernov-Markarian-Zhang (CMZ) structures). Then, as a direct consequence, we obtain that µ MLeb M and dµ M d Leb M ∈ L ∞ , dµ M d Leb M > 0 Leb M -a.s. on M. 2. If R = 1, then X = M, meaning that (M, f, µ M ) is a CMZ structure. Lemma 3. 1 ( 1See Proposition 16.17 of [13]). Lemma 3 . 3 ( 33From quasi-sections to sections). Definition 4. 2 . 2Define point processes on (∆ m , µ ∆m ) so that for any measurable set A ⊆ [0, which are sub-towers of ∆ e,m and ∆ e,m , respectively. Then the dynamics F e,m → ∆ e,m preserve probability measures (z) ≤ Cβ pNe , diam π e F Nep e γ s (z) ≤ Cβ pNe . By Assumption 2.11, we have diam f k π e F Nep e γ u (z) ≤ C 1+α β αpNe , diam f k π e F Nep e γ s (z) ≤ C 1+α β αpNe . F (o) ≤ Cβ Nep−k , diam π e Nep≤ diam π e F Nep−k e γ u (o) + diam π e F Nep−k e γ s (o) ≤ 2Cβ Nep−Ne . Lemma 7. 7 . 7If δ > 0 is sufficiently small, then for a.e. m m m ∈ M, and for any n > N e N m m m Z n/Ne −(1+δ)/d (m m m) ∈ {1, 2, · · · , N e N m m m − N e } (n − N e , ∞). any sufficiently small r ∈ (0, r m m m ), such that r ∈ ( (1+n)/N e −(1+δ)/d , n/N e −(1+δ)/d ), and for a large n > max{N e N m m m , N e r −(1+δ)/d m m m }, we have Z r ≥ n − N e , and lim inf r→0 log Z r (m m m) log r −d/(1+δ) ≥ lim inf n→∞ log(n − N e ) log(n + 1)/N e Lemma 7 . 10 . 710For any m m m ∈ M, for all i ∈ [p, n − l]. Thus the claim holds. Now, using Lemma 5.9, we can continue the estimates as m 1 B B B≥1 1 p≤i≤n−l ∂B ∂B ∂B•U i ≥1 dµ ∆ e,m + µ ∆ e,m (∂B ∂B ∂B ≥ 1) + β p−2k m µ ∆ e,m (B B B ≥ 1) u ). Since N e,m 2k−(i+N e,m k) = N e,m k−i > N e,m (k−1) > N e,m m, then from π • π e,m • F Ne,mk+i e,m (γ u ) to π • π e,m • F Ne,m2k e,m (γ u ), follows that there are at least N e,m times of return to the base X. This means that π•F ut •π e,m •F u ) ∈ X, π e,m • F Ne,mk+i−j e,m(γ u ) = g −q f −j γ u 0 , where q ∈ [ Ne,mk+j −i m+1 , N e,m k + j − i].By Assumption 2.11 and by Definition 2.4 R is constant on the set π e,m • F Ne,mk+i−j e,m (γ u ), and there is µ ∆ e,m (∂B i ) = N e,m µ ∆m (H) + 0≤i≤Ne,m−1 µ ∆ e,m (∂B i ) Lemma 9 . 1 . 91Suppose that regular components of the boundary ∂Q are either flat, or dispersing, or focusing components, such that each focusing component Γ i is an arc of a circle, but not a full circle, and this circle does not intersect any other component of the boundary ∂Q. In what follows we will call this condition the simplest focusing chaos or SFC-condition. C s x := {(dq, dφ) ∈ T x M : −∞ ≤ dφ/dq ≤ −K}, int C s x := {(dq, dφ) ∈ T x M \ {0} : −∞ < dφ/dq < −K}for dispersing and flat boundary components and Figure 1 : 1Sinai Figure 3 : 3Stadium billiard Figure 4: Squashes billiard Figure 5 Figure 6 : 56Semi-dispersing billiard f n γ u (x 0 ) s n |dx n | ≈ f n γ u (x 0 ) s n |ds| 1 n ( f n γ u (x 0 ) |ds|) 2 ≈ [diam f n γ u (x 0 )] 2 n . Since diam γ u (x 0 ) = O(1/n 2 ), then diam f n γ u (x 0 ) [n diam γ u (x 0 )] 1/2 [diam γ u (x 0 )] 1/4 .By combining now all the arguments above we obtaindiam f j γ u (x 0 ) [diam f n γ u (x 0 )] 1/3 [diam γ u (x 0 )] 1/12 .If i ∈ [0, n) \ {j}, then, by making use of the relation diam γ u (x 0 ) = O(1/n 2 ), we get ) is a mixing expanding Young tower. Besides, using (5.4) and (5.3), we haveThe return map for F e,m Ne,m : ∆ e,m → ∆ e,m is F e,m Ne,m Re,m/Ne,m : Λ → Λ, and it has the distortion: log det D F e,m Ne,m Re,m/Ne,m (x) det D F e,m Ne,m Re,m/Ne,m (y) ≤ Cβ s Fe,m Ne,m Re,m/Ne,m (x), Fe,m Ne,m Re,m/Ne,m (y) . Therefore, ( ∆ e,m , F e,m Ne,m α,m µ ∆ (π −1 S r ) + β .Recall that (7.5) µ M (B r (q) × S d ) = Br(q) µ q (S d )d Leb ∂Q . Since dµ M d Leb M > 0 almost surely, then µ q (S d ) > 0 Leb-a.s. q ∈ ∂Q. Therefore by the Lebesgue differentiation theorem,holds for Leb-a.e. q ∈ ∂Q. HenceUsing the estimates above and Lemmas 8.1 and 8.2, choose k = p/4. Thenconverges to zero, when r → 0, and this concludes a proof.By that we finished the estimates of (6.3) and (6.2) in the Lemma 6.3. Thus the Theorem 2.13 holds.ApplicationsA practical scheme for applications of the obtained results to concrete systemsHere we present some criteria to verify the cone conditions in Assumption 2.11 for two-dimensional billiards. The following notations will be used throughout this section: Let ∂R 0 , then B r (q) × S 1 is a quasi-section. Without loss of generality, we may assume that R 0 = [0, 1] 2 , q = (0, c) ∈ ∂R 0 , where c ∈ (0, 1). Unfold now the bounded billiard table Q to R 2 by mirror reflections after collisions with the flat boundary, so that a billiard orbit, which is a broken line in Q, is lifted to become a straight line in R 2 , and scatterers B 1 , · · · , B r are lifted to generate a periodic configuration of scatterers in R 2 . (Note that this trick is the same as the one used for Sinai billiards with infinite horizon. Namely, the analysis of a billiard flow between two reflections off scatterers in R 0 reduces to consideration of straight segments between consecutive reflections off scatterers in R 2 ). Suppose that a point q is lifted in this way to the points {(p, c + k) : p, k ∈ Z}.In order to prove that B r (q) × S 1 is a quasi-section for sufficiently small r > 0, we will study the "non-injective" part, i.e., a measure of the configuration (q 1 , φ 1 ) ∈ B r (q) × S 1 , which satisfies f n (q 1 , φ 1 ) = (q 2 , φ 2 ) for some n ≥ 1, (q 2 , φ 2 ) ∈ B r (q) × S 1 . A billiard orbit, which is moving along the following set of pointsdoes not intersect i B i . The lifting to R 2 has the following property. There are p, k ∈ Z (depend on φ 1 , n) such that the line between q 1 ∈ B r (q) and q 2 + (p, k) ∈ B r (q + (p, k)) does not intersect the periodic configuration of the scatterers, and the slope of this line isSince this line intersects B r (q) and B r (q + (p, k)), then it intersects B r (q + (p/i, k/i)) for any i such that i|p, i|k, in particular, i = gcd(p, k). Therefore every "non-injective" configuration (q 1 , φ 1 ) ∈ B r (q) × S 1 corresponds to unique direction vector (p/ gcd(k, p), k/ gcd(k, p)). Without loss of generality, we assume that gcd(k, p) = 1. Here p means that the line between q 1 and q 2 + (p, k) intersects the vertical boundary ≈ p times, which implies that the billiard flow in Q divides R 0 into several pieces with measures of order O(1/p). Let k means that the line between q 1 and q 2 + (p, k) intersects the horizontal boundary ≈ k times, which implies that the billiard flow in Q divides R 0 into several pieces with measure of order O(1/k). If p or k is larger than M > 0 (which depends on the size of i B i ), then the billiard orbit, which is passing consecutively through the points (q 1 , φ 1 ), f (q 1 , φ 1 ), · · · , f n−1 (q 1 , φ 1 ), f n (q 1 , φ 1 ) = (q 2 , φ 2 ) Note that, if R varies between i and i+1, then a collision must occur at the endpoints of two circular arcs. i≥1 ∂{x ∈ X : R(x) = i} ⊆ SIt follows fromi≥1 ∂{x ∈ X : R(x) = i} ⊆ S. Note that, if R varies between i and i+1, then a collision must occur at the endpoints of two circular arcs. It follows from Consider the map π X : π −1 B r (q) × S 1 → X. If it is not injective. Then there are x 1 , x 2 ∈ π −1 B r (q) × S 1 , such that π X x 1 = π X x 2 = x ∈ X. B r (q) × S 1 is a quasi-section. This implies that there are 0 ≤ n 1 < n 2 < R(x). suchB r (q) × S 1 is a quasi-section. Consider the map π X : π −1 B r (q) × S 1 → X. If it is not injective. Then there are x 1 , x 2 ∈ π −1 B r (q) × S 1 , such that π X x 1 = π X x 2 = x ∈ X. This implies that there are 0 ≤ n 1 < n 2 < R(x), such B r (q) × S 1 is a section if q ∈ ∂Q + . We will show that. if q ∈ ReferencesObviously, B r (q) × S 1 is a section if q ∈ ∂Q + . We will show that, if q ∈ References Two moments suffice for Poisson approximations: the Chen-Stein method. R Arratia, L Goldstein, L Gordon, 0091-1798Ann. Probab. 1711<9:TMSFPA>2.0.CO;2-X&origin=MSNR. Arratia, L. Goldstein, and L. Gordon. Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab., 17(1):9-25, 1989. ISSN 0091-1798. URL http://links.jstor.org/sici?sici=0091-1798(198901) 17:1<9:TMSFPA>2.0.CO;2-X&origin=MSN. Return time statistics via inducing. H Bruin, B Saussol, S Troubetzkoy, S Vaienti, 10.1017/S0143385703000026Ergodic Theory Dynam. Systems. 234H. Bruin, B. Saussol, S. Troubetzkoy, and S. Vaienti. Return time statistics via inducing. Ergodic Theory Dynam. Systems, 23(4):991-1013, 2003. ISSN 0143- 3857. doi: 10.1017/S0143385703000026. URL https://doi.org/10.1017/ S0143385703000026. L Bunimovich, Y Su, arXiv:2103.05418Poisson Approximations and Convergence Rates for Hyperbolic Dynamical Systems. arXiv e-prints, art. L. Bunimovich and Y. Su. Poisson Approximations and Convergence Rates for Hyperbolic Dynamical Systems. arXiv e-prints, art. arXiv:2103.05418, Mar. 2021. M Carney, M Holland, M Nicol, arXiv:1909.04748Extremes and extremal indices for level set observables on hyperbolic systems. arXiv e-prints, art. M. Carney, M. Holland, and M. Nicol. Extremes and extremal indices for level set observables on hyperbolic systems. arXiv e-prints, art. arXiv:1909.04748, Sept. 2019. Decay of correlations and dispersing billiards. N Chernov, 10.1023/A:1004581304939J. Statist. Phys. 943-4N. Chernov. Decay of correlations and dispersing billiards. J. Statist. Phys., 94(3-4):513-556, 1999. ISSN 0022-4715. doi: 10.1023/A:1004581304939. URL https://doi.org/10.1023/A:1004581304939. Chaotic billiards. N Chernov, R Markarian, 10.1090/surv/127Mathematical Surveys and Monographs. 127American Mathematical SocietyN. Chernov and R. Markarian. Chaotic billiards, volume 127 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. ISBN 0-8218-4096-7. doi: 10.1090/surv/127. URL https://doi.org/ 10.1090/surv/127. Billiards with polynomial mixing rates. N Chernov, H.-K Zhang, 10.1088/0951-7715/18/4/006Nonlinearity. 184N. Chernov and H.-K. Zhang. Billiards with polynomial mixing rates. Nonlin- earity, 18(4):1527-1553, 2005. ISSN 0951-7715. doi: 10.1088/0951-7715/18/4/ 006. URL https://doi.org/10.1088/0951-7715/18/4/006. Improved estimates for correlations in billiards. N Chernov, H.-K Zhang, 10.1007/s00220-007-0360-x0010-3616. doi: 10.1007/ s00220-007-0360-xComm. Math. Phys. 2772N. Chernov and H.-K. Zhang. Improved estimates for correlations in billiards. Comm. Math. Phys., 277(2):305-321, 2008. ISSN 0010-3616. doi: 10.1007/ s00220-007-0360-x. URL https://doi.org/10.1007/s00220-007-0360-x. Observation of chaotic and regular dynamics in atom-optics billiards. N Friedman, A Kaplan, D Carasso, N Davidson, Physical review letters. 8681518N. Friedman, A. Kaplan, D. Carasso, and N. Davidson. Observation of chaotic and regular dynamics in atom-optics billiards. Physical review letters, 86(8): 1518, 2001. Certain conditions for the existence of K-decompositions for special flows. B M Gurevič, 0134-8663Trudy Moskov. Mat. Obšč. 17B. M. Gurevič. Certain conditions for the existence of K-decompositions for special flows. Trudy Moskov. Mat. Obšč., 17:89-116, 1967. ISSN 0134-8663. Limiting entry and return times distribution for arbitrary null sets. N Haydn, S Vaienti, 10.1007/s00220-020-03795-0Comm. Math. Phys. 3781N. Haydn and S. Vaienti. Limiting entry and return times distribution for arbitrary null sets. Comm. Math. Phys., 378(1):149-184, 2020. ISSN 0010- 3616. doi: 10.1007/s00220-020-03795-0. URL https://doi.org/10.1007/ s00220-020-03795-0. On the notion of recurrence in discrete stochastic processes. M Kac, 10.1090/S0002-9904-1947-08927-8Bull. Amer. Math. Soc. 53M. Kac. On the notion of recurrence in discrete stochastic pro- cesses. Bull. Amer. Math. Soc., 53:1002-1010, 1947. ISSN 0002-9904. doi: 10.1090/S0002-9904-1947-08927-8. URL https://doi.org/10.1090/ S0002-9904-1947-08927-8. Foundations of modern probability. O Kallenberg, 10.1007/978-1-4757-4015-8Probability and its Applications. Springer-Verlagsecond editionO. Kallenberg. Foundations of modern probability. Probability and its Appli- cations (New York). Springer-Verlag, New York, second edition, 2002. ISBN 0-387-95313-2. doi: 10.1007/978-1-4757-4015-8. URL https://doi.org/10. 1007/978-1-4757-4015-8. Billiards with polynomial decay of correlations. R Markarian, 10.1017/S01433857030002700143-3857. doi: 10.1017/ S0143385703000270Systems. 241Ergodic Theory DynamR. Markarian. Billiards with polynomial decay of correlations. Ergodic The- ory Dynam. Systems, 24(1):177-197, 2004. ISSN 0143-3857. doi: 10.1017/ S0143385703000270. URL https://doi.org/10.1017/S0143385703000270. Optical billiards for atoms. V Milner, J Hanssen, W Campbell, M Raizen, Physical Review Letters. 8681514V. Milner, J. Hanssen, W. Campbell, and M. Raizen. Optical billiards for atoms. Physical Review Letters, 86(8):1514, 2001. Ray and wave chaos in asymmetric resonant optical cavities. J U Nöckel, A D Stone, Nature. 3856611J. U. Nöckel and A. D. Stone. Ray and wave chaos in asymmetric resonant optical cavities. Nature, 385(6611):45-47, 1997. Directional emission from asymmetric resonant cavities. J U Nöckel, A D Stone, G Chen, H L Grossman, R K Chang, Optics letters. 2119J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang. Di- rectional emission from asymmetric resonant cavities. Optics letters, 21(19): 1609-1611, 1996. Back to balls in billiards. F Pène, B Saussol, 10.1007/s00220-009-0911-4Comm. Math. Phys. 2933F. Pène and B. Saussol. Back to balls in billiards. Comm. Math. Phys., 293 (3):837-866, 2010. ISSN 0010-3616. doi: 10.1007/s00220-009-0911-4. URL https://doi.org/10.1007/s00220-009-0911-4. Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing. F Pène, B Saussol, 10.1017/etds.2015.28Ergodic Theory Dynam. Systems. 368F. Pène and B. Saussol. Poisson law for some non-uniformly hyperbolic dy- namical systems with polynomial rate of mixing. Ergodic Theory Dynam. Sys- tems, 36(8):2602-2626, 2016. ISSN 0143-3857. doi: 10.1017/etds.2015.28. URL https://doi.org/10.1017/etds.2015.28. Spatio-temporal Poisson processes for visits to small sets. F Pène, B Saussol, 10.1007/s11856-020-2074-00021-2172. doi: 10.1007/ s11856-020-2074-0Israel J. Math. 2402F. Pène and B. Saussol. Spatio-temporal Poisson processes for visits to small sets. Israel J. Math., 240(2):625-665, 2020. ISSN 0021-2172. doi: 10.1007/ s11856-020-2074-0. URL https://doi.org/10.1007/s11856-020-2074-0. Wave dynamical chaos: An experimental approach in billiards. A Richter, Quantum Chaos Y2K. World ScientificA. Richter. Wave dynamical chaos: An experimental approach in billiards. In Quantum Chaos Y2K, pages 212-222. World Scientific, 2001. Statistical properties of dynamical systems with some hyperbolicity. L.-S Young, 10.2307/120960Ann. of Math. 1472L.-S. Young. Statistical properties of dynamical systems with some hyper- bolicity. Ann. of Math. (2), 147(3):585-650, 1998. ISSN 0003-486X. doi: 10.2307/120960. URL https://doi.org/10.2307/120960. Recurrence times and rates of mixing. L.-S Young, 10.1007/BF02808180Israel J. Math. 110L.-S. Young. Recurrence times and rates of mixing. Israel J. Math., 110: 153-188, 1999. ISSN 0021-2172. doi: 10.1007/BF02808180. URL https: //doi.org/10.1007/BF02808180.
[]
[ "Order Estimation of Markov Chains", "Order Estimation of Markov Chains" ]
[ "Gusztáv Morvai ", "Benjamin Weiss " ]
[]
[ "IEEE Trans. Inform. Theory" ]
We describe estimators χ n (X 0 , X 1 , . . . , X n ), which when applied to an unknown stationary process taking values from a countable alphabet X , converge almost surely to k in case the process is a k-th order Markov chain and to infinity otherwise.
10.1109/tit.2005.844093
[ "https://arxiv.org/pdf/0711.0472v1.pdf" ]
9,025,657
0711.0472
4b43c28179b6191a9e91ff77ecdfb7c15b00686b
Order Estimation of Markov Chains 2005 Gusztáv Morvai Benjamin Weiss Order Estimation of Markov Chains IEEE Trans. Inform. Theory 5142005Stationary processesMarkov chainsorder estimation Mathematics Subject Classifications (2000)62M0560G2560G10 We describe estimators χ n (X 0 , X 1 , . . . , X n ), which when applied to an unknown stationary process taking values from a countable alphabet X , converge almost surely to k in case the process is a k-th order Markov chain and to infinity otherwise. Introduction When faced with an unknown stationary and ergodic stochastic process X 1 , X 2 , . . . , X n , . . . one may try to determine various properties of this process from the successive observations up to time n. For example, one might try to estimate the entropy of the process. Several schemes of the form g n (X 1 , . . . , X n ) are known which will converge almost surely to the entropy of the process {X n } cf. Bailey [1], Csiszár and Shields [2], Csiszár [3], Ornstein and Weiss [8], [7], [9], Kontoyiannis, Algoet, Suhov and Wyner [6] and Ziv [10]. However, if one just wants to determine whether or not the process has positive entropy (often associated with the popular notion of chaos) then there is no sequence of two valued functions e n (X 1 , . . . , X n ) ∈ {ZERO, P OSIT IV E} with the property that almost surely, e n stabilize at ZERO for all zero entropy processes and at P OSIT IV E for all positive entropy processes. (While this result does not appear explicitly in Ornstein amd Weiss [7], it can be readily established using a very simple variant of the construction given there in § 4.) A similar situation obtains in testing for membership in the class of k-th order Markov chains. One can estimate the order of a Markov chain by e.g the method of Csiszár and Shields [2] or Csiszár [3]. They show that the minimum description length Markov estimator will converge almost surely to the correct order if the alphabet size is bounded a priori. Without this assumption they show that this is no longer true. To accomplish their goals they study the large scale typicality of Markov sample paths. A further negative result is that of Bailey [1] who showed that no two valued test exists for testing mixing Markov vs. not mixing Markov. We will present a more direct estimator for the order of a Markov chain which also uses the fact that there are universal rates for the convergence of empirical k-block distributions in this class. Our approach enables us to dispense with the assumption that the alphabet size is bounded, indeed it may even be infinite, as long as there is a finite memory. In addition we will show that if the process is not a Markov chain then the estimate for the order will tend to infinity. This is in complete analogy with the entropy estimation that we mentioned earlier. The Order Estimator Let {X n } ∞ n=−∞ be a stationary and ergodic time series taking values from a discrete (finite or countably infinite) alphabet X . (Note that all stationary time series {X n } ∞ n=0 can be thought to be a two sided time series, that is, {X n } ∞ n=−∞ . ) For notational convenience, let X n m = (X m , . . . , X n ), where m ≤ n. Note that if m > n then X n m is the empty string. Let p(x 0 −k ) and p(y|x 0 −k ) denote the distribution P (X 0 −k = x 0 −k ) and the conditional distribution P (X 1 = y|X 0 −k = x 0 −k ), respectively. A discrete alphabet stationary time series is said to be a Markov chain if for some K ≥ 0, for all y ∈ X , i ≥ 1 and z 0 −K−i+1 ∈ X K+i , if p(z 0 −K−i+1 ) > 0 then p(y|z 0 −K+1 ) = p(y|z 0 −K−i+1 ) . The order of a Markov chain is the smallest such K. In order to estimate the order we need to define some explicit statistics. For k ≥ 0 let S k denote the support of the distribution of X 0 −k as S k = {x 0 −k ∈ X k+1 : p(x 0 −k ) > 0}. Define ∆ k = sup 1≤i sup (z 0 −k−i+1 ,x)∈S k+i p(x|z 0 −k+1 ) − p(x|z 0 −k−i+1 ) . We will divide the data segment X n 0 into two parts: X ⌈ n 2 ⌉−1 0 and X n ⌈ n 2 ⌉ . Let S (1) n,k denote the set of strings with length k + 1 which appear at all in X ⌈ n 2 ⌉−1 0 . That is, S (1) n,k = {x 0 −k ∈ X k+1 : ∃k ≤ t ≤ ⌈ n 2 ⌉ − 1 : X t t−k = x 0 −k }. For a fixed 0 < γ < 1 let S (2) n,k denote the set of strings with length k + 1 which appear more than n 1−γ times in X n ⌈ n 2 ⌉ . That is, S (2) n,k = {x 0 −k ∈ X k+1 : #{⌈ n 2 ⌉ + k ≤ t ≤ n : X t t−k = x 0 −k } > n 1−γ }. Let S n k = S (1) n,k S(2) n,k . For notational convenience, let C(x|z 0 −k+1 : [n 1 , n 2 ]) denote the empirical conditional probability of X 1 = x given X 0 −k+1 = z 0 −k+1 from the samples (X n 1 , . . . , X n 2 ), that is, C(x|z 0 −k+1 : [n 1 , n 2 ]) = #{n 1 + k ≤ t ≤ n 2 : X t t−k = (z 0 −k+1 , x)} #{n 1 + k − 1 ≤ t ≤ n 2 − 1 : X t t−k+1 = z 0 −k+1 } where 0/0 is defined as 0. We define the empirical version of ∆ k as follows: ∆ n k = max 1≤i≤n max (z 0 −k−i+1 ,x)∈S n k+i C(x|z 0 −k+1 : [⌈ n 2 ⌉, n]) − C(x|z 0 −k−i+1 : [⌈ n 2 ⌉, n]) . Observe, that by ergodicity, for any fixed k, lim inf n→∞∆ n k ≥ ∆ k almost surely.(1) We define an estimate χ n for the order from samples X n 0 as follows. Let 0 < β < 1−γ 2 be arbitrary. Set χ 0 = 0, and for n ≥ 1 let χ n be the smallest 0 ≤ k n < n such that∆ n kn ≤ n −β . THEOREM. If the stationary and ergodic time series {X n } taking values from a discrete alphabet happens to be a Markov chain with any finite order then χ n equals to the order eventually almost surely, and if it is not Markov with any finite order then χ n → ∞ almost surely. Application: Let M > 0 be arbitrary. The goal is to decide if the discrete alphabet stationary and ergodic time series is a Markov chain with order less than M or not. One may use χ n and say YES if χ n < M and say NO otherwise. By the Theorem, eventually, the answer will be correct. Proof of the Theorem Proof: If the process is a Markov chain, it is immediate that for all k greater than or equal the order, ∆ k = 0. For k less than the order ∆ k > 0. If the process is not a Markov chain with any finite order then ∆ k > 0 for all k. Thus by (1) if the process is not Markov then χ n → ∞ and if it is Markov then χ n is greater or equal the order eventually almost surely. We have to show that χ n is less or equal the order eventually almost surely provided that the process is a Markov chain. Assume that the process is a Markov chain with order k. Let n ≥ k. We will estimate the probability of the undesirable event as follows: P (∆ n k > n −β |X ⌈ n 2 ⌉ 0 ) ≤ n i=1 P ( max (z 0 −k−i+1 ,x)∈S n k+i C(x|z 0 −k+1 : [⌈ n 2 ⌉, n]) − C(x|z 0 −k−i+1 : [⌈ n 2 ⌉, n]) > n −β |X ⌈ n 2 ⌉ 0 ). We can estimate each probability in the sum as the sum of two terms: P ( max (z 0 −k−i+1 ,x)∈S n k+i C(x|z 0 −k+1 : [⌈ n 2 ⌉, n]) − C(x|z 0 −k−i+1 : [⌈ n 2 ⌉, n]) > n −β |X ⌈ n 2 ⌉ 0 ) ≤ P ( max (z 0 −k−i+1 ,x)∈S n k+i C(x|z 0 −k+1 : [⌈ n 2 ⌉, n]) − p(x|z 0 −k+1 ) > 0.5n −β |X ⌈ n 2 ⌉ 0 ) + P ( max (z 0 −k−i+1 ,x)∈S n k+i p(x|z 0 −k+1 ) − C(x|z 0 −k−i+1 : [⌈ n 2 ⌉, n]) > 0.5n −β |X ⌈ n 2 ⌉ 0 ). We overestimate these probabilities. For any m ≥ 0 and x 0 −m define σ m i (x 0 −m ) as the time of the i-th ocurrence of the string x 0 −m in the data segment X n ⌈ n 2 ⌉ , that is, let σ m 0 (x 0 −m ) = ⌈ n 2 ⌉ + m − 1 and for i ≥ 1 define σ m i (x 0 −m ) = min{t > σ m i−1 (x 0 −m ) : X t t−m = x 0 −m }. Now P ( max (z 0 −k−i+1 ,x)∈S n k+i C(x|z 0 −k+1 : [⌈ n 2 ⌉, n]) − C(x|z 0 −k−i+1 : [⌈ n 2 ⌉, n]) > n −β |X ⌈ n 2 ⌉ 0 ) ≤ P ( max (z 0 −k+1 ,x)∈S (1) n,k sup j>n 1−γ 1 j j r=1 1 {X σ k−1 r (z 0 −k+1 ) =x} − p(x|z 0 −k+1 ) > 0.5n −β |X ⌈ n 2 ⌉ 0 ) + P ( max (z 0 −k−i+1 ,x)∈S (1) n,k+i sup j>n 1−γ 1 j j r=1 1 {X σ k+i−1 r (z 0 −k−i+1 ) =x} − p(x|z 0 −k+1 ) > 0.5n −β |X ⌈ n 2 ⌉ 0 ) Since both S (1) n,k and S (1) n,k+i depend solely on X ⌈ n 2 ⌉ 0 we get P ( max (z 0 −k−i+1 ,x)∈S n k+i C(x|z 0 −k+1 : [⌈ n 2 ⌉, n]) − C(x|z 0 −k−i+1 : [⌈ n 2 ⌉, n]) > n −β |X ⌈ n 2 ⌉ 0 ) ≤ (z 0 −k+1 ,x)∈S (1) n,k ∞ j=⌈n 1−γ ⌉ P ( 1 j j r=1 1 {X σ k−1 r (z 0 −k+1 ) =x} − p(x|z 0 −k+1 ) > 0.5n −β |X ⌈ n 2 ⌉ 0 ) + (z 0 −k−i+1 ,x)∈S (1) n,k+i ∞ j=⌈n 1−γ ⌉ P ( 1 j j r=1 1 {X σ k+i−1 r (z 0 −k−i+1 ) =x} −p(x|z 0 −k+1 ) > 0.5n −β |X ⌈ n 2 ⌉ 0 ). Each of these represents the deviation of an empirical count from its mean. The variables in question are independent since whenever the block z 0 −k+1 occurs the next term is chosen using the same distribution p(x|z 0 −k+1 ). Thus by Hoeffding's inequality (cf. Hoeffding [5] or Theorem 8.1 of Devroye et. al. [4]) for sums of bounded independent random variables and since the cardinality of both S Thus P (∆ n k > n −β |X ⌈ n 2 ⌉ 0 ) ≤ n(n + 2)4e −2n −2β+1−γ . Integrating both sides we get P (∆ n k > n −β ) ≤ n(n + 2)4e −2n −2β+1−γ . The right hand side is summable provided 2β + γ < 1 and the Borel-Cantelli Lemma yields that P (∆ n k ≤ n −β eventually) = 1. Thus χ n ≤ k eventually almost surely provided the process is Markov with order k. The proof of the Theorem is complete. 2n −2β j . D H Bailey, Sequential Schemes for Classifying and Predicting Ergodic Processes. Stanford UniversityPh. D. thesisD. H. Bailey, Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph. D. thesis, Stanford University, 1976. The consistency of the BIC Markov order estimator. I Csiszár, P Shields, Annals of Statistics. 28I. Csiszár and P. Shields, "The consistency of the BIC Markov order estimator," Annals of Statistics., vol. 28, pp. 1601-1619, 2000. Large-scale typicality of Markov sample paths and consistency of MDL order estimators. I Csiszár, IEEE Transactions on Information Theory. 48I. Csiszár, "Large-scale typicality of Markov sample paths and consis- tency of MDL order estimators ," IEEE Transactions on Information Theory, vol. 48, pp. 1616-1628, 2002. A Probabilistic Theory of Pattern Recognition. L L Devroye, G Györfi, Lugosi, Springer-VerlagNew YorkL Devroye, L. Györfi, G. Lugosi, A Probabilistic Theory of Pattern Recognition. Springer-Verlag, New York, 1996. Probability inequalities for sums of bounded random variables. W Hoeffding, Journal of the American Statistical Association. 58W. Hoeffding, "Probability inequalities for sums of bounded random variables ," Journal of the American Statistical Association, vol. 58, pp. 13-30, 1963. Nonparametric entropy estimation for stationary processes and random fields, with application to English text. I Kontoyiannis, P Algoet, Yu M Suhov, A J Wyner, IEEE Transactions on Information Theory. 44I. Kontoyiannis, P. Algoet, Yu.M. Suhov, A.J. Wyner, "Nonparametric entropy estimation for stationary processes and random fields, with ap- plication to English text," IEEE Transactions on Information Theory, vol. 44, pp. 1319-1327, 1998. How sampling reveals a process. D S Ornstein, B Weiss, The Annals of Probability. 18D. S. Ornstein and B. Weiss, "How sampling reveals a process," The Annals of Probability, vol. 18, pp. 905-930, 1990. Entropy and data compression schemes. D S Ornstein, B Weiss, IEEE Transactions on Information Theory. 39D. S. Ornstein and B. Weiss, "Entropy and data compression schemes," IEEE Transactions on Information Theory, vol. 39, pp. 78-83, 1993. Entropy and recurrence rates for stationary random fields. D S Ornstein, B Weiss, IEEE Transactions on Information Theory. 48D. S. Ornstein and B. Weiss, "Entropy and recurrence rates for station- ary random fields," IEEE Transactions on Information Theory, vol. 48, pp. 1699-1697, 2002. Coding theorems for individual sequences. J Ziv, IEEE Transactions on Information Theory. 24J. Ziv, " Coding theorems for individual sequences. IEEE Transactions on Information Theory, vol. 24, pp. 405-412, 1978.
[]
[ "ADAPTIVE ENERGY MINIMISATION FOR hp-FINITE ELEMENT METHODS", "ADAPTIVE ENERGY MINIMISATION FOR hp-FINITE ELEMENT METHODS" ]
[ "Paul Houston ", "Thomas P Wihler " ]
[]
[]
This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of hp-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new hp-refinement technique for both one-and two-dimensional problems.2010 Mathematics Subject Classification. 65N30.
10.1016/j.camwa.2016.01.002
[ "https://arxiv.org/pdf/1507.01333v1.pdf" ]
16,153,786
1507.01333
4ec4be1172c04de249c2c5fac657e9987c592e3d
ADAPTIVE ENERGY MINIMISATION FOR hp-FINITE ELEMENT METHODS Paul Houston Thomas P Wihler ADAPTIVE ENERGY MINIMISATION FOR hp-FINITE ELEMENT METHODS This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of hp-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new hp-refinement technique for both one-and two-dimensional problems.2010 Mathematics Subject Classification. 65N30. Introduction Over the last few decades, tremendous progress has been made on both the mathematical analysis and practical application of finite element methods to a wide range of problems of industrial importance. In particular, significant contributions have been made in the area of a posteriori error estimation and automatic mesh adaptation. For recent surveys and historical background, we refer to [2,8,15,18,30,32], and the references cited therein. Here, adaptive methods seek to automatically enrich the underlying finite element space, from which the numerical solution is sought, in order to compute efficient and reliable numerical approximations. The standard approach used within much of the literature is to simply undertake local isotropic refinement of the elements (h-refinement). However, in recent years, so-called hpadaptive finite element methods have been devised, whereby both local subdivision of the elements and local polynomial-degree-variation (p-refinement) are employed. These ideas date back to the work by Babuška and co-workers (cf. [4][5][6][7]); see also the recent books [11,21,28,29], and the references cited therein. The exploitation of general hp-refinement strategies can produce remarkably efficient methods with high algebraic or even exponential rates of convergence. Moreover, such approaches can also be combined with anisotropic refinement techniques in order to efficiently approximate problems with sharp transition features. These techniques enable the user to perform accurate and reliable computational simulations without excessive computing resources, and with the confidence that complex local features of the underlying solution are accurately captured. Many physical processes can be modelled by locating critical points of a given (in our setting, convex) energy functional, over an admissible space of functions; a typical example includes quasilinear partial differential equations (PDEs). In this article we consider the application of adaptive finite element techniques, employing a combination of hp-mesh refinement, to problems of this type. In particular, we consider a new and widely applicable paradigm for adaptive mesh generation within which we directly seek to construct the hp-finite element space in order to approximate the critical point of the underlying energy functional associated to the problem of interest. The simplest such example is the one-dimensional Poisson equation on the interval (0, 1), subject to a load f ; in this case, we seek to minimise E(u) = 1 /2 1 0 u 2 x dx− 1 0 f u dx over an appropriate solution space V (which naturally incorporates the boundary conditions). The corresponding standard Galerkin finite element approximation of this problem automatically inherits the same energy minimisation property with respect to the underlying finite element space V h ⊂ V , i.e., the finite element solution is the unique minimiser of E(·) over V h . With this idea in mind, a natural approach is to adaptively modify the finite element space V h in a manner which seeks to directly decrease the energy E, i.e., denoting the new finite element solution and finite element space by u h and V h , respectively, we require that E(u h ) ≤ E(u h ). By considering an appropriately defined elementwise energy functionalẼ κ , with κ denoting the current element in the underlying computational mesh, we devise a competitive refinement strategy which marks elements for refinement. More precisely, in the context of our one-dimensional example, consider an h-refinement strategy which subdivides each element into two sub-elements. We may then compute the numerical solution to a local finite element problem posed on a local patch of elements which includes the two sub-elements. On the basis of this local reference approximation, we may then determine the predicted (elemental) energy loss if the proposed refinement (i.e., a bisection of each element into two sub-elements) is undertaken. Once the predicted energy loss has been computed for all elements in the mesh, a percentage of elements with the largest predicted energy loss may be identified and subsequently refined. This idea naturally extends to the p-refinement setting, whereby, additional higher-order modes are used to locally enrich the finite element solution elementwise. With this in mind, we propose a competitive hp-adaptive refinement strategy which computes the maximal predicted energy loss on each element based on comparing a p-refinement of each element (i.e., an isotropic increase of the elemental polynomial degrees by 1) with a collection of h-refinements of the same element (featuring different local polynomial degree distributions), which are selected so as to lead to the same increase in the number of degrees of freedom associated with the current element as the p-enrichment, cf. [11,12,27,29]. A key aspect of this algorithm is the computation of a local elementwise/patchwise reference solution needed for the definition ofẼ κ . In order to illustrate the key ideas, for the purposes of this article, we restrict our discussion to convex optimisation problems posed on a computational domain Ω ⊂ R d , d ≥ 1. However, we point out that this strategy is completely general in the sense that it can be applied to any physical problem which may be modelled as a critical point of a given energy functional E (including saddle point problems, see, e.g., [26]). In particular, one of the key advantages of our proposed approach is that it naturally facilitates the use of hp-mesh adaptation, and indeed even anisotropic hp-mesh refinement. By considering an enrichment of the finite element space locally using any combination of isotropic/anisotropic h-/p-refinement, an element κ can be refined according to the refinement which leads to the maximal predicted energy loss. This is in contrast to standard adaptive techniques, whereby elements are marked for refinement according to the size of a local a posteriori error indicator. Indeed, in this latter setting, such indicators rarely contain information concerning how the local finite element space should be enriched, but only indicate that a refinement should be performed. Thereby, alternative numerical techniques must be devised which are capable of determining the direction of refinement (for anisotropic refinement) or the type of refinement (h-or p-). For the latter case, such strategies include regularity estimation [14,19,22,33], use of a priori knowledge [9,31], and the computation of reference solutions within competitive refinement strategies [3,11,13,16,17,23,25,27,29], for example. This latter class of methods, cf. in particular [11,12,27,29] and [16,17], are very much in the spirit of the proposed competitive refinement algorithm developed in this article. Finally, we refer to [24] for an extensive review and comparison of many of the hp-adaptive refinement techniques proposed within the literature. This article is structured as follows. In Section 2 we briefly present an abstract framework for variational problems, and consider an application to quasilinear partial differential equations. Subsequently, in Section 3, the hp-version finite element discretisation of such problems is presented, and a new hp-adaptivity approach is developed in detail. The theory will be illustrated with a number of numerical experiments on linear and quasilinear boundary value problems in Section 5. Finally, in Section 6 we summarise the work presented in this article and draw some conclusions. Throughout this article, we let L p (D), p ∈ [1, ∞], be the standard Lebesgue space on some bounded domain D, with boundary ∂D, equipped with the norm · L p (D) . Furthermore, for k ∈ N, we write W k,p (D) to signify the Sobolev space of order k, endowed with the norm · W k,p (D) and seminorm | · | W k,p (D) . For p = 2 we write H k (D) in lieu of W k,2 (D); moreover, H 1 0 (D) denotes the subspace of H 1 (D) of functions with zero trace on ∂D. Variational Problems In this section we outline an abstract framework for variational problems in Banach spaces, and consider an application to quasilinear boundary value problems. 2.1. Abstract Minimisation Problem. On a real reflexive Banach space X let us consider the minimisation problem (2.1) min u∈X E(u) ≡ min u∈X {F(u) − l, u } . Here, ·, · is the duality product on X × X , where X signifies the dual space of X, and l ∈ X is given. Furthermore, throughout this manuscript, we suppose that F : X → R is a continuous and strictly convex functional on X, i.e., F(v 1 + t(v 2 − v 1 )) < F(v 1 ) + t(F(v 2 ) − F(v 1 )) ∀t ∈ [0, 1] ∀v 1 , v 2 ∈ X. In addition, we make the assumption that F satisfies the coercivity type condition (2.2) F(u) − l, u → +∞ as u X → ∞, where · X is a norm on X. Then, (2.1) possesses a unique minimiser u ∈ X; see, e.g., [34,Corollary 42.14]. Furthermore, the problem of finding u ∈ X can be written in weak form as F (u ), v = l, v ∀v ∈ X, provided that F has a sufficiently regular Gâteaux derivative F . 2.2. Model Problem. We now consider a specific application of the above abstract setting. To this end, let Ω ⊂ R d , d ≥ 1, be an open bounded Lipschitz domain with boundary ∂Ω. We consider the following quasilinear partial differential equation: −∇ · (µ (∇u )) + g (u ) = f, in Ω, u = 0, on ∂Ω. (2.3) Here, f = f (x), µ = µ(∇u), and g = g(u) are given functions, and u = u (x) is the unknown analytical solution. We suppose that f ∈ L q (Ω), for some q > 1. The corresponding variational problem reads: (2.4) min u∈X E(u) := min u∈X Ω {µ(∇u) + g(u) − f u} dx, where X = W 1,p 0 (Ω) for some suitable p > 1. With this notation, the following proposition holds. Proposition 2.5. Let µ and g from (2.4) be strictly convex and convex, respectively, and both continuous on R d and R, respectively. Furthermore, suppose that, for some constants C 1 , C 2 > 0, p > 1, and c 1 , c 2 ∈ R the lower bounds hold, µ(ξ) ≥ C 1 |ξ| p , (2.6) g(η) ≥ c 1 η + c 2 , (2.7) as well as the growth conditions µ(ξ) ≤ C 2 (1 + |ξ| p ), (2.8) g(η) ≤ C 2 (1 + |η| p ), (2.9) for any ξ ∈ R d and any η ∈ R. Then, for any given f ∈ L q (Ω), where 1 /p + 1 /q = 1, (2.4) has a unique solution in X = W 1,p 0 (Ω) as well as on any linear subspace of X. Proof. Let us define u → F(u) := Ω {µ(∇u) + g(u) − f u} dx. Then, we can cast (2.4) into the abstract framework of (2.1), with l = 0 in X . We check the conditions from Section 2.1 separately. To this end, we follow the proof presented in [34,Example 42.15]. (i) Continuity of F: Let us consider a sequence {u n } ⊂ W 1,p 0 (Ω), with a limit u ∈ W 1,p 0 (Ω), i.e., u n → u as n → ∞. Then, with (2.8), the Nemyckii operator ζ → µ(ζ) is continuous from [L p (Ω)] d to L 1 (Ω); see [35, Proposition 26.6]. Therefore, Ω {µ(∇u) − µ(∇u n )} dx ≤ µ(∇u) − µ(∇u n ) L 1 (Ω) → 0 as n → ∞. Similarly, using (2.9), as n → ∞, we have that Ω {g(u) − g(u n )} dx → 0. The continuity of u → Ω f u dx follows from f ∈ L q (Ω) and from Hölder's inequality. Thus, F is continuous. (ii) Strict convexity of F: This simply follows from the strict convexity of µ and the convexity of g. (iii) Coercivity: According to (2.6) and (2.7), we find that F(u) ≥ C 1 |u| p W 1,p (Ω) + Ω {g(u) − f u} dx ≥ C 1 |u| p W 1,p (Ω) + Ω {c 1 u + c 2 − f u} dx ≥ C 1 |u| p W 1,p (Ω) − |c 1 | u L 1 (Ω) + c 2 |Ω| − Ω f u dx , where |Ω| signifies the volume of Ω. Employing the Poincaré-Friedrich's and Hölder's inequalities, we arrive at F(u) ≥ C u p W 1,p (Ω) − ( c 1 + f L q (Ω) ) u L p (Ω) − c 2 ≥ C u p W 1,p (Ω) − ( c 1 + f L q (Ω) ) u W 1,p (Ω) − c 2 , for some constants c 1 , c 2 > 0 depending on Ω. Therefore, it follows that E(u) = F(u) − l, u ≡ F(u) → ∞, with u W 1,p (Ω) → ∞. This is the coercivity condition (2.2). The result now follows from [34, Corollary 42.14]. hp-Finite Element Discretisation Consider now a linear subspace X n ⊂ X with dim(X n ) = n < ∞. Then, by our previous assumptions on F, solving the finite dimensional convex optimisation problem min u∈Xn E(u) for the unique minimiser u n ∈ X n results in an approximation of u ∈ X from (2.1) with u n ≈ u . This is the well-known Ritz method. Equivalently, in weak form, we may seek u n ∈ X n such that the Galerkin formulation F (u n ), v = l, v ∀v ∈ X n is satisfied; cf. [34, § 42.5]. For the purposes of discretising our model problem (2.3), we will focus on an hp-finite element approach. To this end, let us first introduce some notation: We let T = {κ} be a subdivision of the computational domain Ω into disjoint open simplices such that Ω = κ∈T κ and denote by h κ the diameter of κ ∈ T ; i.e., h κ = diam(κ). In addition, to each element κ ∈ T we associate a polynomial degree p κ , p κ ≥ 1, and collect the p κ in the polynomial degree vector p = [p κ : κ ∈ T ]. With this notation we define the hp-finite element spaces by V(T , p) = v ∈ H 1 (Ω) : v| κ ∈ P pκ (κ) ∀κ ∈ T , V 0 (T , p) = V(T , p) ∩ H 1 0 (Ω), where, for p ≥ 1, we denote by P p (κ) the space of polynomials of total degree p on κ. The hp-version finite element approximation of the variational formulation (2.1) is given by: Find the numerical approximation u hp ∈ V 0 (T , p) such that E(u hp ) = min u∈V0(T ,p) E(u), where E is defined in (2.4), or equivalently, provided that the (weak) derivatives µ and g belong to L 1 loc (Ω), in weak form: Find u hp ∈ V 0 (T , p) such that (3.1) a Ω (u hp , v) = Ω (v) ∀v ∈ V 0 (T , p). Here, a Ω (w, v) := Ω {µ (∇w) · ∇v + g (w)v} dx, w, v ∈ V(T , p) , Ω (v) := Ω f v dx, v ∈ V(T , p). This is the hp-finite element discretisation of (2.3). hp-Adaptivity The goal of this section is to design a procedure that generates sequences of hpadaptively refined finite element spaces in such a manner as to minimise the error in the computed energy functional E. In the context of hp-version finite element methods, the local finite element space on a given element κ, κ ∈ T , may be enriched in a number of ways. In particular, traditional hp-adaptive finite element methods typically make a choice between either: • p-refinement: The local polynomial degree p κ on κ is increased by a given increment, p inc : p κ ← p κ + p inc . Typically, a value of p inc = 1 is selected. • h-refinement: The element κ is divided into a set of n κ new sub-elements, such that κ = nκ i=1 κ i . Here, n κ will depend on both the type of element to be refined, and the type of refinement employed, i.e., isotropic/anisotropic. For isotropic refinement of a triangular element κ in two-dimensions, we have n κ = 4. The polynomial degree may then be inherited from the parent element κ, i.e., we set p κi = p κ , for i = 1, . . . , n κ . Motivated by the work presented in [27], cf. also [11,12,29], for example, in this article, we consider a competitive refinement strategy, whereby on each element κ in T , we estimate the predicted reduction in the local contribution to the energy functional E based on either employing p-refinement, with p inc = 1, together with a series of hp-refinements, which lead to the same number of degrees of freedom as the p-enrichment. In contrast to standard h-refinement, where the subdivided elements inherit the polynomial degree of their parent, cf. above, in this latter case, the distribution of the polynomial degrees on the resulting sub-elements is possibly non-uniform. Motivation. The key to the forthcoming hp-refinement strategy is to estimate the predicted reduction in the energy functional locally on each element in the finite element mesh T . With this in mind, we must first rewrite E as the sum of local contributions on T . Given that E is simply defined as an integral over Ω, then clearly, we may write E(v) = κ∈T E κ (v), where E κ is defined in an analogous fashion to E, with the integrals over Ω being restricted to integrals over κ, κ ∈ T . However, while the above definition of the local energy functionals E κ seems entirely natural, there is no guarantee that the computed error will converge optimally based on locally minimising E κ over each κ in T . In order to investigate this issue further and to motivate the idea proposed in this article, let us consider the following second-order linear self-adjoint partial differential equation: Find u such that −∆u + u = f in Ω, u = 0 on ∂Ω. Thereby, we have that µ(∇u) = 1 /2|∇u| 2 and g(u) = 1 /2 u 2 , and X = H 1 0 (Ω) (i.e., p = q = 2). In this setting, the (global) energy functional from (2.4) may be written in the form E(u) = 1 2 a Ω (u, u) − Ω (u). Moreover, we may define the associated energy norm: u 2 E := a Ω (u, u). Given the energy norm, exploiting the symmetry of the bilinear form a Ω (·, ·), we immediately deduce the following relationship between the error in the computed energy functional E, and the error measured in the terms of the energy norm · E , namely: E(u ) − E(u hp ) = − 1 2 u − u hp 2 E . Thereby, on a global level, reduction of the error in the energy functional E naturally leads to a reduction in the energy norm of the error. In order to repeat this argument on a subset D ⊂ Ω, we now suppose that the boundary datum g is given and seek u ∈ H 1 (D) such that u | ∂D = g and (4.1) a D (u , v) = D (v) ∀v ∈ H 1 0 (D). Here, a D (·, ·) and D (·) are defined in an analogous manner to a Ω (·, ·) and Ω (·), respectively, with the domain of integration restricted to D. In this case, writing T D and p D to denote the finite element sub-mesh and polynomial degree distribution over D, respectively, the finite element approximation is given by: Find u hp ∈ V(T D , p D ) such that u hp | ∂D = Πg and a D (u hp , v) = D (v) ∀v ∈ V 0 (T D , p D ), where Πg denotes a piecewise polynomial approximation in H 1 /2 (∂D) of the Dirichlet datum g. Thereby, writing E D to denote the restriction of the energy functional E over D, i.e., E D (u) := 1 2 a D (u, u) − D (u), we deduce the following identity: E D (u ) − E D (u hp ) = − 1 2 a D (u − u hp , u − u hp ) + a D (u , u − u hp ) − D (u − u hp ). Employing integration by parts we deduce that a D (u , u − u hp ) − D (u − u hp ) = ∂D ∂u ∂n D (u − u hp ) ds = ∂D µ (∇u ) · n D (u − u hp ) ds, where n D denotes the unit outward normal vector on the boundary ∂D of the domain D. Thereby, (4.2) E D (u )−E D (u hp ) = − 1 2 a D (u −u hp , u −u hp )+ ∂D µ (∇u )·n D (u −u hp ) ds. Stimulated by (4.2), we define the local energy functionalẼ D (·) by (4.3)Ẽ D (v) := E D (v) − ∂D µ (∇u ) · n D v ds. With this definition, we immediately deduce the following relationship between the error in the local energy functional and the error measured in terms of the local energy norm, namely,Ẽ D (u ) −Ẽ D (u hp ) = − 1 2 u − u hp 2 E,D , where, for w ∈ H 1 (D), we let w 2 E,D := a D (w, w). Moreover, if we consider the evaluation of the above local energy functional on each element κ, κ ∈ T , then we note the following consistency condition holds E(v) ≡ κ∈TẼ κ (v). Let us now write u hp ∈ V(T D , p D ) and u , hp ∈ V(T D , p D ) to denote two finite element approximations to (4.1) based on employing the computational meshes T D and T D , respectively, with polynomial degree vectors p D and p D , respectively. Assuming the finite element space V(T D , p D ) represents an enrichment of the original one V(T D , p D ), we deduce that the expected reduction in the error in the energy functional defined over D satisfies the equalitỹ E D (u hp ) −Ẽ D (u , hp ) = (Ẽ D (u ) −Ẽ D (u , hp )) − (Ẽ D (u ) −Ẽ D (u hp )) = 1 2 u − u hp 2 E,D − 1 2 u − u , hp 2 E,D . (4.4) Hence, by employing the modified local definition of the energy functionalẼ D defined over the subdomain D, we observe that the expected reduction inẼ D is directly related to the reduction in the energy norm of the error over D. The equality (4.4) will form the basis of the proceeding hp-adaptive refinement algorithm. 4.2. Competitive hp-refinement strategy. In this section, we develop an hpadaptivity algorithm based on employing a competitive refinement strategy on each element κ in the computational mesh T . The essential idea is to compute the maximal predicted energy reductionẼ κ (u hp ) −Ẽ κ (u κ,loc ) on each element κ ∈ T , where u hp is the (global) finite element element solution defined by (3.1), and u κ,loc is the (local) finite element approximation to the analytical solution u evaluated on a local patch of elements neighbouring κ, subject to a given p-/hp-refinement. Employing the forthcoming notation u κ,loc will either represent u κ,p ∈ V(T N κ , p p ), cf. (4.8), or u κ,hp i ∈ V(T N κ,ref , p hp i ), i = 1, . . . , N κ,hp , cf. (4.9), corresponding to either a local por hp-refinement of element κ, respectively. Elements with the largest maximal predicted decrease in the local energy functional are then appropriately refined. However, before we proceed, we first note that the boundary correction term included within the definition of the local energy functionalẼ κ (·), cf. (4.3) with D replaced by κ, is not computable since it directly assumes knowledge of the unknown analytical solution u . With this in mind, we replace u by an approximate reference solution, cf. [11,29]. However, in contrast to these citations, for the purposes of the current article we simply compute local reference solutions, rather than global ones. More precisely, given κ ∈ T , we first construct the local mesh T N κ comprising of κ and its immediate face-wise neighbours, cf. Fig. 1(b). Given (4.5) a D(κ) (u κ,ref , v) = D(κ) (v) ∀v ∈ V 0 (T N κ,ref , p ref ) . On the basis of the computed reference solution, we define the approximate local energy functional on κ, κ ∈ T , as follows: (4.6)Ẽ κ (v) := E κ (v) − ∂κ { {µ (∇u κ,ref )} } · n κ v ds, where n κ denotes the unit outward normal vector to the boundary ∂κ of κ, and { {·} } denotes the average operator. More precisely, given two neighbouring elements κ + and κ − , let x be an arbitrary point on the interior face given by F = ∂κ + ∩ ∂κ − . Given a vector-valued function q which is smooth inside each element κ ± , we write q ± to denote the traces of q on F taken from within the interior of κ ± , respectively. Then, the average of q at x ∈ F is given by { {q} } = 1 2 (q + + q − ). On a boundary face F ⊂ ∂Ω, we set { {q} } = q + . With the definition ofẼ κ (·) given in (4.6), we now outline the proposed competitive refinement strategy on element κ, κ ∈ T . Firstly, we compute the predicted energy functional reduction when p-refinement is employed, i.e., (4.7) ∆Ẽ κ,p :=Ẽ κ (u hp ) −Ẽ κ (u κ,p ), where u κ,p is the solution of the local finite element problem: Find u κ,p ∈ V(T N κ , p p ) such that u κ,p | ∂D(κ) = u hp | ∂D(κ) and (4.8) a D(κ) (u κ,p , v) = D(κ) (v) ∀v ∈ V 0 (T N κ , p p ) ; here, p p | κ = p κ + 1 for all κ ∈ T N κ . Secondly, we also consider a sequence of competitive hp-refinements, such that the number of degrees of freedom associated with the finite element space defined over κ is identical to the case when pure p-refinement has been employed. Here, for each element κ ∈ T , we again exploit the same local mesh T N κ,ref employed for the computation of the local reference solution u κ,ref . Then for the elements which result from the isotropic refinement of κ, we employ local polynomial degrees p κi , i = 1, . . . , n κ ; for the remaining elements stemming from the refinement of the neighbours of κ, we simply set the local polynomial degree equal to p κ , cf. Fig. 2. For example, in one-dimension, following [11,29], given an element κ with polynomial degree p κ , an enrichment of p κ → p κ + 1 gives rise to p κ + 2 degrees of freedom associated with κ. On the other hand, we can now consider the case when κ is uniformly subdivided into two sub-elements κ 1 and κ 2 , i.e., n κ = 2, with associated polynomial degrees p κ1 and p κ2 , respectively. To ensure that the number of degrees of freedom in the underlying hp-refined finite element space defined over κ 1 and κ 2 is identical to the case when pure p-enrichment is undertaken, we require that p κ1 + p κ2 = p κ + 1. Hence, there are N κ,hp = p κ , hp-competitive refinements and one p-refinement in one-dimension. In higher-dimensions, the construction of the competitive hp-refinements is undertaken in an analogous manner. For simplicity, we focus on the two-dimensional case when triangular elements are employed. Then for the elements which result from the isotropic refinement of κ, we employ local polynomial degrees p κi , i = 1, . . . , n κ = 4; as before, the local polynomial degree of the remaining elements stemming from the refinement of the neighbours of κ is set equal to p κ . Let us signify the set of all such polynomial degree distributions on T N κ,ref by P κ,pκ . Given that the full space of polynomials has been employed for the p-refinement, the number of degrees of freedom associated with κ is 1 /2(p κ + 2)(p κ + 3). Then, for an arbitrary polynomial degree distribution {p κi } 4 i=1 for the sub-elements {κ i } 4 i=1 of κ, the number of degrees of freedom associated with κ is 6 + 3 i=1 [min(p κi , p κ4 ) − 1 + 2(p κi − 1)] + 1 2 4 i=1 (p κi − 1)(p κi − 2), where we have assumed that κ 4 is the sub-element located at the interior of κ, cf. Fig. 2(b). Thereby, we select the set of hp-refinements which satisfy the condition 6 + 3 i=1 [min(p κi , p κ4 ) − 1 + 2(p κi − 1)] + 1 2 4 i=1 (p κi − 1)(p κi − 2) = 1 2 (p κ + 2)(p κ + 3). Analogous expressions can also be determined for different element types, other kinds of refinement, e.g., anisotropic refinement, as well as in higher-dimensions. The precise number of competitive hp-refinements, denoted by N κ,hp , is not possible to determine in a simple closed form expression; instead, N κ,hp can be precomputed for any polynomial order. To this end, in Fig. 3 we present the number of combinations of local polynomial degrees {p κi } 4 i=1 with respect to p κ in the above setting, i.e., for the case of isotropic refinement of a triangular element in two-dimensions. We notice that the number N κ,hp of possible p-configurations is, not surprisingly, growing as p κ increases. In view of this observation we remark that, although the subsequent local discrete problems defined on each corresponding (patchwise) hp-finite element space, cf. (4.9) below, are extremely inexpensive to compute, and moreover are trivially parallelisable, from a practical point of view, it might be computationally beneficial to limit the number of samples to a certain preset maximum N max . For example, a random selection of N max samples may be considered, cf. Section 5 below; alternatively, a more sophisticated strategy selecting polynomial degree distributions with limited variations could be employed. We now write V(T N κ,ref , p hp i ), i = 1, . . . , N κ,hp , to denote the finite element space based on employing the local (refined) mesh T N κ,ref and some local polynomial degree distribution p hp i ∈ P κ,pκ . Thereby, the following competitive hp-refinements may be defined: Find u κ,hp i ∈ V(T N κ,ref , p hp i ) such that u κ,hp i | ∂D(κ) = u hp | ∂D(κ) and (4.9) a D(κ) (u κ,hp i , v) = D(κ) (v) ∀v ∈ V 0 (T N κ,ref , p hp i ), for i = 1, . . . , N κ,hp . For each local competitive hp-refinement, we compute the estimated local energy reduction (4.10) ∆Ẽ κ,hp i :=Ẽ κ (u hp ) −Ẽ κ (u κ,hp i ), for i = 1, . . . , N κ,hp . In this way, for each element κ ∈ T , we may compute the maximum local predicted error reduction (4.11) ∆Ẽ κ,max = max ∆Ẽ κ,p , max i=1,...,N κ,hp ∆Ẽ κ,hp i , with ∆Ẽ κ,p from (4.7). Finally, we refine the set of elements κ ∈ T which satisfy the condition (4.12) ∆Ẽ κ,max > θ max κ∈T ∆Ẽ κ,max , where 0 < θ < 1 is a given parameter, cf. [11,29]. On the basis of [11,29], throughout this article, we set θ = 1 /3. The above competitive hp-refinement strategy is summarised in Algorithm 1. Numerical Examples In this section we present a series of numerical experiments to demonstrate the practical performance of the proposed hp-adaptive refinement strategy outlined in Algorithm 1. 5.1. Example 1: Linear Elliptic Problem. In this first example, we consider a one-dimensional problem defined over the domain Ω = (0, 1). Moreover, we set µ(u x ) = 1 /2 ε u 2 x , ε > 0, g(u) = 1 /2 u 2 , and f (x) = 1; this is equivalent to solving the linear elliptic boundary value problem: −εu xx + u = 1, x ∈ Ω, subject to homogeneous Dirichlet boundary conditions. We note that the analytical solution is given by u (x) = e − 1 / √ ε − 1 e 1 / √ ε − e − 1 / √ ε e x / √ ε + 1 − e 1 / √ ε e 1 / √ ε − e − 1 / √ ε e − x / √ ε + 1. In particular, for 0 < ε 1, the analytical solution u contains boundary layers in the vicinity of x = 0 and x = 1, cf. [33]; as in [33], we set ε = 10 −5 . In Fig. 4 we illustrate the performance of the proposed hp-adaptive algorithm, cf. Algorithm 1, based on a starting mesh consisting of 4 elements, with the initial polynomial degree p = [1, 1, 1, 1]. Here, we have plotted the error in the underlying energy functional E, together with the energy norm · E and L 2 (Ω) norm of the Algorithm 1 Competitive hp-adaptive refinement procedure 1: Choose a coarse initial mesh T 0 of Ω and a corresponding low-order starting polynomial degree vector p 0 . Set n = 0. 2: Solve (3.1) for u hp ∈ V(T n , p n ). 3: for each element κ ∈ T n do 4: Construct the local reference mesh T N κ,ref . 5: Compute the local finite element reference solution u κ,ref ∈ V(T N κ,ref , p ref ) satisfying (4.5). 6: Compute the local finite element p-enriched solution u κ,p ∈ V(T N κ , p p ) satisfying (4.8), together with the corresponding predicted energy functional reduction ∆Ẽ κ,p , cf. (4.7). 7: for i = 1, . . . , N κ,hp do 8: Compute the local competitive hp-refined finite element solutions u κ,hp i ∈ V(T N κ,ref , p hp i ) satisfying (4.9), together with their respective predicted energy functional reduction ∆Ẽ κ,hp i defined in (4.10). Compute the maximum local predicted error reduction ∆Ẽ κ,max , cf. (4.11). 11: end for 12: Determine the set of elements K n which are flagged for refinement, based on the criterion (4.12). 13: Perform por hp-refinement on each κ ∈ K n according to which refinement takes the maximum in (4.11). This results in a refined global finite element space V(T n+1 , p n+1 ). 14: Set n ← n + 1, and goto Line 2. 15: After sufficiently many iterations have been performed output the final solution u hp ∈ V(T n , p n ). error, with respect to the total number of degrees of freedom employed within the finite element space V(T , p), on a linear-log scale; here, v 2 E = 1 0 (εv 2 x + v 2 ) dx. From Fig. 4(a), (b), & (c), we observe, that after an initial transient, the convergence lines for each error measure become (on average) straight, thereby indicating exponential convergence of the quantities |E(u ) − E(u hp )|, u − u hp E , and u −u hp L 2 (Ω) , respectively, as V(T , p) is adaptively enriched. Finally, in Fig. 4(d) we show the hp-mesh distribution after 9 adaptive refinements. Here, we observe that the algorithm clearly identifies the location of the boundary layers present in the analytical solution u ; indeed, in these regions, local subdivision of the mesh has first been employed, followed by subsequent p-enrichment, cf. [33]. Thereby, the corresponding Euler-Lagrange equation for the underlying minimisation problem corresponds to the strongly monotone quasilinear PDE given by: −∇ · 1 + e −|∇u | 2 ∇u =f, in Ω. (5.1) We select f and appropriate inhomogeneous Dirichlet boundary conditions so that the analytical solution to (5.1) is given by u = r 2 /3 sin 2 3 ϕ , where (r, ϕ) denote the system of polar coordinates, cf. [10,20], for example. Selecting the energy norm · E to be the standard H 1 (Ω) norm, in Fig. 5 we again present the convergence history of the error in the computed energy functional E, together with u − u hp E , and u − u hp L 2 (Ω) , as the finite element space is hp-adaptively refined. On a linear-log scale (where the horizontal axis measures the third root of the total number of the degrees of freedom, cf. [28]), we again observe exponential rates of convergence, in the sense that asymptotically the convergence lines become roughly straight. In addition, in Fig. 5 we also present analogous results in the case when a Monte Carlo (MC) approach is employed to limit the number N max of hp-refinement samples considered on each element. More precisely, we randomly select samples based on employing N max = 10 and N max = 15; in each case two typical realisations are presented. Here, we observe a slight degradation of the rate of convergence in each of the above error quantities as our hp-refinement procedure progresses, as we would expect; however, in each case exponential convergence is retained when this simple selection principle is exploited. As noted in Section 4.2 more sophisticated selection principles may also be employed. The final hp-mesh distribution is depicted in Fig. 6; here, we see that the computational mesh has been largely refined in the vicinity of the re-entrant corner located at the origin. In addition, we see that the polynomial degrees have been increased away from the origin, since the underlying analytical solution is smooth in this region. In particular, we observe that the refinement algorithm has generated an hp-mesh distribution which is symmetric with respect to the line x 2 = −x 1 . i.e., we have µ(∇u) = 1 /p|∇u| p and g = 0. We select f , and impose suitable inhomogeneous Dirichlet boundary conditions, so that the analytical solution of (5.2) is given by u (x) = r α , α > 0. As in [1], throughout this section, we set p = 3 and α = 3 /4, which implies that u ∈ W β− , 3 (Ω), where β = 13 /6 and > 0 is arbitrarily small. In Fig. 7 we plot |E(u ) − E(u hp )|, u − u hp W 1,3 (Ω) , and u − u hp L 3 (Ω) , with respect to the third root of the number of degrees of freedom in V(T , p). As in the previous examples, we again observe exponential convergence of each of the above error measures, as the finite element space is hp-adaptively modified. Here, we also consider the case when N max = 30 random samples are selected; as in the previous example, we again see that exponential convergence of each of the above error quantities is retained, though the rate of convergence is inferior when compared to the case when all potential trial hp-refinements are considered. The final hp-mesh distribution is shown in Fig. 8; as in the previous examples, the adaptive algorithm clearly identifies the location of the singularity present within the analytical solution u , whereby h-refinement is undertaken. Conclusions In this article, we have proposed a novel hp-adaptive refinement procedure for application to the finite element approximation of convex variational problems. In particular, the underlying adaptive algorithm exploits a competitive refinement technique which seeks to maximise the decrease in the elemental contribution to the total energy based on employing local pand hp-enrichments of the finite element space. Whilst our approach has been successfully applied to a range of secondorder quasilinear problems in both one-and two-dimensions, we emphasise that it is immediately extensible to more general variational-based PDE problems. Future work will be concerned with exploiting anisotropic hp-mesh adaptation. Figure 1 . 1Local element patches in two-dimensions, when triangular elements are employed. (a) Original element κ (assumed to be an interior element); (b) Mesh patch T N κ , which consists of the element κ and its neighbours; (c) Mesh patch T N κ,ref which is constructed based on isotropically refining κ (red refinement) and on a green refinement of its neighbours. T N κ , we then uniformly (red) refine element κ into n k sub-elements; the introduction of any hanging nodes may then be removed by introducing additional (green) refinements, or alternatively, by simply uniformly refining all elements in the submesh T N κ . For the purposes of the article, in two-dimensions, we exploit the former strategy, purely on the basis of reducing the number of degrees of freedom in the underlying local finite element space. Denoting the resulting finite element mesh by T N κ,ref , cf.Fig. 1(c), we construct the finite element spaceV(T N κ,ref , p ref ), where p ref | κ = p κ + 1 for all κ ∈ T N κ,ref . Writing D(κ) = κ ∈T N κ,ref κ , the elementwise reference solution may be computed as follows: Find u κ,ref ∈ V(T N κ,ref , p ref ) such that u κ,ref | ∂D(κ) = u hp | ∂D(κ) and Figure 2 . 2Polynomial degree distribution employed for the competitive hp-refinements: (a) One-dimension; (b) Two-dimensional triangular element. Figure 3 . 3Number of competitive hp-refinements, N κ,hp , versus the local polynomial degree p κ when a triangular element κ is isotropically refined. 5. 2 . 2Example 2: Strongly monotone quasilinear PDE. In this second example, we let Ω be the L-shaped domain (−1, 1) 2 \ [0, 1) × (−1, 0], and set µ(∇u) = 1 2 |∇u| 2 − e −|∇u| 2 . Figure 4 . 4Example 1. Comparison of the error with respect to the number of degrees of freedom: (a) |E(u )−E(u hp )|; (b) u −u hp E ; (c) u − u hp L 2 (Ω) . (d) hp-Mesh distribution after 9 adaptive refinements. Figure 5 . 5Example 2. Comparison of the error with respect to the third root of the number of degrees of freedom: (a) |E(u )−E(u hp )|; (b) u − u hp E ; (c) u − u hp L 2 (Ω) . Figure 6 . 6Example 2. (a) hp-Mesh distribution after 18 adaptive refinements; (b) Zoom of (a).5.3.Example 3: p-Laplacian. In this final example, for p > 1, we consider the p-Laplacian problem(5.2) − ∇ · (|∇u | p−2 ∇u ) = f, in Ω = (0, 1) 2 ,subject to inhomogeneous Dirichlet boundary conditions. We point out that in this setting, (5.2) corresponds to the Euler-Lagrange equation for the energy minimi Figure 7 . 7Example 3. Comparison of the error with respect to the third root of the number of degrees of freedom: (a) |E(u )−E(u hp )|; (b) |u − u hp | W 1,3 (Ω) ; (c) u − u hp L 3 (Ω) . Figure 8 . 8Example 3. (a) hp-Mesh distribution after 23 adaptive refinements; (b) Zoom of (a). The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs. M Ainsworth, D Kay, Numer. Math. 82M. Ainsworth and D. Kay, The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs, Numer. Math. 82 (1999), 351-388. A posteriori error estimation in finite element analysis. M Ainsworth, J T Oden, Series in Computational and Applied Mathematics. ElsevierM. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, Series in Computational and Applied Mathematics, Elsevier, 1996. An adaptive refinement strategy for hp-finite element computations. M Ainsworth, B Senior, Appl. Numer. Math. 261-2M. Ainsworth and B. Senior, An adaptive refinement strategy for hp-finite element computa- tions, Appl. Numer. Math. 26 (1998), no. 1-2, 165-178. Regularity of the solution of elliptic problems with peicewise analytic data. Part I. Boundary value problems for linear elliptic equation of second order. I Babuška, B Q Guo, SIAM J. Math. Anal. 19I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with peicewise analytic data. Part I. Boundary value problems for linear elliptic equation of second order, SIAM J. Math. Anal. 19 (1988), 172-203. The treatment of nonhomogeneous Dirichlet boundary conditions by the p-version of the finite element method. I Babuška, M Suri, RAIRO Anal. Numér. 21SIAM ReviewI. Babuška and M. Suri, The hp-version of the finite element method with quasiuniform meshes, RAIRO Anal. Numér. 21 (1987), 199-238. 6. , The treatment of nonhomogeneous Dirichlet boundary conditions by the p-version of the finite element method, Numer. Math. 55 (1989), 97-121. 7. , The p and h-p versions of the finite element method, basic principles and properties, SIAM Review 36 (1994), 578-632. An optimal control approach to a-posteriori error estimation in finite element methods, Acta Numerica. R Becker, R Rannacher, A. IserlesCambridge University PressR. Becker and R. Rannacher, An optimal control approach to a-posteriori error estimation in finite element methods, Acta Numerica (A. Iserles, ed.), Cambridge University Press, 2001, pp. 1-102. An error indicator for mortar element solutions to the Stokes problem. C Bernardi, N Fiétier, R G Owens, IMA J. Numer. Anal. 214C. Bernardi, N. Fiétier, and R. G. Owens, An error indicator for mortar element solutions to the Stokes problem, IMA J. Numer. Anal. 21 (2001), no. 4, 857-886. Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs. S Congreve, P Houston, T P Wihler, J. Sci. Comput. 552S. Congreve, P. Houston, and T. P. Wihler, Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs, J. Sci. Comput. 55 (2013), no. 2, 471-497. L Demkowicz, Computing with hp-adaptive finite elements. Boca Raton, FLChapman & Hall/CRC1Maxwell problemsL. Demkowicz, Computing with hp-adaptive finite elements. Vol. 1, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007, One and two dimensional elliptic and Maxwell problems. A fully automatic hp-adaptivity. L Demkowicz, W Rachowicz, Ph Devloo, J. Sci. Comput. 171-4L. Demkowicz, W. Rachowicz, and Ph. Devloo, A fully automatic hp-adaptivity, J. Sci. Com- put. 17 (2002), no. 1-4, 117-142. Convergence of an adaptive hp finite element strategy in one space dimension. W Dörfler, V Heuveline, Appl. Numer. Math. 5710W. Dörfler and V. Heuveline, Convergence of an adaptive hp finite element strategy in one space dimension, Appl. Numer. Math. 57 (2007), no. 10, 1108-1124. T Eibner, J M Melenk, An adaptive strategy for hp-FEM based on testing for analyticity. 39T. Eibner and J. M. Melenk, An adaptive strategy for hp-FEM based on testing for analyticity, Comput. Mech. 39 (2007), no. 5, 575-595. Introduction to adaptive methods for differential equations. K Eriksson, D Estep, P Hansbo, C Johnson, Acta Numerica (A. Iserles. Cambridge University PressK. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica (A. Iserles, ed.), Cambridge University Press, 1995, pp. 105-158. E H Georgoulis, E Hall, P Houston, Discontinuous Galerkin methods on hp-anisotropic meshes II: A posteriori error analysis and adaptivity. 59E. H. Georgoulis, E. Hall, and P. Houston, Discontinuous Galerkin methods on hp-anisotropic meshes II: A posteriori error analysis and adaptivity, Appl. Numer. Math. 59(9) (2009), 2179-2194. Anisotropic hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows. S Giani, P Houston, Int. J. Numer. Anal. Model. 94S. Giani and P. Houston, Anisotropic hp-adaptive discontinuous Galerkin finite element meth- ods for compressible fluid flows, Int. J. Numer. Anal. Model. 9 (2012), no. 4, 928-949. Adaptive finite element approximation of hyperbolic problems, Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. P Houston, E Süli, P. Houston and E. Süli, Adaptive finite element approximation of hyperbolic problems, Er- ror Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. . Lect. Notes Comput. Sci. Engrg. T. Barth and H. Deconinck25SpringerLect. Notes Comput. Sci. Engrg. (T. Barth and H. Deconinck, eds.), vol. 25, Springer, 2002, pp. 269-344. A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. P Houston, E Süli, Comput. Methods Appl. Mech. Engrg. 1942-5P. Houston and E. Süli, A note on the design of hp-adaptive finite element methods for elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg. 194(2-5) (2005), 229-243. A posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasi-linear PDEs. P Houston, E Süli, T P Wihler, IMA J. Numer. Anal. 282P. Houston, E. Süli, and T. P. Wihler, A posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasi-linear PDEs, IMA J. Numer. Anal. 28 (2007), no. 2, 245-273. Spectral/hp finite element methods in cfd. G E Karniadakis, S Sherwin, Oxford University PressG. E. Karniadakis and S. Sherwin, Spectral/hp finite element methods in cfd, Oxford Univer- sity Press, 1999. Adaptive mesh strategies for the spectral element method. C Mavriplis, ICOSAHOM'92. Montpellier116C. Mavriplis, Adaptive mesh strategies for the spectral element method, Comput. Methods Appl. Mech. Engrg. 116 (1994), no. 1-4, 77-86, ICOSAHOM'92 (Montpellier, 1992). On residual-based a posteriori error estimation in hp-FEM. J M Melenk, B I Wohlmuth, Adv. Comp. Math. 15J. M. Melenk and B. I. Wohlmuth, On residual-based a posteriori error estimation in hp-FEM, Adv. Comp. Math. 15 (2001), 311-331. A comparison of hp-adaptive strategies for elliptic partial differential equations. W F Mitchell, M A Mcclain, ACM Transactions on Mathematical Software (TOMS). 239W. F. Mitchell and M. A. McClain, A comparison of hp-adaptive strategies for elliptic partial differential equations, ACM Transactions on Mathematical Software (TOMS) 41 (2014), 2:1- 2:39. J T Oden, A Patra, Y S Feng, An hp-adaptive strategy, Adaptive, Multilevel, and Hierarchical Computational Strategies. New YorkASME Publication157J. T. Oden, A. Patra, and Y. S. Feng, An hp-adaptive strategy, Adaptive, Multilevel, and Hierarchical Computational Strategies, vol. 157, ASME Publication, New York, 1992, pp. 23- 26. P H Rabinowitz, Minimax methods in critical point theory with applications to differential equations. Washington, DC; Providence, RIAmerican Mathematical Society65Published for the Conference Board of the Mathematical SciencesP. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Con- ference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. Toward a universal hp-adaptive finite element strategy. Part 3: Design of hp meshes. W Rachowicz, J T Oden, L Demkowicz, Comput. Methods Appl. Mech. Engrg. 77W. Rachowicz, J. T. Oden, and L. Demkowicz, Toward a universal hp-adaptive finite element strategy. Part 3: Design of hp meshes, Comput. Methods Appl. Mech. Engrg. 77 (1989), 181-212. C Schwab, p-and hp-FEM -Theory and application to solid and fluid mechanics. OxfordOxford University PressC. Schwab, p-and hp-FEM -Theory and application to solid and fluid mechanics, Oxford University Press, Oxford, 1998. Higher-order finite element methods, Studies in advanced mathematics. P Solin, K Segeth, I Dolezel, Chapman &Hall/CRCBoca Raton, LondonP. Solin, K. Segeth, and I. Dolezel, Higher-order finite element methods, Studies in advanced mathematics, Chapman &Hall/CRC, Boca Raton, London, 2004. Finite element analysis. B Szabó, I Babuška, Wiley & SonsNew YorkB. Szabó and I. Babuška, Finite element analysis, J. Wiley & Sons, New York, 1991. An h-p adaptive spectral element method for Stokes flow. J Valenciano, R G Owens, Appl. Numer. Math. 331-4J. Valenciano and R. G. Owens, An h-p adaptive spectral element method for Stokes flow, Appl. Numer. Math. 33 (2000), no. 1-4, 365-371. A review of a posteriori error estimation and adaptive mesh-refinement techniques. R Verfürth, B.G. Teubner, StuttgartR. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement tech- niques, B.G. Teubner, Stuttgart, 1996. An hp-adaptive strategy based on continuous Sobolev embeddings. T P Wihler, J. Comput. Appl. Math. 235T. P. Wihler, An hp-adaptive strategy based on continuous Sobolev embeddings, J. Comput. Appl. Math. 235 (2011), 2731-2739. Variational methods and optimization, Translated from the German by L. F. Boron. MR 768749 (90b:49005) 35. , Nonlinear functional analysis and its applications. E Zeidler, Nonlinear functional analysis and its applications. III. New York; New YorkSpringer-VerlagNonlinear monotone operators. Translated from the German by the author and L. F. Boron. MR 1033498 (91b:47002E. Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985, Variational methods and optimization, Translated from the German by L. F. Boron. MR 768749 (90b:49005) 35. , Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990, Nonlinear monotone operators, Translated from the German by the author and L. F. Boron. MR 1033498 (91b:47002) Sidlerstrasse 5, CH-3012 Bern, Switzerland E-mail address: [email protected]. Mathematisches Institut, Universität Bern, chMathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzer- land E-mail address: [email protected]
[]
[ "Performance of DPSK Signals with Quadratic Phase Noise", "Performance of DPSK Signals with Quadratic Phase Noise" ]
[ "Keang-Po Ho " ]
[]
[]
Nonlinear phase noise induced by the interaction of fiber Kerr effect and amplifier noises is a quadratic function of the electric field. When the dependence between the additive Gaussian noise and the quadratic phase noise is taking into account, the error probability for differential phase-shift keying (DPSK) signals is derived analytically. Depending on the number of fiber spans, the signal-to-noise ratio (SNR) penalty is increased by up to 0.23 dB due to the dependence between the Gaussian noise and the quadratic phase noise.
10.1109/tcomm.2005.852831
[ "https://arxiv.org/pdf/physics/0404058v1.pdf" ]
15,684,187
physics/0404058
f333bc29ba64e1586c3023d693986ace7c837d0d
Performance of DPSK Signals with Quadratic Phase Noise 12 Apr 2004 October 15, 2018 Keang-Po Ho Performance of DPSK Signals with Quadratic Phase Noise 12 Apr 2004 October 15, 2018arXiv:physics/0404058v1 [physics.optics]phase modulationerror probabilityfiber Kerr effectsnon- linear phase noise Nonlinear phase noise induced by the interaction of fiber Kerr effect and amplifier noises is a quadratic function of the electric field. When the dependence between the additive Gaussian noise and the quadratic phase noise is taking into account, the error probability for differential phase-shift keying (DPSK) signals is derived analytically. Depending on the number of fiber spans, the signal-to-noise ratio (SNR) penalty is increased by up to 0.23 dB due to the dependence between the Gaussian noise and the quadratic phase noise. Introduction Other than the projection of additive Gaussian noise to the phase, phase noises from other sources can be considered as multiplicative noise that adds directly to the phase of the received signal. When the local oscillator is not locked perfectly into the signal, the noisy reference gives additive phase noise [1,2]. Laser phase noise degrades coherent optical communication systems [3][4][5]. Those types of extra additive phase noise that add directly into the signal phase are independent of the additive Gaussian noise. In this paper, the additive phase noise is quadratic function of the electric field. When the electric field is contaminated with additive Gaussian noise, although the quadratic phase noise is uncorrelated with the linear phase noise, both non-Gaussian distributed, the phase noise weakly depends on the additive Gaussian noise. Differential phase-shift keying (DPSK) signals [6][7][8][9][10][11][12][13][14][15][16] have received renewed attention recently for long-haul or spectrally efficiency lightwave transmission systems. When optical amplifiers are used periodically to compensate the fiber loss, the interaction of optical amplifier noise and fiber Kerr effect induced nonlinear phase noise, often called Gordon-Mollenauer effect [17], or more precisely, nonlinear phase noise induced by self-phase modulation. Added directly into the signal phase, Gordon-Mollenauer effect is a quadratic function of the electric field and degrades DPSK signal [11,14,[17][18][19][20][21][22][23]. Previous studies found the variance or the corresponding Q-factor of the quadratic phase noise [11,17,[24][25][26][27] or the spectral broadening of the signal [14,18,28]. Recently, quadratic phase noise is found to be non-Gaussian distributed both experimentally [20] and theoretically [29,30]. As non-Gaussian random variable, neither the variance nor Q-factor is sufficient to completely characterize the phase noise. The probability density of quadratic phase noise is found in [30] and used in [23] to evaluate the error probability of DPSK signal by assuming that quadratic phase noise and Gaussian noise are independent of each other. However, as shown in the simulation of [22,23], the dependence between Gaussian noise with quadratic phase noise increases the error probability. Using the distributed assumption of infinite number of fiber spans, the joint statistics of nonlinear phase noise and Gaussian noise is derived analytically by [19,21,31]. The characteristic function of nonlinear phase noise becomes a very simple expression with the distributed assumption [29]. The error probability of DPSK signal has been derived with [22] and without [21,32] the assumption that nonlinear phase noise is independent of the Gaussian noise. Based on the distributed assumption, it is found that the dependence between linear and nonlinear phase noise increases both the error probability and SNR penalty [21,32]. The distributed assumption is very accurate when the number of fiber spans is larger than 32 [21,29]. For a typical fiber span length of 80 km, a fiber link of 32 spans has a total distance of over 2500 km. Most terrestrial fiber systems have an overall distance of less than 1000 km, the distributed assumption needs to be verified for small number of fiber spans. Recently, DPSK signals have been used in systems with small number of fiber spans [16,33,34]. Of course, the independence assumption can be used for either small [23] or large [22] number of fiber spans. However, as shown in [21,32], the independence assumption of [22,23] underestimates both the error probability and the required SNR, contradicting to the principles of conservative system design. In this paper, taking into account the dependence between the quadratic phase noise and Gaussian noise, the error probability of DPSK signal is derived for finite number of fiber spans, to our knowledge, the first time. Comparing with the independence approximation of [23], the dependence between the quadratic phase noise and Gaussian noise increases the error probability of the system. In the remaining parts of this paper, Sec. 2 gives the model of the quadratic phase noise, mostly follows the approaches of [30]; Sec. 3 derives the joint statistics of the additive Gaussian noise and the quadratic phase noise; Using the joint statistics, Sec. 4 gives the exact error probability of DPSK signals with quadratic phase noise, taking into account the dependence between the additive Gaussian noise and quadratic phase noise; Sec. 5 calculates the error probability and the SNR penalty of DPSK signals, and compared with the independence approximation of [23]; Sec. 6 is the conclusion of the paper. Quadratic Nonlinear Phase Noise For an N -span systems, for simplicity and without loss of generality, the overall quadratic phase noise is [17,25,26,30] Φ NL = | E 0 + n 1 | 2 + | E 0 + n 1 + n 2 | 2 + · · · + | E 0 + n 1 + · · · + n N | 2 ,(1) where E 0 = (A, 0) is a two-dimensional vector as the baseband representation of the transmitted electric field, n k , k = 1, . . . , N , are independent identically distributed (i.i.d.) zero-mean circular Gaussian random complex numbers as the optical amplifier noise introduced into the system at the k th fiber span. Both electric field of E 0 and amplifier noises of n k in (1) can also be represented as complex number. The variance of n k is E{| n k | 2 } = 2σ 2 0 , k = 1, . . . , N , where σ 2 0 is the noise variance per span per dimension. In (1), the constant factor of the product of fiber nonlinear coefficient and the effective nonlinear length per span, γL eff , is ignored for simplicity. Without affected the SNR, both signal and noise in (1) can be scaled by the same ratio for different mean nonlinear phase shift of <Φ NL >= N A 2 +N (N +1)σ 2 0 except the case without quadratic phase noise of <Φ NL >= 0. After the scaling, the mean nonlinear phase shift is approximately equal to the product of number of fiber spans and the launched power per span, especially for the usual case of large SNR with small noise. In the linear regime, ignoring the fiber loss of the last span and the amplifier gain required to compensate it, the signal received after N spans is E N = E 0 + n 1 + n 2 + · · · + n N(2) with a power of P N = | E N | 2 and SNR of ρ s = A 2 /(2N σ 2 0 ). In Eqs. (1) and (2), the configuration of each fiber spans is assumed to be identical with the same length and launched power. In [30], using the method of [35,36], the characteristic function of the quadratic phase noise (1) is found to be Ψ Φ NL (ν) = N k=1 1 1 − 2jνσ 2 0 λ k exp jνA 2 ( v T k w) 2 /λ k 1 − 2jνσ 2 0 λ k . (3) where w = (N, N − 1, . . . , 2, 1) T , λ k , v k , k = 1, 2, . . . , N are the eigenvalues and eigenvectors of the covariance matrix C, respectively. The covariance matrix C = M T M with M =         1 0 0 · · · 0 1 1 0 · · · 0 1 1 1 · · · 0 . . . . . . . . . . . . . . . 1 1 1 · · · 1         .(4) The characteristic function of (3) is used to find the error probability of a DPSK signal in [23] based on the assumption that the quadratic phase noise of (1) is independent of the received electric field of (2). Joint Statistics of Gaussian Noise and Quadratic Phase Noise To find the dependence between the quadratic phase noise and the received electric field, the joint characteristic function of Ψ Φ NL , E N (ν, ω) = E exp(jνΦ NL + j ω · E N(5) will be derived here with Φ NL and E N given by (1) and (2), respectively. Similar to [21,30], with ω = (ω 1 , ω 2 ) and E N = (e 1 , e 2 ), we obtain jνϕ 1 + jω 1 e 1 = jνN A 2 + jω 1 A + 2jνA w T x + jω 1 w T I x + jν x T C x,(6) where ϕ 1 is given by ϕ 1 = |A + x 1 | 2 + |A + x 1 + x 2 | 2 + · · · + |A + x 1 + · · · + x N | 2 ,(7) with n i = (x i , y i ), i = 1, . . . , N , w I = (1, 1, . . . , 1) T , jω 1 e 1 = jω 1 (A + x 1 + x 2 + · · · + x N ) = jω 1 A + jω 1 w T I x, and x = (x 1 , x 2 , . . . , x N ) T . Similar to [30], using the N -dimensional Gaussian probability density function (p.d.f.) of ( 2πσ 2 0 ) − N 2 exp − x T x/2σ 2 0 for x, we obtain Ψ ϕ 1 ,e 1 (ν, ω 1 ) = e jνN A 2 +jω 1 A (2πσ 2 0 ) N 2 × exp 2jνA w T x + jω 1 w T I x − x T Γ x d x.(8) or Ψ ϕ 1 ,e 1 (ν, ω 1 ) = e jνN A 2 +jω 1 A (2σ 2 0 ) − N 2 det[Γ] − 1 2 × exp − νA w + 1 2 ω 1 w I T Γ −1 νA w + 1 2 ω 1 w I ,(9) where Γ = I/(2σ 2 0 ) − jνC and I is an N × N identity matrix. Similarly, using A = 0 in (9), we get Ψ ϕ 2 ,e 2 (ν, ω 2 ) = exp − 1 4 ω 2 2 w T I Γ −1 w I (2σ 2 0 ) N 2 det[Γ] 1 2 .(10) where ϕ 2 = y 2 1 + |y 1 + y 2 | 2 + · · · + |y 1 + · · · + y N | 2 . The joint characteristic function of Ψ Φ NL , E N (ν, ω) = Ψ ϕ 1 ,e 1 (ν, ω 1 )Ψ ϕ 2 ,e 2 (ν, ω 2 )(12) becomes Ψ Φ NL , E N (ν, ω) = Ψ Φ NL (ν) exp jω 1 m N (ν) − σ 2 N (ν) | ω| 2 2 ,(13) where Ψ Φ NL (ν) = exp jνN A 2 − ν 2 A 2 w T Γ −1 w (2σ 2 0 ) N det[Γ] ,(14)m N (ν) = A + jνA w T Γ −1 w I ,(15)σ 2 N (ν) = 1 2 w T I Γ −1 w I .(16) Based on the eigenvalues and eigenvectors of the covariance matrix C, the characteristic function of Ψ Φ NL (ν) becomes that of (3), and m N (ν) = A + 2jνσ 2 0 A N k=1 ( v T k w)( v T k w I ) 1 − 2jνσ 2 0 λ k = A N k=1 ( v T k w)( v T k w I )/λ k 1 − 2jνσ 2 0 λ k ,(17)σ 2 N (ν) = σ 2 0 N k=1 ( v T k w I ) 2 1 − 2jνσ 2 0 λ k .(18) The characteristic function of (13) is similar to the corresponding characteristic function with the distributed assumption [21]. If the number of spans N approaches infinite, the characteristic function should converge to that of [21]. Based on (13), we obtain F −1 ω Ψ Φ NL , E N = Ψ Φ NL (ν) 2πσ 2 N (ν) exp − | z − ξ ν | 2 2σ 2 N (ν) ,(19) with ξ ν = (m N (ν), 0), and F −1 ω {·} denotes inverse Fourier transform with respect to ω. The partial characteristic function and p.d.f. of (19) is similar to a two-dimensional Gaussian p.d.f. with mean of (m N (ν), 0) and variance of σ 2 N (ν). With the dependence on the quadratic phase noise, the variance of σ 2 N (ν) and the mean of m N (ν) are both complex numbers depending on the "angular frequency" of ν. The marginal p.d.f. of the received electric field E N is a two-dimensional Gaussian distribution with variance of σ 2 N (ν)| ν=0 = N σ 2 0 and mean of m N (ν)| ν=0 = A. With normalization, the corresponding joint characteristic of (19) in [21] has 20) when N → ∞. Based on joint statistics of (19), similar to that of [21,32,37], the exact error probability of DPSK signal can be derived analytically, even for case with linearly compensated nonlinear phase noise [23-25, 27, 38]. As shown in [21], the optimal compensation curve of [26,27] can also be derived using (19). σ 2 ∞ (ν) = 1 2 tan( √ jν) √ jν and m ∞ (ν) = sec( jν) √ ρ s( Exact Error Probability With nonlinear phase noise, assuming zero transmitted phase, the overall received phase is Φ r = Θ n − Φ NL(21) where Θ n is the phase of E N (2). The received phase is confined to the range of [−π, +π). The p.d.f. of the received phase is a periodic function with a period of 2π. If the characteristic function of the received phase is Ψ Φr (ν), the p.d.f. of the received phase has a Fourier series expansion of p Φr (θ) = 1 2π + 1 π +∞ m=1 ℜ {Ψ Φr (m) exp(−jmθ)} ,(22) where ℜ{·} denotes the real part of a complex number. In (22), we use the conjugate symmetry property of Ψ Φr (−ν) = Ψ * Φr (ν). In order to derive the Fourier coefficient of Ψ Φr (m), we need the joint characteristic function of Θ n and Φ NL at integer "angular frequency" of ν = m. Based on (19), using the same method as [19,21,32,39], we obtain Ψ Φ NL ,Θn (ν, m) = √ π 2 Ψ Φ NL (ν) γ(ν)e −γ(ν)/2 × I m−1 2 γ(ν) 2 + I m−1 2 γ(ν) 2 , m ≥ 0,(23) where γ(ν) = 1 2 m N (ν) 2 /σ 2 N (ν) is the complex-valued frequency dependence SNR parameter. When ν = 0, it is obvious that γ(ν)| ν=0 = ρ s . From (21), the Fourier coefficient in (22) is Ψ Φr (m) = Ψ Φ NL ,Θn (m, m). For DPSK signal, the differential received phase is ∆Φ r = Φ r (t) − Φ r (t − T ) in which the p.d.f.'s of Φ r (t) and Φ r (t − T ) are the same as that of (22). The p.d.f. of the differential received phase is the same as (22) with Fourier coefficient equal to |Ψ Φr (m)| 2 , i.e., p ∆Φr (θ) = 1 2π + 1 π +∞ m=1 |Ψ Φr (m)| 2 cos(mθ).(24) Similar to the procedure of [2,3,21,23,32,[39][40][41], the error probability becomes p e = 1 2 − 2 π +∞ k=0 (−1) k 2k + 1 |Ψ Φr (2k + 1)| 2 .(25) or p e = 1 2 − 1 2 ∞ k=0 (−1) k |r k e −r k | 2k + 1 I k r k 2 + I k+1 r k 2 2 × |Ψ Φ NL (2k + 1)| 2 ,(26) where r k = m 2 N (2k + 1) 2σ 2 N (2k + 1)(27) analogous to the "angular frequency" depending SNR as the ratio of complex power of 1 2 m 2 N (ν) to the noise variance of σ 2 N (ν). The error probability expression of (26) is almost the same as that in [21,32] but with a different parameter of (27). The error probability of (26) is also similar to the cases when additive phase noise is independent to Gaussian noise [2,3,23,40,41]. The frequency depending SNR is originated from the dependence between the additional phase noise and the Gaussian noise [19,21,32,37]. Numerical Results For DPSK signals with quadratic phase noise, Figure 1 shows the exact error probability as a function of SNR ρ s for mean nonlinear phase shift of <Φ NL >= 0.5 rad. Figure 2 shows the SNR penalty for an error probability of 10 −9 as a function of mean nonlinear phase shift < Φ NL >. The SNR penalty is defined as the additional required SNR to achieve the same error probability of 10 −9 . Both Figs. 1 and 2 are calculated using (26) and the independence approximation of [23]. The independence approximation of [23] underestimates both the error probability and SNR penalty of a DPSK signal with quadratic phase noise of (1). Both Figs. 1 and 2 also include the exact and approximated error probability for N = ∞ that are the distributed model from [32] and [22], respectively. The distributed model is applicable when the number of fiber spans is larger than 32. In Fig. 1, without quadratic phase noise of <Φ NL >= 0, the error probability is p e = exp(−ρ s )/2 [42]. The required SNR for systems without nonlinear phase noise of <Φ NL >= 0 is ρ s = 20 (13 dB) for an error probability of 10 −9 . From Figs. 1 and 2, for the same mean nonlinear phase shift of <Φ NL >, the SNR penalty is larger for smaller number of fiber spans. When the mean nonlinear phase shift is <Φ NL >= 0.56 rad, the SNR penalty is about 1 dB with large number (N > 32) of fiber spans but up to 3-dB SNR penalty for small number (N = 1, 2) of fiber spans. For 1-dB SNR penalty, the mean nonlinear phase shift is also reduced from 0.56 to 0.35 rad with small number of fiber spans. In [17], the optimal operating point is defined when the variance of quadratic phase noise is approximately equal to the variance of the phase of Gaussian noise. In [22,23], the optimal operating is calculated rigorously at the operation condition in which the increase of launched power does not improve the system performance. The optimal operating point is reduced from 0.97 to 0.55 rad with the decrease of the number of fiber spans. When the exact error probability is compared with the independence approximation of [23]. The independence approximation is closer to the exact error probability for small number of fiber spans. In all cases, the independence assumption of [22,23] underestimates the error probability of the system, contradicting to the conservative principle of system design. The dependence between linear and nonlinear phase noise increases the SNR penalty up to 0.23 dB. From the SNR penalty of Fig. 2, if a prior penalty of about 0.23 dB is added into the system, the independence assumption of [23] can be used to provide a conservative system design guideline. Conclusion For a system with small number of fiber spans, the exact error probability of a DPSK signal with quadratic phase noise is derived analytically the first time when the dependence between linear and nonlinear phase noise is taking into account. For the same mean nonlinear phase shift, the error probability increases for small number of fiber spans. The dependence between linear and nonlinear phase noises increases the error probability for DPSK signals. Depending on the number of fiber spans, the SNR penalty increases by up to 0.23 dB due to the dependence between Gaussian noise and the quadratic phase noise. For the same mean nonlinear phase shifts and SNR, the error probability of the system increases with the decrease of the number of fiber spans. As an example, the optimal operating point for system with large number (N > 32) is a mean nonlinear phase shift of about 1 rad that is reduced to about 0.55 rad for system with small number of fiber spans (N = 1, 2). Figure 1 : 1The error probability of DPSK signal as a function of SNR for N =1, 2, 4, 8, 32, and infinite number of fiber spans and mean nonlinear phase shift of <Φ NL >= 0.5 rad. Figure 2 : 2The SNR penalty vs. mean nonlinear phase shift <Φ NL >. Telecommunication Systems Engineering. W C Lindsey, M K Simon, Prentice-HallEnglewood Cliffs, N.J.W. C. Lindsey and M. K. Simon, Telecommunication Systems Engi- neering. Englewood Cliffs, N.J.: Prentice-Hall, 1973. Error probabilities in binary angle modulation. P C Jain, IEEE Trans. Info. Theory. 201P. C. Jain, "Error probabilities in binary angle modulation," IEEE Trans. Info. Theory, vol. IT-20, no. 1, pp. 36-42, 1974. Probability of error for optical heterodyne DPSK system with quantum phase noise. G Nicholson, Electron. Lett. 2024G. Nicholson, "Probability of error for optical heterodyne DPSK system with quantum phase noise," Electron. Lett., vol. 20, no. 24, pp. 1005- 1007, 1984. Characterizing filtered light waves corrupted by phase noise. G J Foschini, G Vannucci, IEEE Trans. Info. Theory. 346G. J. Foschini and G. Vannucci, "Characterizing filtered light waves corrupted by phase noise," IEEE Trans. Info. Theory, vol. 34, no. 6, pp. 1438-1448, 1988. Optical heterodyne binary-DPSK systems: A review of analysis and performance. P J Smith, M Shafi, C P Kaiser, IEEE J. Sel. Areas Commun. 133P. J. Smith, M. Shafi, and C. P. Kaiser, "Optical heterodyne binary- DPSK systems: A review of analysis and performance," IEEE J. Sel. Areas Commun., vol. 13, no. 3, pp. 557-568, 1995. 2.5 Tb/s (64 × 42.7 Gb/s) transmission over 40 × 100 km NZDSF using RZ-DPSK format and all-Raman-amplified spans. A H Gnauck, G Raybon, S Chandrasekhar, J Leuthold, C Doerr, L Stulz, A Agrawal, S Banerjee, D Grosz, S Hunsche, A Kung, A Marhelyuk, D Maymar, M Movassaghi, X Liu, C Xu, X Wei, D M Gill, Optical Fiber Commun. Conf., OFC '02. Anaheim, CApostdeadline paper FC2A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, A. Agrawal, S. Banerjee, D. Grosz, S. Hunsche, A. Kung, A. Marhelyuk, D. Maymar, M. Movassaghi, X. Liu, C. Xu, X. Wei, and D. M. Gill, "2.5 Tb/s (64 × 42.7 Gb/s) transmission over 40 × 100 km NZDSF using RZ-DPSK format and all-Raman-amplified spans," in Optical Fiber Commun. Conf., OFC '02, Anaheim, CA, 2002. post- deadline paper FC2. 25 40-Gb/s copolarized DPSK transmission over 12 100-km NZDF with 50-GHz channel spacing. A H Gnauck, G Raybon, S Chandrasekhar, J Leuthold, C Doerr, L Stulz, E Burrows, IEEE Photon. Technol. Lett. 153A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25 40-Gb/s copolarized DPSK transmission over 12 100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett., vol. 15, no. 3, pp. 467-469, 2003. DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km Ultra-Wave TM fiber spans. C Rasmussen, T Fjelde, J Bennike, F Liu, S Dey, B Mikkelsen, P Mamyshev, P Serbe, P Van De Wagt, Y Akasaka, D Harris, D Gapontsev, V Ivshin, P Reeves-Hall, Optical Fiber Commun. Conf., OFC '03. Atlanta, GApostdeadline paper PD18C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van de Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, and P. Reeves-Hall, "DWDM 40G trans- mission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km Ultra-Wave TM fiber spans," in Optical Fiber Commun. Conf., OFC '03, Atlanta, GA, 2003. postdeadline paper PD18. . B Zhu, L E Nelson, S Stulz, A H Gnauck, C Doerr, J Leuthold, L Grüner-Nielsen, M O Pederson, J Kim, R Lingle, Y Emori, Y Ohki, N Tsukiji, A Oguri, S Namiki, 6.4-Tb/s (160 × 42.7B. Zhu, L. E. Nelson, S. Stulz, A. H. Gnauck, C. Doerr, J. Leuthold, L. Grüner-Nielsen, M. O. Pederson, J. Kim, R. Lingle, Y. Emori, Y. Ohki, N. Tsukiji, A. Oguri, and S. Namiki, "6.4-Tb/s (160 × 42.7 Gb/s) transmission with 0.8 bit/s/Hz spectral efficiency over 32 × 100 km of fiber using CSRZ-DPSK format. Optical Fiber Commun. Conf., OFC '03. Atlanta, GApostdeadline paper PD19Gb/s) transmission with 0.8 bit/s/Hz spectral efficiency over 32 × 100 km of fiber using CSRZ-DPSK format," in Optical Fiber Commun. Conf., OFC '03, Atlanta, GA, 2003. postdeadline paper PD19. Transmission of 25-Gb/s RZ-DQPSK signals with 25-GHz channel spacing over 1000 km of SMF-28 fiber. P S Cho, V S Grigoryan, Y A Godin, A Salamon, Y Achiam, IEEE Photon. Technol. Lett. 153P. S. Cho, V. S. Grigoryan, Y. A. Godin, A. Salamon, and Y. Achiam, "Transmission of 25-Gb/s RZ-DQPSK signals with 25-GHz channel spacing over 1000 km of SMF-28 fiber," IEEE Photon. Technol. Lett., vol. 15, no. 3, pp. 473-475, 2003. Comparison of returnto-zero differential phase-shift keying and on-off keying long-haul dispersion managed transmission. C Xu, X Liu, L F Mollenauer, X Wei, IEEE Photon. Technol. Lett. 154C. Xu, X. Liu, L. F. Mollenauer, and X. Wei, "Comparison of return- to-zero differential phase-shift keying and on-off keying long-haul dis- persion managed transmission," IEEE Photon. Technol. Lett., vol. 15, no. 4, pp. 617-619, 2003. Transmission of 8 × 20 Gb/s DQPSK signals over 310-km SMF with 0.8 b/s/Hz spectral efficiency. H Kim, R.-J Essiambre, IEEE Photon. Technol. Lett. 155H. Kim and R.-J. Essiambre, "Transmission of 8 × 20 Gb/s DQPSK signals over 310-km SMF with 0.8 b/s/Hz spectral efficiency," IEEE Photon. Technol. Lett., vol. 15, no. 5, pp. 769-771, 2003. High spectral efficiency 1.6-b/s/Hz transmission (8×40 Gb/s with a 25-GHz grid) over 200-km SSMF using RZ-DQPSK and polarization multiplexing. C Wree, N Hecker-Denschlag, E Gottwald, P Krummrich, J Leibrich, E.-D Schmidt, B L W Rosenkranz, IEEE Photon. Technol. Lett. 159C. Wree, N. Hecker-Denschlag, E. Gottwald, P. Krummrich, J. Leibrich, E.-D. Schmidt, and B. L. W. Rosenkranz, "High spectral efficiency 1.6-b/s/Hz transmission (8×40 Gb/s with a 25-GHz grid) over 200-km SSMF using RZ-DQPSK and polarization multiplexing," IEEE Photon. Technol. Lett., vol. 15, no. 9, pp. 1303-1305, 2003. A comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise. T Mizuochi, K Ishida, T Kobayashi, J Abe, K Kinjo, K Motoshima, K Kasahara, J. Lightwave Technol. 219T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, "A comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degra- dations due to nonlinear phase noise," J. Lightwave Technol., vol. 21, no. 9, pp. 1933-1943, 2003. RZ-DPSK field trial over 13,100 km of installed non slope-matched submarine fibers. J.-X Cai, D G Foursa, L Liu, C R Davidson, Y Cai, W W Patterson, A J Lucero, B Bakhshi, G Mohs, P C Corbett, V Gupta, W Anderson, M Vaa, G Domagala, M Mazurczyk, H Li, M Nissov, A N Pilipetskii, N S Bergano, Optical Fiber Commun. Conf., OFC '04. Los Angeles, CApostdeadline paper PDP34J.-X. Cai, D. G. Foursa, L. Liu, C. R. Davidson, Y. Cai, W. W. Pat- terson, A. J. Lucero, B. Bakhshi, G. Mohs, P. C. Corbett, V. Gupta, W. Anderson, M. Vaa, G. Domagala, M. Mazurczyk, H. Li, M. Nissov, A. N. Pilipetskii, and N. S. Bergano, "RZ-DPSK field trial over 13,100 km of installed non slope-matched submarine fibers," in Optical Fiber Commun. Conf., OFC '04, Los Angeles, CA, 2004. postdeadline paper PDP34. 6 × 42.7-Gb/s transmission over ten 200-km EDFAamplified SSMF spans using polarization-alternating RZ-DPSK. A H Gnauck, J Leuthold, C Xie, I Kang, S Chandrasekhar, P Bernasconi, C Doerr, L Buhl, J D Bull, N A F Jaeger, H Kato, A Guest, Optical Fiber Commun. Conf., OFC '04. Los Angeles, CApostdeadline paper PDP35A. H. Gnauck, J. Leuthold, C. Xie, I. Kang, S. Chandrasekhar, P. Bernasconi, C. Doerr, L. Buhl, J. D. Bull, N. A. F. Jaeger, H. Kato, and A. Guest, "6 × 42.7-Gb/s transmission over ten 200-km EDFA- amplified SSMF spans using polarization-alternating RZ-DPSK," in Optical Fiber Commun. Conf., OFC '04, Los Angeles, CA, 2004. post- deadline paper PDP35. Phase noise in photonic communications systems using linear amplifiers. J P Gordon, L F Mollenauer, Opt. Lett. 1523J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic commu- nications systems using linear amplifiers," Opt. Lett., vol. 15, no. 23, pp. 1351-1353, 1990. Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers. S Ryu, J. Lightwave Technol. 1010S. Ryu, "Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers," J. Light- wave Technol., vol. 10, no. 10, pp. 1450-1457, 1992. Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers. A Mecozzi, J. Lightwave Technol. 1211A. Mecozzi, "Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers," J. Lightwave Technol., vol. 12, no. 11, pp. 1993-2000, 1994. Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise. H Kim, A H Gnauck, IEEE Photon. Technol. Lett. 152H. Kim and A. H. Gnauck, "Experimental investigation of the perfor- mance limitation of DPSK systems due to nonlinear phase noise," IEEE Photon. Technol. Lett., vol. 15, no. 2, pp. 320-322, 2003. Statistical properties of nonlinear phase noise. K.-P Ho, Advances in Optics and Laser Research. W. T. ArkinHauppauge, NYNova Science Publishers3K.-P. Ho, "Statistical properties of nonlinear phase noise," in Advances in Optics and Laser Research (W. T. Arkin, ed.), vol. 3, Hauppauge, NY: Nova Science Publishers, 2003. Performance degradation of phase-modulated systems with nonlinear phase noise. K.-P Ho, IEEE Photon. Technol. Lett. 159K.-P. Ho, "Performance degradation of phase-modulated systems with nonlinear phase noise," IEEE Photon. Technol. Lett., vol. 15, no. 9, pp. 1213-1215, 2003. Compensation improvement of DPSK signal with nonlinear phase noise. K.-P Ho, IEEE Photon. Technol. Lett. 159K.-P. Ho, "Compensation improvement of DPSK signal with nonlinear phase noise," IEEE Photon. Technol. Lett., vol. 15, no. 9, pp. 1216- 1218, 2003. Postnonlinearity compensation with data-driven phase modulators in phase-shift keying transmission. C Xu, X Liu, Opt. Lett. 2718C. Xu and X. Liu, "Postnonlinearity compensation with data-driven phase modulators in phase-shift keying transmission," Opt. Lett., vol. 27, no. 18, pp. 1619-1621, 2002. Improving transmission performance in differential phase-shift-keyed systems by use of lumped nonlinear phase-shift compensation. X Liu, X Wei, R E Slusher, C J Mckinstrie, Opt. Lett. 2718X. Liu, X. Wei, R. E. Slusher, and C. J. McKinstrie, "Improving trans- mission performance in differential phase-shift-keyed systems by use of lumped nonlinear phase-shift compensation," Opt. Lett., vol. 27, no. 18, pp. 1616-1618, 2002. The optimal compensator for nonlinear phase noise. K.-P Ho, Opt. Commun. 2114-6K.-P. Ho, "The optimal compensator for nonlinear phase noise," Opt. Commun., vol. 211, no. 4-6, pp. 419-425, 2003. Electronic compensation technique to mitigate nonlinear phase noise. K.-P Ho, J M Kahn, J. Lightwave Technol. 223K.-P. Ho and J. M. Kahn, "Electronic compensation technique to mit- igate nonlinear phase noise," J. Lightwave Technol., vol. 22, no. 3, pp. 779-783, 2004. System performance of coherent transmission over cascaded in-line fiber amplifiers. S Saito, M Aiki, T Ito, J. Lightwave Technol. 112S. Saito, M. Aiki, and T. Ito, "System performance of coherent trans- mission over cascaded in-line fiber amplifiers," J. Lightwave Technol., vol. 11, no. 2, pp. 331-342, 1993. Asymptotic probability density of nonlinear phase noise. K.-P Ho, Opt. Lett. 2815K.-P. Ho, "Asymptotic probability density of nonlinear phase noise," Opt. Lett., vol. 28, no. 15, pp. 1350-1352, 2003. Probability density of nonlinear phase noise. K.-P Ho, J. Opt. Soc. Amer. B. 209K.-P. Ho, "Probability density of nonlinear phase noise," J. Opt. Soc. Amer. B, vol. 20, no. 9, pp. 1875-1879, 2003. Long-distance transmission at zero dispersion: combined effect of Kerr nonlinearity and the noise of the in-line amplifiers. A Mecozzi, J. Opt. Soc. Amer. B. 113A. Mecozzi, "Long-distance transmission at zero dispersion: combined effect of Kerr nonlinearity and the noise of the in-line amplifiers," J. Opt. Soc. Amer. B, vol. 11, no. 3, pp. 462-469, 1994. Exact error probability of phase-modulated signals with nonlinear phase noise. K.-P Ho, J. Lightwave Technol. submitted, included in the online version of [21K.-P. Ho, "Exact error probability of phase-modulated signals with non- linear phase noise," J. Lightwave Technol., 2003. submitted, included in the online version of [21]. S-band WDM coherent transmission of 40 × 43-Gbit/s CS-RZ DPSK signals over 400 km DSF using hybrid GS-TDFAs/Raman amplifiers. Y Miyamoto, H Masuda, A Hirano, S Kuwahara, Y Kisaka, H Kawakami, M Tomizawa, Y Tada, S Aozasa, Electron. Lett. 3824Y. Miyamoto, H. Masuda, A. Hirano, S. Kuwahara, Y. Kisaka, H. Kawakami, M. Tomizawa, Y. Tada, and S. Aozasa, "S-band WDM coherent transmission of 40 × 43-Gbit/s CS-RZ DPSK signals over 400 km DSF using hybrid GS-TDFAs/Raman amplifiers," Electron. Lett., vol. 38, no. 24, pp. 1569-1570, 2002. 1.6 Tbit/s (40 × 40 Gbit/s) DPSK transmission over 3 × 100 km of TeraLight fibre with direct detection. H Bissessur, G Charlet, E Gohin, C Simonneau, L Pierre, W Idler, Electron. Lett. 392H. Bissessur, G. Charlet, E. Gohin, C. Simonneau, L. Pierre, and W. Idler, "1.6 Tbit/s (40 × 40 Gbit/s) DPSK transmission over 3 × 100 km of TeraLight fibre with direct detection," Electron. Lett., vol. 39, no. 2, pp. 192-193, 2003. On the theory of noise in radio receivers with square law detectors. M Kac, A J F Siegert, J. Appl. Phys. 18M. Kac and A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys, vol. 18, pp. 383-397, 1947. The characteristic function of Hermitian quadratic forms in complex normal variables. G L Turin, Biometrika. 471-2G. L. Turin, "The characteristic function of Hermitian quadratic forms in complex normal variables," Biometrika, vol. 47, no. 1-2, pp. 199-201, 1960. Exact error probability of phase-modulated signals with linearly compensated nonlinear phase noise. K.-P Ho, J. Lightwave Technol. submitted, included in the online version of [21K.-P. Ho, "Exact error probability of phase-modulated signals with linearly compensated nonlinear phase noise," J. Lightwave Technol., 2003. submitted, included in the online version of [21]. Compensation of nonlinear self-phase modulation with phase modulators. C Xu, L F Mollenauer, X Liu, Electron. Lett. 3824C. Xu, L. F. Mollenauer, and X. Liu, "Compensation of nonlinear self-phase modulation with phase modulators," Electron. Lett., vol. 38, no. 24, pp. 1578-1579, 2002. Error-rate considerations for digital phase-modulation systems. V K Prabhu, IEEE Trans. Commun. Technol. 171V. K. Prabhu, "Error-rate considerations for digital phase-modulation systems," IEEE Trans. Commun. Technol., vol. COM-17, no. 1, pp. 33- 42, 1969. Detection of a PSK signal transmitted through a hard-limited channel. P C Jain, N M Blachman, IEEE Trans. Info. Theory. 195P. C. Jain and N. M. Blachman, "Detection of a PSK signal transmitted through a hard-limited channel," IEEE Trans. Info. Theory, vol. IT-19, no. 5, pp. 623-630, 1973. The effect of phase error on DPSK error probability. N M Blachman, IEEE Trans. Commun. 293N. M. Blachman, "The effect of phase error on DPSK error probability," IEEE Trans. Commun., vol. COM-29, no. 3, pp. 364-465, 1981. Proakis, Digital Communications. J , McGraw HillNew Yorkfourth ed.J. G. Proakis, Digital Communications. New York: McGraw Hill, fourth ed., 2000.
[]
[ "NATURAL BOUNDARY CONDITIONS IN GEOMETRIC CALCULUS OF VARIATIONS", "NATURAL BOUNDARY CONDITIONS IN GEOMETRIC CALCULUS OF VARIATIONS" ]
[ "Giovanni Moreno ", "Monika Ewa Stypa " ]
[]
[]
In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y -the manifold of dependent and independent variables underlying a given problem-as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in R n+1 , and the Lagrangian is first-order (in particular, the hypersurface area).
10.1515/ms-2015-0105
[ "https://arxiv.org/pdf/1301.2985v2.pdf" ]
118,802,031
1301.2985
98ce1d37d9bbf1537f570d94fe8a5ada935ffc74
NATURAL BOUNDARY CONDITIONS IN GEOMETRIC CALCULUS OF VARIATIONS Giovanni Moreno Monika Ewa Stypa NATURAL BOUNDARY CONDITIONS IN GEOMETRIC CALCULUS OF VARIATIONS In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y -the manifold of dependent and independent variables underlying a given problem-as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in R n+1 , and the Lagrangian is first-order (in particular, the hypersurface area). Introduction Let Y be a smooth (real) manifold of dimension n + 1, with nonempty boundary ∂Y . Definition 0.1. An n-dimensional submanifold L ⊆ Y such that (1) L is connected, compact and oriented; (2) ∂L = L ∩ ∂Y ; (3) L is nowhere tangent to ∂Y , is called admissible; the totality of such submanifolds is denoted by A Y . Introduce a local coordinate system (x, u) on Y , where x := (x 1 , . . . , x n ). Let λ = Ld n x be an r th order Lagrangian, i.e., let L = L(x, u, u I ) and I denote a multi-index of length ≤ r. Suppose that, in such coordinates, an element L ∈ A Y is the graph of a function u = u(x), defined on a connected and bounded domain Ω ⊆ R n : then the integral Indeed, if properly understood in a geometric framework, S λ is a real-valued function on A Y ; the choice of the denomination is justified by (0.1): if L is allowed to vary within the class A Y , then the function u describing L is "free" to take any boundary value, as long as u maps ∂Ω into ∂Y . The main theoretical question addressed in this paper is the following: do the solutions to a variational problem with free boundary values fulfill some extra equation(s) besides the Euler-Lagrange equations? A positive answer has already been given in [12,10,11], but without detailed proofs: Section 4 is devoted to review this result by adding the missing details. Sections 3-4 deal with technical aspects of flag fibrations and relative Cspectral sequences, respectively: the reader not interested in theoretical considerations may skip them, and jump to Corollary 4.15, which summarizes their results. Section 1 explains the key used to obtain the main result (Section 5), namely the natural behavior of the relative Euler map, under boundary-friendly transformations. As certainly know all who work in geometric variational calculus and cohomological theory of nonlinear PDEs, the Euler map is but a small feature of a general theory (comprising, e.g., conservation laws, Helmoltz conditions, hamiltonian structures, recursion operators, etc.), which possesses a natural relative analog: we added Sections 3-4 just to give a glimpse of it. The applicative purpose of this paper is to present explicit examples of natural boundary conditions. In the rather pedagogical Section 2, we review the classical analytic solution, given by van Brunt in a recent (2006) book [4], to one of the simplest examples of variational problems with free boundary values. More involved examples are suggested by real-life circumstances, as, e.g., the problem of finding the equilibrium of a soap film freely sliding along the inner wall of an arbitrarily-shaped pipe, discussed in Subsection 5. 2 The geometric point of view is the backbone of this paper: besides allowing a transparent formulation of the main problem, it provides a key tool to obtain a solution. Analytic formulation (0.1) will be used whenever it is necessary to perform actual computations, as well as a source of valuable insights. For example, the Euler-Lagrange equations (0.2) δL δu = 0, where δL δu is the Euler-Lagrange derivative of L, are obtained by a well-known manipulation of (0.1), under the assumption that the variations of u have a compact support in • Ω: hence, a solution to the main problem should be a stronger condition than the Euler-Lagrange equations themselves. This clue was confirmed by the discovery of the relative Euler map [12], reviewed in Section 4. Generalities on geometric calculus of variations The Euler map E appears, in one form or another, in all geometric frameworks for Variational Calculus that are based on the language of differential forms on jet spaces (called here Grassmann fibrations following the recent paper [13], of which we also adopt the notation). Building the Grassmann fibration 1 G r n Y over Y is just a coordinate-free way to add new coordinates u I , with |I| ≤ r, to the manifold Y , in such a way that λ can be considered as an n-form on G r n Y . In this new perspective, (0.1) can be rewritten without mentioning the local expression of L: indeed, since G r n L ≡ L, being dim L = n, the canonical inclusion ι L : L ⊆ Y is lifted to an immersion j r L := G r n ι L : L −→ G r n Y , which allows to pull any Lagrangian back to L. In other words, (0.1) reads (1.1) S λ : L ∈ A Y −→ L j r L * λ ∈ R. Passing from (0.1) to (1.1) is far from being a mere aesthetic exercise. It deploys powerful tools to attack the main problem: essentially, the possibility of using transformations which mix dependent and independent variables (x and u, respectively, in the above coordinate system). In Subsection 5.1 we show how a suitable change of coordinates can help avoid the lengthy computations 1 In Vinogradov and his school's approach, G r n Y is denoted by J r (Y, n), see [14] proposed in Section 2, and how to obtain some useful formulae which, to the authors' opinion, would be very hard (though not impossible) to discover relying on pure analytic methods. The power of transformation methods descends from the natural character of the Euler map: in the principal geometric frameworks for Variational Calculus (Krupka's variational sequences [8,9], Anderson's variational bicomplex [1], and Vinogradov's C-spectral sequence [15]) the Euler map connects two spaces, say L(Y ) and K(Y ), containing, respectively, the Lagrangians and the Euler-Lagrange expressions for the variational problems on Y . We shall not go into the details, since a lot of excellent literature has been written on the subject; nonetheless, we stress that the natural character of the association Y −→ G r n Y , where r ≤ ∞, i.e., the canonical way to lift transformations of Y to the Grassmann fibrations, makes the associations Y −→ L(Y ), K(Y ) natural as well. Indeed, L(Y ) and K(Y ) are usually defined as quotients of sub-complexes (or sub-sequences) of the de Rham complex of finite (or infinite-order) Grassmann fibrations, and as such they inherit the pull-back from differential forms. In other words, any diffeomorphism F : Y −→ Y , determines a commutative diagram (1.2) L(Y ) E Y / / K(Y ) L(Y ) E Y / / F * O O K(Y ). F * O O If F is a wisely-chosen change of coordinates, then "the long way" from L(Y ) to K(Y ), i.e., F * • E Y • (F * ) −1 may be more convenient concerning computations. But this is just a category-theoretic restatement of the well-known transformation rule for the Euler-Lagrange equations, which was already known to E. Cartan: the purpose of this paper is to extend it to the class of variational problems with free boundary values, where the Euler-Lagrange equations are sided by the so-called natural boundary conditions, or, equivalently, they are replaced by the relative Euler-Lagrange equations. Roughly speaking, the "relative" version of the Euler map arises because of the boundary ∂Y . Definition 1.1. By abuse of notation, 2 we shall put ∂G r n Y := (ρ r,0 ) −1 (∂Y ). Indeed, the canonical inclusion ι : ∂G r n Y ⊆ G r n Y determines a differential algebra epimorphism ι * : Ω(G r n Y ) −→ Ω(∂G r n Y ) whose kernel ker ι * := Ω(G r n Y, ∂G r n Y ) is, by definition, 3 the ideal of relative differential forms on G r n Y . 2 ∂G r n Y is more like a prolongation, or lift, to G r n Y , of the boundary ∂Y . 3 Such a construction is common in Differential Topology (see, e.g., [3]). Much as L(Y ), K(Y ), and E Y are constructed out of (classes of) differential forms on G r n Y and natural morphism connecting them, their "relative counterparts", denoted by L(Y, ∂Y ), K(Y, ∂Y ), and E rel Y , respectively, are built out of relative differential forms on G r n Y . Details of this construction, carried out in the context of C-spectral sequences (meaning, in particular, r = ∞), can be found in [12,10] Section 4 explains why the relative Euler-Lagrange equations (1.3) E rel Y (λ) =(1.4) E Y (λ) = 0, which involve n independent variables, and the natural boundary conditions (1.5) E ∂ Y (λ) = 0, where the number of independent variables involved is n − 1. Besides providing a common environment for such heterogeneous equations, the formalism of flag fibrations, introduced by the first author in [11], and reviewed in Section 3, allows to write down (1.5) in a workable way. 4 The natural character of relative Euler map follows automatically from its very definition: in other words, the "relative" version of diagram (1.2), paraphrased by Lemma 1.1 below, needs not to be proved. Lemma 1.1. Let F be boundary-friendly, i.e., F (∂Y ) ⊆ ∂Y . Then E rel Y = F * • E rel Y • (F * ) −1 . A motivating example Let n = 1, Y = [a, b] × R, and λ = L(x, u, u )dx: in this case, functions can be identified with their graphs, and A := C ∞ ([a, b]) as a subset of A Y . Hence, up to a (noncritical) restriction of A Y to A, the boundary problem with free boundary values determined by λ on Y , entails finding the functions u such that 5 (2.1) lim û−u →0 S λ [û] − S λ [u] û − u = 0. In Chapter 7 of van Brunt's book [4], the above problem is modified by allowinĝ u to be defined on a different interval than u. To fit this new setting, A must give up its linear structure and norm, namely A := ∪ x0<x1 C ∞ ([x 0 , x 1 ]); according, Y := R 2 . Despite this, A keeps a rather obvious metric structure d A . Moreover, with two real numbers X 0 and X 1 and a suitable function ξ, one can construct a variationû of u, whose d A -distance from u is controlled by a parameter > 0. First, use X 0 and X 1 to define a new interval [x 0 ,x 1 ], wherê x k = x k + X k , k = 0, 1, and suppose, without loss of generality, that x 0 = min{x 0 ,x 0 } and x 1 = max{x 1 ,x 1 }. Then, use ξ ∈ C ∞ ([x 0 ,x 1 ]) to construct the variation (2.2)û := u + ξ, of u, where u is the 2 nd order polynomial extension 6 of u to the interval [x 0 ,x 1 ], i.e., u (x) = u, x ∈ [x 0 , x 1 ], u(x 1 ) + (x − x 1 )u (x 1 ) + (x−x1) 2 2 u (x 1 ), x ∈ (x 1 ,x 1 ]. Now d A (u,û) := ||u −û|| + |(x 0 , u 0 ) − (x 0 ,û 0 )| + |(x 1 , u 1 ) − (x 1 ,û 1 )| is a well-defined distance on A, which allows to adapt (2.1) to the case when the domain of definition of u can be altered: the norm û − u has to be replaced by the distance d(u,û). Take the variationû (2.2), and compute (2.3) d A (u,û) ≤ ||u−û||+ X 0 +||u 0 −û 0 ||+ X 1 +||u 1 −û 1 || ≤ (3ξ +X 0 +X 1 ). Inequality (2.3) shows that (2.4) d A (u,û) = o( ). In order to estimate the numerator in (2.1), compute 7 S λ [û] − S λ [u] = x1 x0 L[û]dx − x1 x0 L[u]dx (2.5) = x1+ X1 x0+ X0 L[û]dx − x1 x0 L[u]dx = x1 x0 (L[û] − L[u])dx + x1+ X1 x1 L[û]dx − x0+ X0 x0 L[û]dx. Equality (2.5) shows that, with respect to the "fixed domain case" (2.1), the variation of S λ in u has two additional contributions due to the variations of the 6 To reduce the load of notations, we retain the same symbol u for the extension u of u. 7 It is convenient to write L[u] instead of L(x, u(x), u (x)). endpoints of the domain of u. The main advantage of the geometric approach presented in Subsection 5.1 later on, is that such a distinction between the variations of u and the variation of its domain, simply disappear. For the time being, (2.5) can be just rewritten in a more suggestive form S λ [û] − S λ [u] = δS λ δξ [u] + δS λ δX 1 [u]− δS λ δX 0 [u], where δS λ δξ [u] = x1 x0 (L[û] − L[u])dx = ξ ∂L ∂u [u] x1 x0 + x1 x0 ξ ∂L ∂u − d dx ∂L ∂u [u]dx + o( 2 ), and δS λ δX k [u] = x k + X k x k L[û]dx = X k L(x k , u(x k ), u (x k )) + o( 2 ), k = 0, 1, where we used the fact that d dt t=0 x k +t x k L[u]dx = L(x, u(x k ), u (x k )) and L(x,û(x k ),û (x k )) − L(x, u(x k ), u (x k )) = o( 2 ), for k = 0, 1. Now define real numbers U 0 , U 1 by U k =û(x k ) − u(x k ), for k = 0, 1, and compute U k = (u + ξ)(x k ) − u(x 0 ) = u(x k ) + X k u (x k ) + ξ(x k ) + o( 2 ) − u(x k ) = (X k u (x k ) + ξ(x k ) + o( )) . This shows that (2.6) ξ(x k ) = U k − X k u (x k ) + o( ), k = 0, 1. In view of (2.6), (2.5) reads now S λ [û] − S λ [u]= ξ ∂L ∂u [u] x1 x0 + x1 x0 ξ ∂L ∂u − d dx ∂L ∂u [u]dx + k=0,1 (−1) k −X k L − u ∂L ∂u (x k , u(x k ), u (x k )) −U k ∂L ∂u (x k , u(x k ), u (x k )) + ξ(x k ) ∂L ∂u (x k , u(x k ), u (x k )) + o( 2 ). Equivalently, S λ [û] − S λ [u] =    x1 x0 ξ δL δu dx + k=0,1 (−1) k · −X k L − u ∂L ∂u − U k ∂L ∂u (x k , u(x k ), u (x k )) + o( 2 ). Plugging the last expression into (2.1), and taking into account (2.4), we finally see that u is a critical point for S λ if the above term in vanishes for all variations of u, i.e., for all possible choices of ξ, X 0 and X 1 . In particular, u must satisfy the (2 nd order) Euler-Lagrange equations, δL δu (x, u(x), u (x), u (x)) = 0, on its domain of definition, plus a (1 st order) natural boundary condition at the endpoints, (2.7) k=0,1 (−1) k (U k − u X k ) ∂L ∂u + X k L (x k , u(x k ), u (x k )) = 0. Formula (2.7) is used in van Brunt's book to prove Theorem 2.1 below, which answers the main question for one of the simplest (though nontrivial) examples of a variational problem with free boundary values. Theorem 2.1 (Transversality conditions). Let Y ⊂ R 2 be a closed and connected smooth domain, such that ∂Y is the disjoint union of two curves γ 0 and γ 1 , and R 2 Y is disconnected, and λ be a 1 st order Lagrangian. 8 If an element u ∈ A Y is a solution of the variational problem with free boundary values determined by λ, then (1) u obeys the Euler-Lagrange equations on its domain of definition [x 0 , x 1 ]; (2) u fulfills the following transversality conditions (2.8) dy γ k dσ σ=0 ∂L ∂u − dx γ k dσ σ=0 u ∂L ∂u − L (x k , u(x x ), u (x k )) = 0, k = 0, 1, where γ k (σ) = (x γ k (σ), y γ k (σ)) and γ k (0) = (x k , u(x k )), k = 0, 1. Proof. See [4], Chapter 7. In this Section we observed the lack of robustness of the functional-analytic approach: the slightest change of settings destroyed the norm on the class of admissible functions, and a (in many respects, unnatural) distance appeared in its place, which worked well only after some lengthy tricks. Flag fibrations The main motivation for flag fibrations is that equations (1.4) and (1.5) involve n and n−1 independent variables, respectively: merging them into a unique equation requires a new formalism where the number of independent variables can take (at least) two values: n and n − 1. Recall the fundamental embedding G r n Y ⊆ G 1 n (G r−1 n Y ). It allows to regard a point θ ∈ G r n Y as an n-dimensional tangent plane 9 to G r−1 n Y , and an element of the fibered product (3.1) P := G r n Y × G r−1 n Y G 1 n−1 (G r−1 n Y ) as a pair consisting of an n-dimensional and (n − 1)-dimensional tangent plane to G r−1 n Y (at the same point). Define (3.2) F r n,n−1 Y := {(θ n , θ n−1 ) ∈ P | θ n ⊃ θ n−1 } . In many respects, the theory of flag fibrations parallels that of Grassmann fibrations; it is useful to review here some of its characteristic features. Theorem 3.1. Let r > 0. Then the following results hold: • F r n,n−1 Y is a smooth manifold, called the (r th order) flag fibration of Y (of signature (n, n − 1)): it is fibered over the base Y , as well as all lower-order flag fibrations, i.e., the manifolds F s n,n−1 Y , with s < r. • The flag fibration F r n,n−1 Y is naturally fibered over the corresponding (i.e., with the same order r and the same number of independent vari- ables n) Grassmann fibration G r n Y . • The image of F r n,n−1 Y under the canonical projection over G 1 n−1 (G r−1 n Y ) is a smooth submanifold, naturally understood as 1 st order nonlinear partial differential equation on G r−1 n Y in n − 1 independent variables: the equation of involutive (n − 1)-planes of G r−1 n Y . • The equation of involutive (n − 1)-planes of G r n Y projects naturally over F r n,n−1 Y . • The infinite-order flag fibration F ∞ n,n−1 Y , obtained as the inverse limit of finite-order flag fibrations, identifies with the equation of involutive (n − 1)-planes of G ∞ n Y , and, hence, it can be considered as an equation on G ∞ n Y . • The infinite prolongation 10 (F ∞ n,n−1 Y ) (∞) of the 1 st order differential equation F ∞ n,n−1 Y , understood as a pro-finite leaf space, 11 is naturally interpreted as a space of infinite-order Cauchy data. 9 Called integral element by Bryant&Griffiths [5], or R-plane by Vinogradov and his school [14,2]. 10 See [7] for a definition of infinitely prolonged equations. 11 In the sense of Vinogradov's "Secondary Calculus": see, for instance, the introduction of [16]. • The infinitely-prolonged equation (F ∞ n,n−1 Y ) (∞) fits into a double filtration picture (F ∞ n,n−1 Y ) (∞) n & & p x x G ∞ n Y G ∞ n−1 Y. mimicking the similar diagram in the (linear) theory of flag manifolds. Proof. See [11]. Theorem 3.2 ([11], Theorem 9.1). Let L (resp., Σ) be a leaf (i.e., maximal integral submanifold with respect to the infinite-order contact distribution) of G ∞ n Y (resp., G ∞ n−1 Y ). Then the following identifications p −1 (L) = G ∞ n−1 (L) n −1 (Σ) = J ∞ (N Σ ) (3.3) hold, where N Σ is a pro-finite vector bundle called the infinite-order normal bundle. Moreover, p and n are transverse one to another, in the sense that p −1 (L) (resp., n −1 (Σ)) maps non degenerately onto G ∞ n−1 Y (resp., G ∞ n Y ). Equality (3.3) is the less straightforward of the two, and plays a prominent role in the description of the relative Euler map, which will be introduced in the next section. Relative Euler operator and natural boundary conditions In order to clarify the relationship between relative cohomology and variational problems with free boundary values, recall Definition 1.1, and suppose that λ = dλ 0 , where λ 0 ∈ Ω n−1 (G r n Y ) is such that λ 0 | ∂G r n Y = 0. Then (1.1) reads (4.1) S λ [L] = L j r L * λ = L j r L * dλ 0 = ∂L j r L * λ 0 | ∂L = 0, since ∂L ⊂ ∂Y according to Definition 0.1. Indeed, j r L maps ∂L into ∂G r n Y and, hence, the fact that λ 0 vanishes on the latter implies that its pull-back j r L * λ 0 vanishes on the former. Lemma 4.1. The action S λ on A Y is determined by the equivalence class of λ modulo the subspace dΩ n−1 (G r n Y, ∂G r n Y ) of Ω n (G r n Y ). Proof. A paraphrase of (4.1). In order to simplify further analysis, we shall work, from now on, in the context of infinite Grassmann fibrations and C-spectral sequences; in particular, a Lagrangian will be a horizontal n-form on G ∞ n Y , λ ∈ Ω n h (G ∞ n Y ), where Ω h (G ∞ n Y ) is the quotient differential algebra of Ω(G ∞ n Y ) with respect to the ideal of contact forms. An expert in bicomplexes or C-spectral sequences would say that the next corollary is the "horizontalization" of Lemma 4.1 above. Corollary 4.0.1. The action S λ on A Y is determined by the relative horizontal cohomology class [λ] rel ∈ H n h (G ∞ n Y, ∂G r n Y ) := Ω n h (G ∞ n Y ) d h Ω n−1 h (G ∞ n Y, ∂G ∞ n Y ) , where d h is the horizontal differential. Corollary 4.0.1 says precisely that H n h (G ∞ n Y, ∂G r n Y ) is the space L(Y, ∂Y ) mentioned in Section 1. The space K(Y, ∂Y ) can be obtained in a similar way, using relative forms, contact ideal, and cohomology: we shall rather use an approach based on total differential operators and Spencer cohomology, as in [12]. In the same cohomological framework it will also appear the relative Euler map E rel Y , which allows to obtain the equation ( The first result can be found in [12], but its proof, which is a consequence of Theorem 3.2, was provided later in [11], and it is a consequence of the following structural result, which dictates strong restrictions on the topology of the fibration ∂G ∞ n Y −→ G ∞ n−1 (∂Y ). G ∞ n−1 (Y ) via the embedding ∂Y ⊆ Y . Then ∂G ∞ n Y = J ∞ h (N G ∞ n−1 (∂Y ) ) where N is the infinite-order normal bundle (see Theorem 3.2), and J h means "horizontal jet bundle". 12 It follows form Corollary 4.0.2 that the relative (i.e., constructed with relative forms) C-spectral sequence of ∂G ∞ n Y is particularly simple (i.e., one-line 13 ); in turn, this implies that K(Y, ∂Y ) splits into the sum K(Y ) ⊕ K(∂Y ) (the proof 12 Roughly speaking, the analog of jet bundle where derivatives are replaced by total derivatives. See, e.g., [14,16] for more details. 13 See [14] for the meaning of "one-line" can be found in [10] The second result has been stated in [11] without proof, which is provided by Lemma 4.2 below. Remark 1. Proposition 4.1 below contains a general theoretical result concerning relative C-spectral sequences, so that there is no need to restrict ourselves to the case of one independent variable: in other words, we let Y to be of dimension n + m, where m is arbitrary, i.e., locally, to be fibered over an n-dimensional manifold X with m-dimensional fiber (when needed, such a fibration is called π). Here we recall some terminology. VSym(Y ) is the module of vertical symmetries (denoted by κ in [2,7]) of the infinite-order contact distribution on G ∞ n Y , and D is the sub-algebra of differential operators generated by total derivatives (the C-differential operators, according to [2,7]). Suppose now we work in a local chart (in particular, ∂Y = {x n = 0} and π is trivial): in this case, D C ∞ (G ∞ n Y ), Ω n h (G ∞ n Y )) identifies with D(C ∞ (G ∞ n Y ), C ∞ (G ∞ n Y ) ) by means of the horizontal volume form d n x , and D(C ∞ (∂G ∞ n Y ), Ω n−1 h (∂G ∞ n Y )) with D(C ∞ (∂G ∞ n Y ), C ∞ (∂G ∞ n Y ) ) by means of the horizontal volume form d n−1 n x on ∂G ∞ n Y . Accordingly, the formally adjoint modules (see [16]) VSym † (Y ) and VSym † (∂G ∞ n Y ) are identified with the dual module of VSym(Y ) and VSym(∂G ∞ n Y ), which are still free, with bases {D Recall that D I is the composition of total derivatives D i1 x 1 •D i2 x 2 •· · ·•D in x n , with I = (i 1 , . . . , i n ), and, by our own convention, the difference between the multiindex I and an integer α ≤ i n is the multi-index I − α := (i 1 , . . . , i n−1 , i n − α). (4.2) K(Y, ∂Y ) = D(VSym(Y ), Ω n h (G ∞ n Y )) δ D(VSym(Y ), C ∞ (G ∞ n Y )) ⊗ Ω n−1 h (G ∞ n Y, ∂G ∞ n Y ) , where δ is the Spencer differential. Moreover, the cohomology class of the cocycle = a I j D (j) I ∈ D(VSym(Y ), Ω n h (G ∞ n Y )) is identified with the pair (E( ), E ∂ ( )) ∈ VSym † (Y ) ⊕ VSym † (∂G ∞ n Y ), where E( ) = (−1) |I| D I (a I j )D (j) 0 , and (4.3) E ∂ ( ) = I∈N n 0 (−1) |I|−in m j=1 α<in (−1) α D I−in (D α xn (a I ) |∂G ∞ n Y )D (j,in−α−1) 0 Proof. By the definition of relative C-spectral sequences [12], the space K(Y, ∂Y ) is the n th cohomology space of the subcomplex D(VSym(Y ), C ∞ (G ∞ n Y )) ⊗ Ω h (G ∞ n Y, ∂G ∞ n Y ) of D(VSym(Y ), Ω h (G ∞ n Y )). Expression (4.2) is a consequence of the fact that Ω n h (G ∞ n Y, ∂G ∞ n Y ) equals Ω n h (G ∞ n Y ) . In other words, K(Y, ∂Y ) has the same n-cocycles as K(Y ), but fewer n-coboundaries, which explains why the n th cohomology of the subcomplex turns out to be quite larger than the cohomology of the entire complex: in turn, this explains the appearance of natural boundary conditions. For the sake of simplicity, we shall skip the index j. We now prove that the relative Spencer cohomology of is identified with (C ∞ (G ∞ n Y ), ∂G ∞ n Y ). Accordingly, ele- ments of D(C ∞ (G ∞ n Y ), Ω n−1 h (C ∞ (G ∞ n Y ), ∂G ∞ n Y ))x, with i ∈ D(C ∞ (G ∞ n Y ), C ∞ (G ∞ n Y )) for i = 1, . . . , n. We compute the Spencer differential δ( i ⊗d n−1 Proof. Let r be the order of λ and I = (i 1 , . . . , i n ). Put i x) = (−1) i−1 (D 1i • i )⊗d n x, and observe that (δ( n ⊗ x n d n−1 n x))(f ) = d h ( n (f )x n d n−1 n x) = d h ( n (f )) ∧ (x n d n−1 n x) + n (f )d(x n d n−1 n x) = D 1n ( n (f ))dx n ∧ x n d n−1 n x + n (f )dx n ∧ d n−1 n x = (−1) n−1 (x n D 1n • n + n )(f )d n x, for arbitrary f ∈ C ∞ (G ∞ n Y ), whence δ( n ⊗ x n d n−1 n x) = (−1) n−1 ((x n D 1n + 1) • n ) ⊗ d n x = (−1) n−1 (D 1n • x n • n ) ⊗ d n x,L i1i2···in := ∂L ∂u I = ∂L ∂u x i 1 1 x i 2 2 ...x in n . From the well-known formula of elementary calculus (4.4) f g (i) = i−1 s=0 f (s) g (i−s−1) + (−1) i f (i) g it follows that Ω i1+···+in≤r L i1i2···in η x i 1 1 x i 2 2 ...x in n d n x (4.5) = Ω i1+···+in≤r d dx 1 i1−1 s1=0 (−1) s1 d s1 L i1i2···in dx s1 1 η x i 1 −s 1 −1 1 x i 2 2 ...x in n (4.6) + (−1) i1 d i1 L i1i2···in dx i1 1 η x i 2 2 x i 3 3 ...x in n d n x (4.7) Then, applying again (4.4) to the term (4.7), we obtain (4.5) = (4.6) + Ω i1+···+in≤r d dx 2 i2−1 s2=0 (−1) s2 d s2 dx s2 2 · (−1) i1 d i1 L i1i2···in dx i1 1 η x i 2 −s 2 −1 2 ...x in n (4.8) +(−1) i2 d i2 dx i2 2 (−1) i1 d i1 L i1i2···in dx i1 1 η x i 3 3 x i 4 4 ...x in n d n x (4.9) Again, by (4.4), we develop term (4.9): (4.5) = (4.6) + (4.8) + · · · (4.10) On the other hand, (4.12) is the Euler-Lagrange; it remains just (4.11), i.e., + Ω i1+···+in≤r d dx n in−1 sn=0 (−1) sn d sn dx sn n (−1) in−1 · d in−1 dx in−1 n−1 . . . (−1) i1 d i1 L i1i2···in dx i1 1 . . . η x in −sn−1 n (4.11) + (−1) in d in dx in n . . . (−1) i1 d i1 L i1i2···in dx i1 1 . . . η d n x.(4.13) ∂Ω |I|≤r in−1 sn=0 (−1) |I|−in+sn d |I|−in dx I−in d sn L i1i2···in dx sn n η x in −sn−1 n ∂Ω d n−1 n x. Since (4.13) has to vanish for all variations η, all equations (4.16) must be satisfied. Before providing a proof, it is convenient to cast a bridge between the approach based on total differential operators to the space K(Y ), sketched in Remark 1, and a perhaps more familiar one, based on "1-contact, n-horizontal" (n + 1)-forms, or forms "of type (1, n)". Namely, (1.4) can be written down as (4.14) E Y (λ) = δL δu ω ∧ d n x, where ω is the zero-order contact form, and ω ∧ d n x plays the role of the generator D 0 (see in Remark 1) of the module K(Y ). Equation (4.14) clarifies the above sentence "(1.4) holds on L": it means that (4.14), pulled back to L via j ∞ L, vanishes. Similarly, the results contained in Corollary 4.0.2 and Proposition 4.1 give a solid basis to the sentence "holds on ∂L", since (4.15) E ∂ Y (λ) = E ∂ Y,α (λ)ω α ∧ d n n−1 x, where now the ω α 's are the zero-order contact forms on J ∞ h (N G ∞ n−1 (∂Y ) ), and ω α ∧ d n n−1 x plays the role of the generators D j,α 0 , where j = 1 (see in Remark 1), of the module K(∂G ∞ n Y ). According, E ∂ Y,α (λ) can be obtained as the coefficient of D 1,α 0 , in (4.3), where j = 1, and a I = ∂L ∂u I . Again, the sentence "(1.5) holds on ∂L" means that (4.15), pulled back to ∂L via j ∞ ∂L, vanishes (see also Theorem 11.1 in [11]). Proof. The first fact is obvious: if L a solution of a variational problem with free boundary values, then L ∂L is a solution of the Euler-Lagrange equation determined by the same Lagrangian λ, i.e., equation (1.4) holds. We stress that, in order to prove the second fact, it is necessary to have the result on the structure of equation (1.5) provided by Corollary 4.0.2. Namely, (1.5) is localizable, in the sense that its left-hand side belong to a module of sections, and hence it vanishes locally if and only if it vanishes globally. Then, we can choose a coordinate system (x, u) such that L is the graph of a function u = u(x) on Ω and ∂Y = {x n = 0}. Since L is critical, all equations (4.16) must hold true on ∂Ω; on the other hand, the above discussions showed that equations (4.16) are nothing but E ∂ Y,α = 0: hence, (4.15) vanishes, i.e., (1.5) must be valid on ∂L. For readers not interested in theoretical details, we collect the main result of the last two sections into a convenient (though redundant) Corollary. , E ∂ Y (λ) = E ∂ Y,α (λ)ω α ∧ d n n−1 x, where: i) the ω α 's are the zero-order contact forms on the infinite jet of a suitable pro-finite vector bundle over ∂Y which arises in the theory of flag fibrations over Y ; ii) if α is less than the order of the Lagrangian λ, and L is locally the graph of a function u = u(x) on Ω ⊆ R n , the component E ∂ Y,α (λ) is given by (4.16) E ∂ Y,α (λ) = |I|≤r, in>α (−1) |I|−α−1 d |I|−in dx I−in d in−α−1 dx in−α−1 n ∂L ∂u I ∂Ω . Applications Together, Lemma 1.1 and Corollary 4.2.1 provide a powerful tool for writing down concrete examples of natural boundary conditions. Computations presented in this section will be simplified by some "tricks" based on multi-linear algebra and total differentials (Remarks 2 and 3 below). Remark 2 (Top differential forms). A brute-force attempt to change variables in a multi-dimensional integral may lead to a meaningless formula (5.1) f (x)d n x = f (x(t)) d n x d n t d n t. Nonetheless, since the C ∞ (R n )-module Ω n (R n ) is freely generated by d n t, any n-form can be identified with a function. In particular, d n t identifies with 1, and d n x with the Jacobian of the change of variables x = x(t), thus recovering the meaning of (5.1). From now on, all n-forms will be identified with functions: hence, an expression like Ξ(ω), where Ξ is a vector field and ω an n-form, is not the Lie derivative of ω, but the function Ξ(f ), where f is uniquely defined by ω = f d n t. Remark 3 (Total differentials). Formula (5.1) can be adapted to variational integrals, just by replacing differentials by total differentials, namely f (x, u(x), u 1 (x), . . . , u n (x))d n x = f (x(t, y), u(t, y), u 1 (t, y, y 1 , . . . , y n ), . . . , u n (t, y, y 1 , . . . , y n )) d n x d n t d n t where now d n x = dx 1 ∧ · · · ∧ dx n (i.e., the operator d h used in Section 4). Recall that dx = D t i (x)dt i , x = x(t) where D t i = ∂ t i + y i ∂ u is the total derivative operator with respect to t i . In this context, horizontal n-forms on G 1 n Y , i.e., the space with coordinates (t, y, y 1 . . . , y n ), are identified with functions on the same space, via the horizontal volume form d n t. Accordingly, d n x is the "total Jacobian" associated with the change of variables (x, u) = (x(t, y), u(t, y)). 5.1. A 1 st order, one-dimensional example. Consider again the variational problem with free boundary values of Theorem 2.1, Section 2. Let L = L(t, y, y ) be its Lagrangian, and recall that ∂Y is the disjoint union of two curves in the (t, y)-plane. Then, if γ(σ) = (t γ (σ), y γ (σ)) is one of them, a critical point y = y(x) for S λ must fulfill the natural boundary condition (5.2) y γ (0) ∂L ∂y − t γ (0) y ∂L ∂y − L (t, y(t), y (t)) = 0 where (t, y(t)) = γ(0) (see (2.8)). Equation (5.2) can be obtained in a transparent geometrical way, without introducing ad hoc metrics on the set A. Just use a change of coordinates (t, y) F −→ (x, u), x = x(t, y) u = u(t, y) which "rectifies" the curve γ, i.e., such that F * (γ) is, for instance, the u-axis of the (x, u)-plane. Then, lift F to a contact transformation (t, y, y ) F −→ (x, u, u ) of G 1 1 Y , where (5.3) u = u t + y u y x t + y x y . It is a simple exercise to get (5.3) (see, for instance, [2], Section 1.2); nevertheless, in view of the next generalization, we prefer to justify it, by using the total differential operator d. Recall that (5.4) df = D t (f )dt, for f = f (t, y, y ), with D t = ∂ t + y ∂ y being the total derivative operator in t. As announced in Remark 3, we shall identify horizontal one-forms on the (t, y, y )-space with functions: hence, (5.3) above can be written as (5.5) u = du dx = D t (u)dt D t (x)dt . It is worth noticing that the inverse transformation F −1 is the same as (5.5) (5.6) y = dy dt = D x (y)dx D x (t)dx = y x + u y u t x + u t u , where now the total derivative operator is taken with respect to x. It follows that (5.7) ∂y ∂u = ∂ ∂u D x (y) D x (t) . Finally, since dt = dt (see (5.4)), the Lagrangian λ reads λ = L(t, y, y )dt = L(t, y, y )D x (t)dx in the (x, u, u )-space, where D x (t) plays the role of "total Jacobian" (Remark 3). In other words, (F * ) −1 (λ) = Ldx, where L = L(x, u, u ) is given by It remains to express (5.9) in terms of the coordinates (t, y, y ), i.e., to apply Lemma 1.1. We compute ∂ L ∂u (5.8) L = (F * ) −1 (L)D x (t) = L(t(x, u), y(x, y), y (x, u, u ))D x (t)(x, u, u ).(5.8) = ∂ ∂u (LD x (t)) = ∂L ∂u D x (t) + L ∂D x (t) ∂u (5.7) = ∂L ∂y ∂ ∂u D x (y) D x (t) D x (t) + L ∂D x (t) ∂u = ∂L ∂y D x (t) ∂ ∂u (D x (y)) − D x (y) ∂ ∂u (D x (t)) D x (t) + L ∂D x (t) ∂u = ∂L ∂y ∂ ∂u (D x (y)) − D x (y) D x (t) ∂ ∂u (D x (t)) + L ∂D x (t) ∂u (5.6) = ∂L ∂y ∂ ∂u (D x (y)) − y ∂ ∂u (D x (t)) + L ∂D x (t) ∂u . (5.9) It suffices to observe that ∂D x (t) ∂u = ∂(t x + u t u ) ∂u = t u , (5.10) ∂D x (y) ∂u = ∂(y x + u y u ) ∂u = y u . (5.11) Substituting (5.10) and (5.10) into (5.9), one gets ∂L ∂y (y u − t u y ) + Lt u = 0, which, evaluated at (0, u(0), u (0)), returns (5.2), since, by the choice of F , t γ (0) = t u (γ(0)), y γ (0) = y u (γ(0)). 5.2. A 1 st order, multi-dimensional example. Now we pass to an n-dimen--sional example: as we shall see, the geometric methods used before generalize effortlessly to this case; an analogous generalization of the methods used in Section 2, i.e., defining a metric structure on the space of all functions defined on a compact and connected subset of R n , would introduce a lot of technical difficulties, obscuring the simple solution of the problem. Theorem 5.1 (Generalized transversality conditions). Let Y be a closed smooth domain, with nonempty (smooth) boundary, of R n+1 = (t, y), with t = (t 1 , . . . , t n ), and λ be a 1 st order Lagrangian, locally given by L = L(t, y, y 1 , . . . , y n ). If y = y(t) is a critical point for S λ , then, for any point θ ∈ ∂Y , the normal vector ν θ to the hypersurface ∂Y must be orthogonal to H(t, y, y 1 (t), . . . , y n (t)), where H is the R n+1 -valued function on G 1 n Y given by H := ∂L ∂y 1 , . . . , ∂L ∂y n , L − y i ∂L ∂y i , and θ = (t, y(t)). Proof. Choose a change of coordinates (t, y) F −→ (x, u), x = x(t, y), u = u(t, y), such that F (∂Y ) has equation x n = 0. In analogy with (5.6), (5.12) y i = dy dt i = dy ∧ d n−1 i t dt i ∧ d n−1 i t = ω i d n t , where ω i := dt 1 ∧ · · · ∧ dt i−1 ∧ dy ∧ dt i+1 ∧ · · · dt n , and we use again the convention that horizontal n-forms are identified with functions via the (horizontal) volume form dx n introduced in Remark 3. Developing all total differentials appearing in (5.12), one recovers the familiar formula for the lifting of F , as it can be found, e.g. in [2], Section 1.2. In analogy with (5.7), ∂y i ∂u n = ∂ ∂u n ω i d n t , i = 1, 2, . . . , n. Finally, analogously to (5.8), we obtain (F * ) −1 (λ) = Ld n x, where L = L(x, u, u 1 , . . . , u n ) is given by (5.13) L = (F * ) −1 (L)d n t where, as explained by Remark 3, d n t = d n t d n x is just an unconventional way to write down the "total Jacobian" of F . Again, (F * ) −1 (λ) determines a variational problem with free boundary values on F (Y ) and ∂F (Y ) = F (∂Y ), by construction, is the hyperplane x n = 0: by Corollary 4.2.1, on such a hyperplane the natural boundary conditions read (5.14) ∂ L ∂u n (x 1 , . . . , x n−1 , 0, u(x 1 , . . . , x n−1 , 0), . . . , u n (x 1 , . . . , x n−1 , 0)) = 0. Finally, let us write down (5.14) in terms of the coordinates (t, y, y 1 , . . . , y n ). We compute ∂ L ∂u n ∂u n ∈ R n−1 . Then, (5.15) coincides with ν · H. Notice that (5.15) is formally the same as (5.9): the synthetic language of multi-linear algebra allowed to handle all the "total Jacobian" determinants involved in the proof, without any additional difficulty as compared with the one-dimensional case. 5.3. The soap film. We generalize now a classical example that can be found in Giaquinta and Hildebrandt's book [6] (Section 2.4). Namely, in the hypotheses of Theorem 5.1 above, suppose that λ is the (hypersurface) area Lagrangian, i.e., locally, L = 1 + n i=1 y 2 i . Then H = 1 1 + n i=1 y 2 i (y 1 , y 2 , . . . , y n , −1) is precisely the unit normal vector to the surface y = y(t). This proves the last result of this paper. Corollary 5.1.1. Let Y ⊆ R n+1 be a closed smooth domain with smooth nonempty boundary. If a hypersurface L ∈ A Y is a solution of the variational problem with free boundary values determined by the area functional, then L must intersect orthogonally ∂Y everywhere. In particular, Corollary 5.1.1 shows that a soap film, whose boundary is constrained to slide over the inner surface of a fixed domain (e.g., a pipe of arbitrary shape), tends toward a position of equilibrium where it forms a right angle with the walls of the container (besides, of course, possessing zero mean curvature). Remark 4. If n = 2 and Y = D × R 2 , where D ⊆ R is diffeomorphic to a closed disk, then a surface L from Corollary 5.1.1 above is forced to be the graph of a constant function y = y(t 1 , t 2 ). Indeed, since Y is a surface with zero mean curvature, its maximum is attained on ∂D. Hence, there exists a point θ ∈ ∂L, such that ∂L = L ∩ ∂Y has negative curvature. But L hits ∂Y orthogonally in θ, thus, along the normal direction to ∂L, the surface L must possess positive curvature, i.e., there must exist a point θ ∈ L, in a neighborhood of θ, such that the y-component of θ is greater than the y-component of θ, thus contradicting the fact that θ corresponds to a maximum. It follows that L must be the graph of a constant function. It would be nice to generalize this simple observation to multi-dimensional cases. |I| u ∂x I d n x makes sense; it can also be given a coordinate-free formulation.Definition 0.2. The variational problem with free boundary values determined by the Lagrangian λ on Y consists of finding the elements of A Y which are critical for S λ . 0 represent a solution to the main problem. More precisely, since K(Y, ∂Y ) identifies with the direct sum K(Y ) ⊕ K(∂Y ), the single equation (1.3) captures two equations simultaneously, viz., the Euler-Lagrange equations Lemma 1.1, together with Corollary 4.2.1, will be employed in the last Section 5 to obtain new examples of natural boundary conditions. 1.3) out of the Lagrangian λ. The aim of this section is to prove that (1.3) is indeed equivalent to the pair of equations (1.4)-(1.5) and, furthermore, that either the single equation (1.3), or the two coupled equations (1.4)-(1.5), provide a (nontrivial) answer to the main question stated in the Introduction. Corollary 4.0.2 ([12], Theorem 2). Consider G ∞ n−1 (∂Y ) as a submanifold of the j th component of the free C ∞ (G ∞ n Y )-module C ∞ (G ∞ n Y ) j = 1, .. . , m, and I multi-index of length n, i.e., I ∈ N n 0 . Moreover, D( } j=1,...,m,α∈N0 , respectively. Proposition 4 . 1 ( 41On the structure of K(Y, ∂Y )). Let Y be as in Remark 1. Then can be obtained by summing up the elements i ⊗d n−1 i x and n ⊗x n d n−1 n ) since 1 = [D 1n , x n ].It follows that a•D 1i • is cohomologous to −D 1i (a)• for all i = 1, . . . , n−1,whereas a • D 1n • = (−D 1n (a) + D 1n • a) • is not generally cohomologous to −D 1n (a) • , since D 1n • a • is not a coboundary, unless a factors through x n . Take now = a I D I ∈ D(C ∞ (G ∞ n Y ), Ω n h ), with I ∈ N n 0 . Such an operator is cohomologous to the operator= (−1) |I|−in D I−in (a I ) • D in xn = (−1) |I|−in (−D xn (D I−in (a I )) + D xn • D I−in (a I )) • D in−1 xn = (−1) |I|−(in−1) D I−(in−1) (a I ) • D in−1 xn + (−1) |I|−in D xn • D I−in (a I ) • D in−1 xn . . . i n iterations = (−1) |I| D I (a I ) + (−1) |I|−in D xn • D I−in (a I ) • D in−1 xn + (−1) |I|−(in−1) D xn • D I−(in−1) (a I ) • D |I| D I (a I ) + 0<α≤in (−1) |I|−α D xn • D I−α (a I ) • D α−1 xn ,which turns into a coboundary if and only if the function D I (a I ) is zero and all the functions D I−α (a I ) factor through x n , i.e., they vanish on ∂G ∞ n Y . This means that the cohomology class [ ] is uniquely determined by (−1) |I| D I (a I ) ∈ C ∞ (G ∞ n Y ), i.e., by E( ) and by the set of functions(−1) |I|−α D I−α (a I ) |∂G ∞ n Y ∈ C ∞ (∂G ∞ n Y ), with α = 1, . . . , i n .Notice that the latter ones can be rewritten as D I−in (D α xn (a I ) |∂G ∞ n Y ), with α = 0, . . . , i n − 1, because the multi-index I − i n belongs actually to N n−1 0 and the first n − 1 total derivatives are tangent to ∂G ∞ n Y : hence the last set of functions represent the coordinates of the element E ∂ ( ), according to(4.3). Proposition 4 . 2 ( 42On natural boundary conditions). Let L be locally given by the graph of a function u = u(x), defined on a compact and connected subset Ω ⊂ R n , such that ∂Ω has equation x n = 0. If L is critical for S λ , then the following equations hold on ∂Ω: α = 0, . . . , r − 1. Sincex ∂Ω = {x n = 0}, all terms appearing on line (4.10) disappear, ∂Ω = 0, ∀i = 1, 2, . . . , n − 1. Lemma 4. 2 . 2Let L ∈ A Y be a solution to the variational problem with free boundary values determined by λ on Y . Then equation (1.4) holds on L and equation (1.5) holds on ∂L. Corollary 4.2.1 (A solution of the main problem). Let L ∈ A Y be a solution to the variational problem with free boundary values determined by λ on Y . Then, the natural boundary conditions E ∂ Y (λ) = 0 are satisfied on ∂L. In local coordinates Now everything is ready. (F * ) −1 (λ) determines a variational problem with free boundary values on F (Y ) and ∂F (Y ) = F (∂Y ), by construction, consists of two curves, one of which is the u-axis: by Corollary 4.2.1, on such an axis the natural boundary conditions take a particularly simple form ∂ L ∂u (0, u(0), u (0)) = 0. This paper is based on the talk "A geometrical framework for Lagrangian theories which involve n and n − 1 independent variables simultaneously" delivered by the first author on August 24, 2012, within the conference "Variations on a Theme", dedicated to D. Krupka's seventieth birthday.5 The norm can be either the L ∞ or the H 1 norm on C 2 ([a, b]). Note that A Y is made precisely by all curves lying in Y , such that one endpoint belongs to γ 0 and the other one to γ 1 , without being tangent to any of them. It remains to show that ν is indeed the normal vector to F (∂Y ). To this end, it is convenient to pass to the determinantal notation for n-forms. Namely,· · · y xn−1 + u n−1 y u · · · t n xn−1 + u n−1 t n u t xn + u n t 1 u · · · y xn + u n y u · · · t n xn + u n t n u contains u n only in the last line: hence,Subtracting from the j th row the n th row multiplied by u j , for all j = 1, . . . , n−1, the determinant does not change, i.e.,Observe that (5.16) and (5.17) are the multi-dimensional analogues of (5.10) and(5.11), respectively. In other words, ν is composed of the n × n minors (with sign) of the n × (n + 1) matrixwhere the n vectors T i = (t x i , y x i ), i = 1, . . . , n − 1, T = (t u , y u ), form a basis for the tangent space of F (∂Y ) = {x n = 0}, i.e., ν = T 1 × · · · × T n−1 × T . On the existence of global variational principles. I M Anderson, T Duchamp, Amer. J. Math. 102ANDERSON, I. M.-DUCHAMP, T.: On the existence of global variational principles, Amer. J. Math., 102 (1980), 781-868. Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. A V Bocharov, V N Chetverikov, S V Duzhin, N G Khorkova, I S Shchik, A V Samokhin, Yu N Torkhov, A M Verbovetsky, A M Vinogradov, Amer. Math. SocBOCHAROV, A. V.-CHETVERIKOV, V. N.-DUZHIN, S.V.-KHORKOVA, N.G.- KRASIL'SHCHIK, I.S.-SAMOKHIN, A.V.-TORKHOV, YU.N.-VERBOVETSKY, A.M.-VINOGRADOV, A. M.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Amer. Math. Soc., 1999. R.-Tu Bott, L W , Differential forms in algebraic topology. New YorkSpringer-VerlagBOTT, R.-TU, L. W.: Differential forms in algebraic topology, Springer-Verlag, New York, 1982. The Calculus of Variations. B Brunt, SpringerNew YorkBRUNT, B.: The Calculus of Variations, Springer, New York, 2006. Exterior differential systems. R L Bryant, S S Chern, R B Gardner, H L Goldschmidt, P A Griffiths, Springer-VerlagNew YorkBRYANT, R. L.-CHERN, S. S.-GARDNER, R. B.-GOLDSCHMIDT, H. L.- GRIFFITHS, P. A.: Exterior differential systems, Springer-Verlag, New York, 1991. Calculus of Variations, I. M.-Hildebrandt Giaquinta, S , SpringerBerlin and HeidelbergGIAQUINTA, M.-HILDEBRANDT, S.: Calculus of Variations, I, Springer, Berlin and Heidelberg, 1996. Geometry of Jet Spaces and Integrable Systems. S Krasil&apos;shchik J, A M Verbovetsky, J. Geom. Phys. 61KRASIL'SHCHIK J. S.-VERBOVETSKY, A. M.: Geometry of Jet Spaces and Inte- grable Systems, J. Geom. Phys. 61 (2011), 1633-1674. Of the structure of the Euler mapping. D Krupka, Arch. Math. (Brno). 10KRUPKA, D.: Of the structure of the Euler mapping, Arch. Math. (Brno), 10 (1974), 55-61. Variational sequences on finite order jet spaces. D Krupka, World Sci. Publ. KRUPKA, D.: Variational sequences on finite order jet spaces, World Sci. Publ., Teaneck, NJ, 1990. A C-Spectral Sequence associated with free boundary variational problems. G Moreno, Proceedings of the Eleventh International Conference on Geometry, Integrability and Quantization. M. Mladenov, Gaetano Vilasi and Akira Yoshiokathe Eleventh International Conference on Geometry, Integrability and QuantizationVarna, Bulgaria; SofiaAvangard PrimaMORENO, G.: A C-Spectral Sequence associated with free boundary variational problems, Proceedings of the Eleventh International Conference on Geometry, Integrability and Quantization, June 5-10, 2009, Varna, Bulgaria, Ivaïlo M. Mladenov, Gaetano Vilasi and Akira Yoshioka, Editors, Avangard Prima, Sofia (2010), 146-156. The geometry of the space of Cauchy data of nonlinear PDEs. G Moreno, Central European Journal of Mathematics. in pressMORENO, G.: The geometry of the space of Cauchy data of nonlinear PDEs, Central European Journal of Mathematics (in press), http://arxiv.org/abs/1207.6290. Domains in Infinite Jets: C-Spectral Sequence. G Moreno, A M Vinogradov, Dokl. Math. 752MORENO, G.-VINOGRADOV, A. M.: Domains in Infinite Jets: C-Spectral Sequence, Dokl. Math. 75, nr. 2 (2007), 204-207. Variational sequences in mechanics on Grassmann fibrations. Z.-Krupka Urban, D , Acta Appl. Math. 112URBAN, Z.-KRUPKA, D.: Variational sequences in mechanics on Grassmann fibra- tions, Acta Appl. Math., 112, (2010), 225-249. Cohomological analysis of partial differential equations and secondary calculus. A M Vinogradov, American Mathematical SocietyProvidence, RIVINOGRADOV, A. M.: Cohomological analysis of partial differential equations and sec- ondary calculus, American Mathematical Society, Providence, RI, 2001. The C-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory. A M Vinogradov, J. Math. Anal. Appl. 100VINOGRADOV, A. M.: The C-spectral sequence, Lagrangian formalism, and conserva- tion laws. II. The nonlinear theory, J. Math. Anal. Appl., 100 (1984), 41-129. Secondary calculus and the covariant phase space. L Vitagliano, J. Geom. Phys. 59VITAGLIANO, L.: Secondary calculus and the covariant phase space, J. Geom. Phys., 59, (2009), 426-447.
[]
[ "Recursive Bayesian Filtering in Circular State Spaces", "Recursive Bayesian Filtering in Circular State Spaces" ]
[ "Gerhard Kurz \nInstitute for Anthropomatics and Robotics\nIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nKarlsruhe Institute of Technology (KIT)\nGermany\n", "Igor Gilitschenski \nInstitute for Anthropomatics and Robotics\nIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nKarlsruhe Institute of Technology (KIT)\nGermany\n", "Uwe D Hanebeck \nInstitute for Anthropomatics and Robotics\nIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nKarlsruhe Institute of Technology (KIT)\nGermany\n" ]
[ "Institute for Anthropomatics and Robotics\nIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nKarlsruhe Institute of Technology (KIT)\nGermany", "Institute for Anthropomatics and Robotics\nIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nKarlsruhe Institute of Technology (KIT)\nGermany", "Institute for Anthropomatics and Robotics\nIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nKarlsruhe Institute of Technology (KIT)\nGermany" ]
[]
For recursive circular filtering based on circular statistics, we introduce a general framework for estimation of a circular state based on different circular distributions, specifically the wrapped normal distribution and the von Mises distribution. We propose an estimation method for circular systems with nonlinear system and measurement functions. This is achieved by relying on efficient deterministic sampling techniques. Furthermore, we show how the calculations can be simplified in a variety of important special cases, such as systems with additive noise as well as identity system or measurement functions. We introduce several novel key components, particularly a distribution-free prediction algorithm, a new and superior formula for the multiplication of wrapped normal densities, and the ability to deal with non-additive system noise. All proposed methods are thoroughly evaluated and compared to several state-of-the-art solutions. system measurement publication distribution model noise model noise Azmani, Reboul, Choquel, Benjelloun [9] von Mises identity additive identity additive Markovic, Chaumette, Petrovic [14] von Mises-Fisher identity additive identity additive Kurz, Gilitschenski, Julier, Hanebeck [21] Bingham identity additive identity additive Kurz, Gilitschenski, Hanebeck [15] wrapped normal/von Mises nonlinear additive identity additive Kurz, Gilitschenski, Hanebeck [16] wrapped normal nonlinear additive nonlinear any this paper wrapped normal/von Mises nonlinear any nonlinear anyTable 1: Circular filters based on directional statistics.There has been some work on filtering algorithms based on circular statistics by Azmani et al.[9], which was further investigated by Stienne et al.[10]. Their work is based on the von Mises distribution and allows for recursive filtering of systems with a circular state space. However, it is limited to the identity with additive noise as the system equation and the measurement equation. The filter from[9]has been applied to phase estimation of GPS signals [11], [12] as well as map matching [13]. Markovic et al. have published a similar filter [14] based on the von Mises-Fisher distribution, a generalization of the von Mises distribution to the hypersphere. We have previously published a recursive filter based on the wrapped normal distribution allowing for a nonlinear system equation [15]. The paper [16] extends this approach to make a nonlinear measurement update possible. Both papers rely on a deterministic sampling scheme, based on the first circular moment. This kind of sampling is reminiscent of the well-known unscented Kalman filter (UKF) [2]. We have extended this sampling scheme to the first two circular moments in [17], so the proposed filters are, in a sense, circular versions of the UKF. The developed filters have been applied in the context of constrained tracking [18], bearings-only sensor scheduling [19], as well as circular model predictive control [20].Furthermore, we proposed a recursive filter based on the circular Bingham distribution in[21]. The Bingham distribution is closely related to the von Mises distribution, but has a bimodal density, which makes it interesting for axial estimation problems (i.e., problems with 180 • symmetry).An overview of all of these filters and the considered distributions as well as system and measurement models is given inTable 1.This paper summarizes and combines our results as well as extends the previous work by a number of additional contributions. First of all, we propose a general filtering framework that can be used in conjunction with a variety of system and measurement equations, different types of noise, and both the wrapped normal and the von Mises distributions. Our previous publications[15],[16]as well as the work by Azmani et al.[9]can be seen as special cases of the proposed framework.Furthermore, we introduce a new multiplication formula for wrapped normal distributions that outperforms the solution proposed in[15]. We generalize the prediction step from [15] to a purely moment-based solution that does not need to assume any kind of distribution. Compared to[16], we add the ability to deal with non-additive noise not only in the measurement update but also in the prediction step. Finally, we perform a thorough evaluation, where we compare the proposed techniques to several state-of-the-art approaches.This paper is structured as follows. First, we formulate the problem in Sec. 2. Then, we introduce the necessary fundamentals from circular statistics in Sec. 3. Based on these fundamentals, we propose deterministic sampling schemes in Sec. 4 and derive the operations on circular densities required for the circular filter in Sec. 5. These results are used to introduce circular filtering algorithms in Sec. 6. An evaluation of the proposed algorithms can be found in Sec. 7. Finally, conclusions are given in Sec. 8.Problem FormulationIn this section, we formulate the problems under consideration and summarize some standard approaches that have been used to address the issues associated with periodicity.
10.1109/maes.2016.150083
[ "https://arxiv.org/pdf/1501.05151v1.pdf" ]
14,812,804
1501.05151
29af11c54a6396734d44a104da453be7d6fe40e9
Recursive Bayesian Filtering in Circular State Spaces Gerhard Kurz Institute for Anthropomatics and Robotics Intelligent Sensor-Actuator-Systems Laboratory (ISAS) Karlsruhe Institute of Technology (KIT) Germany Igor Gilitschenski Institute for Anthropomatics and Robotics Intelligent Sensor-Actuator-Systems Laboratory (ISAS) Karlsruhe Institute of Technology (KIT) Germany Uwe D Hanebeck Institute for Anthropomatics and Robotics Intelligent Sensor-Actuator-Systems Laboratory (ISAS) Karlsruhe Institute of Technology (KIT) Germany Recursive Bayesian Filtering in Circular State Spaces For recursive circular filtering based on circular statistics, we introduce a general framework for estimation of a circular state based on different circular distributions, specifically the wrapped normal distribution and the von Mises distribution. We propose an estimation method for circular systems with nonlinear system and measurement functions. This is achieved by relying on efficient deterministic sampling techniques. Furthermore, we show how the calculations can be simplified in a variety of important special cases, such as systems with additive noise as well as identity system or measurement functions. We introduce several novel key components, particularly a distribution-free prediction algorithm, a new and superior formula for the multiplication of wrapped normal densities, and the ability to deal with non-additive system noise. All proposed methods are thoroughly evaluated and compared to several state-of-the-art solutions. system measurement publication distribution model noise model noise Azmani, Reboul, Choquel, Benjelloun [9] von Mises identity additive identity additive Markovic, Chaumette, Petrovic [14] von Mises-Fisher identity additive identity additive Kurz, Gilitschenski, Julier, Hanebeck [21] Bingham identity additive identity additive Kurz, Gilitschenski, Hanebeck [15] wrapped normal/von Mises nonlinear additive identity additive Kurz, Gilitschenski, Hanebeck [16] wrapped normal nonlinear additive nonlinear any this paper wrapped normal/von Mises nonlinear any nonlinear anyTable 1: Circular filters based on directional statistics.There has been some work on filtering algorithms based on circular statistics by Azmani et al.[9], which was further investigated by Stienne et al.[10]. Their work is based on the von Mises distribution and allows for recursive filtering of systems with a circular state space. However, it is limited to the identity with additive noise as the system equation and the measurement equation. The filter from[9]has been applied to phase estimation of GPS signals [11], [12] as well as map matching [13]. Markovic et al. have published a similar filter [14] based on the von Mises-Fisher distribution, a generalization of the von Mises distribution to the hypersphere. We have previously published a recursive filter based on the wrapped normal distribution allowing for a nonlinear system equation [15]. The paper [16] extends this approach to make a nonlinear measurement update possible. Both papers rely on a deterministic sampling scheme, based on the first circular moment. This kind of sampling is reminiscent of the well-known unscented Kalman filter (UKF) [2]. We have extended this sampling scheme to the first two circular moments in [17], so the proposed filters are, in a sense, circular versions of the UKF. The developed filters have been applied in the context of constrained tracking [18], bearings-only sensor scheduling [19], as well as circular model predictive control [20].Furthermore, we proposed a recursive filter based on the circular Bingham distribution in[21]. The Bingham distribution is closely related to the von Mises distribution, but has a bimodal density, which makes it interesting for axial estimation problems (i.e., problems with 180 • symmetry).An overview of all of these filters and the considered distributions as well as system and measurement models is given inTable 1.This paper summarizes and combines our results as well as extends the previous work by a number of additional contributions. First of all, we propose a general filtering framework that can be used in conjunction with a variety of system and measurement equations, different types of noise, and both the wrapped normal and the von Mises distributions. Our previous publications[15],[16]as well as the work by Azmani et al.[9]can be seen as special cases of the proposed framework.Furthermore, we introduce a new multiplication formula for wrapped normal distributions that outperforms the solution proposed in[15]. We generalize the prediction step from [15] to a purely moment-based solution that does not need to assume any kind of distribution. Compared to[16], we add the ability to deal with non-additive noise not only in the measurement update but also in the prediction step. Finally, we perform a thorough evaluation, where we compare the proposed techniques to several state-of-the-art approaches.This paper is structured as follows. First, we formulate the problem in Sec. 2. Then, we introduce the necessary fundamentals from circular statistics in Sec. 3. Based on these fundamentals, we propose deterministic sampling schemes in Sec. 4 and derive the operations on circular densities required for the circular filter in Sec. 5. These results are used to introduce circular filtering algorithms in Sec. 6. An evaluation of the proposed algorithms can be found in Sec. 7. Finally, conclusions are given in Sec. 8.Problem FormulationIn this section, we formulate the problems under consideration and summarize some standard approaches that have been used to address the issues associated with periodicity. Introduction Estimation of circular quantities is an omnipresent issue, be it the wind direction, the angle of a robotic revolute joint, the orientation of a turntable, or the direction a vehicle is facing. Circular estimation is not limited to applications involving angles, however, and can be applied to a variety of periodic phenomena. For example phase estimation is a common issue in signal processing, and tracking objects that periodically move along a certain trajectory is also of interest. Standard approaches to circular estimation are typically based on estimation techniques designed for linear scenarios that are tweaked to deal with some of the issues arising in the presence of circular quantities. However, modifying linear methods cannot only be tedious and error-prone, but also yields suboptimal results, because certain assumptions of these methods are violated. For example, solutions based on Kalman filters [1] or nonlinear versions thereof [2] fundamentally neglect the true topology of the underlying manifold and assume a Gaussian distribution, which is only defined on R n . In the linear case, the use of a Gaussian distribution is frequently justified by the central limit theorem. This justification no longer holds in a circular setting, as the Gaussian is not a limit distribution on the circle. In order to properly deal with circular estimation problems, we rely on circular statistics [3], [4], a subfield of statistics that deals with circular quantities. More broadly, the field of directional statistics [5] considers a variety of manifolds, such as the circle, the hypersphere, or the torus. Unlike standard approaches that assume linear state spaces, methods based on circular statistics correctly use the proper manifold and rely on probability distributions defined on this manifold. Circular statistics has been applied in a variety of sciences, such as biology [4], bioinformatics [6], meteorology [7], and geosciences [8]. Circular Filtering Circular filtering considers estimation problems on the unit circle, which is commonly parameterized as the set of complex numbers with unit length {x ∈ C : |x| = 1}. To allow for a more convenient one-dimensional notation, we identify S 1 with the half-open interval [0, 2π) ⊂ R, while keeping the topology of the circle. Together with the operation + : S 1 × S 1 → S 1 , x + y := x + R y mod 2π, for all x, y ∈ [0, 2π) with standard addition + R on R, the circle S 1 forms an Abelian group. Because S 1 with the topology given above has a manifold structure, (S 1 , +) is a Lie group. We consider a system whose state x k at time step k is a value on the unit circle S 1 . System and measurement models are assumed to be given. In this paper, we propose several methods to deal with different types of models. More complex models necessitate the use of more sophisticated algorithms and conversely, simpler models allow the use of computationally less expensive algorithms. System Model In this work, we consider a system model whose state evolves according to the general system equation x k+1 = a k (x k , w k ) with (nonlinear) system function a k : S 1 × W → S 1 and noise w k ∈ W stemming from some noise space W . Note that W is not necessarily S 1 , but may be an arbitrary set, for example the real-vector space R n , some manifold, or even a discrete set. An interesting and practically relevant special case is a (nonlinear) system with additive noise x k+1 = a k (x k ) + w k mod 2π with a k : S 1 → S 1 and w k ∈ S 1 . More particular, we also consider the special case, where a k is the identity, i.e., x k+1 = x k + w k mod 2π. Measurement Model The system state cannot be observed directly, but may only be estimated based on measurements that are disturbed by noise. A general measurement function is given bŷ z k = h k (x k , v k ) , whereẑ k ∈ Z is the measurement in the measurement space Z, h k : S 1 × V → Z is the measurement function and v k ∈ V is arbitrary measurement noise in a certain noise space V . Note that the measurement and noise space can be arbitrary sets in general. An interesting special case are measurement functions where the measurement noise is additive, i.e., z k = h k (x k ) + v k with measurement function h k : S 1 → Z and v k ∈ Z. In this case, we require Z to have a group structure with + as the operation. Additionally, we consider the more specific case where h k is the identity, i.e.,ẑ k = x k + v k mod 2π withẑ k , v k ∈ S 1 . Remark 1. We do not consider linear system models because linearity is a concept of vector spaces, not manifolds [15]. For this reason, there are no linear functions on the circle. Standard Approaches As circular estimation problems are wide-spread in a variety of applications, a number of standard approaches have been employed. We introduce three of the most common methods and explain their strengths and weaknesses. Gaussian-based approaches Gaussian-based methods (wrongly) assume a Gaussian distribution and use standard filtering techniques for Gaussians in conjunction with certain modifications to allow their application to circular problems. One-Dimensional Methods. One common approach is the use of a standard Kalman filter [1] or, in case of nonlinear system or measurement functions, unscented Kalman filter (UKF) [2] with a scalar state x k containing the angle θ k , i.e., x k = θ k . However, two modifications are necessary before this approach can be used in practice. First, the estimate has to be forced to stay inside the interval [0, 2π) by performing a modulo operation after every prediction and/or update step. Second, if the measurement space is periodic, the measurement needs to be repositioned to be closer to the state mean in certain cases. This problem occurs whenever the measurement and the current state mean are more than π apart. In this case, the measurement needs to be moved by ±2πk to an equivalent measurement that deviates at most π from the state mean. An illustration of this problem is given in Fig. 1. This type of filter is used as a comparison in [15]. When the uncertainty is small, this kind of approach works fairly well, but it tends to produce unsatisfactory results if the uncertainty is high. Two-Dimensional Methods. Another common approach is based on the Kalman filter or the UKF with two-dimensional state subject to a nonlinear constraint. More specifically, an angle θ k is represented by a state vector x k = [cos(θ k ), sin(θ) k ] T and the constraint is ||x k || = 1 to enforce that x k is on the unit circle. In order to enforce this constraint, x k is projected to the unit circle after each prediction and/or update step. More sophisticated approaches increase the covariance to reflect the fact that the projection operation constitutes an increase in uncertainty [22]. This approach has been used in [18], but did not perform as well as the filter based on circular statistics. One of the issues of this approach is the fact that the system and measurement model sometimes become more complicated when the angle θ k is transformed to a two-dimensional vector. Particle Filters Another method that can be applied is particle filtering [23]. Particle filters on nonlinear manifolds are fairly straightforward to implement because each particle can be treated separately. For the particle filter to work, the system function and the measurement likelihood both need to respect x 1 =cos(φ) x 2 =sin(φ) f(x 1 , x 2 ) (a) The wrapped normal distribution (red) is obtained by wrapping a Gaussian distribution (blue) around the circle. The parameters we used for this example are µ = 0, σ = 2. (b) The von Mises distribution (red) arises when restricting a two-dimensional Gaussian with µ = (cos µ, sin µ) T and covariance κ · I 2×2 to the unit circle. The parameters for this example are µ = 0, κ = 1. the underlying topology. The reweighting step as well as the commonly used sequential importance resampling (SIR) are independent of the underlying manifold and can be used in a circular setting as well. However, issues that are typically associated with particle filters arise. If the measurement likelihood function is very narrow, particle degeneration can occur, i.e., (almost) all particles have zero weight after the reweighting step. Furthermore, a large number of particles is required to obtain stable results. Even though these problems are less critical in a one-dimensional setting, there can still be issues if measurements with high certainty occur in areas with few particles. This can, for example, occur when information from sensors with very different degrees of accuracy is fused. It should also be noted that sampling from certain circular distributions can be somewhat involved and require the use of the Metropolis Hastings algorithm [24] or similar approaches, e.g., in case of the von Mises distribution. Circular Statistics Because of the drawbacks of the approaches discussed above, we propose a filtering scheme based on circular statistics [3], [5]. In the following, we introduce the required fundamentals from the field of circular statistics. Circular Distributions Circular statistics considers probability distributions defined on the unit circle. A variety of distributions has been proposed [4]. We give definitions of all distributions that are required for the proposed filtering scheme. Definition 1 (Wrapped Normal Distribution). The wrapped normal (WN) distribution is given by the probability density function (pdf) f (x; µ, σ) = 1 √ 2πσ ∞ k=−∞ exp − (x − µ + 2πk) 2 2σ 2 with x ∈ S 1 , location parameter µ ∈ S 1 , and concentration parameter σ > 0. The WN distribution is obtained by wrapping a one-dimensional Gaussian distribution around the unit circle and adding up all probability mass that is wrapped to the same point (see Fig. 2a). It appears as a limit distribution on the circle [15] in the following sense. A summation scheme of random variables that converges to the Gaussian distribution in the linear case, will converge to the WN distribution if taken modulo 2π. Because of its close relation to the Gaussian distribution, the WN distribution inherits a variety of its properties, for example its normalization constant as well as the formula for the convolution of densities. Even though there is an infinite sum involved, evaluation of the pdf of a WN distribution can be performed efficiently, because only few summands need to be considered [25]. Definition 2 (Von Mises Distribution). The von Mises (VM) distribution is given by the pdf f (x; µ, κ) = 1 2πI 0 (κ) exp(κ cos(x − µ)) with x ∈ S 1 , location parameter µ ∈ S 1 , and concentration parameter κ > 0. I 0 (·) is the modified Bessel function [26] of order 0. The VM distribution, sometimes also referred to as the circular normal (CN) distribution, is obtained by restricting a two-dimensional Gaussian with mean ||µ|| = 1 and covariance κ · I 2×2 (where I 2×2 is the identity matrix) to the unit circle and reparameterizing to [0, 2π) as can be seen in Fig. 2b. For this reason, it also inherits some of the properties of the Gaussian distribution; most importantly, it is closed under Bayesian inference. The VM distribution has been used as a foundation for a circular filter by Azmani et al. [9]. It is closely related to the Bingham distribution and conversion between the two is effectively a matter of shrinking the the interval [0, 2π) by a factor of two two, i.e., f (2x; µ, κ) is a Bingham distribution [27, p. 4]. Unlike the WN and VM distributions, the WD distribution is a discrete probability distribution on a continous domain. It can be obtained by wrapping a Dirac mixture in R around the unit circle. WD distributions can be used as discrete approximations of continuous distributions with a finite set of samples. In this paper, we use the following notation. We denote the density function of a WN distribution with parameters µ and σ with WN (µ, σ). If a random variable x is distributed according to this WN distribution, we write x ∼ WN (µ, σ). The terms VM(µ, κ) and WD(γ 1 , . . . , γ L , β 1 , . . . , β L ) are used analogously to describe the density functions of VM and WD distributions with parameters µ, κ and γ 1 , . . . , γ L , β 1 , . . . , β L , respectively. Circular Moments In circular statistics, there is a concept called circular (or trigonometric) moment. Definition 4 (Circular Moments). For a random variable x ∼ f (x) defined on the circle, its n-th circular moment is given by m n = E(exp(ix) n ) = E(exp(inx)) = 2π 0 exp(inx) · f (x) dx ∈ C with imaginary unit i. The n-th moment is a complex number and, hence, has two degrees of freedom, the real and the imaginary part. For this reason, the first moment includes information about the circular mean arg m 1 = atan2(Im m 1 , Re m 1 ) as well as the concentration |m 1 | = (Re m 1 ) 2 + (Im m 1 ) 2 of the distribution 1 , similar to the first two real-valued moments. It can be shown that WN and VM distributions are uniquely defined by their first circular moment [3]. Remark 2. Any continuous and piecewise continuously differentiable 2π-periodic function f : R → R can be written as a Fourier series f (x) = ∞ k=−∞ c k exp(ikx) where c k = 1 2π 2π 0 f (x) exp(−ikx)dx . If f (·) is the pdf of a circular probability distribution, the Fourier coefficients are closely related to the circular moments according to c k = 1 2π m −k . Lemma 1 (Moments for WN, VM, and WD Distributions). For WN, VM, and WD distributions with given parameters, the n-th circular moment can be calculated according to m W N n = exp inµ − n 2 σ 2 2 , m V M n = exp(inµ) I |n| (κ) I 0 (κ) , m W D n = L j=1 γ j exp(inβ j ) . A proof is given in [5]. Circular Moment Matching As both WN and VM distributions are uniquely defined by their first moment, it is possible to convert between them by matching the first circular moment. Lemma 2 (Circular Moment Matching). We define A(x) = I 1 (x) I 0 (x) as given in [5]. 1. For a given first moment m 1 , the WN distribution with this first moment has the density WN atan2(Im m 1 , Re m 1 ), −2 log (|m 1 |) . For a given first moment m 1 , the VM distribution with this first moment has the density VM atan2(Im m 1 , Re m 1 ), A −1 (|m 1 |) . 3. For a given VM distribution with density VM(µ, κ), the WN distribution with identical first moment has the density WN µ, −2 log I 1 (κ) I 0 (κ) . For a given WN distribution with density WN (µ, σ), the VM distribution with identical first moment has the density VM µ, A −1 exp − σ 2 2 The proof is given in [15]. Calculation of the function A −1 (·) is somewhat tricky. In [15], we use the algorithm by Amos [28] 2 to calculate A(·) and MATLAB's fsolve to invert this function. Stienne et al. have proposed closed-form approximations, which can be calculated very easily but have a large approximation error [12], [10]. A more detailed discussion on approximations of A −1 (·) can be found in [29] and [30, Sec. 2.3]. Deterministic Sampling In order to propagate continous probability densities through nonlinear functions, it is a common technique to use discrete sample-based approximations of the continous densities. A set of samples can easily be propagated by applying the nonlinear function to each sample individually. This approach can be used for both the prediction and the measurement update step. We distinguish between deterministic and nondeterministic sampling. Nondeterministic sampling relies on a randomized algorithm to stochastically obtain samples of a density. Typical examples include the samplers used by the particle filter [23] or the Gaussian particle filter [31]. Deterministic sampling selects samples in a deterministic way, for example to fit certain moments (the sampler used by the UKF, [2]), or to optimally approximate the shape of the density (published in [32], this sampler is used by the S 2 KF, [33]). Deterministic sampling schemes have the advantage of requiring a significantly smaller number of samples, which is why we will focus on this type of solution. A naïve solution for approximating a WN density may be the application of a deterministic sampling scheme for the Gaussian distribution (such as the samplers used in [2], [33]) and subsequently wrapping the samples. Even though this technique is valid for stochastic samples, it does not provide satisfactory results for deterministic samples. In extreme cases, wrapping can cause different samples to be wrapped to the same point, grossly misrepresenting the original density. This problem is illustrated in Fig. 3. In the case of UKF samples (Fig. 3a), it can be seen that for σ ≈ 2.5, one sample is placed at µ and two samples are placed on the opposite side of the circle, i.e., the mode of the approximation is opposite to the true mode. Furthermore, for σ ≈ 5 all three UKF samples are wrapped to the same position, i.e., the sample-based approximation degenerates to a distribution with single Dirac component even though the true distribution is nearly uniform. Similar issues arise in the case of S 2 KF samples (see Fig. 3b). Analytic Solutions First of all, we consider analytic solutions to obtain deterministic samples. These solutions are based on circular moment-matching and only provide a small, fixed number of Dirac delta components, but are extremely fast to calculate, making them a good choice for real-time applications. In [15], we presented a method to obtain a WD approximation with three equally weighted components, which is based on matching the first circular moment (see Algorithm 1). We further extended this scheme to obtain a WD with five components by matching the first as well as second circular moment (see Algorithm 2), which, as we proved in [17], necessitates the use of different weights. Both approaches can approximate arbitrary symmetric circular densities with given circular moments. Algorithm 1: Deterministic approximation with L = 3 components. Input: first circular moment m 1 Output: WD(γ 1 , . . . , γ 3 , β 1 , . . . , β 3 ) /* extract µ */ µ ← atan2(Im m 1 , Re m 1 ); /* obtain Dirac positions */ α ← arccos 3 2 |m 1 | − 1 2 ; β 1 ← µ − α mod 2π; β 2 ← µ mod 2π; β 3 ← µ + α mod 2π; /* obtain weights */ γ 1 , γ 2 , γ 3 ← 1 3 ; Algorithm 2: Deterministic approximation with L = 5 components. Input: first circular moment m 1 , second circular moment m 2 , parameter λ ∈ [0, 1] with default λ = 0.5 Output: [34] for 10 components, e) quantization approach from [35] for 10 components. Note that the result of Algorithm 1 is identical for both densities because only the first circular moment is considered. WD(γ 1 , . . . , γ 5 , β 1 , . . . , β 5 ) /* extract µ */ µ ← atan2(Im m 1 , Re m 1 ); m 1 ← |m 1 |; m 2 ← |m 2 |; /* obtain weights */ γ min 5 ← (4m 2 1 − 4m 1 − m2 + 1)/(4m 1 − m 2 − 3); γ max 5 ← (2m 2 1 − m 2 − 1)/(4m 1 − m 2 − 3); γ 5 ← γ min 5 + λ(γ max 5 − γ min 5 ); γ 1 , γ 2 , γ 3 , γ 4 ← (1 − γ 5 )/4; /* obtain Dirac positions */ c 1 ← 2 1−γ 5 (m 1 − γ 5 ); c 2 ← 1 1−γ 5 (m 2 − γ 5 ) + 1; x 2 ← (2c 1 + 4c 2 1 − 8(c 2 1 − c 2 ))/4; x 1 ← c 1 − x 2 ; φ 1 ← arccos(x 1 ); φ 2 ← arccos(x 2 ); (β 1 , . . . , β 5 ) ← µ + (−φ 1 , +φ 1 , −φ 2 , +φ 2 , 0) mod 2π; Optimization-based Solutions If a larger number of samples is desired and there are more degrees of freedom in the samples than constraints (such as preservation of circular moments), optimization-based solutions can be used. The number of samples can be adjusted by the user and an optimal approximation is derived by minimizing a distance measure. In order to simultaneously calculate optimal locations and weights for the samples, a systematic approach based on VM kernels has been proposed in [34]. For a WD mixture, an induced VM mixture is compared to the true distribution with a quadratic integral distance. A specific kernel width is considered for each component, which depends on the weight of the component and the value of the true distribution at the location of the component. Both the weights and the locations of a fixed even number of WD components are optimized to obtain an optimal symmetric approximation. Constraints in the optimization algorithms are used to maintain a predefined number of circular moments. This approach results in well-distributed Dirac mixtures that fulfill the moment constraints. A quantization approach is discussed in [35]. It is based on computing optimal Voronoi quantizers. In this approach, optimality refers to minimal quadratic distortion. The resulting Voronoi quantizer gives rise to a circular discrete probability distribution on a continuous domain that approximates the original continuous distribution. Use of this approximation is particularly beneficial in the prediction step of stochastic filters, because an error bound for propagation through a non-trivial system function can be obtained without actually knowing that function. It is sufficient to require it to be Lipschitz and to know an upper bound for the Lipschitz constant. Furthermore, circular moment constraints can be introduced in the optimization procedure of the quantization approach. Examples from all discussed methods for deterministic sampling are depicted in Fig. 4. Operations on Densities In order to derive a circular filtering algorithm, we need to be able to perform certain operations on the involved probability densities. Shifting and Mirroring For a given density f (x), we want to obtain the density f (c − x) for a constant c ∈ S 1 . This operation is necessary in certain cases of the update step. We can split this operation into two steps: mirroring to obtain f (−x), and subsequent shifting by c to obtain f (c + (−x)). Mirroring WN (µ, σ) and VM(µ, κ) yields WN (2π − µ, σ) and VM(2π − µ, κ) because the distributions are symmetric around their mean. Shifting WN (µ, σ) and VM(µ, κ) by c yields WN (µ − c mod 2π, σ) and VM(µ − c mod 2π, κ) , so the combined operation results in WN ((2π − µ) − c mod 2π, σ) and VM((2π − µ) − c mod 2π, κ) . Circular Convolution Given two independent circular random variables x 1 ∼ f 1 (x 1 ), x 2 ∼ f 2 (x 2 ), the sum x 1 + x 2 is distributed according to (f 1 * f 2 )(x) = 2π 0 f 1 (t)f 2 (x − t) dt , where * denotes the convolution. This operation is necessary in the prediction step to incorporate additive noise. WN distributions are closed under convolution and the new pdf can be obtained just as in the Gaussian case [15], i.e., µ = µ 1 + µ 2 mod 2π, σ 2 = σ 2 1 + σ 2 2 . VM distributions are not closed under convolution. For this reason, Azmani et al. [9] used the approximation from [5], which is given by µ = µ 1 + µ 2 , κ = A −1 (A(κ 1 ), A(κ 2 ) ). The function A(·) is the same as defined in Lemma 2. This approximation can be derived from an intermediate WN representation [10]. A similar approximation has been used by Markovic et al. for the von Mises-Fisher case [14, (7)]. In this paper, we present a more general result that calculates the convolution based on circular moments. Lemma 3 (Moments After Addition of Random Variables). Assume independent random variables x 1 ∼ f 1 , x 2 ∼ f 2 defined on the circle. For the sum x = x 1 + x 2 , it holds E(exp(inx)) = E(exp(inx 1 ))E(exp(inx 2 )) . Proof. m n =E(exp(inx)) = 2π 0 exp(inx)f (x) dx = 2π 0 2π 0 exp(in(x))f 1 (y)f 2 (x − y) dy dx = 2π 0 2π 0 exp(in(x 1 + x 2 ))f 1 (x 1 )f 2 (x 2 ) dx 1 dx 2 = 2π 0 exp(inx 1 )f 1 (x 1 ) dx 1 2π 0 exp(inx 2 )f 2 (x 2 ) dx 2 =E(exp(inx 1 ))E(exp(inx 2 )) If moment matching of the first circular moment is used to fit a WN or VM to the density that results from convolution, the solutions for WN and VM distributions from [9] and [15] arise as special cases of Lemma 3. Multiplication Multiplication of pdfs is an important operation for filtering algorithms, because it is required for Bayesian inference. In general, the product of two pdfs is not normalized and thus, not a pdf. For this reason, we consider the renormalized product, which is a valid pdf. VM Von Mises densities are closed under multiplication [9]. It holds that VM(µ 1 , κ 1 ) · VM(µ 2 , κ 2 ) ∝ VM(µ, κ), where µ = atan2(Im m 1 , Re m 1 ), κ = |m 1 |, with m 1 = κ 1 exp(iµ 1 ) + κ 2 exp(iµ 2 ) . WN WN densities are not closed under multiplication. In the following, we consider two different methods to approximate the density of the product with a WN density. WN via VM. WN densities are not closed under multiplication. In [15], we proposed a method to use the VM distribution in order to approximate the product of two WN densities. More specifically, we convert the WN densities to VM densities using Lemma 2, multiply according to the VM multiplication formula, and convert back to a WN distribution by applying Lemma 2 again. This method has the disadvantage that, in general, the first circular moment of the resulting WN does not match the first circular moment of the true product. An example can be seen in Fig. 5. WN via Moment Matching. In this paper, we present a new method for approximating the product of WN distributions. This method is based on directly approximating the true posterior moments. Theorem 1. The first circular moment of WN (µ 1 , σ 1 ) · WN (µ 2 , σ 2 ) after renormalization is given by m 1 = ∞ j,k=−∞ w(j, k) 2π 0 exp(ix)N (x; µ(j, k), σ(j, k))dx ∞ j,k=−∞ w(j, k) 2π 0 N (x; µ(j, k), σ(j, k))dx where N (x; µ, σ) is a one-dimensional Gaussian density with mean µ and standard deviation σ, and µ(j, k) = (µ 1 + 2πj)σ 2 2 + (µ 2 + 2πk)σ 2 1 σ 2 1 + σ 2 2 , σ(j, k) = σ 2 1 σ 2 2 σ 2 1 + σ 2 2 , w(j, k) = exp − 1 2 ((µ 1 +2πj)−(µ 2 +2πk)) 2 σ 2 1 +σ 2 2 2π(σ 2 1 + σ 2 2 ) . Proof. The true renormalized product is given by f (x) = c · f (x; µ 1 , σ 1 ) · f (x; µ 2 , σ 2 ), where c renormalizes the product, i.e., c = 2π 0 f (x; µ 1 , σ 1 ) · f (x; µ 2 , σ 2 ) dx −1 We calculate m 1 =c · 2π 0 exp(ix) · f (x; µ 1 , σ 1 ) · f (x; µ 2 , σ 2 ) dx =c · 2π 0 exp(ix) · ∞ j=−∞ N (x; µ 1 + 2πj, σ 1 ) · ∞ k=−∞ N (x; µ 2 + 2πk, σ 2 ) dx =c · ∞ j=−∞ ∞ k=−∞ 2π 0 exp(ix) · N (x; µ 1 + 2πj, σ 1 ) · N (x; µ 2 + 2πk, σ 2 ) dx =c · ∞ j=−∞ ∞ k=−∞ 2π 0 exp(ix) · w(j, k) · ·N (x; µ(j, k), σ(j, k)) dx =c · ∞ j=−∞ ∞ k=−∞ ·w(j, k) · 2π 0 exp(ix) · N (x; µ(j, k), σ(j, k)) dx , where we use the dominated convergence theorem to interchange summation and integration. We use the abbreviations, µ(j, k) = (µ 1 + 2πj)σ 2 2 + (µ 2 + 2πk)σ 2 1 σ 2 1 + σ 2 2 , σ(j, k) = σ 2 1 σ 2 2 σ 2 1 + σ 2 2 , w(j, k) = exp − 1 2 ((µ 1 +2πj)−(µ 2 +2πk)) 2 σ 2 1 +σ 2 2 2π(σ 2 1 + σ 2 2 ) based on the multiplication formula for Gaussian densities given in [36, 8.1.8]. To obtain the renormalization factor c −1 , we use a similar derivation c −1 = 2π 0 f (x; µ 1 , σ) 1 · f (x; µ 2 , σ 2 ) dx = 2π 0 ∞ j=−∞ N (x; µ 1 + 2πj, σ 1 ) · ∞ k=−∞ N (x; µ 2 + 2πk, σ 2 ) dx = ∞ j=−∞ ∞ k=−∞ 2π 0 N (x; µ 1 + 2πj, σ 1 ) · N (x; µ 2 + 2πk, σ 2 ) dx = ∞ j=−∞ ∞ k=−∞ w(j, k) · 2π 0 N (x; µ(j, k), σ(j, k)) dx . The involved integrals can be reduced to evaluations of the complex error function erf [26, 7.1]. This yields 2π 0 exp(ix) · N (x; µ(j, k), σ(j, k)) dx = 1 2 exp iµ(j, k) − σ(j, k) 2 2 · erf µ(j, k) + iσ(j, k) 2 √ 2σ(j, k) − erf µ(j, k) − 2π + iσ(j, k) 2 √ 2σ(j, k) and 2π 0 N (x; µ(j, k), σ(j, k)) dx = 1 2 erf µ(j, k) σ(j, k) √ 2 − erf µ(j, k) − 2π σ(j, k) √ 2 . There are efficient implementations of the complex error function by means of the related Faddeeva function [37]. Furthermore, the infinite sums can be truncated to just a few summands without a significant loss in accuracy. For example, the multiplication in Fig. 5 requires 5 × 5 summands for an error smaller than the accuracy of the IEEE 754 64 bit double data type [38]. Consequently, the proposed method allows for efficient calculation of the approximate multiplication of WN densities. Circular Filtering Based on the results in the previous section, we derive circular filtering algorithms for the scenarios described in Sec. 2.1. All proposed algorithms follow the recursive filtering concept and consist of prediction and measurement update steps. We formulate the necessary steps without requiring a particular density whenever possible such that most methods can be directly applied to WN as well as VM distributions, and might even be generalized to other continous circular distributions. An overview of all considered prediction and measurement update algorithms is given in Table 2. [9] contribution contribution [9] contribution contribution moment-based contribution contribution contribution --- Table 2: Prediction and measurement update algorithms. Entries marked with contribution are contributions of this paper. Prediction The prediction step is used to propagate the estimate through time. Identity System Model The transition density is given according to f (x k+1 |x k ) = 2π 0 f (x k+1 , w k |x k ) dw k = 2π 0 f (x k+1 |x k , w k )f w (w k ) dw k = 2π 0 δ(x k+1 − (x k + w k ))f w (w k ) dw k = f w (x k+1 − x k ) , where f w (·) is the density of the system noise. For the predicted density, according to the Chapman-Kolmogorov equation we obtain f p (x k+1 ) = 2π 0 f (x k+1 |x k )f e (x k ) dx k .(1) In the special case of an identity system model, this yields f p (x k+1 ) = 2π 0 f w (x k+1 − x k )f e (x k ) dx k = (f w * f e )(x k+1 ) , where * denotes convolution as defined in Sec. 5.2. For the VM distribution, this system model has been considered in [9]. If a WN distribution is assumed, (1) is a special case of [15] where we omit the propagation through the nonlinear function. Nonlinear System Model with Additive Noise Similar to the previous case, the transition density is given by f (x k+1 |x k ) = 2π 0 δ(x k+1 − (a k (x k ) + w k ))f w (w k ) dw k . We approximate the prior density f e (x k ) ≈ L j=1 γ j δ(x k − β j ) using, for example, Algorithm 1 or Algorithm 2. Then, the prediction density can be approximated according to f p (x k+1 ) = 2π 0 f (x k+1 |x k )f e (x k ) dx k ≈ 2π 0 2π 0 δ(x k+1 − (a k (x k ) + w k ))f w (w k ) ddw k ·   L j=1 γ j δ(x k − β j )   dx k = 2π 0 f w (w k ) L j=1 γ j 2π 0 δ(x k+1 − (a k (x k ) + w k )) · δ(x k − β j ) dx k dw k = 2π 0 f w (w k ) L j=1 γ j δ(x k+1 − (a k (β j ) + w k )) ≈f (x k+1 −w k ) dw k = (f w * f )(x k+1 ) , wheref is obtained by moment matching of the WD L j=1 γ j δ(x − (a k (β j )) according to Lemma 2. The convolution can be calculated as described in Sec. 5.2. Nonlinear System Model with Arbitrary Noise In this paper, we extend the previous results to deal with arbitrary noise in the prediction step. For arbitrary noise, the transition density is given by f (x k+1 |x k ) = 2π 0 δ(x k+1 − a k (x k , w k ))f w (w k ) dw k . We approximate the prior density f e (x k ) ≈ L j=1 γ j δ(x k − β j ) as well as the noise density f w (w k ) ≈ L w l=1 γ w l δ(w k − β w l ) . It should be noted that the noise is not necessarily a circular quantity and different approximation techniques may be required. If W = R n , the techniques presented in [32] may be applied. Then, the prediction density can be approximated according to f p (x k+1 ) = 2π 0 f (x k+1 |x k )f e (x k ) dx k = 2π 0 W δ(x k+1 − a k (x k , w k ))f w (w k ) dw k f e (x k ) dx k ≈ 2π 0 W δ(x k+1 − a k (x k , w k )) L w l=1 γ w l δ(w k − β w l ) dw k f e (x k ) dx k = L w l=1 γ w l 2π 0 δ(x k+1 − a k (x k , β w l ))f e (x k ) dx k ≈ L w l=1 γ w l 2π 0 δ(x k+1 − a k (x k , β w l )) ·   L j=1 γ j δ(x k − β j )   dx k = L j=1 L w l=1 γ j γ w l δ(x k+1 − a k (β j , β w l )) A continuous density can be fitted to this result by circular moment matching. The algorithm is given in Algorithm 3. It is worth noting that this algorithm can be executed based purely on the circular moments of f (x k ) and f w (w k ) if the deterministic sampling scheme only depends on these moments. In that case, we do not necessarily need to fit a distribution f (x k+1 ) to the resulting circular moments, but can just store the estimate by retaining those circular moments. Algorithm 3: Prediction with arbitrary noise. Input: prior density f (x k ), system noise density f w (w k ), system function a k (·, ·) Output: predicted density f (x k+1 ) /* sample prior density and noise density * / (γ 1 , . . . , γ L , β 1 , . . . , β L ) ← sampleDeterm(f (x k )); (γ w 1 , . . . , γ w L w , β w 1 , . . . , β w L w ) ← sampleDeterm(f w (w k )); /* obtain Cartesian product and propagate */ for j ← 1 to L do for l ← 1 to L w do γ p j+L(l−1) ← γ j · γ w l ; β p j+L(l−1) ← a k (β j , β w l ); end end /* obtain posterior density */ f (x k+1 ) ← momentMatching(γ p 1 , . . . , γ p L·L w , β p 1 , . . . , β p L·L w )); Measurement Update The measurement update step fuses the current estimate with a measurement that was obtained according to the measurement equation. Identity Measurement Model In the case of the identity measurement model and additive noise, the measurement likelihood is given by f (z k |x k ) = f v (z k − x k ) . For the posterior density, application of Bayes' theorem yields f (x k |z k ) = f (z k |x k )f (x k ) f (z k ) ∝ f (z k |x k )f (x k ) , where f (x k ) is the prior density. Thus, we obtain the posterior density f (x k |z k ) ∝ f v (z k − x k )f (x k ) , as the product of the prior density and f v (z k − x k ), which can be obtained as described in Sec. 5.1. The multiplication depends on the assumed probability density and can be performed using the multiplication formulas given in Sec. 5.3. For the VM case, this is equivalent to the measurement update from [9] and for the WN case, this is equivalent to the measurement update from [15]. Nonlinear Model with Additive Noise For a nonlinear measurement function with additive noise, the measurement likelihood is calculated according to f (z k |x k ) = f v (z k − h k (x k )), as given in [16]. The remainder of the measurement update step is identical to the case of arbitrary noise as described in the following section. Nonlinear Model with Arbitrary Noise For a nonlinear update with arbitrary noise, we assume that the likelihood is given. The key idea is to approximate the prior density f (x k ) with a WD mixture and reweigh the components according to the likelihood f (z k |x k ). However, this can lead to degenerate solutions, i.e., most or all weights are close to zero, if the likelihood function is narrow. We have shown in [16] that a progressive solution as introduced in [39] can be used to avoid this issue. For this purpose, we formulate the likelihood as a product of likelihoods f (z k |x k ) = f (z k |x k ) λ 1 · . . . · f (z k |x k ) λs , where λ 1 , . . . , λ s > 0 and s j=1 λ j = 1. This decomposition of the likelihood allows us to perform the measurement update step gradually by performing s partial update steps. Each update step is small enough to prevent degeneration and we obtain a new sample set after each step, to ensure that the differences between the sample weights stay small. In order to determine λ 1 , . . . , λ s and s, we require that the quotient between the smallest weight γ min and the largest weight γ max after reweighing is not below a certain threshold R ∈ (0, 1), i.e., γ min γmax ≥ R. Using the conservative bounds γ min ≥ min j=1,...,L (γ n j ) · min j=1,...,L (f (z k |β n j )) , γ max ≤ max j=1,...,L (γ n j ) · max j=1,...,L (f (z k |β n j )) , this leads to the condition λ n ≤ log R · max j=1,...,L (γ n j ) min j=1,...,L (γ n j ) log min j=1,...,L f (z k |β n j ) max j=1,...,L f (z k |β n j ) , where WD(γ n 1 , . . . , γ n L , β n 1 , . . . , β n L ) is the deterministic approximation at n-th progression step 4 . The progression continues until n λ n = 1. This method can be applied in conjunction with WN as well as VM distributions (see Algorithm 4). Evaluation Propagation Accuracy In order to evaluate the deterministic sampling as introduced in Sec. 4, we investigate the accuracy when performing propagation through the nonlinear function g : S 1 → S 1 , g(x) = x + c × R sin(x) mod 2π , where c ∈ [0, 1) is a parameter controlling the strength of the nonlinearity and × R refers to multiplication in the field of real numbers R. Furthermore, we consider the density WN (0, σ) that we want to propagate through g(·). For this purpose, we sample WN (0, σ) deterministically using the methods described in Sec. 4 and obtain WD(γ 1 , . . . , γ L , β 1 , . . . , β L ). Then, we apply g(·) componentwise, which yields WD(γ 1 , . . . , γ L , g(β 1 ), . . . , g(β L )). The true posterior is given by f true (x) = f (g −1 (x); µ, σ) g (x) Algorithm 4: Progressive measurement update for arbitrary noise. Input: measurementẑ k , likelihood f (z k |x k ), predicted density f p (x k ) as WN or VM, threshold parameter R Output: estimated fensity as WN or VM s ← 0; f (x k ) ← f p (x k ); while s n=1 λ n < 1 do s ← s + 1; /* deterministic sampling from WN or VM density (Sec. 4) */ (γ 1 , . . . , γ L , β 1 , . . . , β L ) ← sampleDeterm(f (x k )); /* calculate size of progression step */ λ s ← min     1 − s−1 n=1 λ n , log R· max j=1,...,L (γ j ) min j=1,...,L (γ j ) log min j=1,...,L f (z k |β j ) max j=1,...,L f (z k |β j )     ; /* execute progression step */ for j ← 1 to L do γ j ← γ j · f (ẑ k |β j ) λs ; end /*D KL f true ||f f itted = 2π 0 f true (x) log f true (x) f f itted (x) dx ,(2) between the true posterior and the fitted WN. The Kullback-Leibler divergence is an information theoretic measure to quantify the information loss when approximating f true with f f itted . This concept is illustrated in Fig. 6. The results for different values of σ are depicted in Fig. 7. We compare several samplers, the analytic methods with L = 3 components (Algorithm 1) and L = 5 components (Algorithm 2, parameter λ = 0.5) from Sec. 4.1 as well as the quantization approach discussed in Sec. 4.2. It can be seen that the analytic solution with L = 5 components is significantly better than the solution with L = 3 components. The quantization-based solution is computationally quite demanding but gives almost optimal results. However, the analytic solution with L = 5 components has comparable performance in terms of the Kullback-Leibler divergence even though the posterior moments are not calculated as accurately. Moment-Based WN Multiplication In this evaluation, we compare the two methods for WN multiplication given in Sec. 5.3.2 and Sec. 5.3.2. For two WN densities WN (µ 1 , σ 1 ) and WN (µ 2 , σ 2 ), we calculate the true product f true = WN (µ 1 , σ 1 ) · WN (µ 2 , σ 2 ) and compare it to the WN approximation f f itted . In order to determine the approximation quality, we compute the Kullback-Leibler divergence (2). Furthermore, we consider the L 2 distance D L 2 f true , f f itted = 2π 0 f true (x) − f f itted (x) 2 dx . The results for different values of σ 1 , µ 2 , and σ 2 are depicted in Fig. 8 and Fig. 9. We keep µ 1 fixed because only the difference between µ 2 and µ 1 affects the result. Multiplication is commutative, so we consider different sets of values for σ 1 and σ 2 to avoid redundant plots. As can be seen, the moment-based approach derived in Sec. 5.3.2 significantly outperforms the approach from Sec. 5.3.2 in almost all cases according to both distance measures. The new approach is particularly superior for small uncertainties. Filtering Scenario System Function Measurement Noise C v s (3) 0.01 · I 2×2 m (3) 0.1 · I 2×2 l(3) 3 · I 2×2 s-non-additive (4) 0.01 · I 2×2 m-non-additive (4) 0.1 · I 2×2 l-non-additive (4) 3 · I 2×2 Table 3: Evaluation scenarios. In order to evaluate the proposed filtering algorithms, we simulated several scenarios. First of all, we distinguish between models with additive and with a more complex noise structure. In the case of additive noise, we consider the system function x k+1 = x k + c 1 × R sin(x k ) + c 2 + w k ,(3) with two parameters c 1 = 0.1, c 2 = 0.15, noise w k ∼ WN (0, 0.2), and × R is multiplication in the field of real numbers R. Intuitively, c 1 determines the degree of nonlinearity and c 2 is a constant angular velocity that is added at each time step. For the case of arbitrary noise, the system function is given by with the same c 1 , c 2 , and w k as above. In both cases, the nonlinear measurement function is given bŷ x k+1 = x k + c 1 × R sin(x k + w k ) + c 2 ,(4)z k = [cos(x k ), sin(x k )] T + v k ∈ R 2 with additive noise v k ∼ N (0, η · I 2×2 ), η ∈ {3, 0.1, 0.01}. An overview of all considered scenarios is given in Table 3. In the scenarios with additive system noise, we compare the proposed filter to all standard approaches described in Sec. 2.2, a UKF with 1D state vector, a UKF with 2D state vector and particle filters with 10 and 100 particles. In order to handle non-additive noise with a UKF, typically state augmentation is used, which is not applicable to arbitrary noise. For this reason, we only compare the proposed approach to the particle filters in the non-additive noise case. The initial estimate is given by x 0 ∼ WN (0, 1), whereas the true initial state is on the opposite side of the circle x true 0 = π, i.e., the initial estimate is poor, which is difficult to handle for noncircular filters. For the circular filtering algorithm, we use the deterministic sampling method given in Algorithm 2 with parameter λ = 0.5. The progression threshold is chosen as R = 0.2. In order to evaluate the performance of different filters, we consider a specific error measure that takes periodicity into account. The angular error is defined as the shortest distance on the circle d : S 1 × S 1 → [0, π] , d(a, b) = min(|a − b|, 2π − |a − b|) . This leads to an angular version of the commonly used root mean square error (RMSE) between estimates x k and true state variables x true k . We simulated the system for k max = 100 time steps and compared the angular RMSE of all estimators. The results from 100 Monte Carlo runs are depicted in Fig. 10. In the scenarios with additive noise, it can be seen that the proposed filter performs very well regardless of the amount of noise. Only the particle filter with 100 particles is able to produce similar results. However, it should be noted that the proposed filter uses just five samples. The particle filter with 10 particles performs a lot worse and fails completely for small noise as a result of particle degeneration issues. Both variants of the UKF perform worse than the proposed filter. Particularly the UKF with two-dimensional state does not work very well, which can be explained by the inaccuracies in the conversion of a one-dimensional into a two-dimensional noise noise. When non-additive noise is considered, the proposed filter even significantly outperforms the particle filter with 100 particles. As a result of the low number of particles and the associated issues regarding particle degeneration, the particle filter with 10 particles has the worst performance. Conclusion In this paper, we presented a framework for recursive filtering on the circle. The proposed filtering algorithms can deal with arbitrary nonlinear system and measurement functions. Furthermore, they can be used in conjunction with different circular probability distributions. We have shown that the prediction step can be performed based on circular moments only, without ever assuming a particular distribution. For the purpose of evaluation, we have considered several aspects of the proposed methods. First of all, the accuracy of deterministic approximations was evaluated by considering the error when using them to propagate a continous distribution through a nonlinear function. We have found that the proposed deterministic approximation with five samples yields good results for most practical scenarios. Second, we evaluated the novel moment-based WN multiplication method and show that it is superior to the previously published method based on fitting a VM distribution. Finally, we evaluated the proposed filtering algorithms in several scenarios and compared it to standard approaches. These simulations show the advantages of using a circular filtering scheme compared to traditional methods intended for the linear case. Future work may include extensions of the proposed methods to other manifolds such as the torus or the hypersphere. Additionally, consideration of multimodal circular distributions may be of interest, for example by means of WN or VM mixtures. Figure 1 : 1This figure illustrates that repositioning of the measurement is necessary to obtain satisfactory performance when using classical filters on circular problems. Figure 2 : 2Relation between WN and VM distributions and the Gaussian distribution. Definition 3 ( 3Wrapped Dirac Mixture Distribution). The wrapped Dirac mixture (WD) distribution with L components is given by f (x; γ 1 , . . . , γ L , β 1 , . . . , β L ) = L j=1 γ j δ(x − β j ) with Dirac delta distribution δ(·), Dirac positions β 1 , . . . , β L ∈ S 1 , and weights γ 1 , . . . , γ L > 0 where L j=1 γ j = 1. Figure 3 : 3Proposed approaches for generating samples of WN distributions with a different concentration parameters σ compared to the naïve approach of wrapping samples of a Gaussian with identical σ. It can be seen that the UKF and S 2 KF samples are eventually wrapped to the same location, which produces an extremely poor approximation. Figure 4 : 4Example of the deterministic sampling of a VM and WN distribution with equal first circular moment. From top to bottom: a) original densities, b) result of Algorithm 1, c) result of Algorithm 2, d) approach based on VM kernels from Remark 3 .Figure 5 : 35In fact, Lemma 3 allows us to calculate the convolution purely moment-based. For this reason, we do not need to assume any particular distribution, but can just calculate the moments of the convolved density based on the products of original moments. Multiplication of two WN densities with parameters µ 1 = 2, σ 1 = 0.7, and µ 2 = 4.95, σ 2 = 1.3. The true product and the results of both proposed approximation methods (VM and moment-based) are depicted. Note that the true product is not a WN density. use moment matching to obtain WN or VM density */ f ←momentMatching(γ 1 , . . . , γ L , β 1 , . . . , β L ); end and can only be calculated numerically. We evaluate the first and the second circular moment m W D i , i = 1, 2 of the resulting WD distribution and compare to the first and the second circular moment m true i , i = 1, 2 of the true posterior, which is obtained by numerical integration 5 . The considered error measure is given by |m W D i − m true i |, i = 1, 2, where | · | is the Euclidean norm in the complex plane. Additionally, we fit a WN density to the posterior WD by circular moment matching and numerically calculate the Kullback-Leibler divergence Figure 6 : 6g(x) = x + c sin(x) mod 2π posterior Propagation of a WN distribution with parameters µ = 0.1, σ = 1 through a nonlinear function g by means of a deterministic WD approximation with five components. In this example, we use c = 0.7. Figure 7 : 7Propagation of WN (0, σ) through a nonlinear function with nonlinearity parameter c. Figure 8 : 8d(x k , x true k ) Kullback-Leibler divergence between the true product of WN densities and the proposed approximations. Figure 9 : 9L 2 distance between the true product of WN densities and the proposed approximations. Figure 10 : 10RMSE (in radians) for different filters obtained from 100 Monte Carlo runs for additive noise (top) and non-additive noise (bottom). The term 1 − |m1| is sometimes referred to as circular variance.2 Pseudocode of this algorithm is given in[15]. In general, for example in the case of mixture densities, the complexity of the density increases with each successive convolution, so considering only a finite number of moments constitutes an approximation. Compared to[16], we extend the progressive scheme to handle discrete approximations with non-equally weighted components. Numerical integration produces very accurate results in this case, but is too slow for use in practical filtering applications. AcknowledgmentThis work was partially supported by grants from the German Research Foundation (DFG) within the Research Training Groups RTG 1126 "Soft-tissue Surgery: New Computer-based Methods for the Future Workplace" and RTG 1194 "Self-organizing Sensor-Actuator-Networks". A New Approach to Linear Filtering and Prediction Problems. R E Kalman, Transactions of the ASME Journal of Basic Engineering. 82R. E. Kalman, "A New Approach to Linear Filtering and Prediction Problems," Transactions of the ASME Journal of Basic Engineering, vol. 82, pp. 35-45, 1960. Unscented Filtering and Nonlinear Estimation. S J Julier, J K Uhlmann, Proceedings of the IEEE. 923S. J. Julier and J. K. Uhlmann, "Unscented Filtering and Nonlinear Estimation," Proceedings of the IEEE, vol. 92, no. 3, pp. 401-422, Mar. 2004. . S R Jammalamadaka, A Sengupta, Topics in Circular Statistics. World Scientific. S. R. Jammalamadaka and A. Sengupta, Topics in Circular Statistics. World Scientific, 2001. . E Batschelet, Circular Statistics in Biology, ser. Mathematics in Biology. Academic PressE. Batschelet, Circular Statistics in Biology, ser. Mathematics in Biology. London: Academic Press, 1981. K V Mardia, P E Jupp, Directional Statistics. Wiley1st ed.K. V. Mardia and P. E. Jupp, Directional Statistics, 1st ed. Wiley, 1999. A Multivariate von Mises Distribution with Applications to Bioinformatics. K V Mardia, G Hughes, C C Taylor, H Singh, Canadian Journal of Statistics. 361K. V. Mardia, G. Hughes, C. C. Taylor, and H. Singh, "A Multivariate von Mises Distribution with Applications to Bioinformatics," Canadian Journal of Statistics, vol. 36, no. 1, pp. 99-109, 2008. Problems with the Current Definitions of the Standard Deviation of Wind Direction. N I Fisher, Journal of Climate and Applied Meteorology. 2611N. I. Fisher, "Problems with the Current Definitions of the Standard Deviation of Wind Direction," Journal of Climate and Applied Meteorology, vol. 26, no. 11, pp. 1522-1529, 1987. Directional Statistics in Geosciences. K V Mardia, Communications in Statistics -Theory and Methods. 1015K. V. Mardia, "Directional Statistics in Geosciences," Communications in Statistics -Theory and Methods, vol. 10, no. 15, pp. 1523-1543, 1981. A Recursive Fusion Filter for Angular Data. M Azmani, S Reboul, J.-B Choquel, M Benjelloun, IEEE International Conference on Robotics and Biomimetics. M. Azmani, S. Reboul, J.-B. Choquel, and M. Benjelloun, "A Recursive Fusion Filter for Angular Data," in IEEE International Conference on Robotics and Biomimetics (ROBIO 2009), 2009, pp. 882-887. A Multi-temporal Multi-sensor Circular Fusion Filter. G Stienne, S Reboul, M Azmani, J Choquel, M Benjelloun, Information Fusion. 18G. Stienne, S. Reboul, M. Azmani, J. Choquel, and M. Benjelloun, "A Multi-temporal Multi-sensor Circular Fusion Filter," Information Fusion, vol. 18, pp. 86-100, Jul. 2013. Cycle Slip Detection And Repair with a Circular On-line Change-Point Detector. G Stienne, S Reboul, J Choquel, M Benjelloun, Signal Processing. G. Stienne, S. Reboul, J. Choquel, and M. Benjelloun, "Cycle Slip Detection And Repair with a Circular On-line Change-Point Detector," Signal Processing, 2014. Circular Data Processing Tools Applied to a Phase Open Loop Architecture for Multi-Channels Signals Tracking. IEEE/ION Position Location and Navigation Symposium. PLANS 2012--, "Circular Data Processing Tools Applied to a Phase Open Loop Architecture for Multi-Channels Signals Tracking," in IEEE/ION Position Location and Navigation Symposium (PLANS 2012), 2012, pp. 633-642. A Circular Interacting Multi-Model Filter Applied to Map Matching. K E Mokhtari, S Reboul, M Azmani, J.-B Choquel, H Salaheddine, B Amami, M Benjelloun, Proceedings of the 16th International Conference on Information Fusion (Fusion 2013). the 16th International Conference on Information Fusion (Fusion 2013)K. E. Mokhtari, S. Reboul, M. Azmani, J.-B. Choquel, H. Salaheddine, B. Amami, and M. Benjelloun, "A Circular Interacting Multi-Model Filter Applied to Map Matching," in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), 2013. Moving Object Detection, Tracking and Following Using an Omnidirectional Camera on a Mobile Robot. I Markovic, F Chaumette, I Petrovic, Proceedings of the 2014 IEEE International Conference on Robotics and Automation. the 2014 IEEE International Conference on Robotics and AutomationHong-KongICRA 2014I. Markovic, F. Chaumette, and I. Petrovic, "Moving Object Detection, Tracking and Following Using an Omnidi- rectional Camera on a Mobile Robot," in Proceedings of the 2014 IEEE International Conference on Robotics and Automation (ICRA 2014), Hong-Kong, Jun. 2014. Recursive Nonlinear Filtering for Angular Data Based on Circular Distributions. G Kurz, I Gilitschenski, U D Hanebeck, Proceedings of the 2013 American Control Conference (ACC 2013). the 2013 American Control Conference (ACC 2013)Washington D. C., USAG. Kurz, I. Gilitschenski, and U. D. Hanebeck, "Recursive Nonlinear Filtering for Angular Data Based on Circular Distributions," in Proceedings of the 2013 American Control Conference (ACC 2013), Washington D. C., USA, Jun. 2013. Nonlinear Measurement Update for Estimation of Angular Systems Based on Circular Distributions. Proceedings of the 2014 American Control Conference (ACC 2014). the 2014 American Control Conference (ACC 2014)Portland, Oregon, USA--, "Nonlinear Measurement Update for Estimation of Angular Systems Based on Circular Distributions," in Proceedings of the 2014 American Control Conference (ACC 2014), Portland, Oregon, USA, Jun. 2014. Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular Moments. Proceedings of the 17th International Conference on Information Fusion (Fusion 2014). the 17th International Conference on Information Fusion (Fusion 2014)Salamanca, Spain--, "Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular Moments," in Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain, Jul. 2014. Constrained Object Tracking on Compact One-dimensional Manifolds Based on Directional Statistics. G Kurz, F Faion, U D Hanebeck, Proceedings of the Fourth IEEE GRSS International Conference on Indoor Positioning and Indoor Navigation (IPIN 2013). the Fourth IEEE GRSS International Conference on Indoor Positioning and Indoor Navigation (IPIN 2013)Montbeliard, FranceG. Kurz, F. Faion, and U. D. Hanebeck, "Constrained Object Tracking on Compact One-dimensional Manifolds Based on Directional Statistics," in Proceedings of the Fourth IEEE GRSS International Conference on Indoor Positioning and Indoor Navigation (IPIN 2013), Montbeliard, France, Oct. 2013. Bearings-Only Sensor Scheduling Using Circular Statistics. I Gilitschenski, G Kurz, U D Hanebeck, Proceedings of the 16th International Conference on Information Fusion (Fusion 2013). the 16th International Conference on Information Fusion (Fusion 2013)Istanbul, TurkeyI. Gilitschenski, G. Kurz, and U. D. Hanebeck, "Bearings-Only Sensor Scheduling Using Circular Statistics," in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013. Nonlinear Stochastic Model Predictive Control in the Circular Domain. G Kurz, M Dolgov, U D Hanebeck, Proceedings of the 2015 American Control Conference (ACC 2015). the 2015 American Control Conference (ACC 2015)Chicago, Illinois, USAto appearG. Kurz, M. Dolgov, and U. D. Hanebeck, "Nonlinear Stochastic Model Predictive Control in the Circular Domain (to appear)," in Proceedings of the 2015 American Control Conference (ACC 2015), Chicago, Illinois, USA, Jul. 2015. Recursive Estimation of Orientation Based on the Bingham Distribution. G Kurz, I Gilitschenski, S J Julier, U D Hanebeck, Proceedings of the 16th International Conference on Information Fusion (Fusion 2013). the 16th International Conference on Information Fusion (Fusion 2013)Istanbul, TurkeyG. Kurz, I. Gilitschenski, S. J. Julier, and U. D. Hanebeck, "Recursive Estimation of Orientation Based on the Bingham Distribution," in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013. On Kalman Filtering With Nonlinear Equality Constraints. S J Julier, J J Laviola, IEEE Transactions on Signal Processing. 556S. J. Julier and J. J. LaViola, "On Kalman Filtering With Nonlinear Equality Constraints," IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2774-2784, 2007. A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking. M Arulampalam, S Maskell, N Gordon, T Clapp, IEEE Transactions on Signal Processing. 502M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, "A Tutorial on Particle Filters for Online Nonlinear/Non- Gaussian Bayesian Tracking," IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174-188, 2002. Monte Carlo Sampling Methods Using Markov Chains and Their Applications. W K Hastings, Biometrika. 571W. K. Hastings, "Monte Carlo Sampling Methods Using Markov Chains and Their Applications," Biometrika, vol. 57, no. 1, pp. 97-109, 1970. Efficient Evaluation of the Probability Density Function of a Wrapped Normal Distribution. G Kurz, I Gilitschenski, U D Hanebeck, Proceedings of the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2014). the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2014)Bonn, GermanyG. Kurz, I. Gilitschenski, and U. D. Hanebeck, "Efficient Evaluation of the Probability Density Function of a Wrapped Normal Distribution," in Proceedings of the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2014), Bonn, Germany, Oct. 2014. M Abramowitz, I A Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New YorkDover10th edM. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed. New York: Dover, 1972. Characterizations of Directional Distributions. K Mardia, A Modern Course on Statistical Distributions in Scientific Work. NetherlandsSpringer17K. Mardia, "Characterizations of Directional Distributions," in A Modern Course on Statistical Distributions in Scientific Work. Springer Netherlands, 1975, vol. 17, pp. 365-385. Computation of Modified Bessel Functions and Their Ratios. D E Amos, Mathematics of Computation. 28125D. E. Amos, "Computation of Modified Bessel Functions and Their Ratios," Mathematics of Computation, vol. 28, no. 125, pp. 239-251, 1974. A Short Note on Parameter Approximation for von Mises-Fisher Distributions: And a Fast Implementation of Is (x). S Sra, Computational Statistics. 271S. Sra, "A Short Note on Parameter Approximation for von Mises-Fisher Distributions: And a Fast Implementation of Is (x)," Computational Statistics, vol. 27, no. 1, pp. 177-190, 2012. Traitements des signaux circulaires appliquésà l'altimétrie par la phase des signaux GNSS. G Stienne, Université du Littoral Côte d'OpalePh.D. dissertationG. Stienne, "Traitements des signaux circulaires appliquésà l'altimétrie par la phase des signaux GNSS," Ph.D. dissertation, Université du Littoral Côte d'Opale, Dec. 2013. Gaussian Particle Filtering. J H Kotecha, P Djuric, IEEE Transactions on Signal Processing. 5110J. H. Kotecha and P. Djuric, "Gaussian Particle Filtering," IEEE Transactions on Signal Processing, vol. 51, no. 10, pp. 2592-2601, 2003. Localized Cumulative Distributions and a Multivariate Generalization of the Cramér-von Mises Distance. U D Hanebeck, V Klumpp, Proceedings of the 2008 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2008). the 2008 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2008)Seoul, Republic of KoreaU. D. Hanebeck and V. Klumpp, "Localized Cumulative Distributions and a Multivariate Generalization of the Cramér-von Mises Distance," in Proceedings of the 2008 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2008), Seoul, Republic of Korea, Aug. 2008, pp. 33-39. S2KF: The Smart Sampling Kalman Filter. J Steinbring, U D Hanebeck, Proceedings of the 16th International Conference on Information Fusion (Fusion 2013). the 16th International Conference on Information Fusion (Fusion 2013)Istanbul, TurkeyJ. Steinbring and U. D. Hanebeck, "S2KF: The Smart Sampling Kalman Filter," in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013. Moment-based Dirac Mixture Approximation of Circular Densities. U D Hanebeck, A Lindquist, Proceedings of the 19th IFAC World Congress (IFAC 2014). the 19th IFAC World Congress (IFAC 2014)Cape Town, South AfricaU. D. Hanebeck and A. Lindquist, "Moment-based Dirac Mixture Approximation of Circular Densities," in Proceedings of the 19th IFAC World Congress (IFAC 2014), Cape Town, South Africa, Aug. 2014. Optimal Quantization of Circular Distributions (submitted). I Gilitschenski, G Kurz, U D Hanebeck, Proceedings of the 53rd IEEE Conference on Decision and Control (CDC 2014). the 53rd IEEE Conference on Decision and Control (CDC 2014)I. Gilitschenski, G. Kurz, and U. D. Hanebeck, "Optimal Quantization of Circular Distributions (submitted)," in Proceedings of the 53rd IEEE Conference on Decision and Control (CDC 2014), 2014. The Matrix Cookbook. K B Petersen, M S Pedersen, K. B. Petersen and M. S. Pedersen, "The Matrix Cookbook," Nov. 2012. Faddeeva Package. S G Johnson, S. G. Johnson, "Faddeeva Package," http://ab-initio.mit.edu/wiki/index.php/Faddeeva Package, Dec. 2012. What Every Computer Scientist Should Know About Floating-point Arithmetic. D Goldberg, ACM Computing Surveys. 231D. Goldberg, "What Every Computer Scientist Should Know About Floating-point Arithmetic," ACM Computing Surveys, vol. 23, no. 1, pp. 5-48, Mar. 1991. PGF 42: Progressive Gaussian Filtering with a Twist. U D Hanebeck, Proceedings of the 16th International Conference on Information Fusion (Fusion 2013). the 16th International Conference on Information Fusion (Fusion 2013)Istanbul, TurkeyU. D. Hanebeck, "PGF 42: Progressive Gaussian Filtering with a Twist," in Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013.
[]
[ "Table Enrichment System for Machine Learning", "Table Enrichment System for Machine Learning" ]
[ "Yuyang Dong [email protected] \nNEC Corporation\nJapan\n", "Masafumi Oyamada [email protected] \nNEC Corporation\nJapan\n" ]
[ "NEC Corporation\nJapan", "NEC Corporation\nJapan" ]
[ "Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '22)" ]
Data scientists are constantly facing the problem of how to improve prediction accuracy with insufficient tabular data. We propose a table enrichment system that enriches a query table by adding external attributes (columns) from data lakes and improves the accuracy of machine learning predictive models. Our system has four stages, join row search, task-related table selection, row and column alignment, and feature selection and evaluation, to efficiently create an enriched table for a given query table and a specified machine learning task. We demonstrate our system with a web UI to show the use cases of table enrichment.CCS CONCEPTS• Information systems → Data extraction and integration; Web searching and information discovery.
10.1145/3477495.3531678
[ "https://arxiv.org/pdf/2204.08235v1.pdf" ]
248,227,402
2204.08235
6469896292d4e8e4a0f6a5cb7a3b9f7f59614228
Table Enrichment System for Machine Learning ACMCopyright ACMJuly 11-15, 2022 Yuyang Dong [email protected] NEC Corporation Japan Masafumi Oyamada [email protected] NEC Corporation Japan Table Enrichment System for Machine Learning Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '22) the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '22)Madrid, Spain; New York, NY, USAACM5July 11-15, 202210.1145/3477495.3531678ACM Reference Format: Yuyang Dong and Masafumi Oyamada. 2022. Table Enrichment System for Machine Learning. Intable discoverytable augmentationmachine learning Data scientists are constantly facing the problem of how to improve prediction accuracy with insufficient tabular data. We propose a table enrichment system that enriches a query table by adding external attributes (columns) from data lakes and improves the accuracy of machine learning predictive models. Our system has four stages, join row search, task-related table selection, row and column alignment, and feature selection and evaluation, to efficiently create an enriched table for a given query table and a specified machine learning task. We demonstrate our system with a web UI to show the use cases of table enrichment.CCS CONCEPTS• Information systems → Data extraction and integration; Web searching and information discovery. INTRODUCTION Given a table and ML prediction task, many recent works on Auto-ML [14] are designed to automatically select the best features and models. However, users still have trouble if their tables do not contain enough signals (features) for training an ML model with satisfying accuracy. Features are always lacking in practical data analysis scenarios, for example, when a data analyst plans to classify the category of products but only has a few basic attributes such as the product name. The description of products may help to better predict global sales, and these important features may sleep in data lakes such as web scraping tables on E-commerce sites. With the trends of open data and the growth of data lake solutions, we can easily access and obtain a huge amount of tabular data from various domains in data lakes (e.g., the WDC Web Table Corpus [24]). This naturally comes out with a research problem: Can we build a Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. SIGIR '22, July 11-15, 2022 system that automatically enriches a query table with external columns from data lakes and improve the ML prediction task? Related works. We need three steps to achieve table enrichment: table retrieval, table join, and ML evaluation. There are many related works on table retrieval [4,30] with different end tasks such as column and row extension [17,20,22,28], schema matching [16,18], table filling [26,27], and knowledge graph building [13,23]. Regarding table retrieval and table join, because the goal of the enrichment is to connect a query table to the external tables in data lakes by joining them with record value matching. The most basic requirement is to retrieve joinable tables from data lakes [9,32] There is also other research to retrieve related tables but not join. Zhang et al. studied the problem of finding related tables in data lakes for interactive data science [31] by a composed criteria with a join rate, new column rate, and null value decrement. Mahdi et al. [11] proposed a COCOA system to efficiently compute Spearman's correlation coefficient after the join tables are mapped. Aécio et al. [19] studied the problem of joining-correlation query, which can efficiently find tables that can be joined and also contains correlated values with a query table. Nevertheless, the above works focus on retrieving tables, and they do not cover the following steps of enriching a table to improve the machine learning accuracy. Regarding ML evaluation, some works target the problem of feature selection in which dozens of candidate joinable tables are given. Kumar et al. [15] proposed a TR-rule based on the bound of the VC dimension [25] to skip unnecessary table joins. However, the ML task and model are limited since the theory of the VC dimension is only for classification. Chepurko et al. [7] studied efficient feature selection over candidate tables with sampling techniques and proposed a feature selection method that ranks features with a random injection of noise. The above works assume a few candidate tables are already given and process them one by one. Hence, they do not consider how to retrieve useful tables from data lakes, so it is not efficient to use them to process a large number of tables. In conclusion, there is no generalized table enrichment system, and this paper fills this gap by proposing an end-to-end system that covers the whole pipeline with table discovery, table augmentation, and ML model evaluation. Problem statement. Figure 1 gives a running example of our TABLE ENRICHMENT SYSTEM Join row search Given a query column . and a collection of target tables , we retrieve candidate join rows to . from the tables in . For each cell in . , we retrieve similar cells in , and the rows of these cells are candidate join rows. For a query cell value in . and a cell value in , even if theirs rows refer to the same entity, the cell values may have different presentation styles, such as "Mario Bros" and "Super Mario Brothers. " Therefore, we leverage Jaccard, BM25, and semantic similarities to cover different levels of fuzzy matching between query cells and target cells. Jaccard similarity treats and as sets of words, and we use it to retrieve join rows that share the same words with a query. ( , ) = | ∩ | | ∪ |(1) BM25 similarity is an optimization of TF-IDF that treats and as documents, and we use it to retrieve join rows on the basis of the word frequency and document frequency. 25( , ) = ∑︁ =1 ( ) · ( , ) · ( + 1) ( , ) + (1 − ℎ + ℎ · | | )(2) where ( , ) is the term frequency in of the term ∈ , and ( ) is the inverse document frequency of the term . and ℎ are constants, and is the average document length in the text collection from which documents are drawn. Semantic similarity involves using a pre-trained language model such as fastText [2] and BERT [8] to encode the string values of and into vector representations, and we compute the Euclidean distance between vectors as similarities. It is used to retrieve join rows that may not be similar in a string but are semantically similar. ( , ) = || ( ) − ( )|| 2(3) Query We first search top-similar cells (rows) from target tables with Jaccard, BM25, and semantic similarity independently. Then, we take the union of all candidate rows as the final output. To efficiently index and retrieve join rows from large tables, we implement the join row search engine with Elasticsearch [10]. Elasticsearch is a powerful search engine that supports BM25 with Lucene [1] implementation, Jaccard similarity with Minhash token and LSH implementation, and kNN search with nmslib [3] implementation. Therefore, a similarity search can be processed very fast with these sophisticated indices. Another reason we used Elasticsearch is because it is a disk-based distributed engine, and it is easy to increase the capacity without extra maintenance costs. Task-related table selection There are large numbers of candidate tables after the join row search. It is very time-consuming to create a large enriched table with them. More seriously, the candidate tables also contain many irrelevant tables that they can join with the query table, but their information cannot contribute to the ML task. Adding these noisy data (columns) will weaken the performance of feature selection algorithms [6] and decrease the accuracy of an ML model. We propose a further selection to pick up task-related tables. We formalize this task-related table selection problem as a toptable retrieval problem with text-based query and semantic matching [29]. The purpose of using a text-based query is to match a related table in accordance with the description of the ML task. The goal of semantic matching is to go beyond lexical matching by using pre-trained language models to encode both query text and target text as vectors into a semantic space. For a query table, we allow users to input the description text of their ML task as query text. We also propose generating multiple query texts if task description text is not available. The key point is to input the target information (e.g., the "category" of products) into the query text. We use the column names concatenation as query text. The names of columns contain both attribute and target information, e.g., "product name, product category". For a target table, we set the attributes . , . , and . _ as target texts. To evaluate the related score of a target table to the query table, we take the maximum semantic similarity score between a query text and all target texts. Figure 2 gives an image of the semantic matching of a query table with a target table. There are many ways to aggregate the pairwise scores between encoded vectors, and we take a maximum aggregate function in our experiment. ( , ) = ( ( , [ ] ))(4) We can control the selected table number to retrieve the toprelated tables after join row search. This helps us to narrow down the number of candidate tables to hundreds of tables. Row and column alignment The next step is to map these rows and join them to the query table. We join the candidate rows and align the rows if they are from the same table. Note that for each query row, multiple join rows may be retrieved from the same target table. In this case, we map the row with the highest similarity score since a target table usually has only one matching row (entity) to a query row. As shown in Figure 1, the enriched table may contain a large number of columns with many empty values. This is because different query rows may not join the same target table, and the row number of a target table may also differ from the query table. The size of enriched columns determines the processing time, and empty values will decrease the performance of the feature selection algorithm and predictive models. To achieve high performance with a short processing time, we propose to aggregate columns to decrease the column numbers and impute most of the empty values. We use two methods: (a) hard aggregation: we aggregate the columns within the exact same column name; (b) soft aggregation: we first encode the column names into vectors with pre-trained language models such as fastText [2] and BERT [8]. Then, we cluster the columns through the k-means algorithm on the encoded vectors. Last, we aggregate the columns in the same cluster. Having more compact enriched columns does not always result in better accuracy in the final ML task, so the cluster number should be tuned in practice. During aggregation, the values may conflict at the same row position, and we aggregate the conflicting values by string concatenation or averaging numerical values. Figure 1 also shows the result after column aggregation. We can see that it is much cleaner with fewer empty values than the result of join row mapping. Feature selection Finally, we process feature selection to filter noisy columns. Our system offers several feature selection methods from the scikit-learn [12] including using F-value, forward selection, backward selection, recursive feature elimination, the random forest model, and linear model with L1 regularization. We let users tune and select the best one for their data. DEMONSTRATION We will demonstrate our table enrichment system on a real-world dataset. There are two query tables, the AMAZON toy dataset [21] and the Car Sales dataset [5], and we enrich them with a data lake that contains 10M WDC web tables [24]. Each step is marked in a red circle in Figure 3. The following description is for the AMAZON toy dataset, and a video of both two datasets with our demonstration can be found here 1 . Step 1. Upload query table. First, the user uploads the query table (train and test tables) in the CSV file format. The uploaded table will be displayed and statistical information such as row and column numbers can be seen. In our guided demonstration, we upload the AMAZON toy dataset [21]. Step 2. Specify the key column and the task column. The user specifies a column as the key column that is used as the join key to connect external tables. Then the user also specifies another column as the task column which contains the values to predict with the ML algorithm. In our guided demonstration, we specify the "product name" column as the key column and the "category" for the task column. Step 3. [24], and use the default parameter setting. Step 4. Run table enrichment and get the enriched table. Then, the user clicks the " Table Enrich" button to run the table enrichment and sees the results. In our guided demonstration, the uploaded AMAZON table with enriched columns of product descriptions. Step 5. Information of retrieved target tables. After table enrichment, the user can see where those enriched columns come from. The information and metadata of the retrieved target table and metadata can be shown. In our guided demonstration, we show the metadata of the title and context of the retrieved target tables. Our metadata also contains the source URL. Step 6. ML performance improvement and feature importance. Then, the user can see the ML prediction accuracy result before (using only the query table itself) and after table enrichment. The user can also confirm the effectiveness of table enrichment by checking the bar chart showing the feature importance under the ML score results. In our guided demonstration, we show the classification evaluation of product category classification with precision, recall, and F1-score. The feature importance is also ranked with a bar chart with different colors to show that the enriched features also contribute to the ML performance. Step 7. Insights for ML performance. Last, the user can see the record level prediction results with the display filter in our system. The user can observe the difference in prediction results before and after table enrichment. This option is for the classification task. In our guided demonstration, we filter the records to show the different classification on AMAZON toy dataset. After this, the user can adjust the parameters for running the next loop of table enrichment and achieving a better ML performance. CONCLUSION We proposed a table enrichment solution to retrieve, select, join, and aggregate candidate tables step by step. We show the use case of our system with a demo on a web UI. As future work, we plan to extend our system with existing data clean techniques to create more valuable enriched tables for ML. Figure 2 : 2Text-based query and semantic matching. Figure 3 : 3Web UI of table enrichment demo. Steps of our guided demonstration are marked in red circle. , Madrid, Spain. © 2022 Association for Computing Machinery. ACM ISBN 978-1-4503-8732-3/22/07. . . $15.00 https://doi.org/10.1145/3477495.3531678 Table enrichment system takes as input the user's query table = ( . , . , . names. . is a target column for prediction, and . is a query column(s) for join. = { 1 , 2 , ..., } is a collection of target tables. Besidesthe target table data itself, we assume a target table also has two types of metadata, the table title and table context, which can be easily retrieved from the source of tables. Therefore,Figure 1: Example of table enrichment for ML._ , . ), where . is the whole tabular data and . _ is a string list of column = ( . , . , . _ , . ). arXiv:2204.08235v1 [cs.IR] 18 Apr 2022 Product Name Category Description Publisher Description Publisher Description Winnie the Pooh, 86cm Characters & Brands Fictional character Super Mario Bros Video Game Platform game … Nintendo Floppy Rabbit 20cm pink Dolls Wii Sports Video Game Five sports … Nintendo Product Name Category Description Publisher Winnie the Pooh, 86cm Characters & Brands Fictional character Super Mario Bros Video Game Platform game … Nintendo Floppy Rabbit 20cm pink Dolls Wii Sports Video Game Five sports … Nintendo Enriched table after row and column alignment Column aggregation Game Title Description Publisher … … Mario Brothers Platform game … Nintendo … … Table 1. Platform games Table 2. Sports games Table 3. Disney animation characters Candidate tables after join row search Table 4. Animals query table query table enriched columns enriched columns Candidate tables after task-related table selection Join row mapping Product Name Category Winnie the Pooh, 86cm Characters & Brands Super Mario Bros Video Game Floppy Rabbit 20cm pink Dolls Wii Sports Video Game query table: toy & games products data lake Feature selection, ML model training and evaluation ML task: predict the sales Table 1 domain Table 2 domain Table 3 domain Name Description Winnie-the-Pooh Fictional character … … Title Description Publisher … … Wii Sports Five sports … Nintendo Name Family Rabbit Leporidae Bowling … … Game Title Description Publisher … … Mario Brothers Platform game … Nintendo … … Table 1. Platform games Table 2. Sports games Table 3. Disney animation characters Table 4. Animals Name Description Winnie-the-Pooh Fictional character … … Title Description Publisher … … Wii Sports Five sports … Nintendo Name Family Rabbit Leporidae Bowling … … Table Enrichment EnrichmentProblem: Given a query table , the goal of table enrichment is to create an enriched table ′ using the data from a collection of target tables . Without loss of generality, we keep the query table unchanged and fix the number of rows, and we only add columns (features) to it. The enriched table ′ will improve the accuracy of predicting . compared to only predicting with , that is, ( ′ , . ) > ( , . ). table enrichment system. There is a query table of toy & game products, and the task is to classify the categories of products First, we retrieve four candidate tables that can join with the query table. Then, Tables 1,2,3 are selected since the table they are related to toy & games, while Table 4 "animal" is an irrelevant one and we filter it. We also aggregate the values in the same column name to make a compact and clean enriched table. Finally, we input this enriched table into the ML evaluation phase with feature selection, model training, and evaluation. Table enrichment parameter setting. Next, the user sets the parameters for table enrichment, including the domain source of the target table, similarity function for join row searching, the methods for alignment column and feature selection. In our guided demonstration, we set the target table dataset as WDC web tables https://youtu.be/HXikNjblUwU ACKNOWLEDGEMENTWe thank Dr. Takuma Nozawa and Masafumi Enomoto (NEC Corporation) for the discussions of this research. . Apache, n.d.Apache. [n.d.]. . Apache Lucene, Apache Lucene. https://lucene.apache.org/. Enriching Word Vectors with Subword Information. Piotr Bojanowski, Edouard Grave, Armand Joulin, Tomás Mikolov, arXiv:1607.04606Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomás Mikolov. 2016. Enriching Word Vectors with Subword Information. CoRR abs/1607.04606 (2016). arXiv:1607.04606 http://arxiv.org/abs/1607.04606 Engineering Efficient and Effective Non-metric Space Library. Leonid Boytsov, Bilegsaikhan Naidan, 10.1007/978-3-642-41062-8_28Similarity Search and Applications -6th International Conference, SISAP 2013, A Coruña. Nieves R. Brisaboa, Oscar Pedreira, and Pavel ZezulaSpainSpringer8199Proceedings (Lecture Notes in Computer ScienceLeonid Boytsov and Bilegsaikhan Naidan. 2013. Engineering Efficient and Effective Non-metric Space Library. In Similarity Search and Applications -6th International Conference, SISAP 2013, A Coruña, Spain, October 2-4, 2013, Proceedings (Lecture Notes in Computer Science, Vol. 8199), Nieves R. Brisaboa, Oscar Pedreira, and Pavel Zezula (Eds.). Springer, 280-293. https://doi.org/10.1007/978-3-642-41062-8_28 WebTables: exploring the power of tables on the web. Michael J Cafarella, Alon Y Halevy, Daisy Zhe Wang, Eugene Wu, Yang Zhang, Proc. VLDB Endow. 1Michael J. Cafarella, Alon Y. Halevy, Daisy Zhe Wang, Eugene Wu, and Yang Zhang. 2008. WebTables: exploring the power of tables on the web. Proc. VLDB Endow. 1, 1 (2008), 538-549. Kaggle Car sales. Car sales. n.d.Kaggle Car sales. [n.d.]. Car sales. https://www.kaggle.com/datasets/gagandeep16/ car-sales. A survey on feature selection methods. Girish Chandrashekar, Ferat Sahin, Comput. Electr. Eng. 40Girish Chandrashekar and Ferat Sahin. 2014. A survey on feature selection methods. Comput. Electr. Eng. 40, 1 (2014), 16-28. ARDA: Automatic Relational Data Augmentation for Machine Learning. Nadiia Chepurko, Ryan Marcus, Emanuel Zgraggen, Raul Castro Fernandez, Tim Kraska, David Karger, Proc. VLDB Endow. VLDB Endow13Nadiia Chepurko, Ryan Marcus, Emanuel Zgraggen, Raul Castro Fernandez, Tim Kraska, and David Karger. 2020. ARDA: Automatic Relational Data Augmentation for Machine Learning. Proc. VLDB Endow. 13, 9 (2020), 1373-1387. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, arXiv:1810.04805Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. CoRR abs/1810.04805 (2018). arXiv:1810.04805 http://arxiv.org/abs/1810.04805 Efficient Joinable Table Discovery in Data Lakes: A High-Dimensional Similarity-Based Approach. Yuyang Dong, Kunihiro Takeoka, Chuan Xiao, Masafumi Oyamada, 10.1109/ICDE51399.2021.0004637th IEEE International Conference on Data Engineering, ICDE 2021. Chania, GreeceIEEEYuyang Dong, Kunihiro Takeoka, Chuan Xiao, and Masafumi Oyamada. 2021. Efficient Joinable Table Discovery in Data Lakes: A High-Dimensional Similarity- Based Approach. In 37th IEEE International Conference on Data Engineering, ICDE 2021, Chania, Greece, April 19-22, 2021. IEEE, 456-467. https://doi.org/10.1109/ ICDE51399.2021.00046 Elasticsearch -The Definitive Guide. n.d.elasticsearch. [n.d.]. Elasticsearch -The Definitive Guide. https://www.elastic.co/ guide/en/elasticsearch/guide/index.html. COCOA: COrrelation COefficient-Aware Data Augmentation. Mahdi Esmailoghli, Jorge-Arnulfo Quiané-Ruiz, Ziawasch Abedjan, 10.5441/002/edbt.2021.30Proceedings of the 24th International Conference on Extending Database Technology, EDBT 2021. Yannis Velegrakis, Demetris Zeinalipour-Yazti, Panos K. Chrysanthis, and Francesco Guerrathe 24th International Conference on Extending Database Technology, EDBT 2021Nicosia, CyprusOpenProceedings.orgMahdi Esmailoghli, Jorge-Arnulfo Quiané-Ruiz, and Ziawasch Abedjan. 2021. COCOA: COrrelation COefficient-Aware Data Augmentation. In Proceedings of the 24th International Conference on Extending Database Technology, EDBT 2021, Nicosia, Cyprus, March 23 -26, 2021, Yannis Velegrakis, Demetris Zeinalipour- Yazti, Panos K. Chrysanthis, and Francesco Guerra (Eds.). OpenProceedings.org, 331-336. https://doi.org/10.5441/002/edbt.2021.30 SKlearn Feature selection. Feature selectionSKlearn Feature selection. [n.d.]. Feature selection. https://scikit-learn.org/stable/ modules/feature_selection.html. TableNet: An Approach for Determining Fine-grained Relations for Wikipedia Tables. Besnik Fetahu, Avishek Anand, Maria Koutraki, WWW 2019. ACM. Besnik Fetahu, Avishek Anand, and Maria Koutraki. 2019. TableNet: An Approach for Determining Fine-grained Relations for Wikipedia Tables. In WWW 2019. ACM, 2736-2742. AutoML: A Survey of the State-of-the-Art. Xin He, Kaiyong Zhao, Xiaowen Chu, arXiv:1908.00709arXiv:1908.00709arXiv e-prints, Articlecs.LGXin He, Kaiyong Zhao, and Xiaowen Chu. 2019. AutoML: A Survey of the State-of-the-Art. arXiv e-prints, Article arXiv:1908.00709 (Aug. 2019), arXiv:1908.00709 pages. arXiv:1908.00709 [cs.LG] To Join or Not to Join?: Thinking Twice about Joins before Feature Selection. Arun Kumar, Jeffrey F Naughton, Jignesh M Patel, Xiaojin Zhu, 10.1145/2882903.2882952Proceedings of the 2016 International Conference on Management of Data, SIGMOD Conference. Fatma Özcan, Georgia Koutrika, and Sam Maddenthe 2016 International Conference on Management of Data, SIGMOD ConferenceSan Francisco, CA, USAACMArun Kumar, Jeffrey F. Naughton, Jignesh M. Patel, and Xiaojin Zhu. 2016. To Join or Not to Join?: Thinking Twice about Joins before Feature Selection. In Proceedings of the 2016 International Conference on Management of Data, SIGMOD Conference 2016, San Francisco, CA, USA, June 26 -July 01, 2016, Fatma Özcan, Georgia Koutrika, and Sam Madden (Eds.). ACM, 19-34. https://doi.org/10.1145/2882903.2882952 Stitching Web Tables for Improving Matching Quality. Oliver Lehmberg, Christian Bizer, Proc. VLDB Endow. VLDB Endow10Oliver Lehmberg and Christian Bizer. 2017. Stitching Web Tables for Improving Matching Quality. Proc. VLDB Endow. 10, 11 (2017), 1502-1513. The Mannheim Search Join Engine. Oliver Lehmberg, Dominique Ritze, Petar Ristoski, Robert Meusel, Heiko Paulheim, Christian Bizer, J. Web Semant. 35Oliver Lehmberg, Dominique Ritze, Petar Ristoski, Robert Meusel, Heiko Paulheim, and Christian Bizer. 2015. The Mannheim Search Join Engine. J. Web Semant. 35 (2015), 159-166. . Fatemeh Nargesian, Erkang Zhu, Ken Q Pu, Renée J Miller, Table Union Search on Open Data. PVLDB. 11Fatemeh Nargesian, Erkang Zhu, Ken Q. Pu, and Renée J. Miller. 2018. Table Union Search on Open Data. PVLDB 11, 7 (2018), 813-825. Correlation Sketches for Approximate Join-Correlation Queries. S R Aécio, Aline Santos, Fernando Bessa, Christopher Chirigati, Juliana Musco, Freire, 10.1145/3448016.3458456SIGMOD '21: International Conference on Management of Data, Virtual Event. Guoliang Li, Zhanhuai Li, Stratos Idreos, and Divesh SrivastavaChinaACMAécio S. R. Santos, Aline Bessa, Fernando Chirigati, Christopher Musco, and Juliana Freire. 2021. Correlation Sketches for Approximate Join-Correlation Queries. In SIGMOD '21: International Conference on Management of Data, Virtual Event, China, June 20-25, 2021, Guoliang Li, Zhanhuai Li, Stratos Idreos, and Divesh Srivastava (Eds.). ACM, 1531-1544. https://doi.org/10.1145/3448016.3458456 Finding related tables. Anish Das Sarma, Lujun Fang, Nitin Gupta, Alon Y Halevy, Hongrae Lee, Fei Wu, Reynold Xin, Cong Yu, SIGMOD. ACM. Anish Das Sarma, Lujun Fang, Nitin Gupta, Alon Y. Halevy, Hongrae Lee, Fei Wu, Reynold Xin, and Cong Yu. 2012. Finding related tables. In SIGMOD. ACM, 817-828. Kaggle Toy Products on Amazon. Toy Products on AmazonKaggle Toy Products on Amazon. [n.d.]. Toy Products on Amazon. https://www. kaggle.com/PromptCloudHQ/toy-products-on-amazon. Concept Expansion Using Web Tables. Chi Wang, Kaushik Chakrabarti, Yeye He, Kris Ganjam, Zhimin Chen, Philip A Bernstein, WWW 2015. ACM. Chi Wang, Kaushik Chakrabarti, Yeye He, Kris Ganjam, Zhimin Chen, and Philip A. Bernstein. 2015. Concept Expansion Using Web Tables. In WWW 2015. ACM, 1198- 1208. Daheng Wang, Prashant Shiralkar, Colin Lockard, Binxuan Huang, X Dong, Meng Jiang, ArXiv abs/2102.09460TCN: Table Convolutional Network for Web Table Interpretation. Daheng Wang, Prashant Shiralkar, Colin Lockard, Binxuan Huang, X. Dong, and Meng Jiang. 2021. TCN: Table Convolutional Network for Web Table Interpretation. ArXiv abs/2102.09460 (2021). Wdc, WDC Web Table Corpus. WDC. 2015. WDC Web Table Corpus 2015. http://webdatacommons.org/ webtables/2015/downloadInstructions.html. . Wikipedia Wikipedia, Com, n.d.Wikipedia Wikipedia.com. [n.d.]. apnik-Chervonenkis dimension. https://en. wikipedia.org/wiki/Vapnik-Chervonenkis_dimension. InfoGather: entity augmentation and attribute discovery by holistic matching with web tables. Mohamed Yakout, Kris Ganjam, Kaushik Chakrabarti, Surajit Chaudhuri, SIGMOD 2012. ACMMohamed Yakout, Kris Ganjam, Kaushik Chakrabarti, and Surajit Chaudhuri. 2012. InfoGather: entity augmentation and attribute discovery by holistic matching with web tables. In SIGMOD 2012. ACM, 97-108. InfoGather+: semantic matching and annotation of numeric and time-varying attributes in web tables. Meihui Zhang, Kaushik Chakrabarti, SIGMOD. ACM. Meihui Zhang and Kaushik Chakrabarti. 2013. InfoGather+: semantic matching and annotation of numeric and time-varying attributes in web tables. In SIGMOD. ACM, 145-156. EntiTables: Smart Assistance for Entity-Focused Tables. Shuo Zhang, Krisztian Balog, SIGIR. ACMShuo Zhang and Krisztian Balog. 2017. EntiTables: Smart Assistance for Entity- Focused Tables. In SIGIR, 2017. ACM, 255-264. Ad Hoc Table Retrieval using Semantic Similarity. Shuo Zhang, Krisztian Balog, 10.1145/3178876.3186067Proceedings of the 2018 World Wide Web Conference on World Wide Web. Pierre-Antoine Champin, Fabien Gandon, Mounia Lalmas, and Panagiotis G. Ipeirotisthe 2018 World Wide Web Conference on World Wide WebLyon, FranceACMShuo Zhang and Krisztian Balog. 2018. Ad Hoc Table Retrieval using Semantic Similarity. In Proceedings of the 2018 World Wide Web Conference on World Wide Web, WWW 2018, Lyon, France, April 23-27, 2018, Pierre-Antoine Champin, Fabien Gandon, Mounia Lalmas, and Panagiotis G. Ipeirotis (Eds.). ACM, 1553-1562. https://doi.org/10.1145/3178876.3186067 Web Table Extraction, Retrieval, and Augmentation: A Survey. Shuo Zhang, Krisztian Balog, ACM Trans. Intell. Syst. Technol. 1135Shuo Zhang and Krisztian Balog. 2020. Web Table Extraction, Retrieval, and Augmentation: A Survey. ACM Trans. Intell. Syst. Technol. 11, 2 (2020), 13:1-13:35. Finding Related Tables in Data Lakes for Interactive Data Science. Yi Zhang, Zachary G Ives, 10.1145/3318464.3389726Proceedings of the 2020 International Conference on Management of Data, SIGMOD Conference 2020, online conference. David Maier, Rachel Pottinger, AnHai Doan, Wang-Chiew Tan, Abdussalam Alawini, and Hung Q. Ngothe 2020 International Conference on Management of Data, SIGMOD Conference 2020, online conferencePortland, OR, USAACMYi Zhang and Zachary G. Ives. 2020. Finding Related Tables in Data Lakes for Interactive Data Science. In Proceedings of the 2020 International Conference on Management of Data, SIGMOD Conference 2020, online conference [Portland, OR, USA], June 14-19, 2020, David Maier, Rachel Pottinger, AnHai Doan, Wang-Chiew Tan, Abdussalam Alawini, and Hung Q. Ngo (Eds.). ACM, 1951-1966. https: //doi.org/10.1145/3318464.3389726 JOSIE: Overlap Set Similarity Search for Finding Joinable Tables in Data Lakes. Erkang Zhu, Dong Deng, Fatemeh Nargesian, Renée J Miller, SIGMOD 2019. ACMErkang Zhu, Dong Deng, Fatemeh Nargesian, and Renée J. Miller. 2019. JOSIE: Overlap Set Similarity Search for Finding Joinable Tables in Data Lakes. In SIGMOD 2019. ACM, 847-864.
[]
[ "MCMC inference for Markov Jump Processes via the Linear Noise Approximation", "MCMC inference for Markov Jump Processes via the Linear Noise Approximation" ]
[ "Vassilios Stathopoulos \nDepartment of Statistical Science\nCentre for Computational Statistics and Machine Learning\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n", "Mark A Girolami \nDepartment of Statistical Science\nCentre for Computational Statistics and Machine Learning\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n" ]
[ "Department of Statistical Science\nCentre for Computational Statistics and Machine Learning\nUniversity College London\nGower StreetWC1E 6BTLondonUK", "Department of Statistical Science\nCentre for Computational Statistics and Machine Learning\nUniversity College London\nGower StreetWC1E 6BTLondonUK" ]
[]
Bayesian analysis for Markov jump processes is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding thus its applicability is limited to a small class of problems. In this paper we describe the application of Riemann manifold MCMC methods using an approximation to the likelihood of the Markov jump process which is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient while the convergence rate and mixing of the chains allows for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology. * 1 [email protected] † 2 [email protected]
10.1098/rsta.2011.0541
[ "https://arxiv.org/pdf/1211.4801v1.pdf" ]
41,100,827
1211.4801
e4619fb97e68632f09b517f15649e72c0198124f
MCMC inference for Markov Jump Processes via the Linear Noise Approximation 20 Nov 2012 November 2012 Vassilios Stathopoulos Department of Statistical Science Centre for Computational Statistics and Machine Learning University College London Gower StreetWC1E 6BTLondonUK Mark A Girolami Department of Statistical Science Centre for Computational Statistics and Machine Learning University College London Gower StreetWC1E 6BTLondonUK MCMC inference for Markov Jump Processes via the Linear Noise Approximation 20 Nov 2012 November 2012arXiv:1211.4801v1 [stat.CO] Bayesian analysis for Markov jump processes is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding thus its applicability is limited to a small class of problems. In this paper we describe the application of Riemann manifold MCMC methods using an approximation to the likelihood of the Markov jump process which is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient while the convergence rate and mixing of the chains allows for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology. * 1 [email protected] † 2 [email protected] Introduction Markov Jump Processes (MJP) provides us with a formal description of the underlying stochastic behaviour of many physical systems and as such they have a wide applicability in many scientific fields. In chemistry and biology, for example, they are applied for modelling reactions between chemical species [1,2]. In ecology and epidemiology, they are used for modelling the population of interacting species in the environment [3] while in telecommunications they describe the population of information packets over a network [4]. In order to introduce some terminology and notation we will give a more concrete example from chemical kinetics. However, the modelling methodology is similar in other applications although different assumptions are needed, depending on the system being modelled, for calculating reaction rates. Consider a model for the population of molecules of two interacting chemical species, X A and X B , in a solution of volume Ω, where X A and X B denote the number of molecules of chemicals A and B respectively. The interactions between the species are modelled using reactions which are specified using the following notation: R 1 : A + B c 1 − → 2A. On the left hand side appear the reactants and on the right hand side the products of the reaction while over the arrow appears the rate constant c 1 which is the probability that a randomly chosen pair of A and B will react according to R 1 . This reaction, for example, specifies that a pair of molecules A, B react with probability c 1 to produce a new molecule of A. For calculating the probability of a reaction taking place given the current state of the system, i.e. the number of molecules of chemicals A and B, several system dependent assumptions must be made. For chemical reactions it is assumed that in a well stirred solution the probability of a reaction is proportional to the populations of its products [5]. For R 1 we can write it as f 1 (X A , X B , c 1 ) = c 1 Ω −1 X A X B . Following the same reasoning additional reactions and species can be added in order to construct large and complex reaction networks. Together, the state of the system X A , X B , the set of reactions and the reaction rates specify a Markov Jump process where the occurrences of reactions are modelled as a Poisson process. For this particular example the probability of the reaction has a simple form and is linear with respect to the populations. However in many real applications this is often not the case while the rate constants, c 1 , are unknown. Given a fully specified MJP, i.e. a MJP with known parameters, rate constants and initial conditions, it is possible to perform exact simulation and obtain samples from the underlying stochastic process using the Stochastic Simulation Algorithm (SSA) of [1]. In many problems there are system parameters which are not specified or are unknown while it is relatively easy to collect partial observations of the physical process at discrete time points. The interest is therefore to obtain statistical estimates of the unknown parameters using the available data. As a consequence of the Markov property, MJPs satisfy the Chapman-Kolmogorov equation from which we can directly obtain the forward master equation describing the evolution of the system's state probability over any time interval. However, even for small and simple systems the master equation is intractable and it is not straightforward as to how partially and discretely observed data from the physical process should be incorporated in order to perform inference over unknown system parameters. Recently, [6] have shown that it is possible to construct a Markov Chain whose stationary probability distribution is the posterior of the unknown parameters without resorting to any approximations of the original MJP. Their method however is computationally expensive while the strong correlation between posterior samples means that a large number of MCMC iterations are required in order to obtain Monte Carlo estimates with sufficient accuracy. An alternative is to consider suitable approximations of the likelihood function. The system size expansion of [7, Chap. 10] provides a systematic method for obtaining approximations of a physical process approaching its thermodynamic limit. The most simple approximation yields the Macroscopic Rate Equation (MRE) which describes the thermodynamic limit of the system with a set of ordinary differential equations neglecting any random fluctuations. Although the MRE has been extensively studied in the literature, see for example, [8,9], it is not applicable for problems where information about the noise and the random fluctuations is necessary or the system is far from its thermodynamic limit. The diffusion approximation [10,11] describes the physical process by a set of non-linear stochastic differential equations with state dependent Brownian motion. Similar to the master equation however, the likelihood is intractable. In [12] a transformation is applied such that the Brownian increments are independent of the system state and thus the system can be easily simulated. However this limits the applicability of the methodology into systems where such a transformation is possible. A more general methodology in presented in [13] where an approximation of the likelihood is used instead. Finally, a less studied approach for the purpose of inference is the Linear Noise Approximation (LNA) which conveniently decouples non-linearity in the diffusion approximation into a nonlinear set of ordinary differential equations in the MRE and a set of linear stochastic differential equations for the random fluctuations around a deterministic state [7, Chap. 10], [14]. Recently, [15] have shown the simple analytic form of the approximate likelihood obtained by the LNA simplifies MCMC inference and can be applied to problems with relatively small number of molecules. A commonly employed algorithm for MCMC is the Metropolis-Hastings algorithm [16], which relies on random perturbations around the current state using a local proposal mechanism. It should be noted here that the state of the Markov chain is different from the state of the stochastic process. In the MCMC context state refers the current values of the unknown system parameters whereas the state of the system refers to the value of the stochastic process at a given time. We will use the term state interchangeably for the rest of this paper and its meaning will be clear from the context. Due to the local nature of the proposal mechanism used by the Metropolis-Hastings algorithm, samples from the posterior exhibit strong random walk behaviour and auto-correlation. Tuning the proposal mechanisms to achieve good mixing and fast convergence is far from straightforward even though some theoretical guidance is provided [17]. MCMC methods, such as the Metropolis Adjusted Langevin Algorithm (MALA) [18] and the Hamiltonian Monte Carlo (HMC) [19], have also been studied in the literature and have been shown to be more efficient than random walk Metropolis-Hastings in terms of Effective Sample Size (ESS) and convergence rates on several problems. However, HMC and MALA also require extensive tuning of the proposal mechanisms, see for example [20] and [21]. For MJPs the problem is compounded further since system parameters, such as probability rate constants of chemical reactions, are often highly correlated and whose values may differ by orders of magnitudes. The resulting posterior distributions have long narrow "valleys" preventing any local proposal mechanism from proposing large moves about the parameter space. More recently [22] proposed exploitation of the underlying Riemann manifold of probability density functions when defining MCMC methods thus exploiting the intrinsic geometry of statistical models, thereby providing a principled framework and systematic approach to the proposal design process. These algorithms rely on the gradient and Fisher Information matrix of the likelihood function to automatically tune the proposal mechanism such that large moves on the parameter space are possible and therefore improve convergence and mixing of the chains. In [9] this approach has been successfully applied for the MRE approximation of chemical reaction networks. For the LNA the Fisher Information and the gradient of the likelihood function can be easily obtained [2]. In this paper we study the application of the Riemann manifold MCMC methods for the LNA approximation and compare the mixing efficiency and computational cost with to the commonly used Metropolis-Hastings algorithm. Moreover we study how the the Markov chains and the resulting Monte Carlo estimates behave for systems which are far from their thermodynamic limit. The aim is to improve the efficiency of MCMC inference for MJPs in order to allow for larger and more complex models frequently encountered in biology and chemistry to be studied in more detail. In the next section we give a brief overview of Markov jump processes. The diffusion and linear noise approximations are presented in section 3. We then discuss MCMC and the Riemann manifold algorithms in section 4. Numerical simulations are presented in section 6 while section 7 concludes the paper. Markov Jump Processes A D-dimensional stochastic process is a family of D random variables X(t) = [X 1 (t), . . . , X D (t)] T indexed by a continuous time variable t with initial conditions X(t 0 ) = x t 0 . A Markov Jump Process (MJP) is a stochastic process satisfying the Markov property such that p[X(t 0 ), . . . , X(t N )] = p[X(t 0 )] N i=1 p[X(t i )|X(t i−1 )], where the dependence on any parameters or other quantities has been suppressed. That is, the conditional probability of the system state at time t i only depends on state of the system at the previous time t i−1 . A MJP is characterised by a finite number, M , of state transitions with rates f j (x, θ, t) and state change vectors s j = (s 1,j , . . . , s D,j ) T with j ∈ [1, . . . , M ]. f j (x, θ, t)dt is the probability, given the state of the system at time t, X(t) = x, of a jump to a new state x + s j in the infinitesimal time interval [t, t + dt). For the problems we consider in this paper the transition rates not only depend on the current state and time but also on unknown rate parameters θ. From the Markov property we can directly obtain the conditional probability of the system being in state x at time t given initial conditions which is characterised by the master equation p(x, t|x 0 , t 0 ) dt = M j=1 [f j (x − s j , θ, t)p(x − s j , t|x 0 , t 0 ) − f j (x, θ, t)p(x, t|x 0 , t 0 )] .(1) Equation (1) in general form is intractable especially when the transition rate functions f j (·) are nonlinear with respect to the system state. Numerical simulation is also prohibitively expensive as the computational cost grows exponentially with D [23]. However, given initial conditions X(t 0 ) = x t 0 and values for the unknown rate parameters θ we can simulate realisations of the MJP by first noting that the time τ to the next state transition is exponentially distributed with rate λ = M j=1 f j (x t 0 , θ, t 0 ) and the new state X(t 0 + τ ) will be x t 0 + s j with probability f j (x t 0 , θ, t)/λ. This results in an iterative algorithm from which we can forward simulate a complete trajectory for the stochastic process X(t), known as the Stochastic Simulation Algorithm (SSA) [1] in the chemical kinetics literature. From the specification of the MJP we can also write the likelihood function with respect to the parameters θ for a completely observed process X(t) at the time interval [0, T ] as p(X|θ) = N i=1 f k i (x i−1 , θ, τ i−1 ) exp   −τ i M j ′ =1 f j ′ (x i−1 , θ, τ i−1 )   where N is the number of transitions occurred in the time interval [0, T ], k i ∈ [1, . . . M ] is the type of the i th transition and τ i , x i are the time and state at the i th transition respectively. Notice that the likelihood function corresponds to the generative process described by the SSA. By specifying a suitable prior and applying Bayes' theorem, we can obtain the posterior distribution p(θ|X) which we can use for inference over the unknown parameters θ [6]. In many problems of interest however we cannot observe the times and types of all transitions in a given time interval. Rather, we can only observe the state of the system X(t i ) = x i at discrete time points t i ∈ [0, T ]. The solution proposed in [6] is to treat the trajectories, as well as the number, times and types of transitions, between observed time points as latent variables. This leads to a data augmentation framework [24] where a Markov Chain is constructed to sample from the joint posterior of the parameters and the latent variables. At each MCMC iteration the complete trajectory of the MJP process has to be simulated conditional on the observed data and the parameters which for some systems can be computationally demanding. Furthermore, due to the high dimensional nature of the simulated trajectory and the strong dependence on the system parameters and observed data the MCMC algorithm has very poor convergence and mixing properties requiring many samples from the posterior in order to obtain sufficiently accurate Monte Carlo estimates. Finally, a further complication that arises is that the number of transitions between two observed time points is also unknown and has to be sampled using a reversible-jumps type algorithm [25]. For more details see [6]. The resulting algorithm therefore is computationally demanding thus limiting its applicability on small and relatively simple MJPs. A more efficient version of the algorithm is also suggested in [6] where instead of simulating the trajectories between observations using the exact MJP an approximate proposal distribution is employed to sample trajectories which are accepted or rejected using the Metropolis-Hastings ratio. Diffusion and Linear Noise Approximations An alternative to working directly with the master equation and the original MJP is to consider approximations which provide for efficient simulation and possibly an easy to evaluate likelihood function for discretely observed data. Although the resulting posterior will also be approximate in nature, it can be sufficient for inferential purposes given that the system under consideration is near its thermodynamic limit. Here we describe the diffusion approximation and from that how we can arrive at the LNA. Our presentation is rather informal and follows [14] and [1]. For a more formal derivation the reader should refer to [7] and [11]. The requirement for these approximations to be consistent is the existence of a proportionality constant Ω which governs the size of the fluctuations such that for large Ω the jumps will be relatively small and as both Ω and x tend to infinity approaching the system's thermodynamic limit then, f j (x, θ, t) → Ωf j (z, θ, t),(2) where z = x/Ω andf j (·) are independent of Ω. For many physical processes where the fluctuations are due to the discrete nature of matter there is a natural Ω parameter with such properties. Examples of such parameters can be the system size in chemical kinetics, the capacity of a condenser in electric circuits or the mass of a particle [7]. Diffusion approximation In order to obtain a Langevin equation which closely matches the dynamics of the MJP it is assumed that there is an infinitesimal time interval dt which satisfies the following conditions f j (x t ′ , θ, t ′ ) ≈ f j (x t , θ, t), ∀t ′ ∈ [t, t + dt), ∀j ∈ [1, M ] (3) f j (x t , θ, t)dt ≫ 1 ∀j ∈ [1, M ].(4) The first condition constrains dt to be small enough such that the transition rate functions remain approximately constant. This implies that the number of transitions of type j is distributed as a Poisson random variable with mean f j (x t , θ, t)dt and is independent from other transitions of type j ′ = j. The second condition constrains dt to be large enough such that the number of transitions for each state is significantly larger than 1, which further implies that the Poisson distribution can be accurately approximated by a Gaussian distribution. It can be shown [26] that we can choose dt and Ω such that both conditions can be satisfied and this generally occurs when the system approaches its thermodynamic limit. Given such a timescale, the state of the system at time t + dt can be computed by x t+dt = x t + M j=1 N [f j (x t , θ, t)dt, f j (x t , θ, t)dt]s j (5) where N [µ, σ 2 ] denotes a Gaussian random variate with mean µ and variance σ 2 . From Equation (5) we can directly obtain a Langevin equation of the form dx t = Sf (x t , θ, t)dt + S diag[f (x t , θ, t)]dB t(6) where we used S to denote the matrix whose columns are the state change vectors s j , f (·) to denote the vector whose elements are the transition rates f j (·), diag(v) a function that returns a diagonal matrix with elements taken from the vector v and dB t an M dimensional Wiener process. Notice that the dimension of x t differs from that of dB t . Due to the nonlinear state dependent drift and diffusion coefficients in Equation (6) the transition density of the stochastic process is also intractable. Therefore a data augmentation approach similar to the one in [6] has to be followed. However, there is no longer the need to sample the number, times and types of state transitions as the MJP is approximated with a continuous process. Moreover, the latent variables corresponding to unobserved states can now be efficiently simulated by an Euler-Maruyama scheme which is computationally more efficient than the SSA. This approach has been followed by [13] and [12] for inference over the unknown parameters θ while in [27] a similar methodology has been applied on a real data from an auto-regulatory gene expression network. Linear noise approximation Substituting equation (2) in the Langevin equation (6) and dividing by Ω we get dz t = Sf (z t , θ, t)dt + 1 √ Ω S diag[f (z t , θ, t)]dB t(7) from which we can see that the fluctuations are of the order of 1/ √ Ω and in the thermodynamic limit (7) reduces to the Macroscopic Rate Equation (MRE) lim Ω→∞ dz t = Sf (z t , θ, t)dt. To obtain the Linear Noise Approximation (LNA) we make the assumption that for sufficiently large Ω a solution to (7) will differ from the MRE by a stochastic term of order 1/ √ Ω. That is z t = φ t + 1 √ Ω ξ t (8) where φ t are deterministic or sure variables satisfying the MRE and ξ t are stochastic variables. Rewriting the transition rate functions using (8) and Taylor expand around φ we get f j (z, θ, t) =f j φ + 1 √ Ω ξ =f j (φ, θ, t) + 1 √ Ω D d=1 ∂f j (φ, θ, t) ∂φ i ξ i + O(Ω −1 ).(9) We can now substitute (8) and (9) back into (7) and collect terms of O(1) to get the expression for the differential of φ which is nothing other than the MRE dφ t = Sf (φ t , θ, t)dt.(10) Finally, collecting remaining terms and neglecting terms of O(1/ √ Ω) and higher we get the differential of ξ as dξ t = SJf (φ t , θ, t)ξ t dt + S diag[f (φ t , θ, t)]dB t(11) where we used Jf (·) to denote the Jacobian of the transition ratesf (·). Equation (11) characterises the fluctuations around the deterministic state φ and its validity depends on the size of Ω. As Ω increases the magnitude of the individual jumps s j becomes negligible relative to the distance in φ over which the non-linearity off j (·) becomes noticeable. A measure of the sufficiency of LNA is the coefficient of variation, i.e. the ratio of the standard deviation to the mean. For a more thorough discussion on the validity of LNA the reader is referred to [23] and the supplementary material of [15]. Solution of the LNA and the approximate likelihood function LNA provides a convenient expression for the approximate likelihood since the MRE (10) can be easily solved numerically and its computational cost is polynomial in D. Moreover, equation (11) is a system of linear stochastic differential equations which has an explicit solution of the form ξ t = Φ(t 0 , t) ξ 0 + t t 0 Φ(s, t) −1 S diag[f (φ s , θ, s)]dB s(12) where the integral is in the Itô sense and Φ(t 0 , t) is the solution of dΦ(t 0 , s) = SJf (φ t , θ, t)Φ(t 0 , s)ds, Φ(t 0 , t 0 ) = I.(13) Since the Itô integral of a deterministic function is a Gaussian random variable [28], equation (12) implies that ξ t has a multivariate normal distribution. To simplify further the analysis assume that the initial condition for z t has a multivariate normal distribution such that z t 0 ∼ N (φ t 0 , V t 0 ) . For the rest of the paper we will assume that φ t 0 and V t 0 are known. In cases where the initial conditions are unknown they can be treated as additional parameters. Equations (8, 10, 11, 12) and the specification of initial conditions further imply that z t ∼ N (φ t , Ω −1 V t )(14) where φ t are solutions of the MRE and V t are solutions of dV t = SJf (φ t , θ, t)V t + V t J T f (φ t , θ, t)S T + Sdiag[f (φ t , θ, t)]S T . Finally, multiplying (14) by Ω we get x t ∼ N (Ωφ t , ΩV t ). Assume that we have observations from the stochastic process X(t) at discrete time points t i ∈ {t 1 , . . . , t N }. Moreover, assume that each observation x t is obtained by a independent realisation of X(t). For example to obtain an observation at t 1 = 10 the SSA is used to simulate a trajectory from t 0 to t 1 and the state of the system at t 1 is kept. For t 2 = 20 the SSA is again used to simulate a new trajectory from t 0 to t 2 keeping only the state of the system at t 2 and the process continues until all necessary observations are gathered. This kind of data are very frequently encountered in biology where in order to obtain a single measurement the sample has to be "sacrificed". This is common in data obtained using Polymerase Chain Reaction reporter assays [29] for example. See also [15] for an example of an inference problem with such data. Due to the independence between different observations and the Markov property the likelihood is simply p(X|θ) = N i=1 N (x t i |Ωφ t i , ΩV t i ).(15) In this paper we only consider observations of this kind. However the methodology is readily applicable when observations from a single realisation of X(t) are available. In this case the likelihood also has a simple form p(X|θ) = N [X|Ωµ(θ), ΩΣ(θ)] where X = (x t 1 , . . . , x t N ) T is an N D vector with all the observations, µ(θ) = (φ t 1 , . . . , φ N ) T , is also a N D vector with solutions of the MRE and Σ(θ) is a N D × N D block matrix Σ(θ) = {Σ(θ) i,j : i, j ∈ [1, . . . , N ]} such that Σ(θ) i,j = V t i , i = j V t i Φ(t i , t j ) T , i = j(16) This stems from the fact that due to the Markov property and equation (14) each x t i can be written as a sum of multivariate normal random variables and therefore X is also a multivariate normal random variable. For more details refer to the supplementary material of [15] and [2]. The only additional complication which arises for time-series data is that the off-diagonal components of the LNA variance in equation (16) need to be estimated by numerically solving the system of ODEs in equation (13). Notice that despite the fact that the variance matrix is full we can still exploit the Markov property and write the likelihood as a product of the conditional likelihoods and therefore avoid the cost of inverting the N D × N D variance matrix. Markov Chain Monte Carlo Methods In this section we give a brief overview of the MCMC algorithms that we consider in this work. Some familiarity with the concepts of MCMC is required by the reader since an introduction to the subject is out of the scope of this paper. Metropolis-Hastings For a random vector θ ∈ R D with density p(θ) the Metropolis-Hastings algorithm employs a proposal mechanism q(θ * |θ t−1 ) and proposed moves are accepted with probability min 1, p(θ * )q(θ t−1 |θ * )/p(θ t−1 )q(θ * |θ t−1 ) . In the context of Bayesian inference the target density p(θ) corresponds to the posterior distribution of the model parameters. Tuning the Metropolis-Hastings algorithm involves selecting the right proposal mechanism. A common choice is to use a random walk Gaussian proposal of the form q(θ * |θ t−1 ) = N (θ * |θ t−1 , Σ), where N (·|µ, Σ) denotes the multivariate normal density with mean µ and covariance matrix Σ. Selecting the covariance matrix however, is far from trivial in most cases since knowledge about the target density is required. Therefore a more simplified proposal mechanism is often considered where the covariance matrix is replaced with a diagonal matrix such as Σ = ǫI where the value of the scale parameter ǫ has to be tuned in order to achieve fast convergence and good mixing. Small values of ǫ imply small transitions and result in high acceptance rates while the mixing of the Markov Chain is poor. Large values on the other hand, allow for large transitions but they result in most of the samples being rejected. Tuning the scale parameter becomes even more difficult in problems where the standard deviations of the marginal posteriors differ substantially, since different scales are required for each dimension, and this is exacerbated when correlations between different variables exist. Adaptive schemes for the Metropolis-Hastings algorithm have also been proposed [30] though they should be applied with care [31]. Parameters such as reaction rate constants often differ orders of magnitude, thus a scaled diagonal covariance matrix will be a bad choice for such problems. In the numerical simulations in the next section we used a Metropolis within Gibbs scheme where each parameter is updated conditional on all others using a univariate normal density with a parameter-specific scale parameter. This allows us to tune the scale for each proposal independently and achieve better mixing. Manifold Metropolis Adjusted Langevin Algorithm Denoting the log of the target density as L(θ) = log p(θ), the manifold MALA (MMALA) method, [22], defines a Langevin diffusion with stationary distribution p(θ) on the Riemann manifold of density functions with metric tensor G(θ). By employing a first order Euler integrator to solve the diffusion a proposal mechanism with density q(θ * |θ t−1 ) = N (θ * |µ(θ t−1 , ǫ), ǫ 2 G −1 (θ t−1 )) is obtained, where ǫ is the integration step size, a parameter which needs to be tuned, and the dth component of the mean function µ(θ, ǫ) d is µ(θ, ǫ) d = θ d + ǫ 2 2 G −1 (θ)∇ θ L(θ) d − ǫ 2 D i=1 D j=1 G(θ) −1 i,j Γ d i,j(17) where Γ d i,j are the Christoffel symbols of the metric in local coordinates [32]. Similarly to MALA [18], due to the discretisation error introduced by the first order approximation, convergence to the stationary distribution is not guaranteed anymore and thus the Metropolis-Hastings ratio is employed to correct this bias. The MMALA algorithm can be simply stated as in Algorithm 1 and more details can be found in [22]. Algorithm 1 MMALA 1: Inititialise θ 0 2: for t = 1 to T do 3: θ * ∼ N (θ|µ(θ t−1 , ǫ), ǫ 2 G −1 (θ t−1 )) 4: r = min 1, p(θ * )q(θ t−1 |θ * )/p(θ t−1 )q(θ * |θ t−1 ) 5: u ∼ U [0,1] 6: if r > u then We can interpret the proposal mechanism of MMALA as a local Gaussian approximation to the target density similar to the adaptive Metropolis-Hastings of [33]. In contrast to [33], the effective covariance matrix in MMALA is the inverse of the metric tensor evaluated at the current position and no samples from the chain are required in order to estimate it, therefore avoiding the difficulties of adaptive MCMC discussed in [31]. Furthermore a simplified version of the MMALA algorithm (SMMALA) can also be derived by assuming a manifold with constant curvature, thus cancelling the last term in Equation (17) which depends on the Christoffel symbols. Finally, the MMALA algorithm can be seen as a generalisation of the original MALA [18] since, if the metric tensor G(θ) is equal to the identity matrix corresponding to an Euclidean manifold, then the original algorithm is recovered. Manifold Hamiltonian Monte Carlo The Riemann manifold Hamiltonian Monte Carlo (RMHMC) method defines a Hamiltonian on the Riemann manifold of probability density functions by introducing the auxiliary variables p ∼ N (0, G(θ)), which are interpreted as the momentum at a particular position θ and by considering the negative log of the target density as a potential function. More formally, the Hamiltonian defined on the Riemann manifold is: H(θ, p) = −L(θ) + 1 2 log (2π|G(θ)|) + 1 2 p T G(θ) −1 p(18) where the terms −L(θ) + 1 2 log (2π|G(θ)|) and 1 2 p T G(θ) −1 p are the potential energy and kinetic energy terms, respectively. Simulating the Hamiltonian requires a time-reversible and volume preserving numerical integrator. For this purpose the Generalised Leapfrog algorithm can be employed and provides a deterministic proposal mechanism for simulating from the conditional distribution, i.e. θ * |p ∼ p(θ * |p). More details about the Generalised Leapfrog integrator can be found in [22]. To simulate a path across the manifold, the Leapfrog integrator is iterated L times which along with the integration step size ǫ are parameters requiring tuning. Again, due to the integration errors on simulating the Hamiltonian, in order to ensure convergence to the stationary distribution the Metropolis-Hastings ratio is applied. Moreover, following the suggestion in [20] the number of Leapfrog iterations L is randomised in order to improve mixing. The RMHMC algorithm is given in Algorithm 2. end if 20: end for Similar to the MMALA algorithm, when the metric tensor G(θ) is equal to the identity matrix corresponding to an Euclidean manifold, then RMHMC is equivalent to the HMC algorithm of [19]. 5 Implementation details 5.1 Gradient and metric tensor for the LNA For the manifold MCMC algorithms discussed in this section we will need the gradient of the log likelihood as well as a metric tensor for the LNA. For density functions the natural metric tensor is the expected Fisher Information, I(θ), [34] and for a multivariate normal with mean µ(θ) and covariance matrix Σ(θ) its general form is solve θ n+1 * = θ n * + ǫ 2 ∇ p H θ n * , p n+ 1 2 * + ∇ p H θ n+1 * , p n+ 1 2 * 10: p n+1 * = p n+ 1 2 * − ǫ 2 ∇ θ H θ n+1 * , pI(θ) i,j = ∂µ(θ) ∂θ i Σ −1 (θ) ∂µ(θ) ∂θ j + 1 2 Tr Σ −1 (θ) ∂Σ(θ) ∂θ i Σ −1 (θ) ∂Σ(θ) ∂θ j . For the likelihood in equation (15) the Fisher Information is then a sum of N matrices I(θ, t), one evaluated at each time point. Similarly the general form of the partial derivatives for the log of a multivariate normal is ∂ ln N [x|µ(θ), Σ(θ)] ∂θ i = 1 2 Tr (cc T − Σ −1 (θ)) ∂Σ(θ) ∂θ i + c T ∂µ(θ) ∂θ i where c = Σ −1 (θ)[x − µ(θ)]. Moreover, during the leap-frog integration for the RMHMC and for the mean function of MMALA the partial derivatives of the Fisher Information are needed. Their general form is ∂I(θ) i,j ∂θ k = ∂ 2 µ(θ) T ∂θ i ∂θ k a j + a T i ∂ 2 µ(θ) ∂θ j ∂θ k − a T i ∂Σ(θ) ∂θ k a j − 1 2 Tr [A k (A i A j + A j A i )] + 1 2 Tr Σ −1 (θ) ∂Σ(θ) ∂θ i ∂θ k A j + ∂Σ(θ) ∂θ j ∂θ k A i where a i = Σ −1 ∂µ(θ) ∂θ i and A i = Σ −1 ∂Σ(θ) ∂θ i . The above quantities require first and second order sensitivities for the φ and V which we obtain by augmenting the ODE systems with the additional sensitivity equations. For an ODE system of n y equations with formẏ = F (y, t, θ), y(t 0 ) = y 0 (θ) and n θ parameters θ, the first and second order forward sensitivity equations are given by (19) and (20) respectively. ∂ẏ ∂θ = F y ∂y ∂θ + F θ , ∂y(t 0 ) ∂θ = ∂y 0 ∂θ (19) ∂ 2ẏ ∂θ∂θ T = [F y ⊗ I n θ ] ∂ 2 y ∂θ∂θ T + I ny ⊗ ∂y T ∂θ F y,y ∂y ∂θ + F y,θ + F θ,y ∂y ∂θ + F θ,θ , ∂ 2 y(t 0 ) ∂θ 2 = ∂ 2 y 0 ∂θ 2(20) We use F θ to denote the n y × n θ matrix where its j th column is the partial derivatives of F with respect to θ j . F θ,y denotes the derivative of F θ with respect to y and is an n θ · n y × n y matrix where its j th column is the partial derivatives of vec(F T θ ) with respect to y j . I ny denotes the n y × n y Identity matrix, ⊗ the Kronecker product and vec(A) an operator that creates a column vector by stacking the columns of matrix A. Re-parameterisation In many problems the parameters θ can be constrained in certain parts of R n θ where n θ is the number of parameters. In models of chemical kinetics for example, rate parameters must be positive and can differ by orders of magnitude. For the MCMC algorithms described in the previous section we will need a re-parameterisation in order to allow the algorithms to operate on an unbounded and unconstrained parameter space. For the numerical simulations in section 6 we use a log 10 re-parameterisation by introducing the variablesθ p = log 10 (θ p ), p ∈ [1, . . . , n θ ]. To ensure that we sample from the correct posterior the joint density is scaled by the determinant of the Jacobian such that p(X|θ)p(θ)|J (θ)| where J(θ) is a n θ × n θ diagonal matrix with elements J (θ) p,p = 10θ p log (10). The gradient and Fisher information along with its partial derivatives follow from the chain rule as ∇θL(θ) = ∇ θ L(θ)J (θ) I(θ) = J (θ) T I(θ)J (θ) ∂I(θ) ∂θ p = 2J (θ) T I(θ) ∂J (θ) ∂θ p + J(θ) T ∂I(θ) ∂θ p J (θ) ∂θ p ∂θ p Choice of priors In Bayesian statistics priors provide the means for incorporating existing knowledge for the parameters in question. The choice of a suitable prior distribution can be informed from knowledge about the process being modelled, the experimental design and empirical observations. For example we might want to restrict rate parameters in chemical kinetics from becoming very high since we assume from the experimental design that reactions are slow enough to be able to be observed. In some cases the model itself can also guide the choice of the prior. For example when a model is only defined for a certain range of values of the parameters, a prior restricting the parameters in that range should be used. In the numerical simulations of the next section we use independent normal priors for the parametersθ. Due to the re-parameterisation introduced earlier, this corresponds to a lognormal prior with base 10 for the parameters θ. This choice allows parameters to differ several orders of magnitude while it ensures they are strictly positive. Moreover, as noted in [22] the negative Hessian of the prior is added to the Fisher information in order to form the metric tensor used during MCMC sampling. This has the added benefit of regularising the Fisher information when it is near-singular [9] although we have not observed such problems in the simulations presented here. 6 Numerical Simulations Chemical kinetics In this section we consider two examples from chemical kinetics [14] and study the effect of the system size parameter on inference using MCMC. The first system consists of three species where an unstable monomer, S 1 , can dimerise to an unstable dimer, S 2 , which is then converted to a stable form, S 3 . The reaction set for this system is and the state of the system at time t will be denoted by X(t) = [S 1 (t), S 2 (t), S 3 (t)] T . The propensity functions, or state transition probabilities are f (X, θ) = [c 1 S 1 (t), c 2 Ω −1 S 1 (t)(S 1 (t)− 1)/2, c 3 S 2 (t), c 4 S 3 (t)] T and the corresponding state change matrix is R1 : S 1 c 1 − → ∅ R2 : 2S 1 c 2 Ω −1 − −−− → S 2 R3 : S 2 c 3 − → 2S 1 R4 : S 2 c 4 − → S 3S =   −1 −2 2 0 0 1 −1 −1 0 0 0 1   .(21) For our experiments we will assume that initial conditions are known and set them to S 1 (t 0 ) = 5Ω, S 2 (t 0 ) = S 3 (t 0 ) = 0, t 0 = 0. Moreover we will set the reaction rate parameters to c 1 = 1, c 2 = 2Ω −1 , c 3 = 0.5 and c 4 = 0.04. Notice that we make explicit the relation between the system size and parameterĉ 2 and we will infer rate c 2 up to a proportionality constant. For all the experiments we simulate data using the SSA of [1] for the time interval t ∈ [0, 10] and we discretise such that t i − t i−1 = 0.1. Each observation X(t i ) is obtained independently by simulating a trajectory from t 0 to t i and keeping only the last state discarding the rest of the trajectory. Moreover for each time point t i we also simulate 10 independent observations. Since each observation is obtained by a different trajectory of the MJP we assume that initial conditions do not have a point mass rather for each trajectory we sample its initial condition from a Poisson with means S 1 (0), S 2 (0), S 3 (0). We use the synthetic data to perform inference for the rate parameters θ = (c 1 ,ĉ 2 , c 3 , c 4 ) T by drawing samples from the posterior p(θ|X) ∝ p(θ) N i=1 10 r=1 N [X r (t i )|Ωφ(t i ), ΩV (t i )] where r indexes independent observations for the same time point. For all simulations in this paper we assume that the means for the initial conditions are known. Following similar arguments as for the derivation of the LNA in Section 3, namely that as the system approaches its thermodynamic limit transition densities become Gaussian, the initial conditions for the ODE systems for the mean and variance of the transition densities are φ(0) = X(0)Ω −1 and V (0) = I, where I is the identity matrix. In a more realistic scenario the initial conditions must be included as additional parameters in θ. For all parameters we used an independent log-normal prior with base 10, zero mean and one standard deviation and chains are initialised by drawing a random sample from the prior. For the Metropolis-Hastings sampler we set the initial proposal scale parameters to ≈ 1e −6 and automatically adapt them every 100 samples during the burn-in phase in order to achieve an acceptance rate of 25% − 30% [17]. The same adaptation strategy was followed for the simplified MMALA and RMHMC algorithms where the initial step size was also set to ≈ 1e −6 and was tuned in order to achieve acceptance rates in the order of 70 − 80% [22]. Finally, the number of leap-frog steps for RMHMC was fixed to 5. We have found that a burn in period of 10,000 to 20,000 samples was adequate for all algorithms to converge to the stationary distribution. Table 1 compares the minimum Effective Sample Size (ESS) and the time normalised ESS obtained by all algorithms for different values of the system size parameter Ω. The SMMALA and RMHMC samplers utilise the gradients and the Fisher Information of the approximate likelihood obtained by the LNA in order to make efficient proposals. As the system size increases and thus the LNA better approximates the true likelihood then mixing of the manifold MCMC algorithms improves. For this particular example we can see that good mixing can be achieved even for very small systems with only ≈ 25 molecules, (Ω = 5). The M.H. sampler is not affected by the system size but its mixing is very poor in all cases. From the time normalised ESS we can also see that despite the improved mixing of RMHMC the computational cost is significant. On the contrary SMMALA provides a good tradeoff between mixing efficiency and computational cost. Finally, Table 2 reports the marginal posterior means and standard deviations for different values of Ω obtained by RMHMC. The marginal posteriors for parameters c 3 and c 4 with Ω >= 5 are also shown in Figure 1. Results from the MH and SMMALA samplers are similar and are omitted. For small system sizes we can observe that there is an increased bias of the Monte Carlo estimate while the posterior standard deviation is higher reflecting the high degree of uncertainty around the mean. The bias however significantly reduces as the system size increases and for Ω >= 5 reasonable estimates can be obtained. The second example from the chemical kinetics literature that we consider is the Schlögl f (X, θ) =     c 1 Ω −1 1 2 S 1 (S 1 − 1), c 2 Ω −2 1 6 S 1 (S 1 − 1)(S 1 − 2), c 3 Ω, c 4 S 1     (23) reaction set. R1 : 2S 1 c 1 Ω −1 − −−− → 3S 1 R2 : 3S 1 c 2 Ω −2 − −−− → 2S 1 R3 : ∅ c 3 Ω − − → S 1 R4 : S 1 c 4 − → ∅.(22) The corresponding state transition rates and state change matrix are given in equations (22) and (23) respectively. The state of the system consists only of the number of molecules of a single species X(t) = S 1 (t). The system is known to have two stable states which appear at different times depending on the size of the system. [14] have shown that the LNA fails to provide a reasonable approximation of this system even for large concentration numbers. Their numerical experiments demonstrate that the LNA approximation can only approximate one of the two modes depending on the initial conditions. Here our aim is to show that using the LNA to obtain an approximate posterior over the unknown reaction rate constants can be very misleading for bi-stable systems. Using the resulting posterior means for the reaction rates gives us an LNA that fails to approximate any of the two stable modes. To demonstrate that we follow the same experimental procedure as in the previous example. That is, we simulate data using the SSA for the time interval t i ∈ [0, 10], t i − t i−1 = 0.1 with fixed rate parameters and then use this data for posterior inference of the rate parameters using MCMC. Values for the true rate parameters and initial conditions where set as in [14]. Namely, c 1 = 0.003, c 2 = 0.0001, c 3 = 200, c 4 = 3.5 and X(t 0 ) = 280Ω, where Ω was fixed to 1. After 10,000 burn-in samples all samplers converged to a posterior distributions with mean The LNA obtained by using the posterior means for the rate constants is shown in Figure 2 along with the data obtained by the SSA and the LNA using the true values for the rate constants. We can see that the LNA approximation obtained by the posterior means fails to approximate any of the two modes. Rather it approximates the empirical mean and variance of the data. Single gene expression Finally, to illustrate the applicability of the methodology to systems biology we also consider a simplified model for the biochemical reactions involved in the expression of a single gene to protein. The model presented in this section is the same with the model used in the study of [15] and we adopt the same notation in order to make comparisons easier. Gene expression is modelled in terms of three biochemical species; DNA, mRNA and protein; and four chemical reactions or state transitions; transcription, mRNA degradation, translation and protein degradation. The model can be written in chemical reaction notation as R1 : DN A k R (t) − −− → DN A + R R2 : R γ R − − → ∅ R3 : R k P − − → R + P R4 : P γ P − − → ∅. The system state at time t is X(t) = [R(t), P (t)] T where R(t) and P (t) are the number of mRNA and protein molecules respectively. The corresponding state dependent transition rates are f (X, t) = [k R (t), γ R R(t), k P R(t), γ P P (t)] T where γ R , k P and γ P are unknown reaction rate constants. k R (t) is the time dependent transcription rate of the gene which for the purposes of this section is modelled as k R (t) = b 0 exp(−b 1 (t − b 2 ) 2 ) + b 3 where all the b i s are also unknown parameters controlling gene transcription. This corresponds to a transcription rate that due to some stimulus (experimental or environmental) increases for t < b 2 and then it drops towards the base line b 3 for t > b 2 . Finally, the state change matrix for this set of reactions is given in equation (24). S = 1 −1 0 0 0 0 1 −1 .(24) As in the study of [15] we also consider a non-linear extension of this model where the transcription rate of the gene k R (t) is a function of the protein concentration that the gene is transcribed to. This is modelled using a Hill function where for the experiments of this section we will set H = b 3 k P /(2γ R γ P ) and n H = 1/2 making the protein an inhibitor of mRNA transcription. A schematic representation of this model is shown in Figure 3. For the rest of this section we will refer to this model as the auto-regulatory single gene expression model. k R (t, P ) = k R (t)/(1 + (P/H) n H ) Using the transition probabilities f (X, t) and matrix S we simulate synthetic data using the Stochastic Simulation Algorithm (SSA) [1] and sample at discrete time points. Values for the unknown rate constants and the parameters controlling gene transcription are shown in Table 3. The time interval is taken to be t i ∈ [0, 25] while the interval between two observations t i − t i−1 = 0.25. Each time point is sampled from an independent trajectory by starting the SSA from t 0 and simulate up to t i keeping only the state X(t i ) and discarding the rest of the trajectory. This resembles the experimental conditions often encountered in biology where in order to make an observation the sample has to be "sacrificed". Finally for each time point we also generate 10 independent observations from different trajectories. Initial conditions X(t 0 ) are simulated from a Poisson distribution with means b 3 /γ R and b 3 k P /(γ R γ P ) for the mRNA and protein molecules respectively. The system size parameter Ω is considered to be unknown and for this experiment is set to 1 such that concentrations are equal to the number of molecules. Figures (4.a) and (4.b) show data simulated from this process from the singe gene expression model as well as the LNA prediction. Simulated data for the auto-regulatory model are presented in Figures (4.c) and (4.d). We use the simulated data to infer the unknown parameters θ = (γ R , k P , γ P , b 0 , b 1 , b 2 , b 5 ) T by sampling using MCMC from the LNA approximate posterior where r indexes independent samples for the same time point and R = 10. Table 3 summarises the results from the MCMC chains for the two models of gene expression. Firstly, we can see that despite the relatively small number of molecules in both systems the LNA approximation provides very accurate estimates for the true parameters. Moreover we can see that the mixing of the Metropolis-Hastings sampler is very poor for both models while RMHMC and simplified Manifold MALA algorithms perform very well. This can be explained by the strong correlations between parameters in the posterior distribution preventing the M.H. sampler to make sufficiently large proposals. For example, the parameters k P , γ P control mRNA translation and protein degradation respectively. The concentration of protein molecules is directly affected by the two rates and they are expected to be heavily correlated. In Figures 5.a 5.b we show the marginal joint posterior for parameters k P , γ P and γ R , b 3 for the single gene expression model which exhibit very strong positive correlation. Finally figure 6 compares the trace plots obtained from MH, SMMALA and RMHMC for parameters γ P and k P of the autoregulatory gene expression model. p(θ|X) ∝ p(X|θ)p(θ) = p(θ) N i=1 R r=1 N [X r (t i )|φ(t i ), V (t i )] Conclusions and Future Work Bayesian inference for Markov jump processes is a challenging problem which has many important practical applications. Previous research [6] has shown that although exact inference is possible, the computational cost and the autocorrelation of the Markov chains is such that limits its applicability to small problems. The main problem stems from the requirement to simulate the MJP for the trajectory of the system between discrete observations. [13] has shown that by considering a diffusion approximation the simulation can be performed in a much more efficient manner. In this paper we considered the linear noise approximation which only requires to simulate a system of ordinary differential equations while the stochastic fluctuations have an exact analytic solution. The linear noise approximation is valid only when the system is sufficiently close to its thermodynamic limit, a condition that is also required for the diffusion Single gene expression model. Table 3: Marginal posterior means and standard deviations for the parameters of the single gene expression model using simulated data. The ESS is calculated for chains of 10,000 samples after a burn-in period of 10,000 iterations with initial parameters randomly sampled from the prior. Average acceptance rate (A.R.) and sampler parameters are shown in parenthesis. Notice that for the Metropolis-Hastings sampler a different proposal is used for each parameter. The prior for all parameters was log 10 N (0.0, 2.0). We have demonstrated here that when the posterior distribution exhibits strong correlation between parameters then the Metropolis-Hastings sampler has strong auto-correlations. Such correlations are very common for chemical reaction and gene regulatory systems. The Riemann manifold MCMC algorithms we considered in this work exploit the geometric structure of the target posterior in order to design efficient proposal mechanisms. In particular the simplified Manifold MALA algorithm is a conceptually simple algorithm which provides a good trade-off between computational cost and sample auto-correlation. Although the problems considered in this work are relatively small, but certainly non-trivial, we believe that the proposed methodology is applicable for larger and more complex systems. The systems we studied in this paper all have a linear dependence on the unknown parameters and we have not observed any local modes in our simulations. The analysis of such systems is the subject of on-going work. Moreover, in real applications it is not possible to observe the populations of all species and there is an additional measurement error term. Extension of the LNA to handle such cases is straight forward, see [15] for example, however the effect of partial observations and measurement error on the MCMC inference is something that needs to be studied in more detail. Parameters γ R γ P k P b 0 b 1 b 2 b 3 * , p * ) = θ N +1 * , p N +1 * {Metropolis-Hastings ratio} 13: r = min 1, exp −H(θ * , p * ) + H(θ t−1 , p :Figure 1 : 1Marginal posterior means and standard deviations calculated from the RMHMC chain for different values of the system size parameter Ω of the decay-dimerisation reaction model. Notice thatĉ 2 parameter is proportional to Ω. Results are calculated from 10,Marginal posteriors for parameters c 3 (left panel) and c 4 right panel for different values of Ω. Results are obtained by 10,000 posterior samples using RMHMC. Figure 2 : 2Simulated time point data using SSA for the Schlögl reaction set and LNA predictions. Dots correspond to simulated data. The bold and dashed red lines correspond to the LNA prediction for the means and standard deviations using the true parameters. Doted blue lines correspond the LNA predictions using the posterior means for the rate parameters. (Online version in colour.) S = 1, −1, 1, −1, Figure 3 : 3E p(θ|X) [θ] ≈ (0.130, 3.3e −4 , 3.5e +3 , 26.22) T and variance var p(θ|X) [θ] ≈ (1.2e −4 , 8.2e −10 , 8.6e +4 , 4.53) T Schematic representation of the auto-regulatory gene expression model with a negative feedback loop. A gene is transcribed into mRNA which is translated to a protein that suppresses gene transcription. (Online version in colour.) Figure 4 : 4Data simulated from the single gene expression model using SSA. Figures (a), (b), for the linear model and Figures (c), (d), for the auto-regulatory model. Dots correspond to 10 independent draws for each time point. The bold line is the mean predicted by LNA with the true model parameters and the dashed lines are the + − 2× standard deviation predicted by LNA. Left column shows the mRNA molecules and right column the protein. (Online version in colour.) Figure 5 : 5Marginal joint posterior for parameters γ P , k P left panel, Figure (a), and γ R , b 3 right panel, Figure (b) for the single gene expression model. Dashed lines are the true values used to generate the synthetic data. Dots are samples from the posterior. Iso-contours and shaded region are obtained by kernel density estimation using posterior samples. (Online version in colour.) Figure 6 : 6Example trace plots from the auto-regulatory gene expression model for parameters γ P and k P . Red solid line denotes the true values. (Online version in colour.) approximation. Previous research on the linear noise approximation [15] has focussed on the Metropolis-Hastings sampler. Table 1 : 1Comparison of minimum Effective Sample Size (ESS) and time normalised min. ESS for different values of the system size parameter Ω of the decay dimerisation reaction model. Time normalised ESS is given in parenthesis. Results are calculated from 10,000 posterior samples.min. ESS vs. Ω Ω M.H. SMMALA RMHMC 1 121 (3.6) 150 (3.9) 245 (0.06) 2 226 (6.7) 2163 (57.2) 4775 (1.3) 5 132 (3.9) 3539 (93.6) 4618 (1.2) 10 180 (5.3) 3397 (89.8) 5954 (1.6) 100 214 (6.4) 3725 (98.5) 6066 (1.7) Auto-regulatory single gene expression model. Simplified Manifold MALA (A.R.= 0.71, ǫ = 1.17)True values 0.44 0.52 10.0 15.0 0.40 7.0 3.0 Metropolis-Hastings (A.R.) (0.28) (0.33) (0.30) (0.34) (0.29) (0.29) (0.34) (ǫ) (0.013) (0.007) (0.008) (0.022) (0.056) (0.007) (0.016) Mean 0.45 0.54 10.54 14.86 0.39 7.03 3.14 S.D. 0.017 0.017 0.336 0.509 0.029 0.056 0.149 ESS 42 34 34 149 117 58 44 ESS/time 1.42 1.15 1.15 5.05 3.96 1.96 1.49 Simplified Manifold MALA (A.R.= 0.79, ǫ = 1.05 ) Mean 0.45 0.54 10.57 14.88 0.39 7.04 3.17 S.D. 0.018 0.016 0.306 0.537 0.030 0.053 0.152 ESS 2891 2911 2958 2787 3310 3183 2878 ESS/time 83.79 84.37 85.73 80.78 95.94 92.26 83.42 Manifold HMC (A.R.= 0.84, ǫ = 0.91, L=5 ) Mean 0.46 0.54 10.57 14.95 0.39 7.04 3.18 S.D. 0.018 0.015 0.300 0.555 0.030 0.052 0.153 ESS 7731 8238 8304 7160 7380 7791 7950 ESS/time 0.52 0.55 0.56 0.48 0.49 0.52 0.53 Metropolis-Hastings (A.R.) (0.26) (0.36) (0.31) (0.33) (0.24) (0.30) (0.35) (ǫ) (0.028) (0.012) (0.016) (0.071) (0.231) (0.019) (0.029) Mean 0.4360 0.52 10.40 14.61 0.40 6.82 3.13 S.D. 0.016 0.018 0.424 1.089 0.076 0.090 0.142 ESS 201 71 73 465 339 420 239 ESS/time 6.12 2.16 2.22 14.17 10.33 12.80 7.28 Mean 0.43 0.52 10.44 14.24 0.38 6.82 3.12 S.D. 0.016 0.018 0.422 1.125 0.075 0.091 0.142 ESS 2990 3270 3454 3124 3164 3316 3195 ESS/time 76.86 84.06 88.79 80.30 81.33 85.24 82.13 Manifold HMC (A.R.= 0.82, ǫ = 0.91, L=5 ) Mean 0.43 0.52 10.43 14.52 0.40 6.82 3.13 S.D. 0.016 0.017 0.412 1.158 0.078 0.089 0.144 ESS 6532 6593 6614 5112 5384 6595 6642 ESS/time 0.41 0.41 0.41 0.32 0.34 0.41 0.42 Stochastic Simulation of Chemical Kinetics. D T Gillespie, DOI10.1146/annurev.physchem.58.032806.104637.)Annual Review of Physical Chemistry. 581Gillespie, D. T. 2007 Stochastic Simulation of Chemical Kinetics. Annual Review of Physical Chemistry 58 (1), 35-55, (DOI 10.1146/annurev.physchem.58.032806.104637.). Sensitivity, robustness, and identifiability in stochastic chemical kinetics models. M Komorowski, M J Costa, D A Rand, M P H Stumpf, DOI10.1073/pnas.1015814108.)Proceedings of the National Academy of Sciences. 10821Komorowski, M., Costa, M. J., Rand, D. A., and Stumpf, M. P. H. 2011 Sensitivity, robustness, and identifiability in stochastic chemical kinetics models. Proceedings of the National Academy of Sciences 108 (21), 8645-8650, (DOI 10.1073/pnas.1015814108.). Continuous-Time Markov Models for Species Interactions. M Spencer, E Susko, DOI10.1890/05-0029.)Ecology. 8612Spencer, M. and Susko, E. 2005 Continuous-Time Markov Models for Species Interactions. Ecology 86 (12), 3272-3278, (DOI 10.1890/05-0029.). Traffic models in broadband networks. A Adas, DOI10.1109/35.601746.)Communications Magazine. 357IEEEAdas, A. 1997 Traffic models in broadband networks. Communications Magazine, IEEE 35 (7), 82-89, (DOI 10.1109/35.601746.). A rigorous derivation of the chemical master equation. D T Gillespie, DOI10.1016/0378-4371(92)90283-V.Physica A: Statistical Mechanics and its Applications. 1881-3Gillespie, D. T. 2005, A rigorous derivation of the chemical master equation. Phys- ica A: Statistical Mechanics and its Applications 188 (1-3), 404-425, (DOI 10.1016/0378- 4371(92)90283-V.). Bayesian inference for a discretely observed stochastic kinetic model. R J Boys, D J Wilkinson, T B Kirkwood, DOI10.1007/s11222-007-9043-x.)Statistics and Computing. 186. Boys, R. J., Wilkinson, D. J., and Kirkwood, T. B. 2008 Bayesian inference for a dis- cretely observed stochastic kinetic model. Statistics and Computing 18, 125-135, (DOI 10.1007/s11222-007-9043-x.). N G Van Kampen, Stochastic Processes in Physics and Chemistry. North-HollandNetherlandsVan Kampen, N. G. 1992 Stochastic Processes in Physics and Chemistry, 3rd ed., Nether- lands: North-Holland. Inferring signaling pathway topologies from multiple perturbation measurements of specific biochemical species. T Xu, V Vyshemirsky, A Gormand, A Von Kriegsheim, M Girolami, G S Baillie, D Ketley, A J Dunlop, G Milligan, M D Houslay, W Kolch, Science Signaling. 3113ra20, (DOI 10.1126/scisignal.2000517.)Xu, T., Vyshemirsky, V., Gormand, A., von Kriegsheim, A., Girolami, M., Baillie, G. S., Ketley, D., Dunlop, A. J., Milligan, G., Houslay, M. D., and Kolch, W. 2010 Inferring sig- naling pathway topologies from multiple perturbation measurements of specific biochemical species. Science Signaling 3 (113), ra20, (DOI 10.1126/scisignal.2000517.). Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods. B Girolami, M , DOI10.1098/rsfs.2011.0051.)Interface Focus. 169. Calderhead, B. and Girolami, M. 2011 Statistical analysis of nonlinear dynamical sys- tems using differential geometric sampling methods. Interface Focus 1 (6), 821-835, (DOI 10.1098/rsfs.2011.0051.). The Diffusion Approximation for Markov Processes. N G Van Kampen, Thermodynamics & kinetics of biological processes. Lamprecht, I. and Zotin, A. I.New York, USAWalter de Gruyter & Co10. van Kampen, N. G. 1982 The Diffusion Approximation for Markov Processes. In Ther- modynamics & kinetics of biological processes (ed. Lamprecht, I. and Zotin, A. I.) New York, USA: Walter de Gruyter & Co.. The chemical Langevin equation. D T Gillespie, DOI10.1063/1.481811.)The Journal of Chemical Physics. 1131[11] 11. Gillespie, D. T. 2000 The chemical Langevin equation. The Journal of Chemical Physics 113 (1), 297-306, (DOI 10.1063/1.481811.). On Inference for Partially Observed Nonlinear Diffusion Models Using the Metropolis-Hastings Algorithm. G O Roberts, O Stramer, DOI10.1093/biomet/88.3.603.)Biometrika. 883Roberts, G. O. and Stramer, O. 2001 On Inference for Partially Observed Nonlinear Diffusion Models Using the Metropolis-Hastings Algorithm. Biometrika 88 (3), 603-621, (DOI 10.1093/biomet/88.3.603.). Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. A Golightly, D J Wilkinson, DOI10.1098/rsfs.2011.0047.)Interface Focus. 1613. Golightly, A. and Wilkinson, D. J. 2011 Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1 (6), 807-820, (DOI 10.1098/rsfs.2011.0047.). The Linear Noise Approximation is valid over limited times for any chemical system that is sufficiently large. E Wallace, D Gillespie, K Sanft, L Petzold, IET Systems Biology. To appear inWallace, E., Gillespie, D., Sanft, K., and Petzold, L. 2012 The Linear Noise Approxima- tion is valid over limited times for any chemical system that is sufficiently large. To appear in IET Systems Biology ?? (??), ??, Bayesian inference of biochemical kinetic parameters using the linear noise approximation. M Komorowski, B Finkenstadt, C Harper, Rand , D , DOI10.1186/1471-2105-10-343.)BMC Bioinformatics. 101Komorowski, M., Finkenstadt, B., Harper, C., and Rand, D. 2009 Bayesian inference of biochemical kinetic parameters using the linear noise approximation. BMC Bioinformatics 10 (1), 343, (DOI 10.1186/1471-2105-10-343.). . Christian Robert, P Casella, G , Robert, Christian, P. and Casella, G. 2005 . Monte Carlo Statistical Methods, Springer Texts in Statistics. Springer-VerlagMonte Carlo Statistical Methods, Springer Texts in Statistics, New York, USA: Springer-Verlag. Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms. G Roberts, O Gelman, A Gilks, W , R , DOI10.1214/aoap/1034625254.)The Annals of Applied Probability. 71Roberts, G., O., Gelman, A., and Gilks, W., R. 1997 Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms. The Annals of Applied Probability 7 (1), 110-120, (DOI 10.1214/aoap/1034625254.). Langevin Diffusions and Metropolis-Hastings Algorithms. G Roberts, O Stramer, O , 10.1023/A:1023562417138.)Methodology And Computing In Applied Probability. 44Roberts, G., O. and Stramer, O. 2003 Langevin Diffusions and Metropolis-Hastings Algorithms. Methodology And Computing In Applied Probability 4 (4), 337-358, (DOI doi:10.1023/A:1023562417138.). Hybrid Monte Carlo. S Duane, A Kennedy, B Pendleton, J , B Roweth, D , DOI10.1016/0370-2693(87)91197-X.)Physics Letters B. 1952Duane, S., Kennedy, A., B., Pendleton, J., B., and Roweth, D. 1987 Hybrid Monte Carlo. Physics Letters B 195 (2), 216-222, (DOI 10.1016/0370-2693(87)91197-X.). Probabilistic inference using Markov chain Monte Carlo methods. M Radford, N , CRG-TR-93-1Dept. of Computer Science, University of TorontoTechnical ReportRadford, M., N. 1993 Probabilistic inference using Markov chain Monte Carlo methods, Technical Report CRG-TR-93-1 Dept. of Computer Science, University of Toronto. Optimal Scaling of Discrete Approximations to Langevin Diffusions. G O Roberts, J S Rosenthal, DOI10.1111/1467-9868.00123.)Journal of the Royal Statistical Society. Series B (Statistical Methodology). 60121. Roberts, G. O. and Rosenthal, J. S. 1998 Optimal Scaling of Discrete Approxima- tions to Langevin Diffusions. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 60 (1), 255-268, (DOI 10.1111/1467-9868.00123.). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. M Girolami, B Calderhead, DOI10.1111/j.1467-9868.2010.00765.x.)Journal of the Royal Statistical Society: Series B (Statistical Methodology). 73222. Girolami, M. and Calderhead, B. 2011 Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Method- ology) 73 (2), 123-214, (DOI 10.1111/j.1467-9868.2010.00765.x.). A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter. L Ferm, P Lötstedt, A Hellander, DOI10.1007/s10915-007-9179-z.)Journal of Scientific Computing. 34Ferm, L., Lötstedt, P., and Hellander, A. 2008 A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter. Journal of Scientific Computing 34, 127-151, (DOI 10.1007/s10915-007-9179-z.). The Calculation of Posterior Distributions by Data Augmentation. M A Tanner, W H Wong, DOI10.2307/2289457.)Journal of the American Statistical Association. 82398Tanner, M. A. and Wong, W. H. 1987 The Calculation of Posterior Distributions by Data Augmentation. Journal of the American Statistical Association 82 (398), pp. 528-540, (DOI 10.2307/2289457.). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. P Green, J , DOI10.1093/biomet/82.4.711Biometrika. 824Green, P., J. 1995 Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82 (4), 711-732, (DOI 10.1093/biomet/82.4.711.). Deterministic Limit of Stochastic Chemical Kinetics. D T Gillespie, DOI10.1021/jp806431b.)The Journal of Physical Chemistry B. 1136Gillespie, D. T. 2009 Deterministic Limit of Stochastic Chemical Kinetics. The Journal of Physical Chemistry B 113 (6), 1640-1644, (DOI 10.1021/jp806431b.). Bayesian inference for dynamic transcriptional regulation; the Hes1 system as a case study. E Heron, A Finkenstädt, B Rand, D , A , DOI10.1093/bioinformatics/btm367.)Bioinformatics. 231927. Heron, E., A., Finkenstädt, B. and Rand, D., A. Bayesian inference for dynamic tran- scriptional regulation; the Hes1 system as a case study. Bioinformatics 23 (19), 2596-2603, (DOI 10.1093/bioinformatics/btm367.). Stochastic differential equations: an introduction with applications. B Oksendal, Springer-Verlag New York, IncNew York, NY, USA3rd ed.Oksendal, B. 1992 Stochastic differential equations: an introduction with applications, 3rd ed., New York, NY, USA: Springer-Verlag New York, Inc.. Quantification of mRNA using real-time RT-PCR. T Nolan, R E Hands, S A Bustin, DOI10.1038/nprot.2006.236.)Nature Protocols. 13[29] 29. Nolan, T., Hands, R. E., and Bustin, S. A., Quantification of mRNA using real-time RT-PCR. Nature Protocols 1 (3), 1559-1582, (DOI 10.1038/nprot.2006.236.). Componentwise adaptation for high dimensional MCMC. H Haario, E Saksman, E Tamminen, DOI10.1007/BF02789703.)Computational Statistics. 20Haario, H., Saksman, E., and Tamminen, E. 2005 Componentwise adaptation for high dimensional MCMC. Computational Statistics 20, 265-273, (DOI 10.1007/BF02789703.). A tutorial on adaptive MCMC. C Andrieu, J Thoms, DOI10.1007/s11222-008-9110-y.)Statistics and Computing. 18431. Andrieu, C. and Thoms, J. 2008 A tutorial on adaptive MCMC. Statistics and Com- puting 18 (4), 343-373, (DOI 10.1007/s11222-008-9110-y.). W Kühnel, Differential Geometry: Curves -Surfaces -Manifolds. USAAmerican Mathematical Society232. Kühnel, W. 2005 Differential Geometry: Curves -Surfaces -Manifolds, 2, Student Mathematical Library, USA: American Mathematical Society. An Adaptive Metropolis algorithm. H Haario, E Saksman, J Tamminen, DOI10.2307/3318737.)Bernoulli. 72Haario, H., Saksman, E., and Tamminen, J. 2001 An Adaptive Metropolis algorithm. Bernoulli 7 (2), 223-242, (DOI 10.2307/3318737.). Methods of Information Geometry. S.-I Amari, H Nagaoka, Translations of Mathematical Monographs. 191Oxford University PressAmari, S.-I. and Nagaoka, H. 2000 Methods of Information Geometry, 191, Translations of Mathematical Monographs, Oxford, UK: Oxford University Press.
[]
[ "New D(2, 1; α) Mechanics with Spin Variables", "New D(2, 1; α) Mechanics with Spin Variables" ]
[ "S Fedoruk [email protected] \nBogoliubov Laboratory of Theoretical Physics\nJINR\n141980Dubna, Moscow regionRussia\n", "E Ivanov [email protected] \nBogoliubov Laboratory of Theoretical Physics\nJINR\n141980Dubna, Moscow regionRussia\n", "O Lechtenfeld [email protected] \nInstitut für Theoretische Physik\nLeibniz Universität Hannover\nAppelstraße 2D-30167HannoverGermany\n" ]
[ "Bogoliubov Laboratory of Theoretical Physics\nJINR\n141980Dubna, Moscow regionRussia", "Bogoliubov Laboratory of Theoretical Physics\nJINR\n141980Dubna, Moscow regionRussia", "Institut für Theoretische Physik\nLeibniz Universität Hannover\nAppelstraße 2D-30167HannoverGermany" ]
[]
We elaborate on a novel superconformal mechanics model possessing D(2, 1; α) symmetry and involving extra U(2) spin variables. It is the one-particle case of the N =4 superconformal matrix model recently proposed in arXiv:0812.4276 [hep-th], and it generalizes to arbitrary α =0 the OSp(4|2) superconformal mechanics of arXiv:0905.4951 [hep-th]. As in the latter case, the U(2) spin variables are described by a Wess-Zumino action and define the first Hopf map S 3 → S 2 in the target space. Upon quantization, they represent a fuzzy sphere. We find the classical and quantum generators of the D(2, 1; α) superalgebra and their realization on the physical states. The super wavefunction encompasses various multiplets of the SU(2) R and SU(2) L subgroups of D(2, 1; α), with fixed isospins. The conformal potential is determined by the external magnetic field in the Wess-Zumino term, whose strength is quantized like in the OSp(4|2) case. As a byproduct, we reveal new invariant subspaces in the enveloping algebra of D(2, 1; α) for our quantum realization.
10.1007/jhep04(2010)129
[ "https://arxiv.org/pdf/0912.3508v2.pdf" ]
118,024,563
0912.3508
84002020e7e9d364e598379ee6841e14874d3d4d
New D(2, 1; α) Mechanics with Spin Variables 29 Apr 2010 S Fedoruk [email protected] Bogoliubov Laboratory of Theoretical Physics JINR 141980Dubna, Moscow regionRussia E Ivanov [email protected] Bogoliubov Laboratory of Theoretical Physics JINR 141980Dubna, Moscow regionRussia O Lechtenfeld [email protected] Institut für Theoretische Physik Leibniz Universität Hannover Appelstraße 2D-30167HannoverGermany New D(2, 1; α) Mechanics with Spin Variables 29 Apr 2010arXiv:0912.3508v2 [hep-th]0365-w0460Ds0470Bw1130Pb Keywords: Superconformal symmetrysuperfieldsblack holes We elaborate on a novel superconformal mechanics model possessing D(2, 1; α) symmetry and involving extra U(2) spin variables. It is the one-particle case of the N =4 superconformal matrix model recently proposed in arXiv:0812.4276 [hep-th], and it generalizes to arbitrary α =0 the OSp(4|2) superconformal mechanics of arXiv:0905.4951 [hep-th]. As in the latter case, the U(2) spin variables are described by a Wess-Zumino action and define the first Hopf map S 3 → S 2 in the target space. Upon quantization, they represent a fuzzy sphere. We find the classical and quantum generators of the D(2, 1; α) superalgebra and their realization on the physical states. The super wavefunction encompasses various multiplets of the SU(2) R and SU(2) L subgroups of D(2, 1; α), with fixed isospins. The conformal potential is determined by the external magnetic field in the Wess-Zumino term, whose strength is quantized like in the OSp(4|2) case. As a byproduct, we reveal new invariant subspaces in the enveloping algebra of D(2, 1; α) for our quantum realization. Introduction The interest in various models of N =4 superconformal mechanics is mainly caused by the possibility of using them for the description of supergravity black-hole solutions within the AdS/CFT correspondence, as was first suggested in [1]. In [2], we constructed a new N =4 superconformal matrix model with U(n) gauge symmetry. This model is described by the following harmonic superspace action, S = − 1 4(1+α) µ H Tr X −1/α + 1 2 µ (−2) A V 0 Z + Z + + i 2 c µ (−2) A Tr V ++ ,(1.1) where α is a real parameter which can take any non-zero value. The first term in (1.1) is the gauged action of the (1,4,3) multiplets which are described by hermitian (n×n)-matrix superfields X = (X b a ), a, b = 1, . . . , n. They are in the adjoint of U(n) and are subject to appropriate gauge-covariant constraints. These constraints involve the gauge connections which are expressed through the analytic harmonic gauge superfield V ++ (ζ, u) [3]. The third term in (1.1) is a Fayet-Iliopoulos (FI) term for V ++ and the real constant c is its strength. The second term in (1.1) is a Wess-Zumino (WZ) action describing n commuting analytic superfields Z + a which represent off-shell N =4 multiplets of type (4,4,0) and are in the fundamental of U(n). The superfield V 0 (ζ, u) is a real analytic gauge prepotential for the U(n) singlet (1,4,3) superfield X 0 ≡ Tr (X) . After passing to the WZ gauge, eliminating auxiliary degrees of freedom and fixing a gauge with respect to the residual gauge group, the model (1.1) involves n bosonic fields x a which are the first components of the diagonal superfields X a a (no sum over a), n 2 fermionic fields ψ b a which are the second components in the θ expansion of X b a , and the lowest commuting components of the superfields Z + a . The latter variables are described by Wess-Zumino-type d = 1 actions and parametrize n independent target spheres S 2 . Thus, they may be interpreted as target harmonic variables. After quantization, they become a sort of non-dynamical spin variables representing n "fuzzy" spheres. The model (1.1) is invariant under the most general N =4 superconformal symmetry D(2, 1; α) (with the more customary OSp(4|2) and SU(1, 1|2) symmetries as particular cases). It contains two SU(2) R-symmetry subgroups one of which acts only on fermions. In the case of D(2, 1; α=− 1 2 ) ≃ OSp(4|2), this model yields a new N =4 supersymmetric extension of the U(2) spin A n−1 Calogero system. Note that for α=−1 we have D(2, 1; α=−1) ≃ SU(1, 1|2)⊂ ×SU (2). It was argued in [4] that the large-n limit of the n-particle SU(1, 1|2) superconformal Calogero model provides a microscopic description of the extreme Reissner-Nordström (RN) black hole in the near-horizon limit. This hypothesis is based on the assertion that for a large number of particles and in a limit when all coordinates of the Calogero model, except for one, are treated as "small", the Calogero model reduces to the conformal mechanics for this "allocated" coordinate. For all values of α =−1/2 , the actions (1.1) yield non-trivial conformal sigma models in the bosonic limit. Therefore, the model (1.1) can hardly be utilized to describe a single black hole along the lines of [4]. Yet, it may be relevant to the multi-black-hole system, since the corresponding moduli spaces of n black holes in four-and five-dimensional supergravities are known to be described by sigma-model-type multi-black-hole quantum mechanics [5]. They become flat precisely in the case of OSp(4|2) superconformal symmetry, i.e. at α=−1/2. Note that the construction of a self-consistent n-body generalization of black-hole quantum mechanics is a rather complicated problem [5] beyond the one-and two-body cases. In order to have a normalizable ground state in the latter cases, one should apply a proper time redefinition, just as in conformal quantum mechanics [6]. If the general multi-black-hole quantum mechanics amounts to supersymmetric Calogero models, one can employ the powerful machinery developed for integrable super-Calogero systems (see e.g. [7,8,9,10]). In the present paper we investigate the n=1 case of the model (1.1), which describes the center-of-mass motion in the general super-Calogero model and, therefore, corresponds to a single black hole. The special case of α=−1/2, both on classical and quantum levels, was considered in detail in [11]. Here, we extend this consideration to all non-zero values of α. 1 We hope that an exhaustive understanding of the n=1 case will be helpful for attacking the quantum D(2, 1; α) model for arbitrary values of n. We use the standard notations of N =4, d=1 supersymmetric theories, following [13,14] and [11]. 2 Superfield setup The one-particle limit of the model (1.1) involves superfields corresponding to three off-shell N =4 supermultiplets: (i) the "radial" multiplet (1,4,3); (ii) the Wess-Zumino ("isospin") multiplet (4,4,0); and (iii) the gauge ("topological") multiplet. The total action has the form S = S X + S F I + S W Z . (2.1) The first term in (2.1) is the standard free action of the (1,4,3) multiplet (α = 0) S X = − 1 4(1+α) µ H X −1/α , (2.2) where the even real superfield X is subjected to the constraints D ++ X = 0 , (2.3) D + D − X = 0 ,D +D− X = 0 , (D +D− +D + D − ) X = 0 . (2.4) The set of conditions (2.3) and (2.4) is equivalent to the standard constraints D i D i X = 0, D iD i X = 0, [D i ,D i ] X = 0 for the superfield X living in the "central basis N =4 superspace" parametrized by the coordinates θ i ,θ i and t. Note that the action (2.2) is in fact non-singular at α = −1 . Indeed, making use of the fact that µ H X is an integral of total derivative, we cast the action (2.2) in the equivalent form S X = − 1 4(1+α) µ H X −1/α − X . Thus in the limit α = −1 we obtain the standard action S X α=−1 = − 1 4 µ H X lnX ,(2.5) The action (2.2) is not defined at α=0, and this special case needs a separate analysis (see Section 5). In what follows we always assume that α = 0 . The second term in (2.1) is FI term S F I = i 2 c µ (−2) A V ++ (2.6) for the gauge supermultiplet. The even analytic gauge superfield V ++ (ζ, u), D + V ++ = 0, D + V ++ = 0 , is subjected to the gauge transformations V ++′ = V ++ − D ++ λ, λ = λ(ζ, u) ,(2.7) which are capable to gauge away, locally, all the components from V ++ . However, the latter contains a component which cannot be gauged away globally. This is the reason why this d = 1 supermultiplet was called "topological" in [3]. Last term in (2.1) is Wess-Zumino (WZ) term S W Z = 1 2 b µ (−2) A V Z + Z + . (2.8) Here, the complex analytic superfield Z + , Z + (D + Z + =D + Z + = 0) , is subjected to the harmonic constraints D ++ Z + ≡ (D ++ + i V ++ ) Z + = 0 , D ++ Z + ≡ (D ++ − i V ++ ) Z + = 0 (2.9) and describes a gauge-covariantized version of the N =4 multiplet (4,4,0). The relevant gauge transformations are Z +′ = e iλ Z + , Z +′ = e −iλ Z + . (2.10) We explicitly included a coupling constant b in (2.8) in order to track the contribution of WZ term to the full component action. Afterwards, this constant will be put equal to 1. The superfield V(ζ, u) in (2.8) is a real analytic gauge superfield (D + V =D + V = 0), which is a prepotential solving the constraints (2.3) and (2.4) for X. It is related to the superfield X in the central basis by the harmonic integral transform [15] X(t, θ i ,θ i ) = du V t A , θ + ,θ + , u ± θ ± =θ i u ± i ,θ ± =θ i u ± i . (2.11) The unconstrained analytic prepotential V possesses its own pregauge freedom δV = D ++ λ −− , λ −− = λ −− (ζ, u) ,(2.12) which can be exploited to show that V describes just the multiplet (1, 4, 3) (after choosing the appropriate Wess-Zumino gauge) [15]. The coupling to the multiplet (1, 4, 3) in (2.8) is introduced for ensuring superconformal invariance. We shall see that, upon passing to components, it gives rise to non-trivial interactions for the physical fields. The invariance of (2.8) under (2.12) is ensured by the constraints (2.9). Besides the gauge U(1) symmetry (2.7), (2.10) and pregauge symmetry (2.12), the action (2.1) respects the rigid N =4 superconformal symmetry D(2, 1; α) . All superconformal transformations are contained in the closure of the supertranslations and superconformal boosts. Invariance of the action (2.1) under the supertranslations (ε i = (ε i )) δt = i(θ kε k − ε kθ k ), δθ k = ε k , δθ k =ε k is automatic because we use the N =4 superfield approach. The coordinate realization of the superconformal boosts of D(2, 1; α) [14,3] is as follows (η i = (η i )): δ ′ t = it(θ kη k +θ k η k ) + (1 + α) θ iθ i (θ kη k +θ k η k ) , (2.13) δ ′ θ i = η i t − 2iα θ i (θ kη k ) + 2i(1 + α) θ i (θ k η k ) − i(1 + 2α) η i (θ kθ k ) , (2.14) δ ′θi =η i t − 2iαθ i (θ k η k ) + 2i(1 + α)θ i (θ kη k ) + i(1 + 2α)η i (θ kθ k ) , (2.15) δ ′ t A = α −1 Λt A , δ ′ u + i = Λ ++ u − i , (2.16) δ ′ θ + = η + t A + 2i(1 + α)η − θ +θ+ , δ ′θ+ =η + t A + 2i(1 + α)η − θ +θ+ , (2.17) δ ′ (dtd 4 θ) = −α −1 (dtd 4 θ) Λ 0 , δ ′ µ H = µ H 2Λ − α −1 (1 + α)Λ 0 , δ ′ µ (−2) A = 0 , (2.18) where Λ =Λ = 2iα(η − θ + − η −θ+ ) , Λ ++ = D ++ Λ = 2iα(η + θ + − η +θ+ ) , D ++ Λ ++ = 0 , (2.19) Λ 0 = 2Λ − D −− Λ ++ = 2iα(θ kη k +θ k η k ) , D ++ Λ 0 = 0 . (2.20) Taking the field transformations in the form (here we use the "passive" interpretation of them) δ ′ X = −Λ 0 X , δ ′ V = −2Λ V , δ ′ Z + = Λ Z + , δ ′ V ++ = 0 ,(2.X = x + θ − ψ + +θ −ψ+ − θ + ψ − −θ +ψ− + θ −θ− N ++ + θ +θ+ N −− + (θ −θ+ + θ +θ− )N + θ − θ +θ− Ω + +θ −θ+ θ −Ω+ + θ −θ− θ +θ+ D . (3.1) Here N ±± = N ik u ± i u ± k , N = iẋ − N ik u + i u − k , D = 2ẍ + 2iṄ ik u + i u − k , (3.2) ψ ± = ψ i u ± i ,ψ ± =ψ i u ± i , Ω + = 2iψ + ,Ω + = −2i˙ψ + (3.3) and x(t A ), N ik = N (ik) (t A ), ψ i (t A ),ψ i (t A ) = (ψ i ) are d=1 fields. Inserting (3.1) in (2.2) we obtain S X = 1 4α 2 dt x − 1 α −2 ẋẋ − i ψ kψ k −˙ψ k ψ k − 1 2 N ik N ik (3.4) − 1 4α 2 ( 1 α + 2) dt x − 1 α −3 N ik ψ (iψk) − 1 12α 2 ( 1 α + 2)( 1 α + 3) dt x − 1 α −4 ψ iψk ψ (iψk) . In the central basis the θ expansion (3.1) takes the form: X(t, θ i ,θ i ) = x + θ i ψ i +ψ iθ i + θ iθk N ik + i 2 (θ) 2ψ iθ i + i 2 (θ) 2 θ iψ i + 1 4 (θ) 2 (θ) 2ẍ , (3.5) where (θ) 2 ≡ θ i θ i = −2θ + θ − , (θ) 2 ≡θ iθ i = 2θ +θ− . Then, from (2.11) we can identify the fields appearing in the WZ gauge for V with the fields in (3.5) V(t A , θ + ,θ + , u ± ) = x(t A ) − 2 θ + ψ i (t A )u − i − 2θ +ψi (t A )u − i + 3 θ +θ+ N ik (t A )u − i u − k . (3.6) This expansion will be used to express the action (2.8) in terms of the component fields. FI and WZ actions Using the U(1) gauge freedom (2.7), (2.10) we can choose WZ gauge V ++ = −2i θ +θ+ A(t A ) . (3.7) Then S F I = c dt A . (3.8) The solution of the constraint (2.9) in WZ gauge (3.7) is Z + = z i u + i + θ + ϕ +θ + φ + 2i θ +θ+ ∇ t A z i u − i , Z + =z i u +i + θ +φ −θ +φ + 2i θ +θ+ ∇ t Az i u −i where ∇z k =ż k + iA z k , ∇z k =ż k − iAz k . (3.9) In (3.9), z i (t A ) and ϕ(t A ), φ(t A ) are d=1 fields, bosonic and fermionic, respectively. The fields z i form a complex doublet of the R-symmetry SU(2) group, while the fermionic fields are singlets of the latter. Another ("mirror") R-symmetry SU (2) is not manifest in the present approach: the bosonic fields are its singlets, while the fermionic fields form a doublet with respect to it. Inserting expressions (3.9) and (3.6) in the action (2.8) and performing integration over θ s and harmonics there, we obtain a component form of the WZ action S W Z = i 2 b dt z k ∇z k − ∇z k z k x − 1 2 b dt N ikz i z k (3.10) + 1 2 b dt ψ k (φ z k +z k φ) +ψ k φ z k −z k ϕ − x φ φ +φ ϕ . The fermionic fields φ, ϕ are auxiliary. The action is invariant under the residual local U(1) transformations A ′ = A −λ 0 , z i′ = e iλ 0 z i ,z i ′ = e −iλ 0z i (3.11) (and similar phase transformations of the fermionic fields). The total component action is a sum of (3.4), (??) and (3.10). Eliminating the auxiliary fields N ik , φ,φ, ϕ,φ, from this sum by their algebraic equations of motion, N ik = −2bα 2 x 1 α +2 z (izk) − ( 1 α + 2) x −1 ψ (iψk) ,(3.12) φ = −ψ k z k x ,φ = ψ kz k x , ϕ = − ψ k z k x ,φ = −ψ kz k x , (3.13) and making the redefinition x ′ = x − 1 2α , ψ ′ k = − 1 2α x − 1 2α −1 ψ k , z ′i = x 1/2 z i , (3.14) we obtain the on-shell form of the action (2.1) in WZ gauge (we omitted the primes on x, ψ and z) S = S b + S f , (3.15) S b = dt ẋẋ + i 2 b z kż k −ż k z k − b 2 α 2 (z k z k ) 2 4x 2 − A bz k z k − c , (3.16) S f = −i dt ψ kψ k −˙ψ k ψ k + 2bα dt ψ iψk z (izk) x 2 + 2 3 (1 + 2α) dt ψ iψk ψ (iψk) x 2 . (3.17) It is still invariant under the gauge transformations (3.11). The d=1 gauge connection A(t) in (3.16) is the Lagrange multiplier for the constraint It is convenient to fully fix the residual gauge freedom by choosing the phases of z 1 and z 2 opposite to each other. In this gauge, the constraint (3.18) is solved by z k z k = c .z 1 = κ cos γ 2 e iβ/2 , z 2 = κ sin γ 2 e −iβ/2 , κ 2 = c . (3.19) In terms of the newly introduced fields the bosonic action (3.16) takes the form 2 S b = dt ẋẋ − α 2 c 2 4 x 2 − c 2 cos γβ . (3.20) As argued in Section 5, this action can be relevant to describing some particular orbits near horizon of the extreme D=5 black holes. The spinor z k provides a parametrization of the angular part of the set of the horizon coordinates. Unconstrained fields in the action (3.15), three bosons x, γ, β and four fermions ψ k ,ψ k , constitute some on-shell supermultiplet with three bosonic and four fermionic fields. As opposed to the off-shell (3,4,1) supermultiplet considered in [13,16,17] the action (3.16) contains "true" kinetic term only for one bosonic component x which also possesses the conformal potential, whereas two other fields parametrizing the coset SU(2) R /U(1) R are described by a WZ term. Taken separately, the WZ term provides an example of Chern-Simons mechanics [18,19,20,21,22]. The variables γ(t) and β(t) (or z k andz k in the manifestly SU(2) covariant formulation) become (iso)spin degrees of freedom (target SU(2) harmonics) upon quantization. The realization of the D(2, 1; α) superconformal transformations on these fields will be given in the next Section. It should be stressed that the considered model realizes a new mechanism of generating conformal potential ∼ 1/x 2 for the field x(t). Before eliminating auxiliary fields, the component action contains no explicit term of this kind. It arises as a result of varying with respect to the Lagrange multiplier A(t) and making use of the arising constraint (3.18). As we shall see, in quantum theory this new mechanism entails a quantization of the constant c . The naive inspection of the bosonic action (3.16) could lead to the conclusion that angular variables completely decouple from a radial variable, and, hence, are superfluous. Moreover, the classical dynamics associated with the WZ term in (3.16) is trivial. However, like in other Chern-Simons-type theories, this term has a non-trivial impact on the quantum properties of the model. Indeed, as we shall see in the quantum case, owing to the non-trivial geometry of the angular space the quantum state vectors necessarily carry quantum numbers of the SU(2) spin. Though in the bosonic limit this symmetry is purely internal (it commutes with the d = 1 conformal group SL(2, R)), the presence of angular variables leads to the property that the wave function encompasses non-trivial SU(2) multiplets. 3 In the supersymmetry case, when the full action (3.15) is considered, the situation becomes even more involved. Now this SU(2) symmetry in addition acts on fermions in parallel with the second SU(2) R-symmetry which from the very beginning is realized only on fermionic fields, and these either SU(2) are an essential part of the superconformal group. Examining the action (3.15), we were not able to find any change of variables which would decouple the angular variables from other ones. Actually, we already observed the same phenomenon in the particular OSp(4|2) case [11]. Now we see that it persists at any choice of the parameter α in D(2, 1; α) . Even at the classical level, the WZ term yields, e.g., a non-trivial additional contribution to the fermionic equations of motion (coming from the term proportional to b in (3.17)). Although in the β, γ parametrization both γ andβ can be expressed through fermions and some integration constant by their classical equations of motion, an essential trace of the WZ couplings still remains in the equations of motion for fermions, producing a mass term for them and modifying the coefficients before the third order terms. 4 The Hamiltonian, N =4 supercharges and other D(2, 1; α) generators also involve important new pieces caused by the WZ term and additional fermionic couplings associated with it (see below). N =4 superconformal symmetry in WZ gauge The transformations and their generators look most transparent in terms of the SU(2) doublet quantities z k andz k . To determine the superconformal transformations of component fields, we should know the appropriate compensating gauge transformations needed to preserve the WZ gauge (3.7). For supertranslations and superconformal boosts the parameter of the compensating gauge transformations is as follows λ = 2i (θ +ε− −θ + ε − ) + t A (θ +η− −θ + η − ) A (3.21) where ε − := ε i u − i , η − := η i u − i . (3.22) 3 In the bosonic case, in accord with the general concept of separating variables, one can postulate that the wave function is a product of the chargeless conformal mechanics wave function by the lowest Landau level wave function associated with the SU(2) WZ term. No such a separation is possible in the generic superconformal case due to the presence of fermions interacting with both types of bosonic variables. 4 We thank S. Krivonos for a discussion on this issue. Taking this into account, we obtain the relevant infinitesimal D(2, 1; α) transformations which leave the action (3.15) invariant (as in (3.15) we omit 'primes' on the newly introduced variables): δx = −ω i ψ i +ω iψ i , (3.23) δψ i = iη i x − iω iẋ − αω k z (izk) x − (1 + 2α)ω k ψ kψi + ω k ψ k ψ i x , (3.24) δψ i = −iη i x + iω iẋ − α ω k z (izk) x + (1 + 2α) ω kψ k ψ i +ω kψ kψi x , (3.25) δz i = −2α ω (i ψ k) +ω (iψk) x z k , δz i = 2α ω (i ψ k) +ω (iψk) xz k , (3.26) δA = 0 ,(3.27) where ω i = ε i + t η i andω i =ε i + tη i . Note that the closure of d=1 Poincarè supersymmetry transformations is a sum of the time translations and residual U(1) gauge transformation with a field-dependent parameter. Such a sum turns out to vanish for the d=1 gauge field A. In Appendix this specifically d=1 phenomenon is expounded on a simple example of toy N =2 supersymmetric model. Now, using the Nöther procedure, we can directly find the classical generators of the supertranslations Q i = p ψ i + 2iα z (izk) ψ k x + i(1 + 2α) ψ k ψ kψi x , (3.28) Q i = pψ i − 2iα z (izk)ψ k x + i(1 + 2α)ψ kψ k ψ i x , (3.29) where p ≡ 2ẋ, as well as of the superconformal boosts: S i = −2 xψ i + t Q i ,S i = −2 xψ i + tQ i . (3.30) The remaining (even) generators of the supergroup D(2, 1; α) can be found by evaluating mutual anticommutators of the odd generators. As follows from the action (3.15), the SU(2) spinor variables are canonically self-conjugate due to the presence of second-class constraints for their momenta. As a result, non-vanishing canonical Dirac brackets (at equal times) have the following form [x, p] D = 1, [z i ,z j ] D = −iδ i j , {ψ ii ′ , ψ kk ′ } D = i 2 ǫ ik ǫ i ′ k ′ {ψ i ,ψ j } D = i 2 δ i j (3.31) where we introduced the notations ψ ii ′ = (ψ i1 ′ , ψ i2 ′ ) = (ψ i ,ψ i ), (ψ ii ′ ) = ψ ii ′ = ǫ ik ǫ i ′ k ′ ψ kk ′ , (ǫ 12 = ǫ 21 = 1). (3.32) Using Dirac brackets (3.31), we arrive at the following closed superalgebra: {Q ai ′ i , Q bk ′ k } D = 2i ǫ ik ǫ i ′ k ′ T ab + αǫ ab ǫ i ′ k ′ J ik − (1 + α)ǫ ab ǫ ik I i ′ k ′ , (3.33) [T ab , T cd ] D = −ǫ ac T bd − ǫ bd T ac , (3.34) [J ij , J kl ] D = −ǫ ik J jl − ǫ jl J ik , [I i ′ j ′ , I k ′ l ′ ] D = −ǫ ik I j ′ l ′ − ǫ j ′ l ′ I i ′ k ′ , (3.35) [T ab , Q ci ′ i ] D = ǫ c(a Q b)i ′ i , [J ij , Q ai ′ k ] D = ǫ k(i Q ai ′ j) , [J i ′ j ′ , Q ak ′ i ] D = ǫ k ′ (i ′ Q aj ′ )i . (3.36) In (3.33)-(3.36) we use the notation Q 21 ′ i = −Q i , Q 22 ′ i = −Q i , Q 11 ′ i = S i , Q 12 ′ i =S i , (3.37) T 22 = H , T 11 = K , T 12 = −D . (3.38) The explicit expressions for the generators are H = 1 4 p 2 + α 2 (z k z k ) 2 4x 2 − 2α ψ iψk z (izk) x 2 − (1 + 2α) ψ i ψ iψkψ k 2x 2 , (3.39) K = x 2 − t xp + t 2 H , (3.40) D = − 1 2 xp + t H , (3.41) J ij = i z (izj) + ψ ik ′ ψ j k ′ = i z (izk) + 2ψ (iψ k) , (3.42) I i ′ j ′ = iψ ki ′ ψ k j ′ I 1 ′ 1 ′ = −iψ k ψ k , I 2 ′ 2 ′ = iψ kψ k , I 1 ′ 2 ′ = −iψ kψ k . (3.43) The relations (3.33)-(3.36) provide the standard form of the superalgebra D(2, 1; α) (see, e.g., [23,24,16]). Bosonic generators T ab = T ba , J ik = J ki , I i ′ k ′ = I k ′ i ′ form mutually commut- ing su(1, 1) , su(2) R and su(2) L algebras, respectively. 5 It is worth pointing out one important feature of the basic relation {Q i ,Q j } D = 2iHδ i j . Although Q andQ contain terms of the third order in ψ with the coefficients (1 + 2α), no quartic fermionic term ∼ (1+2α) 2 appears in the Hamiltonian. This is because of the vanishing Dirac bracket {ψ k ψ kψi ,ψ lψ l ψ j } D = 0 . (3.44) The expression (3.39) coincides with the canonical Hamiltonian associated with the action (3.15). Owing to the A-term in (3.15), there is also the first-class constraint D 0 − c ≡z k z k − c ≈ 0 , (3.45) which should be imposed on the wave functions in quantum case. Casimir operators (on classical level) of the su(1, 1), su(2) R and su(2) L algebras are T 2 ≡ 1 2 T ab T ab = HK − D 2 = 1 4 α 2 (z kz k ) 2 − 2α z (izk) ψ (iψk) − 1 2 (1 + 2α)ψ i ψ iψkψ k ,(3.46) J 2 ≡ 1 2 J ik J ik = 1 4 (z kz k ) 2 − 2z (izk) ψ (iψk) − 3 2 ψ i ψ iψkψ k ,(3.47)I 2 ≡ 1 2 I i ′ k ′ I i ′ k ′ = IĪ − (I 3 ) 2 = 3 2 ψ i ψ iψkψ k . (3.48) Using these expressions and i 4 Q ai ′ i Q ai ′ i = i 2 (Q iS i − S iQ i ) = 4α z (izk) ψ (iψk) + 2(1 + 2α) ψ i ψ iψkψ k , (3.49) we obtain that the second-order (classical) Casimir operator of D(2, 1; α) , C 2 = T 2 + αJ 2 − (1 + α)I 2 + i 4 Q ai ′ i Q ai ′ i ,(3.50) takes the form C 2 = 1 4 α(α + 1) (z kz k ) 2 = 1 4 α(α + 1) (D 0 ) 2 . (3.51) It is important to note that the (iso)spin (angular) variables make significant contributions to D(2, 1; α), su(1, 1) and su(2) R Casimirs (3.46), (3.47), (3.51). Additional terms in these operators are generated by the second and third terms in the Hamiltonian (3.39) and the first terms in the generators (3.42), all arising from the terms ∝ b in the actions (3.16) and (3.17). By inspecting the expressions (3.46)-(3.49), we observe that the following quantity M vanishes identically for this particular realization of the D(2, 1; α) superalgebra: M ≡ T 2 − α 2 J 2 − 1 3 (1 − α 2 ) I 2 + i 8 (1 − α) Q ai ′ i Q ai ′ i = 0 . (3.52) Using this identity together with the expression (3.50), we obtain the constraint (α + 1) T 2 − αJ 2 − 1 3 (α − 1)I 2 − (α − 1)C 2 = 0 , (3.53) which relates the Casimir of D(2, 1; α) to the Casimirs of the three mutually commuting bosonic subgroups SU(1,1), SU(2) L and SU(2) R in our model. Plugging the expression (3.51) for the D(2, 1; α) Casimir in this constraint, we find that (α + 1) T 2 − αJ 2 − 1 3 (α − 1)I 2 − 1 4 α(α − 1) (D 0 ) 2 = 0 . (3.54) Using the expressions (3.46)-(3.48), we can check that the term in the square brackets is vanishing, that is the expression T 2 = αJ 2 + 1 3 (α − 1)I 2 + 1 4 α(α − 1) (D 0 ) 2 (3.55) is valid for all α = 0 , including α= − 1 . Note that the Hamiltonian (3.39) has the standard form of the Hamiltonian of (super)conformal mechanics 6 H = 1 4 p 2 + T 2 x 2 . (3.56) Using the expression (3.55), we can represent the Hamiltonian in the convenient equivalent form H = 1 4 p 2 + α(α − 1) (D 0 ) 2 4x 2 + α J 2 x 2 + (α − 1) I 2 3x 2 . (3.57) The last two terms involve the Casimirs of the groups SU(2) R and SU(2) L . The second term contains the quantity D 0 =z k z k which is the generator of some extra U(1) commuting with D(2, 1; α) . It is worth pointing out that at α=−1, when D(2, 1; α) degenerates into SU(1, 1|2)⊂ ×SU(2) L , the SU(2) L Casimir I 2 drops out from the expression (3.50) for the Casimir C 2 , as it should be. However, since in the model under consideration this SU(2) L is realized only on fermions, the Casimir I 2 reappears in the subsequent formulas from the term i 4 Q ai ′ i Q ai ′ i . Hence, even for a fixed D(2, 1; α) Casimir (3.50), the term i 4 Q ai ′ i Q ai ′ i makes a contribution ∼ I 2 to the SU(1, 1) Casimir (3.55). As a result, the term with the SU(2) L Casimir I 2 is retained in (3.57) even at α= − 1 . Incidentally, the simplest form of the Hamiltonian is achieved at α=1 . In the next section we shall construct a quantum realization of the D(2, 1; α) superalgebra. 6 From H = 1 4 (p 2 + g x 2 ) and the expressions (3.40), (3.41) we obtain T 2 = g/4. [X, P ] = i , [Z i ,Z j ] = δ i j , {Ψ i ,Ψ j } = − 1 2 δ i j . (4.1) Quantum supertranslation and superconformal boost generators are defined by the classical expressions (3.28), (3.29), (3.30). We take Weyl ordering of the fermionic quantities in the last terms of (3.28) and (3.29): Q i = P Ψ i + 2iα Z (iZ k) Ψ k X + i(1 + 2α) Ψ k Ψ kΨi X , (4.2) Q i = PΨ i − 2iα Z (iZk)Ψ k X + i(1 + 2α) Ψ kΨ k Ψ i X , (4.3) S i = −2 XΨ i + t Q i ,S i = −2 XΨ i + tQ i . (4.4) The symbol ... denotes Weyl ordering. Note that Ψ k Ψ kΨi = Ψ k Ψ kΨi + 1 2 Ψ i , Ψ kΨ k Ψ i =Ψ kΨ k Ψ i + 1 2Ψ i andQ i = − (Q i ) + ,S i = − (S i ) + . Evaluating the anticommutators of the odd generators (4.2), (4.4), one determines uniquely the full set of quantum generators of superconformal algebra D(2, 1; α). We obtain 7 H = 1 4 P 2 + α 2 (Z k Z k ) 2 + 2Z k Z k 4X 2 − 2α Z (iZ k) Ψ (iΨk) X 2 (4.6) − (1 + 2α) Ψ i Ψ iΨkΨ k 2X 2 + (1 + 2α) 2 16X 2 , K = X 2 − t 1 2 {X, P } + t 2 H , (4.7) D = − 1 4 {X, P } + t H , (4.8) J ik = i Z (iZ k) + 2Ψ (iΨk) ,(4.9)I 1 ′ 1 ′ = −iΨ k Ψ k , I 2 ′ 2 ′ = iΨ kΨ k , I 1 ′ 2 ′ = − i 2 [Ψ k ,Ψ k ] . (4.10) Note that Ψ i Ψ iΨkΨ k = 1 2 Ψ i Ψ i ,Ψ kΨ k − 1 4 = Ψ i Ψ iΨkΨ k − Ψ iΨ i + 1 4 , 7 It is worth making here an important clarifying remark which refers as well to our previous paper [11]. In (4.1) and below we assign to quantum operators the following Hermitian conjugation properties X + = X, P + = P ,Z i = − Z i + ,Ψ i = − Ψ i + ,(4.5) whereas for classical quantities we still havez i = (z i ),ψ i = (ψ i ). This change of conventions in the quantum case is necessary for ensuring the standard Clifford algebra for quantum fermionic operators and standard quantum supersymmetry algebra with the positive-definte right-hand side of the basic anticommutator (see the comments after (4.11)-(4.16)). As we show in Appendix B, the standard conjugation conventions can be restored by performing the time reversal t → −t in the initial model, thus bringing the opposite (standard) sign to kinetic terms of all involved d=1 spinor fields. Ψ i Ψ l Ψ lΨkΨ k = − Ψ l Ψ lΨkΨ k Ψ i = 1 2 Ψ l Ψ lΨi , Ψ i Ψ l Ψ lΨkΨ k = − Ψ l Ψ lΨkΨ k Ψ i = 1 2 Ψ kΨ k Ψ i . It can be directly checked that the generators (4.2)-(4.10) indeed form the D(2, 1; α) superalgebra which is obtained from the DB superalgebra (3.33)- (3.36) in the standard fashion (changing altogether DB by (anti)commutators and multiplying the right-hand sides by i): {Q ai ′ i , Q bk ′ k } = −2 ǫ ik ǫ i ′ k ′ T ab + αǫ ab ǫ i ′ k ′ J ik − (1 + α)ǫ ab ǫ ik I i ′ k ′ ,(4.11) [T ab , T cd ] = −i ǫ ac T bd + ǫ bd T ac , (4.12) [J ij , J kl ] = −i ǫ ik J jl + ǫ jl J ik , [I i ′ j ′ , I k ′ l ′ ] = −i(ǫ ik I j ′ l ′ + ǫ j ′ l ′ I i ′ k ′ ) , (4.13) [T ab , Q ci ′ i ] = iǫ c(a Q b)i ′ i , [J ij , Q ai ′ k ] = iǫ k(i Q ai ′ j) , [J i ′ j ′ , Q ak ′ i ] = iǫ k ′ (i ′ Q aj ′ )i . (4.14) As in (3.33)-(3.36), in (4.11)-(4.14) we use the notation Note that due to (4.5) we have Q 21 ′ i = −Q i , Q 22 ′ i = −Q i , Q 11 ′ i = S i , Q 12 ′ i =S i ,(4.Q ai ′ i + = −ǫ ik ǫ i ′ k ′ Q ak ′ k (4.17) and, as a result, the basic anticommutator has the standard form {Q, Q + } = H . In the quantum case, the classical relation (3.44) is replaced by { Ψ k Ψ kΨi , Ψ lΨ l Ψ j } = 1 8 δ i j . (4.18) and, due to (4.18), the term (1+2α) 2 16X 2 appears in the quantum Hamiltonian (4.6). This term is necessary also for preserving the basic supersymmetry relations [H, Q] = [H,Q] = 0. The appearance of such a "conformal" term when quantizing N =4 superconformal systems was earlier observed in [10]. The quantization of the pure bosonic limit (3.16) of the classical system (3.15) does not lead to appearance of the additional term (1+2α) 2 16X 2 in the corresponding quantum Hamiltonian which is thus a sum of only first two terms in (4.6). Using the same procedure as in [11] this Hamiltonian can be represented in the form H = 1 4 P 2 + 4α 2 Y a Y a X 2 ,(4.19) where Y a = 1 2Z i (σ a ) i j Z j (4.20) and σ a , a = 1, 2, 3 are Pauli matrices. The quantities Y a , obtained via the first Hopf map from the SU(2) spinors Z i ,Z i , generate SU(2) R transformations in the bosonic sector of the model (the second SU(2) L R-symmetry group of D(2, 1; α) acts in the fermionic sector only). The operator Y a Y a in the second term of (4.19) is the Casimir operator of the group SU(2) R for its realization in the bosonic sector. Due to the constraint (3.45) (for definiteness, we adoptZ k Z kordering in it; see also (4.58) and (4.65)), this Casimir takes the definite value c 2 c 2 + 1 . Thus, in the pure bosonic limit our model describes a conformal particle with the quantum potential α 2 c 2 c 2 + 1 /X 2 which possesses the fixed SU(2) R spin c 2 . In the entire supersymmetric model, with all fermions taken into account, the generators of SU(2) R contain additional fermionic parts (see (4.9)) and the corresponding full SU(2) R Casimir operator proves not to be fixed. A thorough consideration of the pure bosonic case of the α = −1/2 model can be found in our paper [11]. The second-order Casimir operator of the whole supergroup D(2, 1; α) is given by the following expression [27] C 2 = T 2 + α J 2 − (1 + α) I 2 + i 4 Q ai ′ i Q ai ′ i . (4.21) Using the relations T 2 ≡ 1 2 T ab T ab = 1 2 {H, K} − D 2 = 1 4 α 2 (Z k Z k ) 2 + 2Z k Z k − 2αZ (iZ k) Ψ (iΨk) (4.22) − 1 2 (1 + 2α) Ψ i Ψ iΨkΨ k + 1 16 (1 + 2α) 2 − 3 16 , J 2 ≡ 1 2 J ik J ik = 1 4 (Z k Z k ) 2 + 2Z k Z k − 3 2 Ψ i Ψ iΨkΨ k − Ψ iΨ i − 2Z (iZ k) Ψ (iΨk) ,(4.23) I 2 ≡ 1 2 I i ′ k ′ I i ′ k ′ = 1 2 {Ī, I} − (I 3 ) 2 = 3 2 Ψ i Ψ iΨkΨ k − Ψ iΨ i + 3 4 (4.24) together with i 4 Q ai ′ i Q ai ′ i = i 4 [Q i ,S i ] + i 4 [Q i , S i ] (4.25) = 4αZ (iZ k) Ψ (iΨk) + 2(1 + 2α) Ψ i Ψ iΨkΨ k − Ψ iΨ i + (1 + α) , we finally cast C 2 in the form C 2 = 1 4 α(1 + α) (Z k Z k ) 2 + 2Z k Z k + 1 . Invariant spaces in the enveloping algebra of D(2, 1; α) An important property is that the enveloping algebra of D(2, 1; α) superalgebra has several subspaces which are closed under the action of D(2, 1; α). The presence of such subspaces provides an explanation why some bilinear combinations of the D(2, 1; α) generators in the considered realization identically vanish without conflict with the D(2, 1; α) covariance. This phenomenon is encountered already at the classical level (see (3.52)). As we shall see, the realization of the D(2, 1; α) generators in the considered model is such that the operators forming one of the invariant subspaces just mentioned are vanishing. As a result, the physical states form a module of such a restricted representation of D(2, 1; α). One invariant subspace is formed by the bilinear combinations M ≡ T 2 − α 2 J 2 − 1 3 (1 − α 2 ) I 2 + i 8 (1 − α) Q ai ′ i Q ai ′ i , (4.27) M ai ′ i ≡ i 4 {T a b , Q bi ′ i } − α {J i j , Q ai ′ j } + 1 3 (1 − α) {I i ′ j ′ , Q aj ′ i } , (4.28) M ik, i ′ k ′ ≡ α {J ik , I i ′ k ′ } − i 2 Q b ( i ′ (i Q b k ′) k) ,(4.29)M ac, i ′ k ′ ≡ {T ac , I i ′ k ′ } − i 2 Q ( a(i ′ j Q c ) k ′ ) j , (4.30) M ai, i ′ j ′ k ′ ≡ i{I (i ′ j ′ , Q ak ′ )k } , (4.31) M i ′ j ′ k ′ l ′ ≡ {I (i ′ j ′ , I k ′ l ′ ) } . (4.32) On this set a linear finite-dimensional representation of D(2, 1; α) is realized [M, Q ai ′ i ] = (1 + α) M ai ′ i , (4.33) {M ai ′ i , Q ck ′ k } = −i ǫ ac ǫ i ′ k ′ ǫ ik M + i 3 (2 + α) ǫ ac M ik, i ′ k ′ − i 3 (1 + 2α) ǫ ik M ac, i ′ k ′ , (4.34) [M ik, i ′ k ′ , Q bj ′ j ] = 4ǫ j(i ǫ j ′( i ′ M bk ′) k) + (1 + 2α)ǫ j(i M bk), i ′ j ′ k ′ , (4.35) [M ac, i ′ k ′ , Q bj ′ j ] = −4ǫ b(a ǫ j ′( i ′ M c)k ′) j + (2 + α)ǫ b(a M c)j, i ′ j ′ k ′ , (4.36) {M ai, i ′ j ′ k ′ , Q bl ′ l } = −2iǫ ba ǫ l ′ (i ′ M il,j ′ k ′ ) − 2iǫ li ǫ l ′ (i ′ M ab,j ′ k ′ ) + 2i(1 + α)ǫ ba ǫ li M i ′ j ′ k ′ l ′ , (4.37) [M i ′ j ′ k ′ l ′ , Q bn ′ n ] = ǫ n ′ (i ′ M bn, i ′ j ′ k ′ ) . (4.38) The second invariant subspace is formed by the quantities N ≡ T 2 + 1 3 α(2 + α) J 2 − (1 + α) 2 I 2 + i 8 (2 + α) Q ai ′ i Q ai ′ i , (4.39) N ai ′ i ≡ i 4 {T a b , Q bi ′ i } + 1 3 (2 + α) {J i j , Q ai ′ j } + (1 + α) {I i ′ j ′ , Q aj ′ i } , (4.40) N i ′ k ′ , ik ≡ −(1 + α) {J ik , I i ′ k ′ } − i 2 Q b ( i ′ (i Q b k ′) k) , (4.41) N ac, ik ≡ {T ac , J ik } − i 2 Q ( aj ′ (i Q c ) j ′ k) , (4.42) N ai ′ , ijk ≡ i{J (ij , Q ai ′ k) } , (4.43) N ijkl ≡ {J (ij , J kl) } . (4.44) They can also be shown to constitute a basis of a linear finite-dimensional representation of D(2, 1; α) . At last, the third invariant subspace is formed by the bilinear operators L ≡ 1 3 (1 + 2α) T 2 + α 2 J 2 − (1 + α) 2 I 2 + i 8 (1 + 2α) Q ai ′ i Q ai ′ i , (4.45) L ai ′ i ≡ i 4 1 3 (1 + 2α) {T a b , Q bi ′ i } + α {J i j , Q ai ′ j } + (1 + α) {I i ′ j ′ , Q aj ′ i } , (4.46) L i ′ k ′ , ac ≡ −(1 + α) {I i ′ k ′ , T ac , } − i 2 Q ( a(i ′ j Q c ) k ′ ) j , (4.47) L ik, ac ≡ α{J ik , T ac } − i 2 Q ( aj ′ (i Q c ) j ′ k) , (4.48) L ii ′ , abc ≡ i{T (ab , Q c)i ′ i } , (4.49) L abcd ≡ {T (ab , T cd) } . (4.50) As for two previous invariant subspaces, these operators are closed under the action of D(2, 1; α) . These three invariant subspaces in the enveloping algebra have the following properties. First, these subspaces and one-dimensional space formed by the Casimir operator (4.21) exhaust all possible invariant subspaces in the enveloping algebra, such that they are bilinear in the D(2, 1; α) generators and involve singlets of all three bosonic subgroup SL(2, R), SU(2) R and SU(2) L . Second, these subspaces are related to each other via some discrete transformations. Namely, the subspaces (4.27)-(4.32) and (4.39)-(4.44) are dual to each other. That is, the discrete transformation α ↔ −(1 + α) , J ik ↔ I i ′ k ′ ,(4.α → α −1 , T ab ↔ J ik , Q ai ′ i → α −1/2 Q ai ′ i (4.52) the space (4.39)-(4.44) goes over into the space (4.45)-(4.50). Note, however, that the change (4.52) (and its analog taking (4.27)-(4.32) into (4.45)-(4.50)) is ill defined for the real form of the superalgebra D(2, 1; α) since it takes the sl(2, R) generators into the su(2) ones. These transformations present a true automorphism of the complexified D(2, 1; α) algebra. In the case of α = −1/2 (when 1 + 2α = 0) the subspaces (4.27)-(4.32) and (4.39)-(4.44) coincide. Moreover, the subspace formed by M , M ai ′ i , M ik, i ′ k ′ (4.53) (or N, N ai ′ i , N i ′ k ′ , ik ) is invariant under the D(2, 1; α = −1/2) . Just this subspace was exploited in [11]. In the case of α = −1 (when D(2, 1; α)=SU(1, 1|2)⊂ ×SU(2) L ) the operator (4.27) coincides with the Casimir (4.21), C 2 = M = T 2 − J 2 + i 4 Q ai ′ i Q ai ′ i .M = 0 , M ai ′ i = 0 , M ik, i ′ k ′ = 0 , M ac, i ′ k ′ = 0 , M ai, i ′ j ′ k ′ = 0 , M i ′ j ′ k ′ l ′ = 0 . (4.55) As a consequence of these identities, there arises the relation (1 + α)T 2 − α(1 + α)J 2 + 1 3 (1 − α 2 )I 2 = −(1 − α)C 2 . (4.56) In the case of α = −1 the constraint (4.56) leads to the condition C 2 = 0 that agrees with eqs. (4.54) and (4.55), as well as with (4.26). Using the expression (4.26) for the Casimir in r.h.s. of (4.56) we can represent the relation (4.56) in the form T 2 − α J 2 + 1 3 (1 − α)I 2 = −α(1 − α) 1 2 D 0 ( 1 2 D 0 + 1) + 1 4 , (4.57) which is valid for any value of α. Thus, for an irreducible representation of D(2, 1; α) with the fixed C 2 (see (4.69) below), the values of the Casimir operators T 2 , J 2 , I 2 of the three bosonic subgroups sl(2, R), su(2) R , su(2) L prove to be always related according to (4.57). The operator D 0 =Z k Z k , (4.58) entering the right-hand side of (4.57) commutes with all generators of the superalgebra D(2, 1; α) (as in the classical case). Quantum spectrum The Hamiltonian (4.6) and the SL(2, R) Casimir operator (4.22) can be represented as H = 1 4 P 2 +ĝ X 2 , (4.59) T 2 = 1 4ĝ − 3 16 , (4.60) wherê g ≡ 4α 2 1 2Z k Z k 1 2Z k Z k + 1 − 8αZ (iZ k) Ψ (iΨk) − 2(1 + 2α) Ψ i Ψ iΨkΨ k + 1 4 (1 + 2α) 2 . (4.61) The operators (4.59) and (4.60) formally look like those given in the model of [6]. However, there is an essential difference. Whereas the quantityĝ is a constant in the model of [6], in our caseĝ is an operator which takes fixed, but different, constant values on different components of the full wave function. To find the quantum spectrum of (4.59) and (4.60), we make use of the realization Z i = v + i , Z i = ∂/∂v + i (4.62) for the bosonic operators Z k andZ k , as well as the following realization of the odd operators Ψ i ,Ψ i Ψ i = ψ i ,Ψ i = − 1 2 ∂/∂ψ i ,(4.63) where ψ i are complex Grassmann variables. Then, the wave function is defined as Φ = A 1 + ψ i B i + ψ i ψ i A 2 . (4.64) The full wave function is subjected to the same constraints (3.45) as in the bosonic limit (we use the normal ordering for the even SU(2)-spinor operators, with all operators Z i standing on the right) D 0 Φ =Z i Z i Φ = v + i ∂ ∂v + i Φ = c Φ. (4.65) Like in the bosonic limit, requiring the wave function Φ(v + ) to be single-valued gives rise to the condition that the constant c is integer, c ∈ Z. We take c to be positive in order to have a correspondence with the bosonic limit where c becomes SU(2) spin. Then (4.65) implies that the wave function Φ(v + ) is a homogeneous polynomial in v + i of the degree c: Φ = A (c) 1 + ψ i B (c) i + ψ i ψ i A (c) 2 , (4.66) A (c) i ′ = A i ′ ,k 1 ...kc v +k 1 . . . v +kc , (4.67) B (c) i = B ′(c) i + B ′′(c) i = v + i B ′ k 1 ...k c−1 v +k 1 . . . v +k c−1 + B ′′ (ik 1 ...kc) v +k 1 . . . v +kc .Q ai ′ i Q ai ′ i T 2 J 2 I 2 i 4 Q ai ′ i Q ai ′ i A (c) k ′ α 2 (c+1) 2 −1 4 (c+1) 2 −1 4 3 4 1 + α B ′(c) k α 2 (c+1) 2 −2α(c+1) 4 (c+1) 2 −2(c+1) 4 0 α(c + 1) B ′′(c) k α 2 (c+1) 2 +2α(c+1) 4 (c+1) 2 +2(c+1) 4 0 −α(c + 1) On the same states, the Casimir operators (4.22)-(4.24) of the bosonic subgroups SU(1, 1), SU(2) R and SU(2) L take the values given in the Table 1. 9 For different component wave functions, the quantum numbers r 0 , j and i, defined by T 2 = r 0 (r 0 − 1) , J 2 = j(j + 1) , I 2 = i(i + 1) , take the values listed in the Table 2. The fields B ′ i and B ′′ i form doublets of SU(2) R generated Table 2: The SU(1, 1), SU(2) R and SU(2) L quantum numbers r 0 j i A (c) k ′ (x, v + ) |α|(c+1)+1 2 c 2 1 2 B ′(c) k (x, v + ) |α|(c+1)+1 2 − 1 2 sign(α) c 2 − 1 2 0 B ′′(c) k (x, v + ) |α|(c+1)+1 2 + 1 2 sign(α) c 2 + 1 2 0 by J ik , whereas the component fields A i ′ = (A 1 , A 2 ) form a doublet of SU(2) L generated by I i ′ k ′ . If the super-wave function (4.64) is bosonic (fermionic), the fields A i ′ describe bosons (fermions), whereas the fields B ′ i , B ′′ i present fermions (bosons). It is easy to check that the relation (4.56) is valid in all cases. Each of the component wave functions A i ′ , B ′ i , B ′′ i carries an infinite-dimensional unitary representation of the discrete series of the universal covering group of the one-dimensional 9 Here we use that Ψ i Ψ iΨkΨ k − Ψ iΨ i = 1 4 ψ i ψ i ∂ ∂ψ k ∂ ∂ψ k − 2ψ i ∂ ∂ψ i , Z iZ k Ψ (iΨk) = − 1 2 v +i ∂ ∂v + j ψ (i ∂ ∂ψ j) . Therefore, we have Ψ i Ψ iΨkΨ k − Ψ iΨ i Φ = − 1 2 ψ i B i , Z iZ k Ψ (iΨk) Φ = − 1 2 v +i ∂ ∂v + j ψ (i B j) = 1 4 ψ i [(c + 2)B ′ i − c B ′′ i ] . conformal group SU(1,1). Such representations are characterized by positive numbers r 0 [28,29] (for the unitary representations of SU(1,1) the constant r 0 > 0 must be (half)integer). Basis functions of these representations are eigenvectors of the compact SU(1,1) generator R = 1 2 a −1 K + aH , where a is a constant of the length dimension. These eigenvalues are r = r 0 +n, n ∈ N [28,29,6]. Using the expressions (4.6), (4.22)-(4.24) and the values of Casimirs from the Table 1, we can write the Hamiltonian in the unified form: H = 1 4 P 2 + l(l + 1) X 2 (4.70) where the constant l takes, on the separate wave functions, the values listed in the Table 3. k ′ (x, v + ) |α|(c + 1) − 1 2 B ′(c) k (x, v + ) |α|(c + 1) − 1 2 − sign(α) B ′′(c) k (x, v + ) |α|(c + 1) − 1 2 + sign(α) In the above quantization, we took into account all the conditions implied by the initial classical system. Due to the presence of additional invariant spaces in the enveloping algebra, we may try to impose additional conditions on the wave function, e.g. L Φ = 0 (4.71) where L was defined in (4.45). As a result, we could expect to obtain more restricted spectrum at certain values of the parameters α and c. Regrettably, this conjecture fails: in order to preserve the superconformal D(2, 1; α) covariance, we are led to assume that all operators from the set (4.45)-(4.50), on equal footing with L , annihilate the physical states, and these restrictions prove to be too strong. It is an open question whether the constraints of this kind could have a non-trivial solution in some other D(2, 1; α) invariant superconformal mechanics models. Let us focus on some peculiar properties of the D(2, 1; α) quantum mechanics constructed. As opposed to the standard SU(1, 1|2) superconformal mechanics [13,30,8], the construction presented here essentially uses the variables z i (or v + i ) parametrizing the two-sphere S 2 , in addition to the standard (dilatonic) coordinate x. The presence of additional "(iso)spin" S 2 variables in our construction leads to a richer quantum spectrum. Besides, the relevant wave functions involve representations of the two independent SU(2) groups, in contrast to the SU(1, 1|2) models of [13,30,8,10] where only the SU(2) realized on fermionic variables really matters. Also, in a contradistinction to the previously considered models (and in the same way as in our previous paper [11] devoted to the particular α= − 1/2 case), there naturally appears a quantization of the conformal coupling constant which is expressed as a SU(2) Casimir operator, with both integer and half-integer eigenvalues. This happens already in the bosonic sector of the model, and is ensured by the S 2 variables. 10 Note that the variables v + i in the expansions (4.67) and (4.68) can be identified with a half of the target space harmonic-like variables v ± i (though without the standard constraint v +i v − i ∼ const). Within a different quantization scheme used e.g. in [31], we would have even more literal harmonic interpretation of the bosonic isospinor variables. In both schemes, the S 2 constraint (3.18) is not explicitly solved before quantization, it is imposed on the wave functions as in (4.65). An alternative quantization scheme would be to deal with an explicit parametrization of the two-spere S 2 , e.g. the stereographic projection parametrization [22] or the parametrization by the Euler angles β and γ as in (3.20), and then to apply the canonical methods (Gupta-Bleuler quantization or Dirac procedure). 11 An important role in this case is played by the requirement of the square-integrability of the wave function on S 2 , which substitutes the constraint (4.65) of the parametrization-independent quantization schemes. As follows from the consideration in [21,22], this demand ensures the wave function to contain unitary representations of SU (2). General issues of the canonical quantization of Chern-Simons mechanics were addressed in [19]. Comment on the SU(1, 1|2) case Let us here focus on some peculiarities of the case of SU(1, 1|2) superconformal symmetry. In the case of α= − 1 one has D(2, 1; α= − 1) ≃ SU(1, 1|2)⊂ ×SU(2) L , and thus our model is invariant under SU(1, 1|2) superconformal group and an outer automorphism group SU(2) L acting only on the fermions. In general, the supergroup SU(1, 1|2) is known to admit a nonvanishing central charge which breaks this second R-symmetry SU(2) group down to U(1) [13] 12 . Thus, if we require our model to be invariant under SU(2) L (as in the case of generic α) the corresponding SU(1, 1|2) algebra cannot include a central charge. There arises the question as to whether a different version of the N =4 superconformal mechanics model with spin variables exists, such that it possesses SU(1, 1|2) symmetry with a non-vanishing central charge. The answer is affirmative, and it can be derived from the results of refs. [13,8,10]. When only SU(1, 1|2) symmetry is required, while SU(2) L symmetry is allowed to be broken, the constraints (2.3) and (2.4) for the even real superfield X can be weakened [13] by adding nonzero constants in their right-hand sides. The simplest choice is the following set of the constraints (a) D i D i X = 0 ,D iD i X = 0 ; (b) [D i ,D i ] X = m (4.72) where m is a constant. The solution of the constraints (4.72a) is a sum of (3.5) and additional term − 1 4 θθA, where A is some undefined constant. The constraint (4.72b) serves to fix this constant to be m. Then the action (2.2) (with α=−1) will give rise to additional contributions to the physical component Lagrangian (3.15), such that they are proportional to m 2 /x 2 and mψψ/x 2 [13]. These additional terms appear in the Hamiltonian, and they are induced by the appropriate new terms in the Noether supercharges. Comparing these modified SU(1, 1|2) generators with those given in [8,10], one can see that they correspond just to the SU(1, 1|2) algebra with a central charge proportional to m. More detailed analysis of the U(2) spin N =4 superconformal mechanics in which the even real superfield X is subjected to the constraints (4.72) with m = 0 will be given elsewhere. An interesting new feature of such a model is the presence of two complementary mechanisms of generating the conformal potential ∼ x −2 : the on-shell one via coupling to the auxiliary superfields Z + as in the case of generic α, and the off-shell one based on the deformed constraints (4.72) and a non-zero central charge in the SU(1, 1|2) algebra. It should be stressed that such a modification of the constraints is admissible only in the case of α=−1 ; at any other value of α (not belonging to the equivalence class of the choice α=−1) the superconformal invariance requires the constants in the right-hand sides of the constraints to vanish. Summary and outlook In this paper we presented a new version of N =4 mechanics with D(2, 1; α) superconformal symmetry. It is obtained as the one-particle reduction of the many-particle Calogero-type systems proposed in [2]. This system generalizes the OSp(4|2) superconformal mechanics constructed in our previous work [11], and it shares many characteristic features of the latter. In the bosonic sector it involves two complex fields (world-line harmonics) parametrizing the first Hopf map S 3 → S 2 . Due to the presence of spin variables in the superconformal mechanics, the quantum spectrum involves diverse D(2, 1; α) representations characterized by the specific values of the Casimir operator (4.21), (4.26). In these representations, the particle states carry representations of the bosonic subgroups SU(1, 1), SU(2) L and SU(2) R , the Casimirs of which are related to each other by the constraint (4.57). This constraint is identically satisfied for the particular realization of the D(2, 1; α) generators pertinent to our model. The appearance of this constraint is related to the existence of some invariant subspaces in the enveloping algebra of D(2, 1; α). We found that at generic α there exist more invariant subspaces than for the degenerate case of α= − 1/2 corresponding to OSp(4|2) [11], where some invariant subspaces are identified. The D(2, 1; α) superconformal mechanics was considered here for α =0. Formally, we can take the limit α → 0 in the final relations, and we observe that the target harmonic degrees of freedom decouple (see, e.g., (3.16), (3.17) and (3.23)-(3.27)). Nevertheless, the superconformal superfield action of the (1, 4, 3) multiplet is of a special form for α=0, so this case requires a separate study. Here we give a brief comment on the construction of the superfield superconformal action at α=0 . We note that D(2, 1; α→0) reduces to SU(1, 1|2)⊂ ×SU(2) R . The "passive" superconformal variation (2.21) of X disappears in this case, while the integration measure µ H is transformed as (see (2.18 )) δ ′ µ H = −2i(θ kη k +θ k η k ) µ H . (5.1) As suggested in [16,15], in order to ensure the superconformal invariance, it is necessary to modify the transformation law of X and, therefore, of V in the following way, δ ′ mod X = 2i(θ kη k +θ k η k ) , δ ′ mod V = 4i(η − θ + − η −θ+ ) . (5.2) Then the most general D(2, 1; α=0) superconformal action for the (1, 4, 3) multiplet reads [15] S X α=0 = − 1 4 µ H e X + µ (−2) A c +2 V , (5.3) where c +2 = c ij u + i u + j , and c ij are constant parameters. The second FI term in (5.3) is superconformal only at α=0 . It yields a conformal potential for the dilaton field with a strength ∼ c ik c ik , breaks the decoupled SU(2) R down to U(1) and induces a central charge ∼ c ik in SU(1, 1|2) . Actually, this action is dual to the α=−1 action for X with the modified constraints (4.72) [32]: the duality interchanges SU(2) L with SU(2) R and also α with −(1+α) . However, the D(2, 1; α=0) superconformal invariance is not compatible with the presence of V in the WZ term of the action (2.8), still implying the transformation laws (2.21) for Z + and for V ++ . As a consequence, the WZ term and the FI term of V ++ decouple from the X action: S Z,V ++ α=0 = 1 2 µ (−2) A Z + Z + + i 2 c µ (−2) A V ++ . (5.4) i.e. we loose any interaction between the superfields X and Z + . This situation is quite analogous to what happens in the N =1 and N =2 super Calogero models considered in [2], where the center-of-mass supermultiplet X decouples from the WZ and gauge supermultiplets. Note that in the many-particle N =4 super Calogero models the (matrix) X supermultiplet will still interact with the (column) Z supermultiplet via the gauge supermultiplet even in the α=0 case. Based on the duality just mentioned between the cases of α=0 and α=−1, one may expect that in the α=0 case the interaction of the superfield X with the U(2) spin variables can still be gained by placing the latter into a "mirror" (4, 4, 0) multiplet, for which the SU(2) R and SU(2) L R-symmetry groups switch their roles. In this context, it is worth noting that the bi-harmonic N =4 approach [26] achieves a unified description of systems with D(2, 1; α) and D(2, 1; −1−α) invariance. It allows one to naturally incorporate mirror counterparts for all N =4 supermultiplets with four fermions. Hence, it may provide an extension of the D(2, 1; α) superconformal models considered here, by adding such extra supermultiplets. Upon quantization, the mirror (4, 4, 0) auxiliary multiplets would produce a second family of target harmonic-like U(2) variables. For the remainder of this outlook and as a continuation of the discussion in the Introduction, let us illustrate how the models considered in this paper and in [2,11] could be inscribed into the context of D=5 extreme black-hole quantum mechanics. The motion of a test particle with mass m near the horizon of an extremal Tangherlini black hole of charge Q (a straightforward D=5 generalization of the D=4 extremal Reissner-Nordström solution) is described by the simple action [5] S = mQ 2 2 dt |˙ y| 2 , (5.5) where y are the coordinates of Euclidean four-space which are related to the isotropic nearhorizon black-hole coordinates x via y = x/| x| 2 . Making a polar decompostiion of the 4-vector y into a radial part ρ = | y| and an S 3 angular part, we rewrite the action (5.5) in first-order form as S = p ρ dρ + J· ω − dt 1 2mQ 2 p 2 ρ + 4 J· J ρ 2 . (5.6) Appendix A: Toy model with N =2 supersymmetry Here we consider N =2 supersymmetric model describing a "matter" supermultiplet coupled to U(1) gauge background. Matter is represented by two chiral superfields Z k (t L , θ),Z k (t R ,θ) = (Z k ) + , t L,R = t ± iθθ, satisfying irreducible conditionsDZ k = 0, DZ k = 0, k = 1, 2. Here, the covariant spinor derivatives are D = ∂ θ + iθ∂ t ,D = −∂θ − iθ∂ t , {D,D} = −2i∂ t . The gauge prepotential is a real superfield V (t, θ,θ), (V ) + = V . The action has the following form S = dtd 2 θ Z k e 2V Z k + c V . (A.1) It is invariant under the local U(1) transformations: Z k → e −iΛ Z k ,Z k → e iΛZ k , V → V + i 2 Λ −Λ (A.2) where Λ(t L , θ),Λ(t R , θ) = (Λ) + are chiral and antichiral superfield gauge parameters. Supersymmetry transformations of a general N =2 superfield F are defined by δF = −(δt∂ t + δθ∂ θ + δθ∂θ) F = −(ε Q −εQ) F (A.3) where the generators of SUSY transformations are Q = ∂ θ − iθ∂ t ,Q = −∂θ + iθ∂ t . Component contents of the superfields defined above are It takes the form ( d 2 θ (θθ) = 1) Z k = z k + 2iθφ k + iθθż k ,Z k =z k + 2iθφ k − iθθż k , V = v + θχ −S W Z = dt i(z k ∇z k − ∇z k z k ) + c A − 4φ k φ k , where ∇z and ∇z are the gauge-covariant derivatives, ∇z k =ż k − iAz k , ∇z k =ż k + iAz k . δz k = −iλz k , δz k = iλz k , δA = −λ , (A.9) where λ(t) is the d=1 gauge parameter. Supersymmetry transformations (A.5)-(A.6) do not preserve the WZ gauge conditions v = 0, χ = 0,χ = 0 , and we are led to modify these transformations by a field-dependent compensating gauge transformation with the parameter Λ = −2iθεA ,Λ = −2iθεA . Then the supersymmetry transformations leaving invariant the action (A.8) are given by δ W Z z k = −2iεφ k , δ W Zz k = −2iεφ k , δ W Z φ k = −ε∇z k , δ W Zφ k = −ε∇z k , (A.10) δ W Z A = 0 . (A.11) Let us study the closure of these transformations. On the fields z k we have (δ W Z 1 δ W Z 2 − δ W Z 2 δ W Z 1 ) z k = 2i (ε 1ε2 − ε 2ε1 ) ∇z k = 2ia 12ż k − iλ 12 z k , (A.12) where a 12 = ε 1ε2 − ε 2ε1 , λ 12 = 2i (ε 1ε2 − ε 2ε1 ) A . (A.13) Thus, the r.h.s. of (A.12) is the time translation with the parameter a 12 accompanied by a residual gauge transformation with the parameter λ 12 . Clearly, the closure on the gauge field A(t) should be the same. We find δ W Z 12 A = 2ia 12Ȧ −λ 12 = 0, (A.14) in agreement with (A.11). On shell, after eliminating the auxiliary fields φ,φ in the action (A.8), solution of the constraint (2.3), (2.4) is as follows (in the analytic basis): implies c > 0. After varying with respect to A, the action (3.15) is gauge invariant only with taking into account the constraint (3.18) which is gauge invariant by itself. The constant b in (3.16), (3.17) marks the contributions of the superfield WZ term to the physical component action. It can be eliminated by a proper rescaling of the variables z i ,z i , so hereafter we choose b = 1 . of physical coordinates and momenta satisfy the quantum brackets, obtained in the standard way from(3.31) 15)T 22 = H , T 11 = K , T 12 = −D .(4.16) 51)which is an automorphism of the D(2, 1; α) algebra (4.11)-(4.14), takes the space (4.27)-(4.32) into the space (4.39)-(4.44) and vice versa. The subspace (4.45)-(4.50) is a fixed point of the mapping ( this special case the appropriate invariant subspaces degenerate into the singlets of the superconformal group SU(1, 1|2)⊂ ×SU(2) 8 . Actually, in the case of generic α, for the particular representation of generators given by eqs. (4.22)-(4.24) all quantities (4.27)-(4.32) identically vanish: 4.68) we extracted SU(2) irreducible parts B ′ (k 1 ...k c−1 ) and B ′′ (ik 1 ...kc) of the component wave functions, with the SU(2) spins (c − 1)/2 and (c + 1)/2, respectively. On the physical states (4.65), (4.66) Casimir operator (4.26) takes the value C 2 = α(1 + α)(c + 1) 2 /4 . θχ + θθA , (A.4) where φ k ,φ k = (φ k ) and χ,χ = (χ) are fermionic fields. For the component fields the transformations (A.3) yieldδz k = −2iεφ k , δz k = −2iεφ k , δφ k = −εż k , δφ k = −εż k , (A.5) δv = −εχ +εχ , δχ = −ε(A + iv) , δχ = −ε(A − iv) , δA = −i(εχ +εχ) . (A.6) Let us consider the action (A.1) in the WZ gauge, V (t, θ,θ) = θθA(t) , e 2V = 1 + 2θθA . (A.7) (A.8) The action (A.8) is invariant under the residual local U(1) transformations φ k = 0 0,φ k = 0 , (A.15) the action (A.8) and the supersymmetry transformations (A.10), (A.11) becomeS W Z = dt i(z k ∇z k − ∇z k z k ) + c A , (A.16) δ W Z z k = 0 ,δ W Zz k = 0 ,δ W Z A = 0 . (A.17)Taking into account the equations of motion∇z k = ∇z k = 0 ,these on-shell transformations close on the time translations and gauge transformation like their off-shell counterparts (A.10), (A.11). The structure of the component N =4 supersymmetry transformations in the WZ gauge in our D(2, 1; α) superconformal mechanics model is basically the same as in the toy model just considered. transformations with an arbitrary α =0. It is worth pointing out that the action (2.8) is superconformally invariant just due to the presence of the analytic prepotential V .21) it is easy to check the invariance of the action (2.1). Note that the constraints (2.3), (2.4) and (2.9) as well as the actions (2.2), (2.6) and (2.8), are invariant with respect to the D(2, 1; α) Table 1 : 1The values of the Casimirs of the bosonic subgroups and i 4 Table 3 : 3Values of the constant l l A (c) Another view of the D(2, 1; α) superconformal mechanics models with spin variables (based on an su(2) Hamiltonian reduction at the classical component level) was presented in[12]. The fermionic action (3.17) can also be rewritten in terms of β and γ , like its α= − 1/2 prototype[11]. It would be of interest to clarify the precise relation of our realization of D(2, 1; α) derived from the concrete model to the realization found recently in[25] from a different reasoning. Although our mechanical system is ill defined at α=0, the D(2, 1; α) algebra (4.11) -(4.14) as it stands still admits such a choice, and it gives rise to the superalgebra D(2, 1; α=0)=SU(1, 1|2)⊂ ×SU(2) R . In this case the operator (4.39) coincides with the Casimir (4.21),C 2 = N = T 2 − I 2 + i 4 Q ai ′ i Q ai ′ i . Note that the strength of the conformal potential is related to the strength of the WZ term and so is quantized also in the N =4 superconformal mechanics associated with the (3, 4, 1) multiplet (without nondynamical S 2 variables)[16]. However, no direct relation between these parameters and SU(2) Casimirs appears in this case.11 One more approach is to quantize in the oscillator variables[18,21].12 The quotient of the general SU(1, 1|2) over the central charge generator is sometimes denoted as PSU(1, 1|2) . We use the notations which are related to those in[39] through a redefinition. In particular, we define the fermionic momenta as right derivatives of the Lagrangian.14 To be more precise, under the time reversal we also need to change the sign of the overall normalization constant before the invariant action since the integral dt changes its sign. AcknowledgementsWe acknowledge support from a grant of the Heisenberg-Landau Programme, RFBR grants 08-Here, ω i are the invariant one-forms on S 3 ∼ SU(2) , parametrized by the Euler angles (0≤γ≤π, 0≤β≤2π, 0≤φ<4π): ω 1 = − sinφ dγ + cosφ sinγ dβ , ω 2 = cosφ dγ + sinφ sinγ dβ , ω 3 = dφ + cosγ dβ .(5.7)In the Hamiltonian approach, the quantities J generate some SU(2) invariance[33]. It is easy to see that the action (5.5) is indeed reproduced by eliminating p ρ and J in (5.6) by their algebraic equations of motion. Firstly, we obtain the actionHowever,1 4ω· ω is precisely the S 3 metric[34,33]. Therefore secondly, the action takes the formThis is just (5.5) with y = ρ n. Performing in (5.6) a reduction with respect to the variables J[35],and identifying ρ = bx,, we obtain the one-particle bosonic limit (3.20) of the action (1.1) at |α| = 1.The fact that just this particular value of α comes out is not surprising because the action (5.5) was obtained in[5]as the bosonic limit of the SU(1, 1|2) superconformal model. It is interesting that the action (3.20) at arbitrary non-zero value of α can still be reproduced by the same reduction (5.10) from a deformation of the action (5.5) (or, equivalently, of (5.6)).This can be done in two different ways. One option is to substitute 4(J 1 J 1 + J 2 J 2 + α 2 J 3 J 3 )/ρ 2 for 4 J· J/ρ 2 in the last term of (5.6). The action (5.8) deformed in this way involves the metric1 4(ω t1 ω t1 + ω t2 ω t2 + α −2 ω t3 ω t3 ) instead of 1 4 ω t · ω t . Such a system describes the particle motion on a squashed 3-sphere, with α −2 as the squashing parameter. This model may bear a tight relation to D=5 rotating black holes, whose horizon is known to be a squashed 3-sphere[36,34,37]. The O(4) symmetry of (5.5) is broken to O(3) in this situation.Another possibility is to replace 4 J· J /ρ 2 in the last term of (5.6) by 4α 2 J· J/ρ 2 . The Lagrangian in (5.8) is then deformed into [ρρ + α −2 ρ 2 1 4 ω t · ω t ]. This system describes particle motion on a 4-dimensional cone C(S 3 ) over the round sphere S 3 of radius α −2 as the base[38,16]. This cone is conformally flat and exhibits O(4) isometry at any α =0 , including the values α=±1 which correspond to the action (5.5).In both cases, the reduction (5.10), performed in the relevant counterparts of the action (5.6), exactly yields our action(3.20). It is amusing that the parameter α acquires a nice geometric meaning within such a framework.Appendix B: Time reversal in mechanicsLet us consider the simple mechanical model with the LagrangianThe canonical momenta are 13with Poisson bracketsTherefore, the Hamiltonian isThe definition (B.2) implies second-class constraintsIntroducing Dirac bracketsThen, passing to quantum theory, we obtain the following operator algebraThe time-reversed system is described by the Lagrangian 14 Black holes and superconformal mechanics. P Claus, M Derix, R Kallosh, J Kumar, P K Townsend, A Van Proeyen, arXiv:hep-th/9804177Phys. Rev. Lett. 814553P. Claus, M. Derix, R. Kallosh, J. Kumar, P.K. Townsend, A. Van Proeyen, Black holes and superconformal mechanics, Phys. Rev. Lett. 81 (1998) 4553, arXiv:hep-th/9804177. Supersymmetric Calogero models by gauging. S Fedoruk, E Ivanov, O Lechtenfeld, arXiv:0812.4276Phys. Rev. 79105015hep-thS. Fedoruk, E. Ivanov, O. Lechtenfeld, Supersymmetric Calogero models by gauging, Phys. Rev. D79 (2009) 105015, arXiv:0812.4276 [hep-th]. Gauging N=4 supersymmetric mechanics. F Delduc, E Ivanov, arXiv:hep-th/0605211Nucl. Phys. 753211F. Delduc, E. Ivanov, Gauging N=4 supersymmetric mechanics, Nucl. Phys. B753 (2006) 211, arXiv:hep-th/0605211. Black holes and Calogero models. G W Gibbons, P K Townsend, arXiv:hep-th/9812034Phys. Lett. 454187G.W. Gibbons, P.K. Townsend, Black holes and Calogero models, Phys. Lett. B454 (1999) 187, arXiv:hep-th/9812034. The geometry of (super)conformal quantum mechanics, Commun. J Michelson, A Strominger, arXiv:hep-th/9907191arXiv:hep-th/9908044Math. Phys. 2135JHEPJ. Michelson, A. Strominger, The geometry of (super)conformal quantum mechanics, Com- mun. Math. Phys. 213 (2000) 1, arXiv:hep-th/9907191; Superconformal multi-black hole quantum mechanics, JHEP 9909 (1999) 005, arXiv:hep-th/9908044; Superconformal multi-black hole moduli spaces in four dimensions. A Maloney, M Spradlin, A Strominger, arXiv:hep-th/9911001JHEP. 02043A. Maloney, M. Spradlin, A. Strominger, Superconformal multi-black hole moduli spaces in four dimensions, JHEP 0204 (2002) 003, arXiv:hep-th/9911001; Lectures on superconformal quantum mechanics and multi-black hole moduli spaces. R Britto-Pacumio, J Michelson, A Strominger, A Volovich, arXiv:hep-th/9911066Progress in string theory and M-theory. CargeseR. Britto-Pacumio, J. Michelson, A. Strominger, A. Volovich, Lectures on superconformal quantum mechanics and multi-black hole moduli spaces, published in "Cargese 1999, Progress in string theory and M-theory", 235-264, arXiv:hep-th/9911066; Conformal and superconformal mechanics. G Papadopoulos, arXiv:hep-th/0002007Class. Quant. Grav. 173715G. Papadopoulos, Conformal and superconformal mechanics, Class. Quant. Grav. 17 (2000) 3715, arXiv:hep-th/0002007. Conformal invariance in quantum mechanics. V De Alfaro, S Fubini, G Furlan, Nuovo Cim. 34569V. de Alfaro, S. Fubini, G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A34 (1976) 569. An Exactly Solvable N Particle System In Supersymmetric Quantum Mechanics. D Z Freedman, P F Mende, Nucl. Phys. 344317D.Z. Freedman, P.F. Mende, An Exactly Solvable N Particle System In Supersymmetric Quantum Mechanics, Nucl. Phys. B344 (1990) 317. Super)conformal many body quantum mechanics with extended supersymmetry. N Wyllard, arXiv:hep-th/9910160J. Math. Phys. 412826N. Wyllard, (Super)conformal many body quantum mechanics with extended supersymme- try, J. Math. Phys. 41 (2000) 2826, arXiv:hep-th/9910160. Explicit solution to the N body Calogero problem. L Brink, T H Hansson, M A Vasiliev, arXiv:hep-th/9206049Phys. Lett. 286109L. Brink, T.H. Hansson, M.A. Vasiliev, Explicit solution to the N body Calogero problem, Phys. Lett. B286 (1992) 109, arXiv:hep-th/9206049; Anyonic representation, fermionic extension and supersymmetry. L Brink, T H Hansson, S Konstein, M A Vasiliev, arXiv:hep-th/9302023Nucl. Phys. 401591L. Brink, T.H. Hansson, S. Konstein, M.A. Vasiliev, Anyonic representation, fermionic ex- tension and supersymmetry, Nucl. Phys. B401 (1993) 591, arXiv:hep-th/9302023. Calogero models and nonlocal conformal transformations. A Galajinsky, O Lechtenfeld, K Polovnikov, arXiv:hep-th/0607215arXiv:0802.4386N=4 mechanics, WDVV equations and roots. 643113JHEP. hep-thA. Galajinsky, O. Lechtenfeld, K. Polovnikov, Calogero models and nonlocal conformal transformations, Phys. Lett. B643 (2006) 221, arXiv:hep-th/0607215; N=4 supercon- formal Calogero models, JHEP 0711 (2007) 008, arXiv:0708.1075 [hep-th]; N=4 me- chanics, WDVV equations and roots, JHEP 0903 (2009) 113, arXiv:0802.4386 [hep-th]. OSp(4|2) superconformal mechanics. S Fedoruk, E Ivanov, O Lechtenfeld, arXiv:0905.4951JHEP. 090881hep-thS. Fedoruk, E. Ivanov, O. Lechtenfeld, OSp(4|2) superconformal mechanics, JHEP 0908 (2009) 081, arXiv:0905.4951 [hep-th]. ) reduction in N=4 supersymmetric mechanics. S Krivonos, O Lechtenfeld, Su, arXiv:0906.2469Phys. Rev. 8024501hep-thS. Krivonos, O. Lechtenfeld, SU(2) reduction in N=4 supersymmetric mechanics, Phys. Rev. D80 (2009) 04501, arXiv:0906.2469 [hep-th]. Geometric superfield approach to superconformal mechanics. E A Ivanov, S O Krivonos, V M Leviant, J. Phys. 224201E.A. Ivanov, S.O. Krivonos, V.M. Leviant, Geometric superfield approach to superconformal mechanics, J. Phys. A22 (1989) 4201. N=4 supersymmetric mechanics in harmonic superspace. E Ivanov, O Lechtenfeld, arXiv:hep-th/0307111JHEP. 030973E. Ivanov, O. Lechtenfeld, N=4 supersymmetric mechanics in harmonic superspace, JHEP 0309 (2003) 073, arXiv:hep-th/0307111. Gauging N=4 supersymmetric mechanics II: (1,4,3) models from the (4,4,0) ones. F Delduc, E Ivanov, arXiv:hep-th/0611247Nucl. Phys. 770179F. Delduc, E. Ivanov, Gauging N=4 supersymmetric mechanics II: (1,4,3) models from the (4,4,0) ones, Nucl. Phys. B770 (2007) 179, arXiv:hep-th/0611247. New variant of N=4 superconformal mechanics. E Ivanov, S Krivonos, O Lechtenfeld, arXiv:hep-th/0212303JHEP. 030314E. Ivanov, S. Krivonos, O. Lechtenfeld, New variant of N=4 superconformal mechanics, JHEP 0303 (2003) 014, arXiv:hep-th/0212303. N=4, d=1 supermultiplets from nonlinear realizations of D(2, 1; α). E Ivanov, S Krivonos, O Lechtenfeld, arXiv:hep-th/0310299Class. Quant. Grav. 211031E. Ivanov, S. Krivonos, O. Lechtenfeld, N=4, d=1 supermultiplets from nonlinear realiza- tions of D(2, 1; α), Class. Quant. Grav. 21 (2004) 1031, arXiv:hep-th/0310299. Hamiltonian Reduction of Unconstrained and Constrained Systems. L Faddeev, R Jackiw, Phys. Rev. Lett. 601692L. Faddeev, R. Jackiw, Hamiltonian Reduction of Unconstrained and Constrained Systems, Phys. Rev. Lett. 60 (1988) 1692; Topological" (Chern-Simons) quantum mechanics. G V Dunne, R Jackiw, C A Trugenberger, Phys. Rev. 41661G.V. Dunne, R. Jackiw, C.A. Trugenberger, "Topological" (Chern-Simons) quantum me- chanics, Phys. Rev. D41 (1990) 661. Sigma Models With Purely Wess-Zumino-Witten Actions. R Floreanini, R Percacci, E Sezgin, Nucl. Phys. 322255R. Floreanini, R. Percacci, E. Sezgin, Sigma Models With Purely Wess-Zumino-Witten Ac- tions, Nucl. Phys. B322 (1989) 255. P S Howe, P K Townsend, Chern-Simons Quantum Mechanics. 71655P.S. Howe, P.K. Townsend, Chern-Simons Quantum Mechanics, Class. Quant. Grav. 7 (1990) 1655. Integrable systems from gauged matrix models. A P Polychronakos, Phys. Lett. 26629A.P. Polychronakos, Integrable systems from gauged matrix models, Phys. Lett. B266 (1991) 29. Fuzzy CP (n|m) as a quantum superspace, Contribution to. E Ivanov, L Mezincescu, P K Townsend, arXiv:hep-th/0311159Symmetries in Gravity and Field Theory", conference for Jose-Adolfo de Azcarraga's 60th birthday. Salamanca, SpainE. Ivanov, L. Mezincescu, P.K. Townsend, Fuzzy CP (n|m) as a quantum superspace, Con- tribution to "Symmetries in Gravity and Field Theory", conference for Jose-Adolfo de Azcarraga's 60th birthday, June 2003, Salamanca, Spain, arXiv:hep-th/0311159; L Mezincescu, arXiv:hep-th/0405031Super Chern-Simons Quantum Mechanics, Proceedings of the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'03. Dubna, RussiaL. Mezincescu, Super Chern-Simons Quantum Mechanics, Proceedings of the Interna- tional Workshop "Supersymmetries and Quantum Symmetries" (SQS'03, 24-29 July, 2003), Dubna, Russia, arXiv:hep-th/0405031. L Frappat, A Sciarrino, P Sorba, arXiv:hep-th/9607161Dictionary on Lie Algebras and Superalgebras. Academic PressL. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Algebras and Superalgebras, Academic Press, 2000, arXiv:hep-th/9607161. On the superconformal flatness of AdS superspaces. I A Bandos, E Ivanov, J Lukierski, D Sorokin, arXiv:hep-th/0205104JHEP. 020640I.A. Bandos, E. Ivanov, J. Lukierski, D. Sorokin, On the superconformal flatness of AdS superspaces, JHEP 0206 (2002) 040, arXiv:hep-th/0205104. Hidden symmetries of integrable conformal mechanical systems. T Hakobyan, S Krivonos, O Lechtenfeld, A Nersessian, arXiv:0908.3290Phys. Lett. 374801hep-thT. Hakobyan, S. Krivonos, O. Lechtenfeld, A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A374 (2010) 801, arXiv:0908.3290 [hep-th]. Biharmonic Superspace for N=4 Mechanics. E Ivanov, J Niederle, arXiv:0905.3770Phys. Rev. 8065027hep-thE. Ivanov, J. Niederle, Biharmonic Superspace for N=4 Mechanics, Phys. Rev. D80 (2009) 065027, arXiv:0905.3770 [hep-th]. Irreducible representations of the exceptional Lie superalgebras D(2, 1; α). J Van Der, Jeugt , J. Math. Phys. 26913J. Van der Jeugt, Irreducible representations of the exceptional Lie superalgebras D(2, 1; α), J. Math. Phys. 26 (1985) 913. Representations of the Lorentz group. V Bargmann, Ann. Math. 48568V. Bargmann, Representations of the Lorentz group, Ann. Math. 48 (1947) 568. Algebraical approach to the solution of one-dimensional model of n interacting particles. A M Perelomov, Teor. Mat. Fiz. 6364in RussianA.M. Perelomov, Algebraical approach to the solution of one-dimensional model of n inter- acting particles, Teor. Mat. Fiz. 6 (1971) 364 (in Russian). Superconformal mechanics and nonlinear realizations. J A De Azcarraga, J M Izquierdo, J C Bueno, P K Townsend, arXiv:hep-th/9810230Phys. Rev. 5984015J.A. de Azcarraga, J.M. Izquierdo, J.C. Perez Bueno, P.K. Townsend, Superconfor- mal mechanics and nonlinear realizations, Phys. Rev. D59 (1999) 084015, arXiv:hep-th/9810230. ) 1136]; Particle mechanics in harmonic superspace. V P Akulov, I A Bandos, D P Sorokin, Sov. J. Nucl. Phys. 471633Mod. Phys. Lett.V.P. Akulov, I.A. Bandos, D.P. Sorokin, Particle in harmonic N=2 superspace, Sov. J. Nucl. Phys. 47 (1988) 724 [Yad. Fiz. 47 (1988) 1136]; Particle mechanics in harmonic superspace, Mod. Phys. Lett. A3 (1988) 1633. Partial supersymmetry breaking in N=4 supersymmetric quantum mechanics. E A Ivanov, S O Krivonos, A I Pashnev, Clas. Quant. Grav. 819E.A. Ivanov, S.O. Krivonos, A.I. Pashnev, Partial supersymmetry breaking in N=4 super- symmetric quantum mechanics, Clas. Quant. Grav. 8 (1990) 19. All supersymmetric solutions of minimal supergravity in five-dimensions. J P Gauntlett, J B Gutowski, C M Hull, S Pakis, H S Reall, arXiv:hep-th/0209114Class. Quant. Grav. 204587J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis, H.S. Reall, All supersymmetric solu- tions of minimal supergravity in five-dimensions, Class. Quant. Grav. 20 (2003) 4587, arXiv:hep-th/0209114. Black holes of D = 5 supergravity. J P Gauntlett, R C Myers, P K Townsend, arXiv:hep-th/9810204Class. Quant. Grav. 161J.P. Gauntlett, R.C. Myers, P.K. Townsend, Black holes of D = 5 supergravity, Class. Quant. Grav. 16 (1999) 1, arXiv:hep-th/9810204. Integrable Systems of Classical Mechanics and Lie Algebras. A M Perelomov, Russian: Nauka, Fizmatlit. BostonBirkhauser Verlag240A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhauser Verlag, Boston, 1990, 307pp. [in Russian: Nauka, Fizmatlit, 1990, 240pp.]. D-branes and spinning black holes. J C Breckenridge, R C Myers, A W Peet, C Vafa, arXiv:hep-th/9602065Phys. Lett. 39193J.C. Breckenridge, R.C. Myers, A.W. Peet, C. Vafa, D-branes and spinning black holes, Phys. Lett. B391 (1997) 93, arXiv:hep-th/9602065. Killing spinors, supersymmetries and rotating intersecting branes. P K Townsend, arXiv:hep-th/0211008the Proceedings of 22nd Johns Hopkins Workshop on Novelties of String Theory. Goteborg, Sweden4183Surprises with angular momentum, Annales Henri PoincareP.K. Townsend, Killing spinors, supersymmetries and rotating intersecting branes, in the Proceedings of 22nd Johns Hopkins Workshop on Novelties of String Theory, Goteborg, Sweden, 20-22 Aug 1998, arXiv:hep-th/990110; Surprises with angular momentum, An- nales Henri Poincare 4 (2003) S183, arXiv:hep-th/0211008 Cones, Tri-Sasakian Structures and Superconformal Invariance. G W Gibbons, P Rychenkova, arXiv:hep-th/9809158Phys. Lett. 443138G.W. Gibbons, P. Rychenkova, Cones, Tri-Sasakian Structures and Superconformal Invari- ance, Phys. Lett. B443 (1998) 138, arXiv:hep-th/9809158. On the Quantization of Systems with Anticommutating Variables. R Casalbuoni, Nuovo Cim. 33389Nuovo Cim.R. Casalbuoni, On the Quantization of Systems with Anticommutating Variables, Nuovo Cim. A33 (1976) 115, The Classical Mechanics for Bose-Fermi Systems, Nuovo Cim. A33 (1976) 389.
[]
[ "IS OPERA NEUTRINO SUPERLUMINAL PROPAGATION SIMILAR TO GAIN-ASSISTED SUPERLUMINAL LIGHT PROPAGATION", "IS OPERA NEUTRINO SUPERLUMINAL PROPAGATION SIMILAR TO GAIN-ASSISTED SUPERLUMINAL LIGHT PROPAGATION" ]
[ "Vladan Panković [email protected] \nDepartment of Physics\nFaculty of Sciences\n21000 Novi Sad, Trg Dositeja Obradovića 4Serbia\n" ]
[ "Department of Physics\nFaculty of Sciences\n21000 Novi Sad, Trg Dositeja Obradovića 4Serbia" ]
[]
In this work we consider a possible conceptual similarity between recent, amazing OPERA experiment of the superluminal propagation of neutrino and experiment of the gain-assisted superluminal light propagation realized about ten years ago. Last experiment refers on the propagation of the light, precisely laser pulse through a medium, precisely caesium atomic gas, with characteristic anomalous dispersion and corresponding negative group-velocity index with very large amplitude between two closely spaced gain lines (that is in some way similar to quantum theory of the ferromagnetism). It implies superluminal propagation of the light through this medium. Nevertheless all this, at it has been pointed out by authors, "is not at odds with causality or special relativity", since it simply represents "a direct consequence of the classical interference between different frequency components". We suggest that OPERA experiment can be in some way conceptually similar to the gain-assisted superluminal light propagation experiment. For this reason we suppose too that OPERA experiment can be simply explained in full agreement with causality and special relativity if there is some medium, precisely a scalar field (e.g. dark matter field, Higgs field or similar) through which neutrino propagates. We prove that, according to OPERA experiment data, supposed medium must be non-dispersive while its refractive index must be positive, smaller but relatively close to 1 (that is in some way similar to quantum theory of the diamagnetism). If it is true OPERA experiment results do not mean that special theory of relativity is broken, but they mean detection of suggested medium, i.e. a scalar field (e.g. dark matter field, Higgs field or similar).
null
[ "https://arxiv.org/pdf/1109.6121v2.pdf" ]
118,486,273
1109.6121
14cd0614adcad634cab291d7659f869d3f9561d7
IS OPERA NEUTRINO SUPERLUMINAL PROPAGATION SIMILAR TO GAIN-ASSISTED SUPERLUMINAL LIGHT PROPAGATION 3 Oct 2011 Vladan Panković [email protected] Department of Physics Faculty of Sciences 21000 Novi Sad, Trg Dositeja Obradovića 4Serbia IS OPERA NEUTRINO SUPERLUMINAL PROPAGATION SIMILAR TO GAIN-ASSISTED SUPERLUMINAL LIGHT PROPAGATION 3 Oct 2011arXiv:1109.6121v2 [physics.gen-ph] In this work we consider a possible conceptual similarity between recent, amazing OPERA experiment of the superluminal propagation of neutrino and experiment of the gain-assisted superluminal light propagation realized about ten years ago. Last experiment refers on the propagation of the light, precisely laser pulse through a medium, precisely caesium atomic gas, with characteristic anomalous dispersion and corresponding negative group-velocity index with very large amplitude between two closely spaced gain lines (that is in some way similar to quantum theory of the ferromagnetism). It implies superluminal propagation of the light through this medium. Nevertheless all this, at it has been pointed out by authors, "is not at odds with causality or special relativity", since it simply represents "a direct consequence of the classical interference between different frequency components". We suggest that OPERA experiment can be in some way conceptually similar to the gain-assisted superluminal light propagation experiment. For this reason we suppose too that OPERA experiment can be simply explained in full agreement with causality and special relativity if there is some medium, precisely a scalar field (e.g. dark matter field, Higgs field or similar) through which neutrino propagates. We prove that, according to OPERA experiment data, supposed medium must be non-dispersive while its refractive index must be positive, smaller but relatively close to 1 (that is in some way similar to quantum theory of the diamagnetism). If it is true OPERA experiment results do not mean that special theory of relativity is broken, but they mean detection of suggested medium, i.e. a scalar field (e.g. dark matter field, Higgs field or similar). PACS number: 12.15.-y, 13.15.-g In this work we shall consider a possible conceptual similarity between recent, amazing OPERA experiment of the superluminal propagation of neutrino [1] and experiment of the gain-assisted superluminal light propagation [2] realized about ten years ago. Last experiment refers on the propagation of the light, precisely laser pulse through a medium, precisely caesium atomic gas, with characteristic anomalous dispersion and corresponding negative group-velocity index (very large amplitude between two closely spaced gain lines) that implies superluminal propagation of the light through this medium. (It is in some way similar to quantum theory of the ferromagnetism according to which relative magnetic permeability becomes large in a narrow boundary between two domains.) Nevertheless all this, at it has been pointed out by authors, "is not at odds with causality or special relativity", since it simply represents "a direct consequence of the classical interference between different frequency components". We shall suggest that OPERA experiment can be conceptually similar to the gain-assisted superluminal light propagation. For this reason we shall suppose that OPERA experiment can be simply explained in full agreement with causality and special relativity if there is some medium (a scalar field, e.g. dark matter field, Higgs field or similar) through which neutrino propagates. We shall prove that, according to OPERA experiment data, supposed medium must be nondispersive while its refractive index must be positive, smaller but relatively close to 1. (It is in some way similar to quantum theory of the diamagnetism which needs that orbital momentum of the diamagnetic atoms must be exactly equivalent to zero and which predicts that relative magnetic permeability is smaller but relatively close to 1.) If it is true OPERA experiment results do not mean that special theory of relativity is broken, but they mean detection of suggested medium, i.e. a scalar field (e.g. dark matter field, Higgs field or similar). About ten years ago Wang, Kuzmich and Dogariu [2] realized experiment of the gain-assisted superluminal light propagation whose basic elements (including corresponding theory) we shall now shortly repeat. As it is well-known in a dispersive linear medium with optical refractive index n(ν) depending of the optical frequency ν light pulse with this frequency propagates with the group velocity v g = c ng where n g = n(ν) + ν dn(ν) dν represents the group-velocity refractive index and c = 299792km/s -speed of light. In domain between two closely spaced gain lines there is an anomalous dispersion region where ν dn(ν) dν is negative with extremely large amplitude. (It is in some way similar to quantum theory of the ferromagnetism according to which relative magnetic permeability becomes large in a close space between two domains.) In this situation expression ∆T = L v g − L c = (n g − 1) L c (1) that represents time difference between propagation time of the light pulse through medium with length L and propagation time of the light pulse through vacuum with the same length L, becomes negative too. It means that light pulse propagates through medium effectively superluminally, i.e. faster than propagation of this pulse through vacuum (pulse time advancement shift). Nevertheless all this, at it has been pointed out by authors, "is not at odds with causality or special relativity", since it simply represents "a direct consequence of the classical interference between different frequency components" [2]. "Remarkably, the signal velocity of a light pulse, defined as the velocity at which half point of the pulse front travels, also exceed the speed of light in vacuum, c, in present experiment. It has also been suggested that the true speed at which information is carried by a light pulse should be defined as the "frontal" velocity of the step-function-shaped signal which has been shown not to exceed c." [2] In experimental realization of this theory Wang, Kuzmich and Dogariu used gaseous medium of the caesium atoms any of which has one excited state and two (close) ground states and corresponding polarized laser beam and obtained pulse advancement shift ∆T = 62ns or n g = −310 for L = 6cm. In recent OPERA experiment [1], in agreement with some other earlier experiments on the superluminal neutrino propagations [3], there is a pulse of muon neutrinos that propagates along base line L = 730534.61m (2) , with time difference with respect to the one assuming the speed of light in vacuum ∆T = −60.7ns It represents an extremely unexpected result whose theoretical explanation in this moment is unknown. For example Amelino-Camelia group [4] supposes that OPERA data can be explained by special-relativistic tachyons, etc. We shall originally and simply suppose that OPERA experiment is in some way conceptually similar to the experiment of the gain-assisted superluminal light propagation. Really, many characteristics of the neutrinos are very similar to the characteristic of the photons. But it can be observed that in the experiment of the gain-assisted superluminal light propagation there is characteristic medium, precisely caesium atomic gas, with characteristic anomalous dispersion and corresponding negative group-velocity index, while in the OPERA experiment similar medium, at the first sight, does not exist. However, it can be observed that a scalar field, e.g. dark matter field, Higgs field or similar, can exist. This field seems practically identical to any observer, moving or rest, and mimics vacuum. Moreover, as it has been pointed out by Linde (in the chaotic inflation cosmology [5], [6]) such field can during time occupy practically whole space or, at least, our galaxy, Sun system and Earth. Finally, a quantized scalar field has quantums with zero spin representing bosons. If such suggested scalar field really exists and if it has some group velocity refractive index for neutrino, this refractive index in OPERA experiment, according to (1), (2), (3), equals n g = 0.975. It represents a positive refractive index smaller but relatively close to 1. All this is in some way similar to quantum theory of the diamagnetism which predicts that relative magnetic permeability is smaller but relatively close to 1 and which needs that orbital momentum of the diamagnetic atoms must be exactly equivalent to zero. If this similarity has any sense it would mean that quantums of mentioned scalar field hold zero spin in full agreement with general quantum theory of the scalar fields. Also, according to OPERA experiment data, there is no dependence between ∆T or n g and neutrino energy or (de Broglie) frequency, which means that suggested medium must be non-dispersive. Finally, it can be observed that introduced hypothesis on the scalar field representing nondispersive medium for neutrinos, admits that, like in case of the gain-assisted superluminal light propagation, superluminal propagation of the neutrinos be explained in simple way that "is not at odds with causality or special relativity", since it simply represents, we paraphrase, "a direct consequence of the wave characteristics" of neutrino. In conclusion we shall shortly repeat and point out the following. In this work we consider a possible conceptual similarity between recent, amazing OPERA experiment of the superluminal propagation of neutrino and experiment of the gain-assisted superluminal light propagation realized about ten years ago. Last experiment refers on the propagation of the light, precisely laser pulse through a medium, precisely caesium atomic gas, with characteristic anomalous dispersion and corresponding negative group-velocity index (with very large amplitude between two closely spaced gain lines) that implies superluminal propagation of the light through this medium. (It is in some way similar to quantum theory of the ferromagnetism according to which relative magnetic permeability becomes large in a narrow boundary between two domains.) Nevertheless all this, at it has been pointed out by authors, "is not at odds with causality or special relativity", since it simply represents "a direct consequence of the classical interference between different frequency components". We suggest that OPERA experiment can be in some way conceptually similar to the gain-assisted superluminal light propagation. For this reason we suppose too that OPERA experiment can be simply explained in full agreement with causality and special relativity if there is some medium, precisely a scalar field (e.g. dark matter field, Higgs field or similar) through which neutrino propagates. We prove that, according to OPERA experiment data, supposed medium must be non-dispersive while its refractive index must be positive, smaller but relatively close to 1. (It is in some way similar to quantum theory of the diamagnetism which needs that orbital momentum of the diamagnetic atoms must be exactly equivalent to zero and which predicts that relative magnetic permeability is smaller but relatively close to 1.) If it is true OPERA experiment results do not mean that special theory of relativity is broken, but they mean detection of suggested medium, i.e. a scalar field (e.g. dark matter field, Higgs field or similar). relative difference of the muon neutrino velocity v with respect to the speed of light Measurement of the Neutrino Velocity with the OPERA Detector in the CNGS Beam. T Adam, N Agafonova, A , hep-ex/1109.4897T. Adam, N. Agafonova, A. Aleksandrov et al, Measurement of the Neutrino Velocity with the OPERA Detector in the CNGS Beam, hep-ex/1109.4897 . L J Wang, A Kuzmich, A Dogariu, Nature. 406277L. J. Wang, A. Kuzmich, A. Dogariu, Nature 406 (2000), 277 . P Adamson, MINOS collaborationhep- ex/0706.0437Phys. Rev. 7672005P. Adamson et al, [MINOS collaboration], Phys. Rev. D76 (2007) 072005; hep- ex/0706.0437 OPERA-reassessing Data on the Energy Dependence of Speed of Neutrinos. G Amelino-Camelia, hep-ph/1109.5172G. Amelino-Camelia et al, OPERA-reassessing Data on the Energy Dependence of Speed of Neutrinos, hep-ph/1109.5172 300 Years of Gravitation. A D Linde, S. W. Hawking and W. IsraelCambridge University PressCambridge, EnglandA. D. Linde, in, 300 Years of Gravitation, eds. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1989) A D Linde, Inflation and Quantum Cosmology. BostonAcademic PressA. D. Linde, Inflation and Quantum Cosmology (Academic Press, Boston, 1990)
[]
[ "A Galerkin FE method for elliptic optimal control problem governed by 2D space-fractional PDEs", "A Galerkin FE method for elliptic optimal control problem governed by 2D space-fractional PDEs" ]
[ "X G Zhu \nSchool of Science\nShaoyang University\n422000ShaoyangHunanP.R. China\n" ]
[ "School of Science\nShaoyang University\n422000ShaoyangHunanP.R. China" ]
[]
In this paper, we propose a Galerkin finite element method for the elliptic optimal control problem governed by the Riesz space-fractional PDEs on 2D domains with control variable being discretized by variational discretization technique. The optimality condition is derived and priori error estimates of control, costate and state variables are successfully established. Numerical test is carried out to illustrate the accuracy performance of this approach.
null
[ "https://arxiv.org/pdf/2202.09616v1.pdf" ]
247,012,071
2202.09616
1d056344a2b5030542009ef771d91c3ab167613b
A Galerkin FE method for elliptic optimal control problem governed by 2D space-fractional PDEs 19 Feb 2022 X G Zhu School of Science Shaoyang University 422000ShaoyangHunanP.R. China A Galerkin FE method for elliptic optimal control problem governed by 2D space-fractional PDEs 19 Feb 2022fractional optimal control problemfinite element methodpriori error estimate In this paper, we propose a Galerkin finite element method for the elliptic optimal control problem governed by the Riesz space-fractional PDEs on 2D domains with control variable being discretized by variational discretization technique. The optimality condition is derived and priori error estimates of control, costate and state variables are successfully established. Numerical test is carried out to illustrate the accuracy performance of this approach. Introduction The optimal control problems (OCPs) governed by fractional partial differential equations (PDEs) forms a new branch in the area of optimal control, which recently have gained explosive interest and enjoy great potential in the applications as diverse as temperature control, environmental engineering, crystal growth, disease transmission and so forth [11,12]. In this study, we consider the distributed quadratic fractional OCPs: subjected to the 2D elliptic Riesz fractional PDEs          κ 1 ∂ α u(x, y) ∂|x| α + κ 2 ∂ α u(x, y) ∂|y| α = g(x, y) + q(x, y), (x, y) ∈ Ω, u(x, y) = 0, (x, y) ∈ ∂Ω, (1. 2) where Ω = (a, b) × (c, d), κ 1 , κ 2 , γ ∈ R + , 1 < α < 2, K is a closed convex set and u d (x, y) is the desired state. The fractional derivatives have the weakly singular convolution form: ∂ α u(x, y) ∂|x| α = −1 2 cos( πα 2 ) L a D α x u(x, y) + R x D α b u(x, y) , L a D α x u(x, y) = 1 Γ(2 − α) d 2 dx 2 x a (x − ω) 1−α u(ω, y)dω, R x D α b u(x, y) = 1 Γ(2 − α) d 2 dx 2 b x (ω − x) 1−α u(ω, y)dω, and so is ∂ α u(x,y) ∂|y| α with regard to y. In the past decades, the OCPs governed by PDEs have been widely investigated and a large collection of works on their numerical algorithms have been done, which cover spectral method [17], FE method [3,8,7,18], mixed FE method [4,5], least square method [16], variational discretization method [10,13] and some other niche methods. However, the discussions on fractional OCPs have been rarely reported. The difficulty consisting in finding their numerical solutions not only lies in the nonsmoothness caused by the inequality constraints on control or state, but also the vectorial convolution in fractional derivatives, which bring enormous challenge in the endeavor of numerical schemes and theoretical analysis. Hence, it is of great significance to study the numerical methods for fractional OCPs. In [14], Mophou studied the first-order optimality condition for the OCPs governed by time-fractional diffusion equations. In [19], Ye and Xu derived the optimality condition for the time-fractional OCPs with state integral constraint and developed a spectral method. Zhou and Gong proposed a fully discrete FE scheme to solve the time-fractional OCPs [21]. Du et al. combined the finite difference method and gradient projection algorithm to obtain a fast scheme for the OCPs governed by space-fractional PDEs [6]. Zhou and Tan addressed a fully discrete FE scheme for the space-fractional OCPs [22]. Zhang et al. proposed the space-time discontinuous Galerkin FE methods for the time-fractional OCPs [20,23]. Gunzburger and Wang proposed a fully discrete FE scheme along with convolution quadrature for the time-fractional OCPs [9]. However, these works are limited to 1D or time-fractional OCPs. Due to the difficulty in constructing algorithm and theoretical analysis, there is no study reported on multi-dimensional space-fractional OCPs. Inspired by this, we propose a Galerkin FE scheme for the elliptic OCPs governed by 2D space-fractional PDEs, where the control variable is discretized by variational discretization technique because the inequality constraints always lead to low regularity. The first-order optimality condition is derived and the priori error estimates for the control, costate and state variables are rigorously analyzed. The rest of this paper are organized as follows. In Section 2, we derive the first-order optimality condition for Eqs. (1.1)-(1.2) and in Secton 3, we propose a fully discrete FE scheme for the optimality system. In Section 4, we establish the priori error estimates for the control, costate and state and finally, numerical tests are included to confirm our results. Optimality condition To begin with, we define H µ 0 (Ω) by the closure in C ∞ 0 (Ω) with respect to the fractional Sobolev norm || · || H µ (Ω) defined by ||u|| H µ (Ω) = ||u|| 2 L 2 (Ω) + |u| 2 H µ (Ω) 1/2 , |u| H µ (Ω) = || |ω| µ F [ũ]|| L 2 (Ω) with 1 < µ < 2 and F [ũ] being the Fourier transform of zero extension of u outside Ω. Consider the model of the 2D space-fractional OCPs: Minimize J(u, q) subjected to Eq. (1.2), (q, u) ∈ K × L 2 (Ω),(2.3) with the pointwise constraints on control variable, i.e., K = {q ∈ L 2 (Ω) : v 1 ≤ q(x, y) ≤ v 2 a.e. in Ω, v 1 , v 2 ∈ R}. Lemma 2.1. [15] If 1 < µ < 2, u, χ ∈ H µ 0 (Ω), then we have ( L a D µ x u, χ) = ( L a D µ 2 x u, R x D µ 2 b χ), ( R x D µ b u, χ) = ( R x D µ 2 b u, L a D µ 2 x χ), and the similar results exist for the fractional derivatives in y-direction. Theorem 2.1. The fractional OCPs (2.3) have a unique pair (q, u) and there is a costate state p such that the triplet (q, p, u) fulfills the first-order optimality condition as follow:          κ 1 ∂ α u(x, y) ∂|x| α + κ 2 ∂ α u(x, y) ∂|y| α = g(x, y) + q(x, y), (x, y) ∈ Ω, u(x, y) = 0, (x, y) ∈ ∂Ω, (2.4)          κ 1 ∂ α p(x, y) ∂|x| α + κ 2 ∂ α p(x, y) ∂|y| α = u(x, y) − u d (x, y), (x, y) ∈ Ω, p(x, y) = 0, (x, y) ∈ ∂Ω, (2.5) Ω (γq + p)(δq − q)dxdy ≥ 0, ∀δq ∈ K . (2.6) Proof. Due to the strictly convex J(·, ·), we easily know that the OCPs (2.3) admit a unique pair (q, u) by standard arguments. Next, we prove the first-order optimality condition (2.4)-(2.6). Suppose that v(x, y) is the state with respect to δq(x, y) − q(x, y), i.e.,          κ 1 ∂ α v(x, y) ∂|x| α + κ 2 ∂ α v(x, y) ∂|y| α = δq(x, y) − q(x, y), (x, y) ∈ Ω, v(x, y) = 0, (x, y) ∈ ∂Ω. (2.7) Define the reduced cost functionalĴ(q) := J(q, u(q)), which maps q from K to R. Then the first-order optimality condition reads aŝ J ′ (q)(δq − q) ≥ 0, ∀δq ∈ K . which leads to Ω γq(δq − q)dxdy + Ω v(u − u d )dxdy ≥ 0, ∀δq ∈ K . (2.8) On the other hand, we present the adjoint state equation          κ 1 ∂ α p(x, y) ∂|x| α + κ 2 ∂ α p(x, y) ∂|y| α = u(x, y) − u d (x, y), (x, y) ∈ Ω, p(x, y) = 0, (x, y) ∈ ∂Ω, (2.9) with the costate p. Multiplying by v and using Lemma 2.1, there holds Ω v(u − u d )dxdy = κ 1 Ω v · ∂ α p(x, y) ∂|x| α dxdy + κ 2 Ω v · ∂ α p(x, y) ∂|y| α dxdy = −κ 1 2 cos( πα 2 ) Ω L a D α x p · v + R x D α b p · vdxdy − κ 2 2 cos( πα 2 ) Ω L c D α y p · v + R y D β d p · vdxdy = −κ 1 2 cos( πα 2 ) Ω p · R x D α b v + p · L a D α x vdxdy − κ 2 2 cos( πα 2 ) Ω p · R y D α d v + p · L c D α y vdxdy = κ 1 Ω p · ∂ α v(x, y) ∂|x| α dxdy + κ 2 Ω p · ∂ α v(x, y) ∂|y| α dxdy. Combing with Eq. (2.7), we obtain Ω v(u − u d )dxdy = Ω p(δq − q)dxdy,(2.10) and substituting Eq. (2.10) into (2.8) finally leads to the above results. Fully discrete Galerkin FE scheme In order to derive the FE scheme, divide Ω by triangle meshes T h and for each triangle K, let h K = diam K and h = max K∈T h h K . Define the FE subspace V h = {v : v| K ∈ P linear , ∀K ∈ T h } and V h ∈ H α 2 0 (Ω), where P linear is the linear polynomial space. Using fractional variational principle, the FE scheme for state Eq. (1.2) is to find u h ∈ V h such that Λ h (u h , χ h ) = (g + q, χ h ), ∀χ h ∈ V h , (3.11) where Λ h (u, v) = κ 1 2 cos( πα 2 ) ( L a D α 2 x u, x R D α 2 b v) + ( R x D α 2 b u, L a D α 2 x v) + κ 2 2 cos( πα 2 ) ( L c D α 2 y u, R y D α 2 d v) + ( R y D α 2 d u, L c D α 2 y v) , which satisfies Λ h (u, v) ≤ C||u|| eng ||v|| eng , Λ h (u, u) ≥ C||u|| 2 eng with the energy norms ||u|| eng = ||u|| 2 L 2 (Ω) + |u| 2 eng 1 2 , |u| eng = |Λ h (u, u)| 1 2 . which is equivalent to ||u|| H α 2 (Ω) [15]. Denote the L 2 projection of u by R h u and the piecewise polynomial interpolant of u by Π h u, which have the below properties [1]: ||u − R h u|| L 2 (Ω) ≤ Ch r ||u|| H r (Ω) , (3.12) ||u − Π h u|| H s (Ω) ≤ Ch r−s ||u|| H r (Ω) , 0 ≤ s ≤ r. (3.13) In addition, we define the elliptic projection P h : H α 2 0 (Ω) → V h by Λ h (u, χ h ) = Λ h (P h u, χ h ), ∀χ h ∈ V h , which satisfies the following approximate property. with a constant C independent of h. We can derive the following convergent result for the above FE scheme. (Ω). Then there exists a constant C unrelated to h such that ||u − u h || H α 2 (Ω) ≤ Ch||u|| H 1+ α 2 (Ω) . (3.15) Proof. Using Galerkin orthogonality, we have C||u − u h || eng ≤ Λ h (u − u h , u − u h ) = Λ h (u − u h , u − χ h ) ≤C||u − u h || eng ||u − χ h || eng . 4 withC independent of h. Since the equivalence of || · || eng and || · || H α 2 (Ω) , it implies that ||u − u h || H α 2 (Ω) ≤ C inf χ h ∈V h ||u − χ h || H α 2 (Ω) . Taking χ h = Π h u and noticing (3.13), we finally obtain ||u − u h || H α 2 (Ω) ≤ C||u − Π h u|| H α 2 (Ω) ≤ Ch||u|| H 1+ α 2 (Ω) , which ends the proof. Letting q 0, the FE scheme for Eqs. (1.1)-(1.2) is to find a pair (q h , u h ) such that Minimize J(u h , q h ) subjected to Eq. (3.11), (q h , u h ) ∈ K × V h ,(3.16) which is equivalent to find the triplet (q h , p h , u h ) fulfilling the discrete optimality condition: Λ h (u h , χ h ) = (g + q h , χ h ), ∀χ h ∈ V h , u h (x, y) = 0, (x, y) ∈ ∂Ω, (3.17) Λ h (p h , χ h ) = (u h − u d , χ h ), ∀χ h ∈ V h , p h (x, y) = 0, (x, y) ∈ ∂Ω, (3.18) Ω (γq h + p h )(δq h − q h )dxdy ≥ 0, ∀δq h ∈ K . (3.19) Due to the variational inequality, the control variable always has low regularity. To overcome this drawback, we use the variational discretization method to treat q, i.e., (3.19) is recast as q h = P K − 1 γ p h = max v 1 , min − 1 γ p h , v 2 ,(3.20) where P K is termed by pointwise projection operator. Error estimates In this section, we establish the convergent analysis for the above FE scheme (3.17)-(3.20) and to this end, we introduce the auxiliary variational equations: Λ h (u h (q), χ h ) = (g + q, χ h ), ∀χ h ∈ V h , (4.21) Λ h (p h (q), χ h ) = (u h (q) − u d , χ h ), ∀χ h ∈ V h .(4.(q − q h , χ h ) = Λ h (u h (q) − u h , χ h ), (u h (q) − u h , χ * h ) = Λ h (p h (q) − p h , χ * h ), ∀χ h , χ * h ∈ V h , and letting χ h = p h (q) − p h , χ * h = u h (q) − u h leads to (q − q h , p h (q) − p h ) = (u h (q) − u h , u h (q) − u h ) ≥ 0. From the above inequality, it follows that γ||q − q h || 2 L 2 (Ω) = (γq + p h (q), q − q h ) − (γq h + p h , q − q h ) − (p h (q) − p h , q − q h ) ≤ (γq + p h (q), q − q h ) − (γq h + p h , q − q h ) ≤ (γq + p, q − q h ) + (p h (q) − p, q − q h ) − (γq h + p h , q − R h q) + (γq h + p h , q h − R h q). Meanwhile, by virtue of (2.6) and (3.19), we have (γq + p, q − q h ) ≤ 0, (γq h + p h , q h − R h q) ≤ 0, and then it suffices to prove that γ||q − q h || 2 L 2 (Ω) ≤ (p h (q) − p, q − q h ) − (γq h + p h , q − R h q) = (p h (q) − p, q − q h ) + γ(q − q h , q − R h q) + (p − p h (q), q − R h q) + (p h (q) − p h , q − R h q) − (γq + p, q − R h q) = (p h (q) − p, q − q h ) + γ(q − q h , q − R h q) + (p − p h (q), q − R h q) − (γq + p, q − R h q). (4.25) Furthermore, by using the properties of R h and q ∈ H 1 (Ω), there exists (γq + p, q − R h q) = (γq + p − R h (γq + p), q − R h q) ≤ ||γq + p − R h (γq + p)|| L 2 (Ω) ||q − R h q|| L 2 (Ω) ≤ Ch 2 . (4.26) Applying (3.12), (4.26) and Young's inequality to (4.25), we have γ||q − q h || 2 L 2 (Ω) ≤ ε||q − q h || 2 L 2 (Ω) + C ε ||p − p h (q)|| 2 L 2 (Ω) + C ε ||q − R h q|| 2 L 2 (Ω) − (γq + p, q − R h q) ≤ ε||q − q h || 2 L 2 (Ω) + Ch 2 + C ε ||p − p h (q)|| 2 L 2 (Ω) . By taking ε < γ, we finally obtain ||q − q h || L 2 (Ω) ≤ Ch + C||p − p h (q)|| L 2 (Ω) , and this completes the proof. 6 To derive the error bounds, we further give the auxiliary equation Λ h (p h (u), χ h ) = (u − u d , χ h ), ∀χ h ∈ V h ,(4.27) and obviously, p h (u) is the FE solution of costate p, which satisfies ||p − p h (u)|| H α 2 (Ω) ≤ Ch||p|| H 1+ α 2 (Ω) . (4.28) Based on the above discussions, we have the following priori error estimates. (Ω), then we have ||q − q h || L 2 (Ω) + ||p − p h || H α 2 (Ω) + ||u − u h || H α 2 (Ω) ≤ Ch, (4.29) where C is a constant unrelated to h. Proof. Multiplying Eq. (2.5) by χ h ∈ V h and subtracting Eq. (4.22), we have C||p − p h (q)|| 2 eng ≤ Λ h (p − p h (q), p − p h (q)) ≤ Λ h (p − p h (q), P h p − p h (q)) + Λ h (p − p h (q), p − P h p) = (u − u h (q), P h p − p h (q)) = (u − u h (q), p − p h (q)) + (u − u h (q), P h p − p), by taking χ h = p − p h (q) in both two equations. Using the equivalence of || · || eng and || · || H α 2 (Ω) and Lemma 3.1, there exists ||p − p h (q)|| 2 H α 2 (Ω) ≤ δ||p − p h (q)|| 2 L 2 (Ω) + C δ ||u − u h (q)|| 2 L 2 (Ω) + C δ ||p − P h p|| 2 H α 2 ≤ δ||p − p h (q)|| 2 L 2 (Ω) + C δ ||u − u h (q)|| 2 L 2 (Ω) + Ch 2 ||p|| 2 H 1+ α 2 (Ω) , with 0 < δ ≪ 1, which implies ||p − p h (q)|| H α 2 (Ω) ≤ Ch||p|| H 1+ α 2 (Ω) + C||u − u h (q)|| H α 2 . Combining with (4.23) and Lemma 4.1, we obtain ||q − q h || L 2 (Ω) ≤ Ch + C||p − p h (q)|| H α 2 (Ω) ≤ Ch + C||u − u h (q)|| L 2 (Ω) ≤ Ch.C||u h (q) − u h || 2 eng ≤ Λ h (u h (q) − u h , u h (q) − u h ) = (q − q h , u h (q) − u h ) ≤ ||q − q h || L 2 (Ω) ||u h (q) − u h || eng . Then, based on the error bound of q − q h , we can get ||u h (q) − u h || H α 2 (Ω) ≤ C||q − q h || L 2 (Ω) ≤ Ch,(4.||p h (u) − p h || H α 2 (Ω) ≤ C||u − u h || L 2 (Ω) . (4.32) Using (4.23), (4.28), (4.31), (4.32) and triangle inequality, we obtain ||u − u h || H α 2 (Ω) ≤ ||u − u h (q)|| H α 2 (Ω) + ||u h (q) − u h || H α 2 (Ω) ≤ Ch||u|| H 1+ α 2 (Ω) + C||q − q h || L 2 (Ω) ≤ Ch, (4.33) ||p − p h || H α 2 (Ω) ≤ ||p − p h (u)|| H α 2 (Ω) + ||p h (u) − p h || H α 2 (Ω) ≤ Ch||p|| H 1+ α 2 (Ω) + C||u − u h || H α 2 ≤ Ch. (4.34) Consequently, combining (4.30), (4.33) and (4.34), we obtain the priori error estimate. Illustrative test In this section, to illustrate the accuracy performance of the proposed FE scheme, numerical tests are carried out and numerical results are presented. For solving the coupled system (3.17)-(3.20), we adopt the fixed-point iterative algorithm and terminate iterative loop by reaching a solution q h with tolerant error 1.0e-12. We employ piecewise linear interpolation to approximate p, u and variational discretization method to discretize q. Meanwhile, denote Cov. order = log 2 e h 1 e h 2 log 2 h 1 h 2 , where e h k is the global error corresponding to the meshsize h k , k = 1, 2. Example 1. Letting κ 1 = κ 2 = 1, γ = 1 and K = {q ∈ L 2 (Ω) : −3 ≤ q(x, y) ≤ −0.1}, consider the problem on Ω = (0, 1) × (0, 1) with the analytic solutions u = 10x(1 − x)y(1 − y), p = 5x(1 − x)y(1 − y), q = max − 3, min{−p, −0.1} , where g, u d are determined by u, p and q via the model of OCPs. We evaluate the global error at coarse mesh and then refine the mesh several times. In Fig. 1, we show the decline behavior of error for the control, state and adjoint state variables with different α in log-log scale. In Fig. 2, we present an unstructured mesh of h = 1/25 and compare the analytic solution with the approximation of state when α = 1.9. To obtain more insight about accuracy, letting α = 1.3, we compute the error with different h and report the convergent order for control, state and adjoint state variables in Table 1. From the above results, we observe that our method is almost convergent with theoretical order and yields the solution indistinguishable from the analytic solution, which confirm the convergent accuracy and theoretical analysis. (x, y) − u d (x, y)|| 2 L 2 (Ω) + γ 2||q(x, y)|| 2 L 2 (Ω) , (1.1) Lemma 3. 2 . 2Let q = 0 and u ∈ H Theorem 4. 1 . 1If (q, p, u) are the analytical solutions of the OCPs (2.3), (q h , p h , u h ) are the FE solutions obtained by (3.17)-(3.20) and q ∈ H 1 (Ω), p, u ∈ H .(3.17) from Eq. (4.21) and taking χ h = u h (q) − u h , it holds that Figure 1 :Figure 2 : 12Evolution of error versus the change of h for different α: (i) α = 1.1; (ii) α = 1.5; (iii) α = 1.Unstructured triangular mesh, analytical and numerical solutions of state: (i) h = 1/25; (ii) u; (iii) u h . Table 1 : 1The global error and convergent order for q h , p h and u h when α = 1.3. ||q − q h || L 2 (Ω) Cov. order ||p − p h || H Cov. order ||u − u h || Hh α 2 (Ω) α 2 (Ω) Cov. order 1/10 1.4265e-02 - 4.0697e-02 - 5.8055e-02 - 1/15 8.4741e-03 1.28 2.6194e-02 1.09 3.9514e-02 0.95 1/20 5.8169e-03 1.31 1.9352e-02 1.05 3.0027e-02 0.96 Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. W P Bu, Y F Tang, J Y Yang, J. Comput. Phys. 276W.P. Bu, Y.F. Tang, J.Y. Yang, Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys. 276 (2014) 26-38. Superconvergence analysis of finite element methods for optimal control problems of the stationary Benard type. Y Z Chang, D P Yang, J. Comput. Math. 26Y.Z. Chang, D.P. Yang, Superconvergence analysis of finite element methods for optimal control problems of the stationary Benard type, J. Comput. Math. 26 (2008) 660-676. Superconvergence of mixed finite element methods for optimal control problems. Y P Chen, Math. Comput. 77Y.P. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comput. 77 (2008) 1269-1291. Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Y P Chen, Z L Lu, Finite Elem. Anal. Des. 46Y.P. Chen, Z.L. Lu, Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods, Finite Elem. Anal. Des. 46 (2010) 957-965. A fast gradient projection method for a constrained fractional optimal control. N Du, H Wang, W B Liu, J. Sci. Comput. 68N. Du, H. Wang, W.B. Liu, A fast gradient projection method for a constrained fractional optimal control, J. Sci. Comput. 68 (2016) 1-20. Approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold. W Gong, G S Wang, N N Yan, SIAM J. Control. Optim. 52W. Gong, G.S. Wang, N.N. Yan, Approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold, SIAM J. Control. Optim. 52 (2014) 2008-2035. Finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold. W Gong, N N Yan, SIAM J. Numer. Anal. 54W. Gong, N.N. Yan, Finite element approximations of parabolic optimal control problems with controls acting on a lower dimensional manifold, SIAM J. Numer. Anal. 54 (2016) 1229-1262. Error analysis of fully discrete finite element approximations to an optimal control problem governed by a time-fractional PDE. M Gunzburger, J L Wang, SIAM J. Control. Optim. 57M. Gunzburger, J.L. Wang, Error analysis of fully discrete finite element approximations to an optimal control problem governed by a time-fractional PDE, SIAM J. Control. Optim. 57 (2019) 241-263. Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. M Hinze, C Meyer, Comput. Optim. Appl. 46M. Hinze, C. Meyer, Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems, Comput. Optim. Appl. 46 (2010) 487-510. Fractional control of heat diffusion systems. I S Jesus, J A T Machado, Nonlinear Dynam. 54I.S. Jesus, J.A.T. Machado, Fractional control of heat diffusion systems, Nonlinear Dynam. 54 (2008) 263-282. J L Lions, Optimal Control of Systems Governed by Partial Differential Equations. BerlinSpringer-VerlagJ.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. A variational discretization concept in control constrained optimization: the linear-quadratic case. M M Hinze, Comput. Optim. Appl. 30M. M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005) 45-61. Optimal control of fractional diffusion equation. G M Mophou, Comput. Math. Appl. 61G.M. Mophou, Optimal control of fractional diffusion equation, Comput. Math. Appl. 61 (2011) 68-78. Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2. J P Roop, J. Comput. Appl. Math. 193J.P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , J. Comput. Appl. Math. 193 (2006) 243-268. First-order system least-squares methods for an optimal control problem by the Stokes flow. S Ryu, H C Lee, S D Kim, SIAM J. Numer. Anal. 47S. Ryu, H.C. Lee, S.D. Kim, First-order system least-squares methods for an optimal control problem by the Stokes flow, SIAM J. Numer. Anal. 47 (2009) 1524-1545. Legendre spectral-collocation method for solving some types of fractional optimal control problems. N H Sweilam, T M Al-Ajami, J. Adv. Res. 6N.H. Sweilam, T.M. Al-Ajami, Legendre spectral-collocation method for solving some types of fractional optimal control problems, J. Adv. Res. 6 (2015) 393-403. Finite element approximation of elliptic Dirichlet optimal control problems. V Vexler, Numer. Func. Anal. Opt. 28V. Vexler, Finite element approximation of elliptic Dirichlet optimal control problems, Numer. Func. Anal. Opt. 28 (2007) 957-973. A spectral method for optimal control problems governed by the abnormal diffusion equation with integral constraint on state. X Y Ye, C J Xu, Sci. Sin. Math. 46X.Y. Ye, C.J. Xu, A spectral method for optimal control problems governed by the abnormal diffusion equation with integral constraint on state, Sci. Sin. Math. 46 (2016) 1053-1070. A priori error analysis for time-stepping discontinuous Galerkin finite element approximation of time fractional optimal control problem. C Y Zhang, H P Liu, Z J Zhou, J. Sci. Comput. 80C.Y. Zhang, H.P. Liu, Z.J. Zhou, A priori error analysis for time-stepping discontinuous Galerkin finite element approximation of time fractional optimal control problem, J. Sci. Comput. 80 (2019) 993-1018. Finite element approximation of optimal control problems governed by time fractional diffusion equation. Z J Zhou, W Gong, Comput. Math. Appl. 71Z.J. Zhou, W. Gong, Finite element approximation of optimal control problems governed by time fractional diffu- sion equation, Comput. Math. Appl. 71 (2016) 301-318. Finite element approximation of optimal control problem governed by space fractional equation. Z J Zhou, Z Y Tan, J. Sci. Comput. 78Z.J. Zhou, Z.Y. Tan, Finite element approximation of optimal control problem governed by space fractional equa- tion, J. Sci. Comput. 78 (2019) 1840-1861. Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation. Z J Zhou, C Y Zhang, Numer. Algor. 79Z.J. Zhou, C.Y. Zhang, Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation, Numer. Algor. 79 (2018) 437-455.
[]
[ "Photoproduction of light vector mesons in Xe-Xe ultraperipheral collisions at the LHC and the nuclear density of Xe-129", "Photoproduction of light vector mesons in Xe-Xe ultraperipheral collisions at the LHC and the nuclear density of Xe-129" ]
[ "V Guzey \nNational Research Center \"Kurchatov Institute\"\nPetersburg Nuclear Physics Institute (PNPI)\n188300GatchinaRussia\n\nDepartment of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland\n", "E Kryshen \nNational Research Center \"Kurchatov Institute\"\nPetersburg Nuclear Physics Institute (PNPI)\n188300GatchinaRussia\n", "M Zhalov \nNational Research Center \"Kurchatov Institute\"\nPetersburg Nuclear Physics Institute (PNPI)\n188300GatchinaRussia\n" ]
[ "National Research Center \"Kurchatov Institute\"\nPetersburg Nuclear Physics Institute (PNPI)\n188300GatchinaRussia", "Department of Physics\nUniversity of Jyväskylä\nUniversity of Jyväskylä\nP.O. Box 3540014Finland", "Helsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 6400014Finland", "National Research Center \"Kurchatov Institute\"\nPetersburg Nuclear Physics Institute (PNPI)\n188300GatchinaRussia", "National Research Center \"Kurchatov Institute\"\nPetersburg Nuclear Physics Institute (PNPI)\n188300GatchinaRussia" ]
[]
We make predictions for cross sections of ρ and φ vector meson photoproduction in ultraperipheral Xe-Xe collisions at √ sNN = 5.44 TeV. Analyzing the momentum transfer distribution of ρ mesons in this process, we explore the feasibility of extracting the nuclear density of 129 Xe, which is needed in searches for dark matter with Xenon-based detectors.
10.1016/j.physletb.2018.05.058
[ "https://arxiv.org/pdf/1803.07638v2.pdf" ]
56,144,992
1803.07638
a5681a391d724211bca7e8755ebdeb5b87f51278
Photoproduction of light vector mesons in Xe-Xe ultraperipheral collisions at the LHC and the nuclear density of Xe-129 29 May 2018 V Guzey National Research Center "Kurchatov Institute" Petersburg Nuclear Physics Institute (PNPI) 188300GatchinaRussia Department of Physics University of Jyväskylä University of Jyväskylä P.O. Box 3540014Finland Helsinki Institute of Physics University of Helsinki P.O. Box 6400014Finland E Kryshen National Research Center "Kurchatov Institute" Petersburg Nuclear Physics Institute (PNPI) 188300GatchinaRussia M Zhalov National Research Center "Kurchatov Institute" Petersburg Nuclear Physics Institute (PNPI) 188300GatchinaRussia Photoproduction of light vector mesons in Xe-Xe ultraperipheral collisions at the LHC and the nuclear density of Xe-129 29 May 2018 We make predictions for cross sections of ρ and φ vector meson photoproduction in ultraperipheral Xe-Xe collisions at √ sNN = 5.44 TeV. Analyzing the momentum transfer distribution of ρ mesons in this process, we explore the feasibility of extracting the nuclear density of 129 Xe, which is needed in searches for dark matter with Xenon-based detectors. I. INTRODUCTION Collisions of ultrarelativistic ions at large impact parameters -the so-called ultraperipheral collisions (UPCs) -provide opportunities to explore photon-photon, photon-proton, and photon-nucleus interactions at previously unattainable high energies [1]. In particular, a test run of collisions of Xenon ions was performed at the Large Hadron Collider (LHC) in Fall 2017. The experiments have collected several µb −1 of statistics, which is sufficient to study photoproduction of light ρ and φ vector mesons. Extending the formalism [2], which reasonably describes coherent ρ meson photoproduction in Au-Au UPCs [3] at the Relativistic Heavy Ion Collider (RHIC) and Pb-Pb UPCs [4] at the LHC, we consider coherent and incoherent ρ and φ meson photoproduction in Xe-Xe UPCs in the LHC kinematics at √ s N N = 5.44 TeV and predict the corresponding UPC cross sections as functions of the vector meson rapidity y and the momentum transfer t. These predictions combined with the earlier results for Pb-Pb UPCs provide the nuclear mass number A dependence of our approach to nuclear shadowing in light vector meson photoproduction on nuclei and can be compared to the future LHC data. In the past, based on the vector meson dominance (VMD) model, photoproduction of light vector mesons on nuclei was used to determine the meson-nucleon cross section and to constrain the nuclear matter density distribution of the target [5]. To our knowledge, only the root-mean-square (rms) charge radii of Xe isotopes have been extracted from isotope shift measurements [6] and the charge density distribution of 132 Xe was recently determined [7] from electron-xenon elastic scattering at SCRIT facility. At the same time the nuclear matter distribution, the effective nuclear radius, and the structure factors of Xe isotopes are of key importance for Dark Matter experiments searching for weakly interacting massive particles (WIMP) with Xenon-based detectors (for details, see, e.g. [8][9][10]). In this note we demonstrate that the measurement of ρ photoproduction in Xe-Xe UPC at the LHC can be used to gain information on the nuclear matter distribution in 129 Xe. II. COHERENT AND INCOHERENT CROSS SECTIONS OF ρ AND φ PHOTOPRODUCTION IN NUCLEUS-NUCLEUS UPCS The cross section of coherent and incoherent (the target nucleus breaks up) cross section of vector meson V (V = ρ, φ) photoproduction in symmetric nucleus-nucleus UPCs reads [1]: dσ AA→V AA ′ (y) dy = N γ/A (y)σ γA→V A ′ (y) + N γ/A (−y)σ γA→V A ′ (−y) ,(1) where N γ/A is the photon flux; y is the rapidity of the produced vector meson V ; σ γA→V A ′ (y) is the photoproduction cross section. The target nucleus label A ′ stands for both coherent A ′ = A and incoherent A ′ = A cases. The presence of two terms with the opposite rapidities in Eq. (1) reflects the fact that each colliding ion can serve as a source of photons and as a target. The photon flux N γ/A (y) produced by an ultrarelativistic ion in nucleus-nucleus UPCs in Eq. (1) can be very well approximated by the photon flux due to a point-like charge Z: N γ/A (y) = 2Z 2 α e.m. π ζK 0 (ζ)K 1 (ζ) − ζ 2 2 K 2 1 (ζ) − K 2 0 (ζ) ,(2) where α e.m. is the fine-structure constant; K 0,1 are Bessel functions of the second kind; ζ = ωb min /γ L ; ω = (M V /2)e y is the photon energy for given y, where M V is the vector meson mass; γ L is the nucleus Lorentz factor in the laboratory frame; b min is the minimal transverse distance between the centers of the colliding nuclei specifying the ultraperipheral collision. Its value b min ≈ 2R A (R A is the radius of the nucleus) is found by requiring that Eq. (2) reproduces the photon flux, which is calculated as convolution over impact parameters of the flux of equivalent photons produced by the charge distribution of the radiating nucleus with the probability to not have the strong inelastic interactions in a given nucleus-nucleus collision. In high-energy UPCs of heavy ions with the large charge Z, the photoproduction process can be accompanied by additional photon exchanges between colliding ions because the parameter α 2 e.m. Z 2 is not small. These additional photon exchanges may lead to excitations of one or both colliding nuclei [11,12], which typically decay by emission of one or more neutrons moving along the direction of ion beams and detected by zero-degree calorimeters (ZDCs). The low-energy electromagnetic excitation of nuclei and the high-energy vector meson photoproduction in UPC can be considered as independent processes because of the large difference in time scales. Hence, one can account for the additional photon exchanges by modifying the photon flux and, thus, selecting photoproduction of vector mesons in nucleus-nucleus UPCs in different channels i, which are specified by emission of various number of neutrons i = (0n0n, 1n1n, 0nXn, XnXn, . . .) [13]. In particular, the photon flux for channel i reads: N i γ/A (y) = ∞ 2RA d 2 b N γ/A (y, b)P i ( b)(3) where N γ/A (y, b) is the photon flux at the transverse distance b (impact parameter) from the center of the nucleus, which produces it; P i ( b) is the probability to emit a given number of neutrons corresponding to channel i. This approach describes very well the ALICE data on electromagnetic dissociation in Pb-Pb UPCs [14] and is implemented in the Starlight Monte Carlo generator [15], which is commonly used for calculations and simulations of various UPC processes. Note that an alternative approach to electromagnetic excitation of nuclei with neutron emission in UPCs, which is based on the Hauser-Feshbach formalism and which provides a good description of the RHIC and LHC data on electromagnetic excitations in UPCs, was developed in [16]. The coherent γA → V A cross section σ γA→ρA in Eq. (1) can be calculated using the combination of the Gribov-Glauber model for nuclear shadowing and a model for hadronic fluctuations for the γN → V N cross section [2,17]. This approach provides a good description of the data on coherent ρ photoproduction on heavy nuclei in UPCs at RHIC and the LHC (Run 1). It is based on the observation that at high energies, the real photon interacts with hadronic targets by means of its long-lived hadronic components (fluctuations). Each fluctuation is characterized by the cross section σ and interacts independently with nucleons of a nuclear target; the probability distribution of these fluctuations P (σ) is constrained using the experimental data on the elastic γp → V p and the diffraction dissociation γp → Xp cross sections, see details in Ref. [2,18]. Thus, the γA → V A cross section in the large W γN -limit (W γN is the invariant photon-nucleus energy per nucleon) is given by the following expression: σ mVMD−GGM γA→V A (W γN ) = e f V 2 d 2 b dσP (σ) 1 − e − σ 2T A( b) 2 ,(4)where f V is the γ − V coupling constant (f 2 ρ /4π = 2.01 for ρ and f 2 φ /4π = 13.7 for φ);T A ( b) = ∞ −∞ dzρ A ( b, z) − (l c σ)/2 ∞ −∞ dzρ 2 A ( b, z) is the nuclear optical density, which also takes into account short-range nucleon-nucleon (N N ) correlations in the nuclear wave function, where ρ A (b, z) is the nuclear density and l c = −0.74 fm is the N N correlation length. For lower values of W γN ≤ O( √ 2R A m N M V ) ≈ 5 GeV, the expression in Eq. (4) should be corrected by including the effects of the non-zero longitudinal momentum transfer in the γN → V N amplitude (the effect of nuclear coherence). It suppresses the dσ AA→V AA /dy UPC cross section (1) at forward and backward rapidities but does not affect it near y ≈ 0. In the case of incoherent nuclear scattering, the γA → V A ′ quasi-elastic cross section σ γA→ρA ′ can be calculated using completeness of final nuclear states A ′ , see, e.g. [5]. Applying the photon fluctuations to the nuclear scattering amplitudes, one obtains: σ mVMD−GGM γA→V A ′ (W γN ) = σ γN →V N (W γN ) d 2 bT A ( b) dσP (σ) σ σ e − σ 2T A( b) 2 ,(5) where σ = dσP (σ)σ. Equation (5) has a clear physical interpretation: quasi-elastic photoproduction of ρ mesons on a nuclear target corresponds to elastic ρ production on any from all A target nucleons with the condition that interactions with remaining nucleons do not lead to inelastic production. The probability to not have inelastic processes describes the effect of nuclear shadowing for individual fluctuations and depends on the distribution P (σ). In the absence of fluctuations, it reduces to the familiar Glauber model expression. In collider kinematics of ion UPCs at ALICE, it is problematic to separate the quasielastic incoherent process γA → V A ′ and photoproduction of vector mesons γA → V A ′ Y with nucleon dissociation γN → V Y into not too large masses M Y < 10 GeV. All fragments Y are going in the very forward direction along the beams. We estimate the contribution of this process within the Gribov-Glauber model. In particular, an examination of corresponding multiple scattering graphs shows that the γN → V Y cross section σ γN →V Y factorizes out and the remaining nuclear shadowing suppression is the same as in the case of Eq. (5). It can also be shown formally by generalizing the derivation of Eq. (5) to include color fluctuations [19] in target nucleons (nucleon shape fluctuations [20]) and keeping the leading power of the variance of these fluctuations. To this accuracy, the form of these fluctuations is not important; the variance is expressed in terms of dσ γN→V Y (t = 0)/dt. Therefore, the incoherent cross section of light vector meson V photoproduction on nuclei with target nucleon dissociation is given by the following expression: σ mVMD−GGM γA→V A ′ Y (W γN ) = σ γN →V Y (W γN ) d 2 b T A ( b) dσP (σ) σ σ e − σ 2T A ( b) 2 .(6) Calculations of the cross section of ρ photoproduction with dissociation of the proton target in the non-perturbative domain of small values of |t| are strongly model-dependent. Instead, for the kinematical region of ALICE measurements, we use the HERA results [21,22] and obtain for the ratio of the forward target-dissociative and elastic ρ photoproduction cross sections on the proton: dσ γp→ρY (t ≈ 0)/dt dσ γp→ρp (t ≈ 0)/dt ≈ 0.1 − 0.12 .(7) Combining this value with the ratio of corresponding slope parameters of the t dependence B diss /B el ≈ 0.25 (it is assumed that the t dependence is exponential), we obtain the following relation: σ γp→V Y (W γp ) ≈ 0.5 σ γp→V p (W γp ) .(8) Note that this relation also agrees very well with the ratio of the elastic and proton-dissociation cross sections of ρ electroproduction in a wide range of Q 2 and at W γp = 75 GeV measured by the H1 collaboration at HERA [23], which shows that Eq. (8) is approximately Q 2 -independent. It should be emphasized that the t-integrated cross section of vector meson photoproduction with target nucleon dissociation is comparable with the incoherent quasielastic cross section due to the contribution of large |t| and a smaller slope parameter. The contribution of this process in the case of a heavy nuclear target is not essential for small |t|, in particular, in the region |t| < 0.02 GeV 2 , where coherent photoproduction dominates the momentum transfer distribution. Before discussing results of our calculations presented in Figs. 1-4, we note that the detailed description of our approach, the modified VMD-GGM, is given in [2]. Here we only emphasized the feature specific for the current calculations. The main part of our results is obtained with the nuclear matter density distribution of 129 Xe calculated within the standard spherical Hartree-Fock-SkyrmeIII model with accounting for the BCS pairing (no special fit specific to the Xe nucleus was done). This model gives the values of rms mass radius of 129 Xe equal to 4.818 fm and the charge rms radius equal to 4.77 fm. The latter can be compared to the experimental value 4.7831 ± 0.0043 [6] obtained from the isotope shift measurements. The effective radius R A = 5.8 fm, which is determined as the distance from the center to the point, where the mass density decreases by a factor of two compared to its maximal value, is somewhat larger than the value R A = 5.45 fm, which is given by the commonly used parametrization R A = 1.112 A 1/3 − 0.86 A −1/3 fm. III. RESULTS The calculated rapidity distributions for ρ and φ mesons produced in ultraperipheral Xe-Xe collisions at √ s N N = 5.44 TeV are presented in Figs. 1 and 2, respectively. In four panels we show the total rapidity distribution and the rapidity distributions in different channels: "0n0n" is the case without additional photon exchanges, whose experimental signature is no neutrons detected in zero-degree calorimeters (ZDCs), the "0nXn" channel corresponds to electromagnetic excitation with subsequent neutron decay of only one of colliding ions, and "XnXn" is the case of mutual excitation of both ions. presents the summed contribution of quasielastic and nucleon target-dissociative photoproduction calculated as described above. Specifically, the black dot-dashed line is obtained by adding the contributions of Eqs. (5) and (6), where the proton-dissociation cross section σ γp→V Y (W γp ) is calculated using Eq. (8). As one can see from Eq. (8), the inclusion of the proton dissociation contribution increases the t-integrated nuclear incoherent cross section by approximately 50%. All cross sections in Figs. 1 and 2 have been integrated over the momentum transfer. The shape of the rapidity distributions for coherent photoproduction reflects an interplay of several phenomena. Bumps at forward and backward rapidities in upper panels of Fig. 1 are due to an enhanced contribution of low-energy photoproduction related to the secondary Reggeon exchange in the ρ-N interaction; the inelastic Gribov shadowing at low energies is still small. Since only the Pomeron exchange contributes to the φ−N interaction, such bumps are absent in φ photoproduction (Fig. 2). The one-side contribution demonstrates how an interplay of the energy dependence of the elementary cross section, suppression due to nuclear shadowing, and drop of the flux of high-energy equivalent photons determine the distribution in the central and forward rapidity regions. A comparison of the distributions in different channels shows that additional photon exchanges resulting in neutron decays of excited nuclei enhance the role of smaller impact parameters of the collision [13] and, correspondingly, enhance the high-energy contribution to the UPC cross section. The calculated transverse momentum distributions at the rapidity y = 0 are presented in Fig. 3. The coherent cross section is shown by the red dashed line, the incoherent one -by the blue dotted line, and the summed cross section is given by the solid black curve. Here we neglected the contribution of the photoproduction process with nucleon dissociation, whose contribution is at the level of a few percent in the region of small |t| < 0.1 GeV 2 . We also neglected washing out of the diffractive dip in the coherent cross section due to a small, but non-vanishing transverse momentum of quasireal photons [24] and the real part of the meson-nucleon amplitude. It is well known that the position of diffractive dips is very sensitive to the radius of the target nucleus. Hence, to reveal them more clearly one needs to suppress the incoherent contribution. It can be achieved by selecting the "0n0n" channel of photoproduction, where one requires no forward neutron emission. It was shown in [25] that incoherent events of high energy photoproduction of vector mesons for |t| > 0.03 GeV 2 are predominantly accompanied by neutrons from the decay of the excited residual nucleus. In Fig. 4 we show the momentum transfer distributions for the "0n0n" channel in photoproduction of ρ meson in Xe-Xe UPCs at the rapidity y = 0. To reveal the possibility of study the influence of the nuclear radius, we performed these calculations with the two-parameter Fermi distribution of the nuclear density ρ(r) = ρ 0 [1 + exp((r − R A )/a)] −1 with the parameters a = 0.54 and R A = 6.1 fm and with R A = 5.45 fm. It is seen that in measurements with high statistics and momentum resolution, by fitting the shape of the momentum transfer distribution in the region up to |t| ≈ 0.1 GeV 2 , one can determine the nuclear radius with rather high accuracy. Note that the theoretical uncertainty of our calculation of inelastic nuclear shadowing, which is shown by shaded bands in Fig. 1, affects primarily the cross section magnitude and not the shape of the t dependence. Hence, this theoretical uncertainty does not affect positions of the minima in Figs. 3 and 4. There are also alternative approaches to calculation of light vector meson photoproduction in UPCs, notably, the one based on the color dipole framework and the phenomenological data-driven approach, which is used in Starlight Monte-Carlo generator. Briefly, photoproduction of light vector mesons in heavy ion Pb-Pb UPCs at the LHC has been extensively studied in the framework of the color dipole model including saturation effects [26][27][28][29]. Since the cross section of the discussed process is very sensitive to the non-perturbative contribution of large dipoles, the dipole model predictions strongly depend on the choice of the dipole cross section in this region and the final light vector meson wave function. In addition, due to the sub-leading Reggeon contribution to the γp → ρp cross section, we predict a two-bump shape of the rapidity distribution dσ AA→ρAA /dy, while the dipole models naturally lead to the distribution, which is bell-shaped. Another framework to describe photoproduction of vector mesons in ion UPCs is based on the Starlight Monte-Carlo generator [15]. It combines phenomenological parameterizations of the γp → V p cross sections on the proton with the optical theorem and the classical expression for the interaction of vector mesons with nuclei and rather successfully describes the available data on ρ photoproduction in Au-Au UPCs at RHIC and Pb-Pb UPCs at the LHC [30,31]. In the context of present analysis, it is important to note that while the t dependence of dσ γA→V A /dt is given by the nuclear form factor squared F 2 A (t) in Starlight (which is true in the limit of small nuclear shadowing), it is shifted towards smaller |t| in our case. An indication of this trend is seen in the ALICE data on coherent ρ photoproduction in Pb-Pb UPCs at √ s N N = 2.76 TeV [4]. For further critical discussion of treatment of light vector meson photoproduction in ion UPCs using the color dipole model and Starlight Monte-Carlo generator, see Ref. [2]. One should note that the short Xe-Xe run did not allow one to collect high enough statistics of events with photoproduction of quarkonia in Xe-Xe UPCs. At the same time, studies of coherent photoproduction of vector mesons interacting with nuclear medium with different strengths should be very informative for the precise determination of nuclear density parameters from the transverse momentum distribution. In particular, the position of the dips depends on the strength of absorption of the produced mesons by the nucleus. We plan to perform in the near future such an analysis for nuclear targets of interest in light of future experiments at a planned electron-ion collider. IV. CONCLUSION In this paper we presented our predictions for photoproduction of light vector mesons in ultraperipheral Xe-Xe collisions at the LHC. We showed that the analysis of the data on this process will provide useful information on nuclear shadowing, in particular, on the nuclear mass number A dependence of nuclear shadowing in light vector meson photoproduction on nuclei. We argue that the measured momentum transfer distributions can be used to gain new information on the density distribution of nuclear matter in 129 Xe and, hence, to constrain the elastic form factor of this nucleus, which is essential in the search for WIMP with Xenon-based detectors. FIG. 1 : 1Red solid lines with strips show the coherent cross section with the uncertainty of our model in accounting for Gribov inelastic shadowing. Blue dashed lines show the contribution to the rapidity distribution from one target nucleus corresponding to the first term in Eq. (1). Finally, the black dot-dashed line Rapidity distributions for ρ photoproduction in ultraperipheral Xe-Xe collisions at √ sNN = 5.44 TeV at the LHC. See explanations in text. FIG. 2 : 2Rapidity distributions for φ photoproduction in ultraperipheral Xe-Xe collisions at √ sNN = 5.44 TeV at the LHC. See explanations in text. FIG. 3 : 3The t-dependence of cross section of light vector meson (ρ on the left and φ on the right) photoproduction in ultraperipheral Xe-Xe collisions at the LHC at √ sNN = 5.44 TeV. FIG. 4 : 4The momentum transfer distribution of ρ photoproduction in Xe-Xe UPCs calculated using the mVMD-GGM approach with the two-parameter Fermi distribution of the Xenon nuclear density and two values of 129 Xe radius. . A J Baltz, arXiv:0706.3356Phys. Rept. 458nucl-exA. J. Baltz et al., Phys. Rept. 458, 1 (2008) [arXiv:0706.3356 [nucl-ex]]. . L Frankfurt, V Guzey, M Strikman, M Zhalov, arXiv:1506.07150Phys. Lett. B. 75251hep-phL. Frankfurt, V. Guzey, M. Strikman and M. Zhalov, Phys. Lett. B 752, 51 (2016) [arXiv:1506.07150 [hep-ph]]. . C Adler, STAR Collaboration10.1103/PhysRevLett.89.272302nucl-ex/0206004Phys. Rev. Lett. 89272302C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 89, 272302 (2002) doi:10.1103/PhysRevLett.89.272302 [nucl-ex/0206004]. . J Adam, ALICE Collaboration10.1007/JHEP09(2015)095arXiv:1503.09177JHEP. 150995nucl-exJ. Adam et al. [ALICE Collaboration], JHEP 1509, 095 (2015) doi:10.1007/JHEP09(2015)095 [arXiv:1503.09177 [nucl-ex]]. . T H Bauer, R D Spital, D R Yennie, F M Pipkin, Rev. Mod. Phys. 50407Rev. Mod. Phys.T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin, Rev. Mod. Phys. 50, 261 (1978) Erratum: [Rev. Mod. Phys. 51, 407 (1979)]. . J Libert, B Roussire, J Sauvage, Nucl. Phys. A. 78647J. Libert, B. Roussire and J. Sauvage, Nucl. Phys. A 786, 47 (2007). . K Tsukada, arXiv:1703.04278Phys. Rev. Lett. 11826262501nucl-exK. Tsukada et al., Phys. Rev. Lett. 118, no. 26, 262501 (2017) [arXiv:1703.04278 [nucl-ex]]. . H Uchida, XMASS-I CollaborationarXiv:1401.4737PTEP. 20146astro-ph.COH. Uchida et al. [XMASS-I Collaboration], PTEP 2014, no. 6, 063C01 (2014) [arXiv:1401.4737 [astro-ph.CO]]. . L Vietze, P Klos, J Menndez, W C Haxton, A Schwenk, arXiv:1412.6091Phys. Rev. D. 91443520nucl-thL. Vietze, P. Klos, J. Menndez, W. C. Haxton and A. Schwenk, Phys. Rev. D 91, no. 4, 043520 (2015) [arXiv:1412.6091 [nucl-th]]. . A Fieguth, M Hoferichter, P Klos, J Menndez, A Schwenk, C Weinheimer, arXiv:1802.04294hep-phA. Fieguth, M. Hoferichter, P. Klos, J. Menndez, A. Schwenk and C. Weinheimer, arXiv:1802.04294 [hep-ph]. . M Vidovic, M Greiner, G Soff, Phys. Rev. C. 482011M. Vidovic, M. Greiner and G. Soff, Phys. Rev. C 48 (1993) 2011. . I A Pshenichnov, J P Bondorf, I N Mishustin, A Ventura, S Masetti, nucl-th/0101035Phys. Rev. C. 6424903I. A. Pshenichnov, J. P. Bondorf, I. N. Mishustin, A. Ventura and S. Masetti, Phys. Rev. C 64 (2001) 024903 [nucl-th/0101035]. . A J Baltz, S R Klein, J Nystrand, nucl-th/0205031Phys. Rev. Lett. 8912301A. J. Baltz, S. R. Klein and J. Nystrand, Phys. Rev. Lett. 89, 012301 (2002) [nucl-th/0205031]. . B Abelev, ALICE CollaborationarXiv:1203.2436Phys. Rev. Lett. 109252302nucl-exB. Abelev et al. [ALICE Collaboration], Phys. Rev. Lett. 109 (2012) 252302 [arXiv:1203.2436 [nucl-ex]]. . S R Klein, J Nystrand, J Seger, Y Gorbunov, J Butterworth, arXiv:1607.03838Comput. Phys. Commun. 212258hep-phS. R. Klein, J. Nystrand, J. Seger, Y. Gorbunov and J. Butterworth, Comput. Phys. Commun. 212 (2017) 258 [arXiv:1607.03838 [hep-ph]]. . M Klusek-Gawenda, M Ciemala, W Schafer, A Szczurek, arXiv:1311.1938Phys. Rev. C. 89554907nucl-thM. Klusek-Gawenda, M. Ciemala, W. Schafer and A. Szczurek, Phys. Rev. C 89 (2014) no.5, 054907 [arXiv:1311.1938 [nucl-th]]. . V Guzey, M Strikman, M Zhalov, arXiv:1312.6486Eur. Phys. J. C. 7472942hep-phV. Guzey, M. Strikman and M. Zhalov, Eur. Phys. J. C 74, no. 7, 2942 (2014) [arXiv:1312.6486 [hep-ph]]. . V Guzey, E Kryshen, M Zhalov, arXiv:1602.01456Phys. Rev. C. 93555206nucl-thV. Guzey, E. Kryshen and M. Zhalov, Phys. Rev. C 93, no. 5, 055206 (2016) [arXiv:1602.01456 [nucl-th]]. . L Frankfurt, M Strikman, D Treleani, C Weiss, arXiv:0808.0182Phys. Rev. Lett. 101202003hep-phL. Frankfurt, M. Strikman, D. Treleani and C. Weiss, Phys. Rev. Lett. 101, 202003 (2008) [arXiv:0808.0182 [hep-ph]]. . H Mäntysaari, B Schenke, arXiv:1603.04349Phys. Rev. Lett. 117552301hep-phH. Mäntysaari and B. Schenke, Phys. Rev. Lett. 117, no. 5, 052301 (2016) [arXiv:1603.04349 [hep-ph]]. . J Breitweg, ZEUS Collaborationhep-ex/9910038Eur. Phys. J. C. 14213J. Breitweg et al. [ZEUS Collaboration], Eur. Phys. J. C 14, 213 (2000) [hep-ex/9910038]. . R M Weber, DISS-ETH-16709R. M. Weber, DISS-ETH-16709, June 2006. . F D Aaron, H1 CollaborationarXiv:0910.5831JHEP. 100532hep-exF. D. Aaron et al. [H1 Collaboration], JHEP 1005 (2010) 032 [arXiv:0910.5831 [hep-ex]]. . S R Klein, J Nystrand, hep-ph/9909237Phys. Rev. Lett. 842330S. R. Klein and J. Nystrand, Phys. Rev. Lett. 84, 2330 (2000) [hep-ph/9909237]. . M Strikman, M Tverskoy, M Zhalov, hep-ph/0505023Phys. Lett. B. 62672M. Strikman, M. Tverskoy and M. Zhalov, Phys. Lett. B 626, 72 (2005) [hep-ph/0505023]. . Y P Ivanov, B Z Kopeliovich, I Schmidt, arXiv:0706.1532hep-phY. P. Ivanov, B. Z. Kopeliovich and I. Schmidt, arXiv:0706.1532 [hep-ph]. . V P Goncalves, M V T Machado, arXiv:1106.3036Phys. Rev. C. 8411902hep-phV. P. Goncalves and M. V. T. Machado, Phys. Rev. C 84 (2011) 011902 [arXiv:1106.3036 [hep-ph]]. . G Sampaio, M V T Santos, Machado, arXiv:1407.4148Phys. Rev. C. 91225203hep-phG. Sampaio dos Santos and M. V. T. Machado, Phys. Rev. C 91 (2015) no.2, 025203 [arXiv:1407.4148 [hep-ph]]. . M Klusek-Gawenda, A Szczurek, arXiv:1609.04355EPJ Web Conf. 1305009hep-phM. Klusek-Gawenda and A. Szczurek, EPJ Web Conf. 130 (2016) 05009 [arXiv:1609.04355 [hep-ph]]. . L Adamczyk, STAR CollaborationarXiv:1702.07705Phys. Rev. C. 96554904nucl-exL. Adamczyk et al. [STAR Collaboration], Phys. Rev. C 96 (2017) no.5, 054904 [arXiv:1702.07705 [nucl-ex]]. . S R Klein, arXiv:1704.04715Nucl. Phys. A. 967249nucl-exS. R. Klein, Nucl. Phys. A 967 (2017) 249 [arXiv:1704.04715 [nucl-ex]].
[]
[ "Nuclear Forces from Lattice Quantum Chromodynamics 1", "Nuclear Forces from Lattice Quantum Chromodynamics 1" ]
[ "Martin J Savage \nInstitute for Nuclear Theory\nBox 35155098195-1550SeattleWAUSA\n" ]
[ "Institute for Nuclear Theory\nBox 35155098195-1550SeattleWAUSA" ]
[]
A century of coherent experimental and theoretical investigations have uncovered the laws of nature that underly nuclear physics. The standard model of strong and electroweak interactions, with its modest number of input parameters, dictates the dynamics of the quarks and gluons -the underlying building blocks of protons, neutrons, and nuclei. While the analytic techniques of quantum field theory have played a key role in understanding the dynamics of matter in high energy processes, they encounter difficulties when applied to low-energy nuclear structure and reactions, and dense systems. Expected increases in computational resources into the exa-scale during the next decade will provide the ability to numerically compute a range of important strong interaction processes directly from QCD with quantifiable uncertainties using the technique of Lattice QCD. These calculations will refine the chiral nuclear forces that are used as input into nuclear many-body calculations, including the three-and four-nucleon interactions. I discuss the state-of-the-art Lattice QCD calculations of quantities of interest in nuclear physics, progress that is expected in the near future, and the impact upon nuclear physics.
null
[ "https://arxiv.org/pdf/1309.4752v1.pdf" ]
117,894,913
1309.4752
3fa27591f0deff6b7f0619da6b62990ee43cdf11
Nuclear Forces from Lattice Quantum Chromodynamics 1 Martin J Savage Institute for Nuclear Theory Box 35155098195-1550SeattleWAUSA Nuclear Forces from Lattice Quantum Chromodynamics 1 Nuclear Forces; Lattice QCD A century of coherent experimental and theoretical investigations have uncovered the laws of nature that underly nuclear physics. The standard model of strong and electroweak interactions, with its modest number of input parameters, dictates the dynamics of the quarks and gluons -the underlying building blocks of protons, neutrons, and nuclei. While the analytic techniques of quantum field theory have played a key role in understanding the dynamics of matter in high energy processes, they encounter difficulties when applied to low-energy nuclear structure and reactions, and dense systems. Expected increases in computational resources into the exa-scale during the next decade will provide the ability to numerically compute a range of important strong interaction processes directly from QCD with quantifiable uncertainties using the technique of Lattice QCD. These calculations will refine the chiral nuclear forces that are used as input into nuclear many-body calculations, including the three-and four-nucleon interactions. I discuss the state-of-the-art Lattice QCD calculations of quantities of interest in nuclear physics, progress that is expected in the near future, and the impact upon nuclear physics. Introduction A nucleus is at the heart of every atom, and loosely speaking, is a collection of protons and neutrons that interact pairwise, with much smaller, but significant, threebody interactions. We are fortunate to know that the underlying laws governing the strong interactions result from a quantum field theory called quantum chromodynamics (QCD). It is constructed in terms of quark and gluon fields with interactions determined by a local SU(3) gauge-symmetry and, along with quantum electrodynamics (QED), underpins all of nuclear physics when the five relevant input parameters, the scale of strong interactions Λ QCD , the three light-quark masses m u , m d and m s , and the electromagnetic coupling α e , are set to their values in nature. It is remarkable that the complexity of nuclei emerges from "simple" gauge theories with just five input parameters. Perhaps even more remarkable is that nuclei resemble collections of nucleons and not collections of quarks and gluons. By solving QCD, we are expecting to predict, with arbitrary precision, nuclear processes and the properties of multi-baryon systems. The fine-tunings observed in the structure of nuclei, and in the interactions between nucleons, are peculiar and fascinating aspects of nuclear physics. For the values of the input parameters that we have in our universe, the nucleon-nucleon (NN) interactions are fine-tuned to produce unnaturally large scattering lengths in both s-wave channels (described by non-trivial fixed-points in the low-energy effective field theory (EFT)), and the energy levels in the 8 Be-system, 12 C and 16 O are in "just-so" locations to produce enough 12 C to support life, and the subsequent emergence and evolution of the human species. At a fundamental level it is important for us to determine the sensitivity of the abundance of 12 C to the light-quark masses and to ascertain the degree of their fine-tuning. Being able to solve QCD for the lightest nuclei, using the numerical technique of Lattice QCD (LQCD), would allow for a partial unification of nuclear physics. It would be possible to "match" the traditional nuclear physics techniques -the solution of the quantum many-body problem for neutrons and protons using techniques such as No-Core Shell Model (NCSM), Greens function Monte Carlo (GFMC), and others, to make predictions for the structure and interactions of nuclei for larger systems than can be directly calculated with LQCD. By placing these calculations on a fundamental footing, reliable predictions with quantifiable uncertainties can then be made for larger systems. Chiral Nuclear Forces During the 1990's, the nuclear forces were systematized by the hierarchy emerging from the spontaneously broken chiral symmetries of QCD. The resulting small expansion parameters are powers of the external momenta and powers of the light-quark masses normalized to the scale of chiral symmetry breaking, as pioneered by Weinberg, first in the meson sector and then the multi-nucleon sector [1,2,3]. In addition to generating nuclear forces that are consistent with QCD, this construction provides the calculational advantage of parametric estimates of the systematic uncertainty introduced by the truncation of the nuclear interactions at a given order in the expansion. The actual ordering of contributions remains a subject of debate even today, with Weinberg's chiral expansion of the potential having its peculiar difficulties, as does the KSW expansion of scattering amplitudes [4,5]. Calculations are being performed at a sufficiently high order where the size of truncation errors are quite small. Weinberg's ordering of operators based upon a chiral expansion of the n-body potentials between nucleons has been carried out to N 3 LO, which includes contributions to the three-body (starting at N 2 LO) and the leading four-body interactions (starting at N 3 LO) (for a recent review see Ref. [6]). During the last several years, nuclear structure calculations have been performed with the chiral nuclear forces, leading to both postdictions and predictions for nuclei to a given order in the expansion, and compared with experiment, e.g. see Fig. 1. The nuclear forces that are presently used in such calculations are constrained by ex- Figure 1: NCSM calculations of lowest-lying levels in 7 Li and 8 B using chiral nuclear forces [7]. [Image is reproduced with the permission of P. Maris.] perimental measurements of NN scattering and light nuclei. As the desired precision increases, which requires working to higher orders in the expansion, the number of required experimental constraints increases. Eventually, there are too few experimental constraints to practically reduce the systematic uncertainty below some level in any given calculation. However, LQCD calculations are expected to provide a way to constrain the nuclear forces beyond what is possible with experiment, and hence to further reduce the systematic uncertainties in nuclear structure calculations. Beyond providing direct calculations of important quantities, LQCD calculations of the light nuclei and nuclear forces can 1. verify experimental constraints and/or reduce the uncertainties in the constraints imposed by experiment, 2. constrain components of the nuclear forces that are inaccessible to experiment, for instance the light-quark mass dependences which dictates some of the multipion vertices, and multi-neutron forces, 3. constrain counterterms at higher orders in the expansion to further reduce the systematic uncertainties. Lattice QCD LQCD is a technique in which space-time is discretized into a four-dimensional grid and the QCD path integral over the quark and gluon fields at each point in the grid is performed in Euclidean space-time using Monte Carlo methods. A LQCD calculation of a given quantity will deviate from its value in nature because of the finite volume of the space-time (with L 3 × T lattice points) over which the fields exist, and the finite separation between space-time points (the lattice spacing, b). However, such deviations can be systematically removed by performing calculations in multiple volumes with multiple lattice spacings, and extrapolating using the theoretically known functional dependences on each. Supercomputers are needed for such calculations due to the number of space-time points and the Monte Carlo evaluation of the path integral over the dynamical fields. In order for a controlled continuum extrapolation, the lattice spacing must be small enough to resolve structures induced by the strong dynamics, encapsulated by bΛ χ 1 where Λ χ is the scale of chiral symmetry breaking. Further, in order to have the hadron masses, and also the scattering observables, exponentially close to their infinite-volume values, the lattice volume must be large enough to contain the lightest strongly interacting particle, encapsulated by m π L > ∼ 2π where m π is the mass of the pion and L is the extent of the spatial dimension of the cubic lattice volume (this, of course, can be generalized to non-cubic volumes). Effective field theory (EFT) descriptions of these observables exist for bΛ χ < ∼ 1 (the Symanzik action and its translation into chiral perturbation theory (χPT) and other frameworks) and m π L > ∼ 2π (the p-regime of χPT and other frameworks). The low-energy constants in the appropriate EFT are fit to the results of the LQCD calculations, which are then used to take the limit b → 0 and L → ∞. Computational resources devoted to LQCD calculations are becoming sufficient to be able to perform calculations at the physical values of the light quark masses in large enough volumes and at small enough lattice spacings to be relevant, but the majority of present day calculations are performed with pion masses of m π > ∼ 200 MeV. Therefore, most calculations require the further extrapolation of m q → m phys q , but do not yet include strong isospin breaking or electromagnetism. In principle, the gauge-field configurations that are generated in LQCD calculations can be used to calculate an enormous array of observables, spanning the range from particle to nuclear physics. In practice, this is becoming less common, largely due to the different scales relevant to particle physics and to nuclear physics. Calculations of quantities involving the pion with a mass of m π ∼ 140 MeV are substantially different from those of, say, the triton with a mass of M ( 3 H) ∼ 3 GeV, and with the typical scale of nuclear excitations being ∆E ∼ 1 MeV. Present day dynamical LQCD calculations of nuclear physics quantities are performed with m π ∼ 400 MeV, lattice spacings of b ∼ 0.1 fm and volumes with spatial extent of L ∼ 4 fm. LQCD calculations are approached in the same way that experimental efforts use detectors to measure one or more quantities -the computer is equivalent to the accelerator and the algorithms, software stack, and parameters of the LQCD calculation(s) are the equivalent of the detector. The parameters, such as lattice spacing, quark masses and volume, are selected based upon available computational resources, and simulations of the precision of the calculation(s) required to impact the physical quantity of interest, i.e. simulations of the LQCD Monte Carlos are performed. The size of the computational resources required for cutting edge calculations are such that you only get "one shot at it". A typical work-flow of a LQCD calculation consists of three major components. The first component is the production of an ensemble of gauge-field configurations which contain statistically independent samplings of the gluon fields resulting from the LQCD action. The production of gauge-fields requires the largest partitions on the leadership class computational facilities, typically requiring > ∼ 128K compute cores. Present-day calculations have n f = 0, 2, 2 + 1, 3, 2 + 1 + 1 dynamical light-quark flavors and use the Wilson, O(b)-improved-Wilson, staggered (Kogut-Susskind), domain-wall or overlap discretizations, each of which have their own "features". It is the evaluation of the light-quark determinant (the determinant of a sparse matrix with dimensions > ∼ 10 8 × 10 8 ) that consumes the largest fraction of the resources. Roughly speaking, > ∼ 10 4 Hybrid Monte Carlo (HMC) trajectories are required to produce an ensemble of 10 3 decorrelated gauge fields, but in many instances this is an under estimate. For observables involving quarks, a second component of production is the determination of the light-quark propagators on each of the configurations. The light-quark propagator from a given source point (an example of which is shown in Fig. 2) is determined by an iterative inversion of the quark two-point function, using the conjugate-gradient (CG) algorithm or variants thereof such as BiCGSTAB, or the most recently developed multi-grid (MG). During the last couple of years, the propagator production codes have been ported to run on GPU machines in parallel. GPU's can perform propagator calculations faster than standard CPU's by an order of magnitude, and have led to a major reduction in the statistical uncertainties in many calculations. There have been numerous algorithm developments that have also reduced the resources required for propagator production, such as the implementation of deflation techniques and the use of multi-grid methods. The third component of a LQCD calculation is the production of correlation functions from the light-quark propagators. This involves performing all of the Wick contractions that contribute to a given quantity. The number of contractions required for computing a single hadron correlation function is small. However, to acquire long plateaus in the effective mass plots (EMPs) that persist to short times, Lüscher-Wolff type methods involve the computation of a large number of correlation functions resulting from different interpolating operators, and the number of contractions can become large. In contrast, the naive number of contractions required for a nucleus quickly becomes astronomically large (∼ 10 1500 for uranium), but symmetries in the contractions, and new algorithms (e.g. Ref. [8]) greatly reduce the number of operations that must be performed. A further consequence of the hierarchy of mass scales is that there is an asymptotic signal-to-noise problem in nuclear correlation functions. The ratio of the mean value of the correlation function to the variance of the sample from which the mean is evaluated degrades exponentially at large times. However, this is absent at short and intermediate times and the exponential degradation of the signal-to-noise in the correlation functions can be avoided. Cold Nuclear Physics with Lattice QCD Capability computing resources provided by leadership class computing facilities are used to produce ensembles of gauge-field configurations, while capacity computing resources, both those operated by USQCD and elsewhere are used to perform observabledependent calculations of correlation functions using these configurations. Thus the capability resources enable a multitude of physics calculations to be accomplished with the capacity resources. In the area of cold nuclear physics there is currently a well-defined set of goals, and a program in place to accomplish these goals, as described in one of the 2013 USQCD Whitepapers [9]: Hadron Structure, Hadron Spectroscopy, Hadronic Interactions, Nuclear Forces and Nuclei, and Fundamental Symmetries. The Spectra and Structure of the Hadrons Before calculations of nuclei can be sensibly undertaken, the mass and structure of the nucleon must be reproduced in LQCD calculations. The spectrum of the lowest-lying hadrons calculated with LQCD is shown in Fig. 3, from which we observe that indeed LQCD is postdicting all of the light-hadron masses within uncertainties. Beyond its mass, one property of the nucleon that is well known experimentally is the forward-matrix element of the isovector axial current, g A . Significant effort has been put into calculating g A with LQCD, a summary of which is shown in Fig. 4, but the extrapolated LQCD value has consistently been smaller than the experimental value. With calculations beginning to be performed at the physical pion mass, the community is focused on understanding and quantifying the systematic uncertainties in these calculations. A central element of the physics program at JLab is to determine the excited spectra of mesons and baryons, including searching for exotic states that are beyond the naive nonrelativistic quark model of hadrons, but arise naturally in QCD. A critical component of this program is the LQCD calculations of the spectra. They will play a central role in interpreting and understanding the experimental measurements. The spectra of such states is complicated by the presence of open multi-hadron channels and significant formal developments remain to be put in place before rigorous statements about the spectra can be made. Calculations at unphysical pion masses have been performed by the JLab LQCD group, examples of which are shown in Fig. 5, Figure 4: A summary of LQCD calculations of g A . [12]. [Image is reproduced with the permission of H.-W. Lin.] and remarkable progress has been made in the identification of states in these calculations. The aim is to have LQCD predict the exotic spectra of hadrons before, or at the same time as, the GlueX experiment at JLab runs, targeting the 2018 milestone HP15. ρ K K * η φ N Λ Σ Ξ ∆ Σ * Ξ * Ω π η′ Meson-Meson Scattering Multi-hadron LQCD calculations are significantly more challenging than single-hadron calculations for a number of reasons, and systems involving baryons are even more challenging. Meson-meson systems are the simplest multi-hadron systems, and impressive progress has been made in the recent past, particularly when the LQCD calculations are combined with χPT. There is little or no signal-to-noise problem in such calculations and therefore highly accurate LQCD calculations of stretchedisospin states can be performed with modest computational resources. Moreover, the EFTs which describe the low-energy interactions of pions and kaons, including lattice-spacing and finite-volume effects, have been developed to non-trivial orders in the chiral expansion. The I = 2 pion-pion (π + π + ) scattering length serves as a Figure 6: Constraints on threshold s-wave ππ scattering [16]. [Image in the left panel is reproduced with the permission of H. Leutwyler.] benchmark calculation with an accuracy that can only be aspired to in other systems. The scattering lengths for ππ scattering in the s-wave are uniquely predicted at LO in χPT [14]: m π + a I=0 ππ = 0.1588 , m π + a I=2 ππ = −0.04537 .(1) While experiments do not directly provide stringent constraints on the scattering lengths, a determination of s-wave ππ scattering lengths using the Roy equations has reached a remarkable level of precision [15,16]: m π + a I=0 ππ = 0.220 ± 0.005 , m π + a I=2 ππ = −0.0444 ± 0.0010 .(2) The Roy equations [17] use dispersion theory to relate scattering data at high energies to the scattering amplitude near threshold. At present, LQCD can compute ππ scattering only in the I = 2 channel with precision as the I = 0 channel contains disconnected diagrams which require large computational resources. It is of great interest to compare the precise Roy equation predictions with LQCD calculations, and Fig. 6 summarizes theoretical and experimental constraints on the s-wave ππ scattering lengths [16]. This is clearly a strong-interaction process for which theory has somewhat out-paced the challenging experimental measurements. Mixed-action n f = 2 + 1 LQCD calculations, employing domain-wall valence quarks on a rooted staggered sea and combined with mixed-action χPT, have predicted [18] m π + a I=2 ππ = −0.04330 ± 0.00042 , at the physical pion mass. The agreement between this result and the Roy equation determination is a striking confirmation of the lattice methodology, and a powerful demonstration of the constraining power of chiral symmetry in the meson sector. However, LQCD calculations at one or more smaller lattice spacings, and with different discretizations, are required to verify and further refine this calculation. The ETM collaboration has performed a n f = 2 calculation of the I = 2 ππ scattering length [19], producing a result extrapolated to the physical pion mass of m π + a I=2 ππ = −0.04385 ± 0.00028 ± 0.00038 .(4) It is interesting to compare the pion mass dependence of the meson-meson scattering lengths to the current algebra predictions. In Fig. 7 (left panel) one sees that the I = 2 ππ scattering length is consistent with the current algebra result up to pion masses that are expected to be at the edge of the chiral regime in the two-flavor sector. While in the two-flavor theory one expects fairly good convergence of the chiral expansion and, moreover, one expects that the effective expansion parameter is small in the channel with maximal isospin, the LQCD calculations clearly imply a degree of cancellation between chiral logs and counterterms. However, as one sees in Fig. 7 (right panel), the same phenomenon occurs in K + K + where the chiral expansion is governed by the strange quark mass and is therefore expected to be much more slowly converging. This remarkable conspiracy between chiral logs and counterterms for the meson-meson scattering lengths remains mysterious. Figure 7: m π + a π + π + vs. m π + /f π + (left panel) and m K + a K + K + vs. m K + /f K + (right panel). The solid (red) curves are the current algebra predictions. LQCD calculations of the meson-meson scattering phase-shifts are much less advanced than of the scattering length. This is because the calculation of the phase shift, δ(E), at a given energy, E, requires a LQCD calculation of the two-meson correlation function at the energy E. Generally speaking, a given calculation can determine the lowest few two-hadron energy eigenvalues for a given momentum of the center-of-mass, and that multiple lattice volumes will allow for additional values of E at which to determine δ(E). The first serious calculation of the s-wave (l = 0) I = 2 ππ phase-shift was done by the CP-PACS collaboration with n f = 2 at a relatively large pion mass [20], and more recently two groups have performed calculations at lower pion masses [21,22], the results of which are shown in Fig. 8. Further, in some Figure 8: The π + π + scattering phase-shift. The left panel shows the results of the LQCD calculations below the inelastic threshold (|k| 2 = 3m 2 π ) at a pion mass of m π ∼ 390 MeV [22]. The vertical (blue) line denotes the start of the t-channel cut. The shaded region in the right panel shows the results of the LQCD calculation extrapolated to the physical pion mass using NLO χPT, while the points and uncertainties corresponds to the existing experimental data. The vertical (red) line corresponds to the inelastic threshold. nice work by the Hadron Spectrum Collaboration (HSC), the first efforts have been made to extract the d-wave (l = 2) I = 2 ππ phase shift [21]. One of the more exciting recent results is the mapping out of the ρ-resonance at m π ∼ 390 MeV from the π + π 0 energy-levels using Luscher's method, as shown in Fig. 9 [23]. Nuclear Interactions Calculations of the nucleon-nucleon scattering lengths have been successfully underway for the last decade [25,26,27,28,29,30,31,32,33,34,35,36,37,38] for a range of pion masses. Recently, LQCD calculations have been performed at m π ∼ 800 MeV that also provide the effective ranges [24], the results of which are shown in Fig. 10. Also shown are fits to the effective range expansion (ERE), including the shape parameter. The scattering length and effective range in the 3 S 1 channel determined from the NLO fit to the ERE are m π a ( 3 S1) = 7.45 +0.57 The shape parameter obtained from the NNLO fit to the ERE expansion is: P m 3 π = 2 +5 −6 +5 −6 . An interesting aspect of this result is that the ratio of scattering length to effective range, a measure of the naturalness of the system, is ∼ 2, which is to be compared with ∼ 3 at the physical quark masses. This leads one to speculate that the deuteron might be unnatural over a large range of quark masses and not just close to the physical values, indicating that it is not finely tuned. This speculation requires precise calculations at lighter quark masses to determine if this is, in fact, the situation. Nuclei Perhaps some of the most important LQCD calculations of late are those of the ground states of the light nuclei, including the deuteron, 3 He, 4 He and light hypernuclei. range of the light-quark masses, such calculations are crucial in dissecting and refining the chiral nuclear forces. However, it is clear that calculations at lighter pion masses are required, including at the physical pion mass. A summary of the energy-levels at the flavor SU(3) symmetry point found in the s-shell nuclei and hypernuclei [37] is shown in Fig. 12. These energy levels are elements of SU(3) irreps. which allowed, in some cases, e.g. the H-dibaryon, the hypertriton and 4 ΛΛ He, for distinct energy levels with the same spin and parity to be determined. Such calculations will become somewhat more complicated at lighter quark masses when the up and down quarks are not degenerate with the strange quark. The (3) symmetry (with spin and parity J π ) calculated by NPLQCD [37] at a pion mass of m π ∼ 800 MeV. and become less bound as the quarks become lighter. In the case of the dineutron, which is bound at m π ∼ 800 MeV, it becomes unbound at some intermediate value of the pion mass, giving rise to a neutron-neutron system with an infinite scattering length. One of the interesting aspects of the nuclear forces to explore is the tensor interaction, responsible for the mixing between the S-wave and D-wave channels in the deuteron channel. There are a series of LQCD calculations that can be performed that will permit an extraction of the SD mixing parameter, 1 using Lüscher's method [39,40,41], see Ref. [45]. Roadblocks of the Past It is important to understand how a few of the past roadblocks to progress in this area have been recently overcome. One of the roadblocks of the past was/is the "signalto-noise problem" that afflicts states other than the pion. This problem is seen most simply in the single-nucleon correlation function, generated with a three-quark source and a three-quark sink. The variance of this correlation function is dictated by a 3quark 3-anti-quark source and a 3-quark 3-anti-quark sink, which overlaps with both the NN and 3π intermediate states (and all others with the appropriate quantum numbers). At large times, the variance correlation function is dominated by the 3π intermediate state, while the single nucleon correlation function is dominated by the single nucleon, giving rise to an exponentially degrading signal. However, at intermediate times, the behavior of the "signal-to-noise" is determined by the overlap of the variance sinks and sources onto the intermediate hadronic states. The momentum projection onto single nucleon blocks, that NPLQCD is currently using, provides a volume suppression of the 3π intermediate state compared to the NN state. Thus, there is an intermediate time interval in which the signal-to-noise ratio is not exponentially degrading. It is in this time interval, dubbed the "Golden Window", that plateaus for the low-lying energy levels in light nuclei can be identified. Unfortunately, the window shrinks as the number of nucleons is increased, and so further developments will be required to go to much larger nuclei. A second roadblock that inhibited progress in LQCD calculations of nuclei was the number of Wick contractions required to form a correlation function. A system containing N u up quarks and N d down quarks requires N u !N d ! Wick contractions, which is a rapidly growing number as one moves beyond the nucleon. It was recognized that recursion relations relating the Wick contractions in systems with N mesons can be related to those with N − 1 mesons [42]. Further, somewhat more sophisticated algorithms [43,8] have been developed for the multi-baryon systems that greatly reduce the computing resources required to perform the contractions. These have led to very efficient calculations of the s-shell nuclei and hypernuclei, moving beyond the s-shell requires extensions of these works, and new ideas are required to calculate heavier nuclei. The Bridge Between LQCD and Nuclear Structure One of the points of discussion that came up during this presentation was how to optimally couple the results of LQCD calculations to nuclear structure calculations. Given the expertise in the nuclear structure community, it makes little sense for LQCD theorists to "go it alone" and attempt to calculate the entire periodic table. It makes much more sense for the LQCD theorists to produce sets of quantities that can be handed to the nuclear structure theorists who use them in their machinery to determine the periodic table. The question is what are the optimal quantities to pass along from LQCD. It seems that the minimal set of quantities that could be passed along are the energy eigenvalues for a given system. LQCD calculations of the energy spectrum of an A-nucleon system could be performed in multiple lattice volumes, with multiple lattice spacings and at multiple light-quark masses, and handed to the the nuclear structure theorists who in turn reproduce the energies by tuning the chiral interactions. These tuned interactions are then used to calculate processes in the continuum. This methodology was used to calculate the nΣ − interactions at the physical pion mass using χPT [44]. The chiral interactions were tuned to reproduce the finitevolume energy levels determined in a series of LQCD calculations, and then used to calculate the scattering phase shift at the physical pion mass. Progress in this direction is starting to be made, as demonstrated in recent calculations by Nir Barnea and collaborators [46], by using the ground state energies of the deuteron, dineutron and 3 He at m π ∼ 800 MeV to reproduce the 4 He ground state using the pionless EFT. Summary and Final Comments I have summarized the rapid progress that is being made in developing LQCD into a reliable calculational tool for low-energy nuclear physics. It holds the promise to directly connect the structure and properties of nuclei with QCD, and to enable a refinement of the chiral nuclear forces that are used as input into nuclear structure calculations. At present, the ground states of the s-shell nuclei and hypernuclei are being calculated at unphysically heavy light-quark masses, but within the next few years, such calculations at m π ∼ 140 MeV will be performed (if hardware and software resources increase as expected). Within the next five years, the spectrum and interactions of the lightest nuclei and hypernuclei will be postdicted or predicted with fully-quantified uncertainties. It is worth emphasizing that the LQCD effort in the US relies heavily on Sci-DAC funding to support the scientists who develop and optimize the software to run on the rapidly evolving computational hardware, e.g. GPU-accelerated compute nodes that comprise Titan at ORNL, or the BG/Qs at ANL and LLNL. Further, the effort requires ongoing access to both capability computing resources on leadershipclass computing facilities, and capacity computing obtained from NERSC, XSEDE, through USQCD and at local compute clusters. Ongoing software (see Fig. 13) and hardware support are critical to progress in this area. Figure 13: Multigrid is a recent algorithmic development to be implemented in LQCD calculations [47]. The horizontal (orange) cost estimates (that I have added to the original figure) provide one example of what is possible for a given production scenario. [Parts of this image [48] are reproduced with the permission of B. Joo.] Ideally, one would start with a LQCD calculation and predict all of the quantities of interest in low-energy nuclear physics. Presently, we are not in a position to do this, even if significantly more computing resources were provided to the program. While Lüscher provided the formalism to relate the two-body S-matrix directly to two-particle energy levels inside a cubic volume with the fields subject to periodic boundary conditions [39,40], which has since been understood and generalized to the two-nucleon systems, e.g. Ref. [41], such formalism is complicated to apply in coupled-channels systems [49,50,51]. Further, the formalism is not in place for the three-and higher-body sectors, but progress is being made in such systems [52,53]. In closing, great progress is being made to reliably determine and refine the nuclear forces directly from QCD using Lattice QCD. Figure 2 : 2An example of (the real part of one component of) a light-quark propagator. The (blue) "wall" corresponds to the anti-periodic boundary conditions imposed in the time direction. [Image is reproduced with the permission of R. Gupta.] Figure 3 : 3A summary of the low-lying hadron masses calculated with LQCD[10,11].[Image is reproduced with the permission of A.Kronfeld.] Figure 5 : 5The spectra of isoscalar mesons calculated at m π ∼ 396 MeV by the JLab LQCD group[13] [Image is reproduced with the permission of R. Edwards.] Figure 9 : 9The ρ-resonance at a pion mass of m π ∼ 390 MeV[23] [Image is reproduced with the permission of R. Edwards.] Figure 10 : 10.49 , m π r ( 3 S1) = 3.71 +0.28 −0.31 +0.28 −0.35 , a ( 3 S1) = 1.82 +0.14 −0.13 +0.17 −0.12 fm , r ( 3 S1) = 0.906 +0.068 The left panel shows the NN scattering phase shift in the 3 S 1 channel extracted from LQCD calculations at the SU(3) symmetric point, including the fit to the ERE at N 2 LO. The right panel shows the ratio of the scattering length to effective range, a quantity that is a measure of the naturalness of the system. Figure 11 : 11Fig. 11shows the binding energy of the deuteron,3 He and 4 He[35,37,38] as a function of the pion mass. Not only is it exciting to see nuclei emerge from QCD for a The deuteron (left panel),3 He (middle panel) and 4 He (right panel) binding energies from n f = 2 + 1 LQCD calculations[35,37,38]. calculations of NPLQCD and those of Yamazaki et al are already shedding light on how the ground-state energies of the light nuclei approach their values at the physical light-quark masses. They are all bound at the heavier light- Figure 12 : 12A compilation of the energy levels in light nuclei and hypernuclei in the limit of flavor SU Happy Birthday James: James Vary is one of the first nuclear theorists I met when I arrived in the United States to enter the PhD program at Caltech in the mid 1980's. I recall James taking the time to talk physics with me during his stay. His detailed knowledge of, and passion for, important problems of the day left a lasting impression on me. Despite having been able to chat with, and even collaborate with, James since that time, when I learned that this conference was in part to celebrate James's 70th birthday, I was taken aback as it seems like yesterday that he was in his early 40's (and I was in my early 20's), and he has retained the same passion and energy for science. I should also add that James is responsible for me remembering the value of c! Happy 70 th !! . S Weinberg, Phys. Lett. B. 251288S. Weinberg, Phys. Lett. B 251, 288 (1990). . S Weinberg, Nucl. Phys. B. 3633S. Weinberg, Nucl. Phys. B 363, 3 (1991). . S Weinberg, hep-ph/9209257Phys. Lett. B. 295114S. Weinberg, Phys. Lett. B 295, 114 (1992) [hep-ph/9209257]. . D B Kaplan, M J Savage, M B Wise, nucl-th/9801034Phys. Lett. B. 424390D. B. Kaplan, M. J. Savage and M. B. Wise, Phys. Lett. B 424, 390 (1998) [nucl-th/9801034]. . D B Kaplan, M J Savage, M B Wise, nucl-th/9802075Nucl. Phys. B. 534329D. B. Kaplan, M. J. Savage and M. B. Wise, Nucl. Phys. B 534, 329 (1998) [nucl-th/9802075]. . E Epelbaum, U. -G Meissner, arXiv:1201.2136Ann. Rev. Nucl. Part. Sci. 62159nucl-thE. Epelbaum and U. -G. Meissner, Ann. Rev. Nucl. Part. Sci. 62, 159 (2012) [arXiv:1201.2136 [nucl-th]]. . P Maris, J P Vary, P Navratil, arXiv:1205.5686Phys. Rev. C. 8714327nucl-thP. Maris, J. P. Vary and P. Navratil, Phys. Rev. C 87, 014327 (2013) [arXiv:1205.5686 [nucl-th]]. . W Detmold, K Orginos, arXiv:1207.1452Phys. Rev. D. 87114512hep-latW. Detmold and K. Orginos, Phys. Rev. D 87, 114512 (2013) [arXiv:1207.1452 [hep-lat]]. . A S Kronfeld, arXiv:1203.1204Ann. Rev. Nucl. Part. Sci. 62heplatA. S. Kronfeld, Ann. Rev. Nucl. Part. Sci. 62, 265 (2012) [arXiv:1203.1204 [hep- lat]]. . A S Kronfeld, arXiv:1209.3468physics.hist-phA. S. Kronfeld, arXiv:1209.3468 [physics.hist-ph]. . H. -W Lin, arXiv:1212.6849PoS. 201213hep-latH. -W. Lin, PoS LATTICE 2012, 013 (2012) [arXiv:1212.6849 [hep-lat]]. . J J Dudek, R G Edwards, B Joo, M J Peardon, D G Richards, C E Thomas, arXiv:1102.4299Phys. Rev. D. 83111502hep-latJ. J. Dudek, R. G. Edwards, B. Joo, M. J. Peardon, D. G. Richards and C. E. Thomas, Phys. Rev. D 83, 111502 (2011) [arXiv:1102.4299 [hep-lat]]. . S Weinberg, Phys. Rev. Lett. 17616S. Weinberg, Phys. Rev. Lett. 17, 616 (1966). . G Colangelo, J Gasser, H Leutwyler, arXiv:hep-ph/0103088Nucl. Phys. B. 603125G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B 603, 125 (2001) [arXiv:hep-ph/0103088]. . H Leutwyler, arXiv:0812.4165PoS C. 868hepphH. Leutwyler, PoS C CONFINEMENT8, 068 (2008) [arXiv:0812.4165 [hep- ph]]. . S M Roy, Phys. Lett. B. 36353S. M. Roy, Phys. Lett. B 36, 353 (1971). . S R Beane, T C Luu, K Orginos, A Parreno, M J Savage, A Torok, A Walker-Loud, arXiv:0706.3026Phys. Rev. D. 7714505hep-latS. R. Beane, T. C. Luu, K. Orginos, A. Parreno, M. J. Savage, A. Torok and A. Walker-Loud, Phys. Rev. D 77, 014505 (2008) [arXiv:0706.3026 [hep-lat]]. . X Feng, K Jansen, D B Renner, arXiv:0909.3255Phys. Lett. B. 684268hep-latX. Feng, K. Jansen and D. B. Renner, Phys. Lett. B 684, 268 (2010) [arXiv:0909.3255 [hep-lat]]. . T Yamazaki, CP-PACS CollaborationarXiv:hep-lat/0402025Phys. Rev. D. 7074513T. Yamazaki et al. [CP-PACS Collaboration], Phys. Rev. D 70, 074513 (2004) [arXiv:hep-lat/0402025]. . J J Dudek, R G Edwards, M J Peardon, D G Richards, C E Thomas, arXiv:1011.6352Phys. Rev. D. 8371504hep-phJ. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas, Phys. Rev. D 83, 071504 (2011) [arXiv:1011.6352 [hep-ph]]. . S R Beane, NPLQCD CollaborationarXiv:1107.5023hep-latS. R. Beane et al. [NPLQCD Collaboration], arXiv:1107.5023 [hep-lat]. . J J Dudek, R G Edwards, C E Thomas, arXiv:1212.0830Phys. Rev. D. 87334505hep-phJ. J. Dudek, R. G. Edwards and C. E. Thomas, Phys. Rev. D 87, no. 3, 034505 (2013) [arXiv:1212.0830 [hep-ph]]. . S R Beane, E Chang, S D Cohen, W Detmold, P Junnarkar, H W Lin, T C Luu, K Orginos, arXiv:1301.5790arXiv:1301.5790Phys. Rev. C. 8824003hep-lat. hep-latS. R. Beane, E. Chang, S. D. Cohen, W. Detmold, P. Junnarkar, H. W. Lin, T. C. Luu and K. Orginos et al., Phys. Rev. C 88, 024003 (2013) [arXiv:1301.5790 [hep-lat], arXiv:1301.5790 [hep-lat]]. . M Fukugita, Y Kuramashi, H Mino, M Okawa, A Ukawa, arXiv:hep-lat/9407012Phys. Rev. Lett. 732176M. Fukugita, Y. Kuramashi, H. Mino, M. Okawa and A. Ukawa, Phys. Rev. Lett. 73, 2176 (1994) [arXiv:hep-lat/9407012]. . M Fukugita, Y Kuramashi, M Okawa, H Mino, A Ukawa, arXiv:hep-lat/9501024Phys. Rev. D. 523003M. Fukugita, Y. Kuramashi, M. Okawa, H. Mino and A. Ukawa, Phys. Rev. D 52, 3003 (1995) [arXiv:hep-lat/9501024]. . S R Beane, P F Bedaque, K Orginos, M J Savage, arXiv:hep-lat/0602010Phys. Rev. Lett. 9712001S. R. Beane, P. F. Bedaque, K. Orginos and M. J. Savage, Phys. Rev. Lett. 97, 012001 (2006) [arXiv:hep-lat/0602010]. . S R Beane, NPLQCD CollaborationarXiv:0912.4243Phys. Rev. D. 8154505hep-latS. R. Beane et al. [NPLQCD Collaboration], Phys. Rev. D 81, 054505 (2010) [arXiv:0912.4243 [hep-lat]]. . N Ishii, S Aoki, T Hatsuda, arXiv:nucl-th/0611096Phys. Rev. Lett. 9922001N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99, 022001 (2007) [arXiv:nucl- th/0611096]. . S Aoki, T Hatsuda, N Ishii, arXiv:0805.2462Comput. Sci. Dis. 115009hep-phS. Aoki, T. Hatsuda and N. Ishii, Comput. Sci. Dis. 1, 015009 (2008) [arXiv:0805.2462 [hep-ph]]. . S Aoki, T Hatsuda, N Ishii, arXiv:0909.5585Prog. Theor. Phys. 12389hep-latS. Aoki, T. Hatsuda and N. Ishii, Prog. Theor. Phys. 123, 89 (2010) [arXiv:0909.5585 [hep-lat]]. . T Yamazaki, Y Kuramashi, A Ukawa, arXiv:1105.1418hep-latT. Yamazaki, Y. Kuramashi, A. Ukawa, arXiv:1105.1418 [hep-lat]. . T Yamazaki, Y Kuramashi, A Ukawa, arXiv:0912.1383Phys. Rev. D. 81111504hep-latT. Yamazaki, Y. Kuramashi, A. Ukawa, Phys. Rev. D 81, 111504 (2010) [arXiv:0912.1383 [hep-lat]]. . P De Forcrand, M Fromm, arXiv:0907.1915Phys. Rev. Lett. 104112005hep-latP. de Forcrand and M. Fromm, Phys. Rev. Lett. 104, 112005 (2010) [arXiv:0907.1915 [hep-lat]]. . S R Beane, NPLQCD CollaborationarXiv:1109.2889Phys. Rev. D. 8554511hep-latS. R. Beane et al. [NPLQCD Collaboration], Phys. Rev. D 85, 054511 (2012) [arXiv:1109.2889 [hep-lat]]. . T Inoue, HAL QCD CollaborationarXiv:1112.5926Nucl. Phys. A. 88128hep-latT. Inoue et al. [HAL QCD Collaboration], Nucl. Phys. A 881, 28 (2012) [arXiv:1112.5926 [hep-lat]]. . S R Beane, E Chang, S D Cohen, W Detmold, H W Lin, T C Luu, K Orginos, A Parreno, arXiv:1206.5219hep-latS. R. Beane, E. Chang, S. D. Cohen, W. Detmold, H. W. Lin, T. C. Luu, K. Orginos and A. Parreno et al., arXiv:1206.5219 [hep-lat]. . T Yamazaki, K. -I Ishikawa, Y Kuramashi, A Ukawa, arXiv:1207.4277Phys. Rev. D. 8674514hep-latT. Yamazaki, K. -I. Ishikawa, Y. Kuramashi and A. Ukawa, Phys. Rev. D 86, 074514 (2012) [arXiv:1207.4277 [hep-lat]]. . M Luscher, Commun. Math. Phys. 105153M. Luscher, Commun. Math. Phys. 105, 153 (1986). . M Luscher, Nucl. Phys. B. 354531M. Luscher, Nucl. Phys. B 354, 531 (1991). . R A Briceno, Z Davoudi, T C Luu, arXiv:1305.4903Phys. Rev. D. 8834502Phys. Rev. D. hep-latR. A. Briceno, Z. Davoudi and T. C. Luu, Phys. Rev. D 88, 034502 (2013) [Phys. Rev. D 88, 034502 (2013)] [arXiv:1305.4903 [hep-lat]]. . W Detmold, M J Savage, arXiv:1001.2768Phys. Rev. D. 8214511hep-latW. Detmold and M. J. Savage, Phys. Rev. D 82, 014511 (2010) [arXiv:1001.2768 [hep-lat]]. . T Doi, M G Endres, arXiv:1205.0585Comput. Phys. Commun. 184hep-latT. Doi and M. G. Endres, Comput. Phys. Commun. 184, 117 (2013) [arXiv:1205.0585 [hep-lat]]. . S R Beane, E Chang, S D Cohen, W Detmold, H. -W Lin, T C Luu, K Orginos, A Parreno, arXiv:1204.3606Phys. Rev. Lett. 109172001hep-latS. R. Beane, E. Chang, S. D. Cohen, W. Detmold, H. -W. Lin, T. C. Luu, K. Orginos and A. Parreno et al., Phys. Rev. Lett. 109, 172001 (2012) [arXiv:1204.3606 [hep-lat]]. . R A Briceno, Z Davoudi, T Luu, M J Savage, arXiv:1309.3556hep-latR. A. Briceno, Z. Davoudi, T. Luu and M. J. Savage, arXiv:1309.3556 [hep-lat]. Nuclear Physics with heavy pions -EFT for LQCD, a talk presented at the Institute for Nuclear Theory. N Barnea, N. Barnea, Nuclear Physics with heavy pions -EFT for LQCD, a talk presented at the Institute for Nuclear Theory, July 31, 2013. . J C Osborn, R Babich, J Brannick, R C Brower, M A Clark, S D Cohen, C Rebbi, arXiv:1011.2775PoS. 201037hep-latJ. C. Osborn, R. Babich, J. Brannick, R. C. Brower, M. A. Clark, S. D. Cohen and C. Rebbi, PoS LATTICE 2010, 037 (2010) [arXiv:1011.2775 [hep-lat]]. Computational Challenges in Cold QCD at the Computational Nuclear Physics Meeting. B. JooWashington D.C.Computational Challenges in Cold QCD at the Computational Nuclear Physics Meeting, Washington D.C., July 23-24, 2012, by B. Joo. . M T Hansen, S R Sharpe, arXiv:1204.0826Phys. Rev. D. 8616007hep-latM. T. Hansen and S. R. Sharpe, Phys. Rev. D 86, 016007 (2012) [arXiv:1204.0826 [hep-lat]]. . R A Briceno, Z Davoudi, arXiv:1212.3398Phys. Rev. D. 8794507hep-latR. A. Briceno and Z. Davoudi, Phys. Rev. D 87, 094507 (2013) [arXiv:1212.3398 [hep-lat]]. . P Guo, J Dudek, R Edwards, A P Szczepaniak, arXiv:1211.0929Phys. Rev. D. 8814501hep-latP. Guo, J. Dudek, R. Edwards and A. P. Szczepaniak, Phys. Rev. D 88, 014501 (2013) [arXiv:1211.0929 [hep-lat]]. . R A Briceno, Z Davoudi, arXiv:1204.1110hep-latR. A. Briceno and Z. Davoudi, arXiv:1204.1110 [hep-lat]. . K Polejaeva, A Rusetsky, arXiv:1203.1241Eur. Phys. J. A. 4867hep-latK. Polejaeva and A. Rusetsky, Eur. Phys. J. A 48, 67 (2012) [arXiv:1203.1241 [hep-lat]].
[]
[ "Shaken Granular Lasers", "Shaken Granular Lasers" ]
[ "Viola Folli \nISC-CNR\nUOS Sapienza\nP.le Aldo Moro 500185RomeItaly\n\nDepartment of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)\n", "Andrea Puglisi \nISC-CNR\nUOS Sapienza\nP.le Aldo Moro 500185RomeItaly\n\nDepartment of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)\n", "Luca Leuzzi \nDepartment of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)\n\nIPCF-CNR\nUOS\n", "Claudio Conti \nISC-CNR\nUOS Sapienza\nP.le Aldo Moro 500185RomeItaly\n\nDepartment of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)\n", "Roma Kerberos ", "P Le ", "Aldo Moro ", "Rome " ]
[ "ISC-CNR\nUOS Sapienza\nP.le Aldo Moro 500185RomeItaly", "Department of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)", "ISC-CNR\nUOS Sapienza\nP.le Aldo Moro 500185RomeItaly", "Department of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)", "Department of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)", "IPCF-CNR\nUOS", "ISC-CNR\nUOS Sapienza\nP.le Aldo Moro 500185RomeItaly", "Department of Physics\nUniversity Sapienza\nP.le Aldo Moro 500185RomeIT)" ]
[]
Granular materials have been studied for decades, also driven by industrial and technological applications. These very simple systems, composed by agglomerations of mesoscopic particles, are characterized, in specific regimes, by a large number of metastable states and an extreme sensitivity (e.g., in sound transmission) on the arrangement of grains; they are not substantially affected by thermal phenomena, but can be controlled by mechanical solicitations. Laser emission from shaken granular matter is so far unexplored; here we provide experimental evidence that it can be affected and controlled by the status of motion of the granular, we also find that competitive random lasers can be observed. We hence demonstrate the potentialities of gravity affected moving disordered materials for optical applications, and open the road to a variety of novel interdisciplinary investigations, involving modern statistical mechanics and disordered photonics.
10.1103/physrevlett.108.248002
[ "https://arxiv.org/pdf/1205.5977v1.pdf" ]
19,021,253
1205.5977
23d005f3877b775abe673e7737e899ae0f9e85cf
Shaken Granular Lasers 27 May 2012 Viola Folli ISC-CNR UOS Sapienza P.le Aldo Moro 500185RomeItaly Department of Physics University Sapienza P.le Aldo Moro 500185RomeIT) Andrea Puglisi ISC-CNR UOS Sapienza P.le Aldo Moro 500185RomeItaly Department of Physics University Sapienza P.le Aldo Moro 500185RomeIT) Luca Leuzzi Department of Physics University Sapienza P.le Aldo Moro 500185RomeIT) IPCF-CNR UOS Claudio Conti ISC-CNR UOS Sapienza P.le Aldo Moro 500185RomeItaly Department of Physics University Sapienza P.le Aldo Moro 500185RomeIT) Roma Kerberos P Le Aldo Moro Rome Shaken Granular Lasers 27 May 2012arXiv:1205.5977v1 [physics.optics] Granular materials have been studied for decades, also driven by industrial and technological applications. These very simple systems, composed by agglomerations of mesoscopic particles, are characterized, in specific regimes, by a large number of metastable states and an extreme sensitivity (e.g., in sound transmission) on the arrangement of grains; they are not substantially affected by thermal phenomena, but can be controlled by mechanical solicitations. Laser emission from shaken granular matter is so far unexplored; here we provide experimental evidence that it can be affected and controlled by the status of motion of the granular, we also find that competitive random lasers can be observed. We hence demonstrate the potentialities of gravity affected moving disordered materials for optical applications, and open the road to a variety of novel interdisciplinary investigations, involving modern statistical mechanics and disordered photonics. Introduction -In random lasers (RLs) stimulated emission is achieved by disorder-induced light scattering [1][2][3][4][5][6][7][8][9][10][11], as observed in colloidal systems, composed by small particles suspended in thermal equilibrium in a solution, or in materials exhibiting a fixed disorder, achieved, e.g., by nano-fabrication. RLs in shaken grains were not reported. Granular materials (sands, powders, seeds, cements, etc.) [12,13] are an extensively studied branch of statistical mechanics, with several important applications in chemistry or engineering. These systems are not affected by temperature, and are mostly dominated by dynamical effects, while being one of the paradigms of the statistical mechanics of disordered systems and still lacking general and universal theoretical descriptions. Granular gases [14,15], i.e., massive particles in rapid movement with inelastic collisions, are obtained by putting grains under mechanical oscillation. By a driving solicitation, a gravity-sedimented ensemble of grains switches, above a critical mechanical energy, from a solid-like state to a gaseous one, whose essential feature is the strong enhancement of fluctuations and the non-equilibrium character [16,17]: even in such a dilute configuration, regions with high density may appear. Such a state can only be maintained by continuously furnishing mechanical energy. This circumstance may have relevant implications when considering random lasing in shaken granulars, which happens when energy is furnished to the system not only mechanically, but also optically, by employing a lightemitting active medium. The specific and characteristic arrangements of the shaken grains not only can alter the RL features, but, as we demonstrate in this work, in the gaseous-like phase, may lead to the occurrence of competing RL emissions, which can be controlled by acting on the external mechanical solicitation. Such a situation is not achievable in formerly considered RL: in the fixed disorder case [4], the structure cannot be externally changed; while in the colloidal RL [6] the considered dielectric nano-particles, [18][19][20][21] are too light to exhibit a switchable granular behavior. Various authors also investigated RL by metallic nano-particles [22][23][24][25], however, so far, only particles with diameters of tens of nanometers were considered, which do not exhibit granular behavior because are not substantially affected by gravity. In these systems, thermal equilibrium largely limits the observable fluctuations with respect to out-of-equilibrium granulars. Here we consider a gravity-affected granular system composed by metallic grains with millimeter size, able to macroscopically change its structural features when the status of motion is altered. We study how the dynamic structural phases affect the RL emission. We find that an alteration of the state of motion of the grains forming the disordered laser cavity dramatically changes the RL emission, and sustains competitive forms of RL. Setup -Our sample is composed by about 1500 amagnetic-steel spherical metallic grains with diameter of 1 mm dispersed in a liquid RhodamineB solution (whose fluorescence display a broad peak around 590 nm). As sketched in Figure 1A, the sample is put on a vertically (z−direction) vibrating plate, driven by a sound woofer, oscillating with amplitude a chosen in the normalized range [0, 1] and calibrated by an accelerometer (a = 0.1 corresponds to an acceleration of 15g and oscillation amplitude of 0.76mm). All the structure is placed on a vertical motorized 25 mm translational stage. We first consider the trasmission of a continuous wave (CW) laser to determine the critical oscillation amplitude a for the gaseous phase: for any value of a, we make a vertical scan of the sample. This allows to determine the CW transmitted power at any vibrational regime and versus the position of the input laser (height z from the bottom of the sample). The measured transmission is averaged over several temporal periods of the driving sinusoidal signal. As shown in Fig.1B, for small a, the transmission makes an abrupt changes versus z, as the grains are deposited in the bottom, and the sample transmits only when the beam is above the grains. When the amplitude is greater than a threshold value a ∼ = 0.1, the transmission profile becomes independent on a and z; note that the region around z = 5 mm is affected by the transverse beam size (beam waist 6 mm). This change in the trend of the transmission signals the transition to the gaseous state, as the grains in the shaken regime have sufficient energy to uniformly explore the whole available volume, and correspondingly intercept the beam and lower the time-averaged transmission. The CW beam hence simply allows to determine the onset of the gaseous state, and the corresponding critical value for a. Shaken granular laser -We then use an high energy pump beam (532 nm Nd:YAG, 10 Hz repetition rate and 7 ns pulse duration) to study RL spectra in the granular dynamic phases. We fix the vertical position z of the 532 nm pump with sample at rest (a = 0) and then vary the input pump energy E and the vibrational amplitude a. The interest here is to show the way spectra are affected by the mechanical energy furnished to the granular by the sound woofer. We show in Fig.2A-F, spectra obtained with exposure time of 1s, corresponding to 10 single-shot average spectra; single shot spectra are shown in Fig.2G. We start considering the bottom layers of granular (z = 3 mm). As shown in Fig.2A, at the pumping energy E = 1.6mJ, RL is above threshold, a peak is visible around 600 nm, which gets narrower when increasing the pump energy (compare panel A with panel B) [6,8]. The peak is strongly affected by the amplitude a (note that for a = 0 a peak at 600nm is visible, which is not present in the fluorescence of the Rhodamine solutions, shown, e.g., in 2E for a = 0, see also the discussion below). Such a mechanism is also found at higher energies E = 3.7mJ (Fig.2B), however we find that above a threshold oscillation amplitude (corresponding to the onset of the gaseous phase in 1B,C), an additional peak aroud 620nm appears in the spectrum (more evident when pumping in the "central" and "top" regions, discussed below). This peak becomes more pronounced when increasing a. As further detailed below, this additional RL, typical of the considered granular sample, competes with the other RL at 600nm, when the pumping optical and mechanical energies are sufficiently high. We repeat the experiments by placing the pump beam at the edge between the grains and the liquid region of Rhodamine (z = 7mm), when the sample is at rest (a = 0). As shown in Fig.2C,D, the previously described phenomena are qualitatively reproduced, when changing a. However, the RL at 600nm only appears in the presence of shaking (which allows the grains to uniformly distribute within the pumped volume) and is more pronounced because of the higher amount of Rhodamine in the central region with respect to the bottom region discussed above. At higher energies and higher oscillation amplitudes (Fig.2D), the additional peak at λ = 620nm is observed in the gaseous phase. When pumping in the top region (z = 13mm) in the absence of vibration no grains are present, and RL is not observed. As shown in Fig.2E, at energy E = 1.6mJ, at a = 0 and a = 0.07, only the fluorescence of the Rhodamine is retrieved with the characteristic broad peak around 590nm. When the sample is put into vibration, the grains progressively fill the liquid region and the RL at λ = 600nm observed. At higher energy levels, the additional peak at λ = 620 nm is more evident with respect to cases considered before. At sufficiently pronounced vibration, this peak is as much intense as that at 600nm (see also Fig.3); however, at higher oscillations, the amplitude of this peak decreases again, such that an optimal amplitude exists for its observation (Fig.3D). To address the origin of this additional peak, we show in figure Fig.2G a number of single shot measurements, which unveil that, at variance with the smooth broad peak at λ = 600nm, also present in the single-short regime, the peak at 620nm is actually composed by many narrow peaks radically changing from shot to shot, and appear smoother in figures 2B,D,F because these correspond to averages over ten shots (exposure time 1s). We interpret these additional peaks as due to optical cavities formed by few metallic grains when the sample is put under shaking because of the instantaneous arrangement of the reflecting metallic spheres. These arrangements involve few grains and change from shot to shot, hence peaks are retrieved only for shots sustaining such specific configurations. Experiments involving a limited number of scatterers, namely segments of dye doped fibers, have shown that Lévy flights of photons may give rise to resonant like emission [26]. In our case, we are dealing with a vibrated granular system, and our results look to be another manifestation of large fluctuations often observed in granular systems, due to both the high degree of non-equilibrium correlations and, at the same time, to their intrinsic nature of "small systems" (indeed, as stated above, the number of grains is of the order of 10 3 ). In this respect, our experimental results can also be interpreted as the occurrence of large amplified photon paths ("lucky photons") [27] due to the specific density fluctuations occurring in a granular under shaking. In Fig.3, we show the trend of the peak intensity versus a for the two RL for z = 13mm. The peak of the RL at 600nm (panel A of Fig.3) grows with the amplitude of oscillations. This is explained by observing that increasing the amplitude of oscillations implies increasing the interstitial holes; correspondingly, the available active medium is larger and the threshold for lasing decreases. For the RL at 620nm, panel B shows a peak for a = 0.1. In panel 3C, the relative peak intensities are reported: for a specific oscillation a the peak at 620nm becomes more pronounced than that at 600nm. To investigate the coherent origin of the peak at 620nm, we also considered a sample composed by the metallic granular system with a small concentration of dielectric ZnO particles with diameter much smaller than the grains (d small =300nm); the results (not reported) show that a small scattering inhibits the onset of the additional peak even in the presence of a very small concentration of the nanometric dielectric particles. To verify that the observed localized emission is not due to the single metallic sphere, we also repeated the experiments by considering a two dimensional system, realized by a thin sample (section 1mm×1cm), such that only one layer of grain is formed, and no peak at 620nm was observed, but only that at 600nm (not reported). We hence infer that the emission at 620nm can be ascribed to a specific three-dimensional arrangement of spheres during shaking. This fact also allows to rule out the effect of plasmonic resonances, which are expected to play a negligible rule for the considered size of the grain (1 mm), which is much greater than the wavelength. Plasmonic resonances are indeed know to be relevant for nanoparticles [22][23][24][25]. These resonances are also ruled out by the fact that the peak at 620nm changes from shot to shot (Fig.2G), and hence depends on the config-uration of several spheres, and not on the single sphere. Conclusions -We have reported on the first experimental evidence of laser emission in shaken granular gases. Our results, placed between modern photonics and statistical mechanics, demonstrate that RL in granulars are sensitive to the specific grain distribution and their emission can be controlled by the status of motion of the system. It is clear that a granular configuration -even in regimes of rapid shaking -is frozen at laser time-scales. Nevertheless the statistics of many emissions allow to probe those of granular configurations visited in a particular dynamical regime, which depends on shaking conditions, boundaries, grains, etc. An analytical assessment of such connection is beyond the scope of this Letter, we believe however that our observation is already sufficient to raise the case of a (statistical) connection between different granular dynamical regimes and RL: figures 2 and 3 provide a quantitative evidence by assessing a limited parameter region with competing RL. Configurations optimal the RL at 620nm seem more frequent at those working points where the "density" profile ( fig. 1C) is not too high and has a gradient large enough. Note that large density gradients in granular materials are typically associated with clustering and shear instabilities [28]. These instabilities are likely to support wide local fluctuations allowing the system to explore a much larger space of configurations for cavities, accessing also those able to sustain lasing effects. Our results open the way to a variety of further investigations, as RL in matter under common granular processes like compactification, metastable granular states, mixed systems, accelerated flow under gravity, supercontinuum generation, and the interaction of light with granular waves. The possibility of achieving a controlled nonequilibrium scenario in granular systems provides a variety of novel tools for the mechanical control of random photonic devices, and ultimately, for assessing the interplay between the status of motion of mesoscopic matter and light. FIG. 1 : 1Experimental setup: (A) Sketch of the granular sample placed on a vibrating woofer and a vertical translational stage, the light beam is also shown; (B) transmission of 1064nm continuous-wave beam for different amplitudes a of the driving force; above a critical amplitude the transmission is independent on the value of a for an increasing range in z, denoting the transition to a gaseous region; (C) as in (B) with vertical log scale. FIG. 2 : 2Spectra of shaken granular lasers. (A-F) Emission at pumping energy of 1.6mJ (left column), and 3.7mJ (right column), for the bottom region (z = 3mm, panels A,B), the central region (z = 7 mm, panels CD) and the top region (z = 13mm, panels E,F). As indicated in the legend, the blue line corresponds to a=0, green line to a=0.07, red line to a=0.1, sky-blue line to a=0.2; (G) single shot spectra at energy 5.5mJ, z = 13mm liquid region, and a = 0.2. The insets show a sketch of the sample and the arrow indicates the position of the beam with respect to the bottom. The spectra are arbitrarily shifted in the vertical direction. FIG. 3 : 3Competion of random lasers. In the liquid region z = 13mm, peak intensity for the RL at 600nm (A) and at 620nm (B) versus the oscillation amplitude a; the ratio of peak at 620nm and that at 600nm is shown in panel (C). The research leading to these results has received funding from the European Research Council under the European's Seventh Framework Program (FP7/2007-2013)/ERC grant agreement n. 201766, project Light and Complexity; from the Italian MIUR under the FIRB-IDEAS grant number RBID08Z9JE and under the Basic Research Investigation Fund (FIRB/2008) program grant code RBFR08M3P4. We thank MD Deen Islam for help in the development of the experimental setup, and Dr. Neda Ghofraniha for the support in the laboratory. . D S Wiersma, M P Vanalbada, A Lagendijk, Nature. 373203D. S. Wiersma, M. P. Vanalbada, and A. Lagendijk, Na- ture 373, 203 (1995). . D S Wiersma, Nature Physics. 4359D. S. Wiersma, Nature Physics 4, 359 (2008). . H Cao, J Y Xu, E W Seelig, R P H Chang, Applied Physics Letters. 762997H. Cao, J. Y. Xu, E. W. Seelig, and R. P. H. Chang, Applied Physics Letters 76, 2997 (2000). H Cao, Waves in Random Media and Complex Media. 131H. Cao, Waves in Random Media and Complex Media 13, R1 (2003). . R M Balachandran, N M Lawandy, J A Moon, Opt. Lett. 22319R. M. Balachandran, N. M. Lawandy, and J. A. Moon, Opt. Lett. 22, 319 (1997). . N M Lawandy, R M Balachandran, A S L Gomes, E Sauvain, Nature. 368436N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994). . W Guerin, F Michaud, R Kaiser, Phys. Rev. Lett. 10193002W. Guerin, F. Michaud, and R. Kaiser, Phys. Rev. Lett. 101, 093002 (2008). . C Conti, M Leonetti, A Fratalocchi, L Angelani, G Ruocco, Phys. Rev. Lett. 101143901C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008). . L Leuzzi, C Conti, V Folli, L Angelani, G Ruocco, Phys. Rev. Lett. 10283901L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009). . J Andreasen, A A Asatryan, L C Botten, M A Byrne, H Cao, L Ge, L Labonté, P Sebbah, A D Stone, H E Türeci, Adv. Opt. Photon. 388J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, et al., Adv. Opt. Photon. 3, 88 (2011). . M Leonetti, C Conti, C Lopez, 1749-4885Nat Photon. 5M. Leonetti, C. Conti, and C. Lopez, Nat Photon 5, 615 (2011), ISSN 1749-4885. . H M Jaeger, S R Nagel, R Sydney, Science. 2551523H. M. Jaeger, S. R. Nagel, and R. Sydney, Science 255, 1523 (1992). . H M Jaeger, S R Nagel, R P Behringer, Rev. Mod. Phys. 681259H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod. Phys. 68, 1259 (1996). . T Pöschel, S Luding, Granular Gases, Springer VerlagBerlinT. Pöschel and S. Luding, Granular Gases (Springer Ver- lag, Berlin, 2000). Granular Gas Dynamics. T. Pöschel and N. BrilliantovBerlinSpringer624T. Pöschel and N. Brilliantov, eds., Granular Gas Dy- namics (Springer, Berlin, 2003), Lecture Notes in Physics 624. J A Mclennanm, Introduction to Non Equilibrium Statistical Mechanics. Prentice-HallJ. A. McLennanm, Introduction to Non Equilibrium Sta- tistical Mechanics (Prentice-Hall, 1988). . A Puglisi, P Visco, A Barrat, E Trizac, F Van Wijland, Phys. Rev. Lett. 95110202A. Puglisi, P. Visco, A. Barrat, E. Trizac, and F. van Wijland, Phys. Rev. Lett. 95, 110202 (2005). . C Goeuedard, D Husson, C Sauteret, F Auzel, A Migus, J. Opt. Soc. Am. B. 102358C. Goeuedard, D. Husson, C. Sauteret, F. Auzel, and A. Migus, J. Opt. Soc. Am. B 10, 2358 (1993). . C Briskina, V Markushev, N E Ter-Gabrielyan, Quantum Electron. 26923C. Briskina, V. Markushev, and N. E. Ter-Gabrielyan, Quantum Electron. 26, 923 (1996). . H Cao, Y G Zhao, S T Ho, E W Seelig, Q H Wang, R P H Chang, Phys. Rev. Lett. 822278H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999). M Ryzhkov, V Markushev, C Briskina, H Cao, Advanced Optoelectronics and Lasers, 2005. Proceedings of CAOL 2005. Second International Conference on. 1M. Ryzhkov, V. Markushev, C. Briskina, and H. Cao, in Advanced Optoelectronics and Lasers, 2005. Proceed- ings of CAOL 2005. Second International Conference on (2005), vol. 1, pp. 72-75. . G Dice, S Mujumdar, A Elezzabi, Applied Physics Letters. 86131105G. Dice, S. Mujumdar, and A. Elezzabi, Applied Physics Letters 86, 131105 (2005). . O Popov, A Zilbershtein, D Davidov, Applied Physics Letters. 89191116O. Popov, A. Zilbershtein, and D. Davidov, Applied Physics Letters 89, 191116 (2006). . X Meng, K Fujita, Y Zong, S Murai, K Tanaka, Applied Physics Letters. 92201112X. Meng, K. Fujita, Y. Zong, S. Murai, and K. Tanaka, Applied Physics Letters 92, 201112 (2008). . X Meng, K Fujita, S Murai, K Tanaka, Physical Review A. 7953817X. Meng, K. Fujita, S. Murai, and K. Tanaka, Physical Review A 79, 053817 (2009). . D Sharma, H Ramachandran, N Kumar, arXiv:physics/0503059ArXiv Physics e-printsD. Sharma, H. Ramachandran, and N. Kumar, ArXiv Physics e-prints (2005), arXiv:physics/0503059. . S Mujumdar, V Turck, R Torre, D S Wiersma, Phys. Rev. A. 7633807S. Mujumdar, V. Turck, R. Torre, and D. S. Wiersma, Phys. Rev. A 76, 033807 (2007). . I Goldhirsch, G Zanetti, Phys. Rev. Lett. 701619I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993).
[]
[ "A DIRECT MEASUREMENT OF SENSE OF ROTATION OF PSR J0737-3039A", "A DIRECT MEASUREMENT OF SENSE OF ROTATION OF PSR J0737-3039A" ]
[ "Nihan Pol [email protected] \nDepartment of Physics and Astronomy\nWest Virginia University\n26506MorgantownWest VirginiaUSA\n\nCenter for Gravitational Waves and Cosmology\nChestnut Ridge Research Building\nWest Virginia University\n26505MorgantownWest Virginia\n", "Maura Mclaughlin \nDepartment of Physics and Astronomy\nWest Virginia University\n26506MorgantownWest VirginiaUSA\n\nCenter for Gravitational Waves and Cosmology\nChestnut Ridge Research Building\nWest Virginia University\n26505MorgantownWest Virginia\n", "Michael Kramer \nMax-Planck Institut für Radioastronomie\nAuf dem Hügel 69D-53121BonnGermany\n", "Ingrid Stairs \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada\n", "Benetge B P Perera \nJodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nThe University of Manchester\nM13 9PLManchesterUK\n", "Andrea Possenti \nINAF-Osservatorio Astronomico di Cagliari\nLoc. Poggio dei PiniI-09012CapoterraCAItaly\n" ]
[ "Department of Physics and Astronomy\nWest Virginia University\n26506MorgantownWest VirginiaUSA", "Center for Gravitational Waves and Cosmology\nChestnut Ridge Research Building\nWest Virginia University\n26505MorgantownWest Virginia", "Department of Physics and Astronomy\nWest Virginia University\n26506MorgantownWest VirginiaUSA", "Center for Gravitational Waves and Cosmology\nChestnut Ridge Research Building\nWest Virginia University\n26505MorgantownWest Virginia", "Max-Planck Institut für Radioastronomie\nAuf dem Hügel 69D-53121BonnGermany", "Department of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBCCanada", "Jodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nThe University of Manchester\nM13 9PLManchesterUK", "INAF-Osservatorio Astronomico di Cagliari\nLoc. Poggio dei PiniI-09012CapoterraCAItaly" ]
[]
We apply the algorithm published byLiang et al. (2014)to describe the Double Pulsar system J0737-3039 and extract the sense of rotation of first born recycled pulsar PSR J0737-3039A. We find that this pulsar is rotating prograde in its orbit. This is the first direct measurement of the sense of rotation of a pulsar with respect to its orbit and a direct confirmation of the rotating lighthouse model for pulsars. This result confirms that the spin angular momentum vector is closely aligned with the orbital angular momentum, suggesting that kick of the supernova producing the second born pulsar J0737-3039B was small.
10.3847/1538-4357/aaa1a0
[ "https://arxiv.org/pdf/1712.04360v1.pdf" ]
119,074,107
1712.04360
83e7480ab84715ed077aa2538dd1cda2fb71ae94
A DIRECT MEASUREMENT OF SENSE OF ROTATION OF PSR J0737-3039A July 23, 2018 12 Dec 2017 Nihan Pol [email protected] Department of Physics and Astronomy West Virginia University 26506MorgantownWest VirginiaUSA Center for Gravitational Waves and Cosmology Chestnut Ridge Research Building West Virginia University 26505MorgantownWest Virginia Maura Mclaughlin Department of Physics and Astronomy West Virginia University 26506MorgantownWest VirginiaUSA Center for Gravitational Waves and Cosmology Chestnut Ridge Research Building West Virginia University 26505MorgantownWest Virginia Michael Kramer Max-Planck Institut für Radioastronomie Auf dem Hügel 69D-53121BonnGermany Ingrid Stairs Department of Physics and Astronomy University of British Columbia 6224 Agricultural RoadV6T 1Z1VancouverBCCanada Benetge B P Perera Jodrell Bank Centre for Astrophysics School of Physics and Astronomy The University of Manchester M13 9PLManchesterUK Andrea Possenti INAF-Osservatorio Astronomico di Cagliari Loc. Poggio dei PiniI-09012CapoterraCAItaly A DIRECT MEASUREMENT OF SENSE OF ROTATION OF PSR J0737-3039A July 23, 2018 12 Dec 2017Draft version Typeset using L A T E X twocolumn style in AASTeX61 2 Pol et al.pulsars: general -pulsars: individual (PSR J0737-3039A) We apply the algorithm published byLiang et al. (2014)to describe the Double Pulsar system J0737-3039 and extract the sense of rotation of first born recycled pulsar PSR J0737-3039A. We find that this pulsar is rotating prograde in its orbit. This is the first direct measurement of the sense of rotation of a pulsar with respect to its orbit and a direct confirmation of the rotating lighthouse model for pulsars. This result confirms that the spin angular momentum vector is closely aligned with the orbital angular momentum, suggesting that kick of the supernova producing the second born pulsar J0737-3039B was small. INTRODUCTION The Double Pulsar, PSR J0737-3039 (Burgay et al. 2005;Lyne et al. 2004) is the first and only neutron star binary system that has had two detectable radio pulsars. The recycled PSR J0737-3039A (hereafter 'A') has a period of 22.7 ms and the younger, non-recycled PSR J0737-3039B (hereafter 'B') has a spin period of 2.8 s. The 2.45-hour orbit makes this system the most relativistic binary known, providing a unique laboratory to conduct the most stringent tests of Einstein's theory of general relativity in the strong-field regime . In addition to strong-field tests of gravity, the Double Pulsar also offers a unique laboratory to test plasma physics and magnetospheric emission from pulsars (Breton et al. 2012;Perera et al. 2012;Lyutikov 2004). We originally were able to detect bright single pulses from B in two regions of its orbit, 190 • ∼ 230 • , referred to as bright phase I (BP I) and 260 • ∼ 300 • , referred to as bright phase II (BP II) Perera et al. 2010). McLaughlin et al. (2004) discovered drifting features in the sub-pulse structure from pulsar B. They showed that this phenomenon was due to the direct influence of the magnetic-dipole radiation from A on B. These drifting features (henceforth referred to as the 'modulation signal') are only visible in BP I when the electromagnetic radiation from A meets the beam of B from the side (Lomiashvili & Lyutikov 2014). These modulation features were observed to have a frequency of ≈ 44 Hz which suggests that this emission is not from the beamed emission of A, which has a frequency of ≈ 88 Hz due to the visibility of emission from both the magnetic poles of A. Freire et al. (2009) proposed a technique to measure, among other things, the sense of rotation of A with respect to its orbit using the time of arrival of pulsed radio emission from A and the modulation feature from B. A complementary technique was proposed by Liang et al. (2014) (henceforth LLW2014) to uniquely determine the sense of rotation of A using an approach based on the frequency of the modulation signal. LLW2014 argued that we should be able to observe an effect similar to the difference between solar and sidereal periods observed in the Solar System in the Double Pulsar. Thus, if pulsar A is rotating prograde with respect to its orbit, the modulation signal should have a period slightly greater than that if it were not rotating, and it would have a slightly smaller period for the case of retrograde motion. LLW2014 provide an algorithm to apply this concept to the observations of the Double Pulsar, and we refer the reader to that paper for more details on the calculations and details of the algorithm. In this paper, we implement this algorithm on the Double Pulsar data and determine the sense of rotation of pulsar A. In Sec. 2, we briefly describe the data used and the implementation of the algorithm from LLW2014. We present our results in Sec. 3 and we discuss the implications of these results in Sec. 4. PROCEDURE Observations and Data Preparation We have carried out regular observations of the Double Pulsar since December 23, 2004 (MJD 52997) with the Green Bank Telescope (GBT). The radio emission of B has shown a significant reduction in flux density (0.177 mJy yr −1 ) and evolution from a single peaked profile to a double peaked profile due to relativistic spin precession with B's radio emission disappearing in March 2008 (Perera et al. 2010). As a result, we choose the data where B's emission is brightest, which also corresponds to the modulation signal being the brightest, for this analysis, i.e. the data collected on MJD 52997. These data on MJD 52997 were taken at a center frequency of 800 MHz with the GBT spectrometer SPIGOT card (Kaplan et al. 2005). This observation had a sampling time of 40.96 µs and the observation length was 5 hours, covering more than two complete orbits. We barycenter these data using the barycenter program from the SIGPROC 1 software package. We decimate the data from its native resolution of 24.41 kHz to 2048 × f B,0 ≈ 738.43 Hz where f B,0 is the rest-frame frequency of B. This is equivalent to splitting up a single rotational period of B into 2048 bins. We do this to increase the signal-to-noise ratio (S/N) of the modulation signal. Since B's drifting pulses are observed only in BP I, we focus only on these orbital phases. Since this data set covers more than two orbits, we obtain two such BP I time series. The first of these BP I time series is shown in Fig. 1. Since the drifting pulses are seen only at the beginning of this phase range, we analyze orbital phase range (defined as the longitude from ascending node, i.e. the sum of longitude of periastron and true anomaly) 195 • to 210 • . This final data set is approximately 344 seconds long. With the data set prepared as described above, and noting that the time series length, T = N ∆t where N is total number of samples and ∆t is the sampling time, our Fourier spectra have a frequency resolution: f s = 1 T = 1 N ∆t = 2.93 mHz(1) Transformation We apply the algorithm from LLW2014 (see Sec. 3.4 therein) to the time series. We found a typographic error in the algorithm from LLW2014. We describe this error and how we fixed it in Appendix A. All programming was done in Python. We wrote a function that would return values for ω B (longitude of periastron of B's orbit) and θ (true anomaly for B) as a function of time (see Chapter 8 in Lorimer & Kramer 2004). All orbital parameters such as eccentricity e, orbital inclination angle i and semi-major axis a B were obtained from the timing solution of B . With all these parameters in place, the implementation of the algorithm was straight-forward. For completeness, we briefly list the transformation described in LLW2014, and refer the reader to that paper for more details. The basic idea of the transformations is to remove the Doppler smearing produced by eccentric orbits in the Double Pulsar by suitably resampling the data and obtain the time at which the modulation signal left A. This can be done by first computing the resampled time series t B [k] which represents the time of the k th sample measured at B, by correcting for B's orbital motion (Eq. 10 in LLW2014): t B [k] = t[k] − L c − a B sin i (1 − e 2 ) sin(ω B + θ) c (1 + e cos θ)(2) where t[k] is the time corresponding to the k th sample measured at the solar system barycenter (ssb), L is the distance to the Double Pulsar and a B is the semi-major axis of B's orbit. The L/c term is a constant offset which can be neglected without loss of information. Now, we can calculate t A [k], the time when the signal causing the modulation features left A, by correcting for an additional propagation time delay along the path length from A to B: t A [k] = t B [k] − (a A + a B ) 1 − e 2 1 + e cos(θ) (3) where a A is the semi-major axis of A. Eq. 3 can be used to transform I[k], the intensity data sampled at the ssb at time t[k], into a frame where the time-variable Doppler shifts have been removed from the data. Using the resampled time series, we can compute a Fourier power spectrum (Eq. 19 in LLW2014): P n (zf A,0 ) s = k I[k] exp(−i n |Φ m [k, z]| s ) 2 (4) where z is a frequency scaling factor, P n (zf A,0 ) s is the power in the n th harmonic of the modulation corresponding to the trial spin frequency zf A,0 , and 5) is the "modulation phase" which is simply the rotational (or pulsational) phase of A corrected for the sense of its rotation, with s = 1, −1, 0 corresponding to prograde, retrograde and no rotation (pulsation) respectively. Here, f A,0 is the sidereal frequency of A's rotation or pulsation, which we know from timing measurements to be 44.05406 Hz ) on MJD 52997. |Φ m [k, z]| s = 2π (z f A,0 ) t A [k] − s θ(t B [k]) ( If we compute the power spectrum in Eq. 4 for each value of s, then, based on the arguments in LLW2014, we should observe a peak at a frequency corresponding to z = 1, i.e. at f = f A,0 and the value of s with the highest power in this peak will determine the sense of rotation of A. RESULTS Since emission from A is stimulating emission in B, we can think of A as the "carrier" signal which modulates the magnetosphere of B. This interpretation implies we would see a signal at the fundamental frequency of the carrier (in this case f A,0 ) and sidebands of this signal separated by the modulation frequency (in this case f B,0 ). Thus, we would expect to see a signal at frequencies f A,0 ± m × f B,0 where m = 0, 1, 2, .... We see this structure in the fourier power spectra for the three different cases of s, with the power spectrum for s = 0 shown in Fig. 2 for reference. We can see the peak at the fundamental frequency f A,0 (marked by a blue solid vertical line) and its sidebands (marked by vertical dashed blue lines). We have also marked the harmonics from B's signal (fundamental frequency of f B,0 = 0.3605 Hz). They are visible as distinct peaks in the power spectrum indicating different origins for the two signals. In addition to these signals, we also see a strong signal close to f A,0 , marked in Fig. 2 by a solid vertical cyan line. This is the relic of the signal from A's emission, but shifted away from its native frequency of f A,0 and reduced in amplitude by the transformations that we have applied. This is a key part of the analysis which allows us to distinguish the signal generated by A's intrinsic emission and the modulation signal. In Fig. 2, we also do not detect any power in the sidebands associated with A's intrinsic signal. The presence of A's intrinsic signal, without the presence of sidebands, serves as evidence that the signal we see at f A,0 after applying the transformations is from the modulation feature rather than from A itself, and that we have successfully managed to separate the modulation signal from A's intrinsic emission. Finally, we want to compare the power in the signals at frequencies f A,0 ± n × f B,0 for all three cases of s, with the value of s with the highest power indicating the direction of rotation of A with respect to its orbit. Note that, as described in Sec. 2.1, we have observed two complete orbits of the Double Pulsar on MJD 52997. Thus, we plot the power at the fundamental frequency and its sidebands for the BP I of the first and second orbit in Figs. 3 and 4 respectively, and plot the BP II (where B is visible, but the modulation features are not visible) for the first orbit in Fig. 5. For completeness, we also plot the power at the fundamental frequency and its sidebands for some randomly selected weak phase (40 • to 52 • , where weak or no emission is observed in B's spectrum and no modulation signal is visible) for the first orbit in Fig. 6. In both the power spectra for BP I (see Figs. 3 and 4), we can see that the fundamental frequency and its sidebands have consistently higher power in s = 1 than for s = 0, −1. The power at these frequencies is also significantly higher than the mean noise floor of the respective power spectra. By comparison, there is very little to no power in these frequencies for the other orbital phase ranges of B's orbit (see Figs. 5 and 6). This is consis- for BP II and the weak phase respectively. Here we can see that none of the values of s dominate over the other values in terms of their total power. To test the significance of this result, we took the data from the two BP I sections and scrambled them such that we had a time series that resembled noise. Next, we passed this scrambled time series through the same analysis described above for the real data. This process of scrambling the data and passing it through the analysis pipeline was repeated 1000 times so that we had a collection of values for the total powers in different cases of s. We compared these total powers across all sidebands (including the fundamental frequency) for s = 1, 0, −1 in the scrambled time series with the total power in s = 1, 0, −1 obtained from the real time series respectively to obtain the standard deviation: σ = A × B C (6) where A is the average standard deviation between total powers in the scrambled data, B is the mean of the total power for s = 1, 0, −1 for the real data, and C is the mean of the total power for s = 1, 0, −1 for the scrambled data. Using the above σ, we can compute the difference in total power for s = 1, 0, −1 for the real data. For the first BP I, we find that σ = 7.4 and the total power in s = 1 is 11.6σ above s = 0 and 23.9σ above s = −1. Similarly, for the second BP I, we have σ = 4.6 and total power in s = 1 is 7.6σ above s = 0 and 16.0σ above s = −1. We perform a similar analysis for the BP II time series and find that none of the differences in total powers exceeds 1.5σ. The observation of consistently higher power in s = 1 over s = 0, −1 along with the high significance of the s = 1 signal in two BP I time series leads us to the conclusion that s = 1 represents the true direction of rotation of A with respect to its orbit. DISCUSSION AND CONCLUSION Based on this analysis, we can conclude that A is rotating in a prograde direction with respect to its orbit. This is the first time, in 50 years of pulsar studies, that such a direct confirmation of the sense of rotation of a pulsar has been obtained. This is additional empirical evidence for the rotating lighthouse model (earlier evidence was presented by Stairs et al. 2004, using special relativistic aberration of the revolving pulsar beam due to orbital motion in the B1534+12 system). This model describes pulsars as rapidly rotating neutron stars emitting magnetic-dipole radiation from their polar cap region. This rapid rotation of the pulsar resulted in the periodic pulses of light that are characteristic of pulsar emission. This work provides direct confirmation of this model. This result will help constrain evolutionary theories of binary systems (Alpar et al. 1982) as well as improve constraints on B's supernova kick. Ferdman et al. (2013) computed a mean 95% upper limit on the misalignment angle between the spin and orbital angular momentum axes of A to be 3.2 • and concluded that the A's spin angular momentum vector is closely aligned with the orbital angular momentum.This work validates that and earlier hypotheses (Willems et al. 2006;Stairs et al. 2006;Ferdman et al. 2013;Tauris et al. 2017) that the kick produced by B's supernova was small. Furthermore, knowing the direction of spin angular momentum of A will allow us to compute the sign of the relativistic spin-orbit coupling contribution to the post-Keplerian parameterω, which in turn will allow us to determine A's moment of inertia ). The moment of inertia of A, along with the welldetermined mass of A will provide us with a radius, which will introduce fundamental constraints on the equation of state for dense matter (Lattimer & Schutz 2005). This measurement of the sense of rotation of A was made using the frequency of the modulation signal and the rotational frequencies of A and B. An alternative way to measure the same effect is using times of arrivals of the pulses from the modulation signal and A's radio emission. Freire et al. (2009) constructed a geometric model for the double pulsar system and used it to exploit the times of arrivals to measure the sense of rotation of A along with determining the height in B's magnetosphere at which the modulation signal originates. Their model will also provide another measurement of the mass ratio of A and B which will affect the precision of some of the tests of general relativity carried out in this binary system. Our preliminary results implementing the Freire et al. (2009) model also indicate prograde rotation for A, and will be published in a future work. We would like to thank Joel Weisberg, Yi Liang, and Zhu-Xing Liang for useful discussions through the course of the analysis. MAM and NP were supported by NSF award #1517003. IHS received support from an NSERC Discovery Grant and from the Canadian Institute for Advanced Research. APPENDIX A. ERRATA While performing the analysis for this paper, we discovered a typographic error in LLW2014. Eq. 10 in LLW2014 reads as: t B [k] = t[k] − L c − z B c (A1) where t[k] is the barycentric time at which the k th sample of the time series with intensity I[k] was observed, L is the distance to the pulsar, and z B is the projection of the position of B with respect to the binary barycentre onto the line of sight: z B = a B sin i (1 − e 2 ) sin(ω B + θ) c (1 + e cosθ)(A2) where a B is the semi-major axis of pulsar B's orbit, e is the eccentricity of B's orbit, ω B is the longitude of periastron passage for B and θ is the true anomaly. For Eq. A1 to be dimensionally consistent, z B should have dimensions of length. Looking at Eq. A2, we can see that z B will have dimensions of length if we remove the factor of c in the denominator. Thus, Eq. A2 should read as: z B = a B sin i (1 − e 2 ) sin(ω B + θ) (1 + e cosθ) (A3) Figure 1 . 1Single pulses of B for MJD 52997 for orbital phase 190 • − 240 • in the first BP I. The drifting features are most prominent in the orbital phase range of 195 • ∼ 210 • . A's pulses are also visible in the background, and are most visible at ∼ 225 • . Note that only a fraction of the pulse phase is shown for clarity and the units on the amplitude are arbitrary. This figure is adapted from McLaughlin et al. (2004). Figure 2 . 2Fourier power spectrum for s = 0. Vertical dashed red lines mark the harmonics from B's intrinsic signal. The fundamental frequency of the modulation signal (fA,0) is shown by a vertical solid blue line and vertical dashed blue lines mark the sidebands of the modulation signal. The emission from A's intrinsic signal is visible as the prominent peak close to the fundamental frequency of the modulation signal, marked by a solid vertical cyan line, which is not at fA,0 due to the transformations applied above (see text). Similar to the sidebands of the modulation signal, we mark the positions of the sidebands for A's intrinsic signal. We can see that there is no power in the sidebands for A's intrinsic signal, indicating that we have successfully separated A's intrinsic signal from the modulation signal. Figure 3 . 3Fourier power in the modulation signal's fundamental frequency and sidebands for the first BP I on MJD 52997. The power for all three cases of s = 1, 0, −1 are plotted together for comparison. The location of the fundamental frequency is shown by a vertical dashed line, while the mean noise level is shown by a horizontal dashed line. The case of s = 1 has consistently high power over all components which indicates this is the true sense of rotation of A. Figure 4 .Figure 5 .Figure 6 . 456Fourier power in the modulation signal's fundamental frequency and sidebands for the second BP I on MJD 52997. The power for all three cases of s = 1, 0, −1 are plotted together for comparison. The location of the fundamental frequency is shown by a vertical dashed line, while the mean noise level is shown by a horizontal dashed line. The power in s = 1 is consistently higher than other values of s. Fourier power in the modulation signal's fundamental frequency and sidebands for the first BP II on MJD 52997. The power for all three cases of s = 1, 0, −1 are plotted together for comparison. The location of the fundamental frequency is shown by a vertical dashed line, while the mean noise level is shown by a horizontal dashed line. In this orbital phase range, the modulation signal is not visible. Hence, we do not see any significant power for any value of s at any of the sidebands of the modulation signal. Fourier power in the modulation signal's fundamental frequency and sidebands for a weak phase (40 • to 52 • ) on MJD 52997. The power for all three cases of s = 1, 0, −1 are plotted together for comparison. The location of the fundamental frequency is shown by a vertical dashed line, while the mean noise level is shown by a horizontal dashed line. In this orbital phase range, B and the modulation signal is not visible. Hence, we do not see any significant power for any value of s at any of the sidebands of the modulation signal. Figure 7 .Figure 8 .Figure 9 .Figure 10 . 78910Cumulative power across all sidebands for the first BP I on MJD 52997. We can see that the power in s = 1 is consistently higher than the power in the other values of s. tent with observations of the Double Pulsar where the modulation driftbands are not visible in any other orbital phase range apart from BP I.To illustrate the consistently higher power in BP I, we plot the cumulative power across all sidebands of the modulation signal in Figs. 7 and 8. In these plots, we Cumulative power across all sidebands for the second BP I on MJD 52997. Similar to the first BP I, we can see that the power in s = 1 is consistently higher than the power in the other values of s.begin with the power in the −64 th sideband inFig. 3and add it with the power in the next sideband and so on until we reach the 64 th sideband. These plots clearly indicate that the power in s = 1 is always greater than that in other values of s. For comparison, we plot in Figs. 9 and 10 the cumulative power across all sidebands Cumulative power across all sidebands for the first BP II on MJD 52997. We can see that no value of s dominates over the other values in terms of total power. This is consistent with results shown inFig.Cumulative power across all sidebands for the weak phase on MJD 52997. We can see that no value of s dominates over the other values in terms of total power. This is consistent with results shown inFig. 6. This software can be found at: http://sigproc.sourceforge. net/ . M A Alpar, A F Cheng, M A Ruderman, J Shaham, Nature. 300728Alpar, M. A., Cheng, A. F., Ruderman, M. A., & Shaham, J. 1982, Nature, 300, 728 . R P Breton, V M Kaspi, M A Mclaughlin, ApJ. 74789Breton, R. P., Kaspi, V. M., McLaughlin, M. A., et al. 2012, ApJ, 747, 89 M Burgay, N D&apos;amico, A Possenti, Astronomical Society of the Pacific Conference Series. Binary Radio Pulsars, ed. F. A. Rasio & I. H. Stairs32853Burgay, M., D'Amico, N., Possenti, A., et al. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 328, Binary Radio Pulsars, ed. F. A. Rasio & I. H. Stairs, 53 . R D Ferdman, I H Stairs, M Kramer, ApJ. 76785Ferdman, R. D., Stairs, I. H., Kramer, M., et al. 2013, ApJ, 767, 85 . P C C Freire, N Wex, M Kramer, MNRAS. 3961764Freire, P. C. C., Wex, N., Kramer, M., et al. 2009, MNRAS, 396, 1764 . D L Kaplan, R P Escoffier, R J Lacasse, Publications of the Astronomical Society of the Pacific. 117643Kaplan, D. L., Escoffier, R. P., Lacasse, R. J., et al. 2005, Publications of the Astronomical Society of the Pacific, 117, 643. http://stacks.iop.org/1538-3873/117/i=832/a=643 . M Kramer, N Wex, Classical and Quantum Gravity. 2673001Kramer, M., & Wex, N. 2009, Classical and Quantum Gravity, 26, 073001. http://stacks.iop.org/0264-9381/26/i=7/a=073001 . M Kramer, I H Stairs, R N Manchester, Science. 97Kramer, M., Stairs, I. H., Manchester, R. N., et al. 2006, Science, 314, 97. . J M Lattimer, B F Schutz, The Astrophysical Journal. 629979Lattimer, J. M., & Schutz, B. F. 2005, The Astrophysical Journal, 629, 979. http://stacks.iop.org/0004-637X/629/i=2/a=979 . Z.-X Liang, Y Liang, J M Weisberg, MNRAS. 4393712Liang, Z.-X., Liang, Y., & Weisberg, J. M. 2014, MNRAS, 439, 3712 . D Lomiashvili, M Lyutikov, MNRAS. 441690Lomiashvili, D., & Lyutikov, M. 2014, MNRAS, 441, 690 . D R Lorimer, M Kramer, Handbook of Pulsar Astronomy. Lorimer, D. R., & Kramer, M. 2004, Handbook of Pulsar Astronomy . A G Lyne, M Burgay, M Kramer, Science. 3031153Lyne, A. G., Burgay, M., Kramer, M., et al. 2004, Science, 303, 1153 . M Lyutikov, MNRAS. 3531095Lyutikov, M. 2004, MNRAS, 353, 1095 . M A Mclaughlin, M Kramer, A G Lyne, ApJL. 61357McLaughlin, M. A., Kramer, M., Lyne, A. G., et al. 2004, ApJL, 613, L57 . B B P Perera, D Lomiashvili, K N Gourgouliatos, M A Mclaughlin, M Lyutikov, ApJ. 750130Perera, B. B. P., Lomiashvili, D., Gourgouliatos, K. N., McLaughlin, M. A., & Lyutikov, M. 2012, ApJ, 750, 130 . B B P Perera, M A Mclaughlin, M Kramer, ApJ. 7211193Perera, B. B. P., McLaughlin, M. A., Kramer, M., et al. 2010, ApJ, 721, 1193 . I H Stairs, S E Thorsett, Z Arzoumanian, https:/link.aps.org/doi/10.1103/PhysRevLett.93.141101Phys. Rev. Lett. 93141101Stairs, I. H., Thorsett, S. E., & Arzoumanian, Z. 2004, Phys. Rev. Lett., 93, 141101. https: //link.aps.org/doi/10.1103/PhysRevLett.93.141101 . I H Stairs, S E Thorsett, R J Dewey, M Kramer, C A Mcphee, +http:/dx.doi.org/10.1111/j.1745-3933.2006.00241.xMonthly Notices of the Royal Astronomical Society: Letters. 373Stairs, I. H., Thorsett, S. E., Dewey, R. J., Kramer, M., & McPhee, C. A. 2006, Monthly Notices of the Royal Astronomical Society: Letters, 373, L50. +http: //dx.doi.org/10.1111/j.1745-3933.2006.00241.x . T M Tauris, M Kramer, P C C Freire, The Astrophysical Journal. 846Tauris, T. M., Kramer, M., Freire, P. C. C., et al. 2017, The Astrophysical Journal, 846, 170. http://stacks.iop.org/0004-637X/846/i=2/a=170 . B Willems, J Kaplan, T Fragos, V Kalogera, K Belczynski, https:/link.aps.org/doi/10.1103/PhysRevD.74.043003Phys. Rev. D. 7443003Willems, B., Kaplan, J., Fragos, T., Kalogera, V., & Belczynski, K. 2006, Phys. Rev. D, 74, 043003. https: //link.aps.org/doi/10.1103/PhysRevD.74.043003
[]
[]
[ "Wen-Long Sang \nSchool of Physical Science and Technology\nSouthwest University\n400700ChongqingChina\n\nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n", "Feng Feng \nChina University of Mining and Technology\n100083BeijingChina\n", "Yu Jia \nInstitute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina\n\nCenter for High Energy Physics\nPeking University\n100871BeijingChina\n", "Shuang-Ran Liang \nInstitute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina\n", "Wen-Long Sang \nSchool of Physical Science and Technology\nSouthwest University\n400700ChongqingChina\n\nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n", "Feng Feng \nChina University of Mining and Technology\n100083BeijingChina\n", "Yu Jia \nInstitute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina\n\nCenter for High Energy Physics\nPeking University\n100871BeijingChina\n", "Shuang-Ran Liang \nInstitute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina\n" ]
[ "School of Physical Science and Technology\nSouthwest University\n400700ChongqingChina", "Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina", "China University of Mining and Technology\n100083BeijingChina", "Institute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina", "Center for High Energy Physics\nPeking University\n100871BeijingChina", "Institute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina", "School of Physical Science and Technology\nSouthwest University\n400700ChongqingChina", "Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina", "China University of Mining and Technology\n100083BeijingChina", "Institute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina", "Center for High Energy Physics\nPeking University\n100871BeijingChina", "Institute of High Energy Physics\nTheoretical Physics Center for Science Facilities\nChinese Academy of Sciences\n100049BeijingChina" ]
[]
We calculate the next-to-next-to-leading-order (NNLO) perturbative corrections to P -wave quarkonia annihilation decay to two photons, in the framework of nonrelativistic QCD (NRQCD) factorization. The order-α 2 s short-distance coefficients associated with each helicity amplitude are presented in a semi-analytic form, including the "light-by-light" contributions. With substantial NNLO corrections, we find disquieting discrepancy when confronting our state-of-the-art predictions with the latest BESIII measurements, especially fail to account for the measured χc2 → γγ width. Incorporating the effects of spin-dependent forces would even exacerbate the situation, since it lifts the degeneracy between the nonperturbative NRQCD matrix elements of χc0 and χc2 toward the wrong direction. We also present the order-α 2 s predictions to χ b0,2 → γγ, which await the future experimental test. PACS numbers: 12.38.Bx, 13.20.Gd, 14.40.PqCharmonium decay has historically played an important role in establishing the asymptotic freedom of QCD, and served as a clean platform to probe the interplay between pertubative and nonperturbative dynamics [1, 2]. Among them, the electromagnetic decay χ c0,2 → γγ provide a particularly interesting, and, rich testing ground of QCD[3,4]. In the past decades, these decay channels have been extensively studied from various theoretical angles, such as nonrelativistic potential model[5,6], relativistic quark model[7][8][9], Bethe-Salpeter approach [10], nonrelativistic QCD (NRQCD) factorization[11,12], as well as lattice QCD[13]. On the experimental side, they were previously measured by . BESIII experiment[15]has recently reported their high precision results, Γ γγ (χ c0 ) = (2.33 ± 0.20 ± 0.13 ± 0.17) keV, (1a) Γ γγ (χ c2 ) = (0.63 ± 0.04 ± 0.04 ± 0.04) keV. (1b)
10.1103/physrevd.94.111501
[ "https://arxiv.org/pdf/1511.06288v2.pdf" ]
119,247,259
1511.06288
a26f0e4da4210b2cb0c5c30e065c61677c09af10
22 Dec 2015 Wen-Long Sang School of Physical Science and Technology Southwest University 400700ChongqingChina Institute of Theoretical Physics State Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina Feng Feng China University of Mining and Technology 100083BeijingChina Yu Jia Institute of High Energy Physics Theoretical Physics Center for Science Facilities Chinese Academy of Sciences 100049BeijingChina Center for High Energy Physics Peking University 100871BeijingChina Shuang-Ran Liang Institute of High Energy Physics Theoretical Physics Center for Science Facilities Chinese Academy of Sciences 100049BeijingChina 22 Dec 2015(Dated: December 23, 2015)Next-to-next-to-leading-order QCD corrections to χ c0,2 → γγ We calculate the next-to-next-to-leading-order (NNLO) perturbative corrections to P -wave quarkonia annihilation decay to two photons, in the framework of nonrelativistic QCD (NRQCD) factorization. The order-α 2 s short-distance coefficients associated with each helicity amplitude are presented in a semi-analytic form, including the "light-by-light" contributions. With substantial NNLO corrections, we find disquieting discrepancy when confronting our state-of-the-art predictions with the latest BESIII measurements, especially fail to account for the measured χc2 → γγ width. Incorporating the effects of spin-dependent forces would even exacerbate the situation, since it lifts the degeneracy between the nonperturbative NRQCD matrix elements of χc0 and χc2 toward the wrong direction. We also present the order-α 2 s predictions to χ b0,2 → γγ, which await the future experimental test. PACS numbers: 12.38.Bx, 13.20.Gd, 14.40.PqCharmonium decay has historically played an important role in establishing the asymptotic freedom of QCD, and served as a clean platform to probe the interplay between pertubative and nonperturbative dynamics [1, 2]. Among them, the electromagnetic decay χ c0,2 → γγ provide a particularly interesting, and, rich testing ground of QCD[3,4]. In the past decades, these decay channels have been extensively studied from various theoretical angles, such as nonrelativistic potential model[5,6], relativistic quark model[7][8][9], Bethe-Salpeter approach [10], nonrelativistic QCD (NRQCD) factorization[11,12], as well as lattice QCD[13]. On the experimental side, they were previously measured by . BESIII experiment[15]has recently reported their high precision results, Γ γγ (χ c0 ) = (2.33 ± 0.20 ± 0.13 ± 0.17) keV, (1a) Γ γγ (χ c2 ) = (0.63 ± 0.04 ± 0.04 ± 0.04) keV. (1b) In addition, BESIII presents the ratio of the decay rates between χ c2 and χ c0 . For the first time, they also measured the ratio of the two polarized decay rates for χ c2 : R = Γ γγ (χ c2 ) Γ γγ (χ c0 ) = 0.271 ± 0.029 ± 0.013 ± 0.027, (2a) f 0/2 = Γ λ=0 γγ (χ c2 ) Γ λ=2 γγ (χ c2 ) = 0.00 ± 0.02 ± 0.02, where λ = |λ 1 −λ 2 |, and λ 1 , λ 2 = ±1 denote the helicities of the outgoing photons. The precise data thereby calls for the full-fledged theoretical inspection. In parallel with positronium decay, the leading-order NRQCD prediction to χ c0,2 → γγ in the nonrelativistic limit is extremely simple, yields R = 4/15 ≈ 0.27 [16]. This is impressively consistent with the measurement (2a). Nevertheless, these processes are sensitive to the next-to-leading-order (NLO) radiative correction [17,18], with the predicted R scattered in the range from 0.09 to 0.36 [6,7]. To date, the next-to-next-to-leading-order (NNLO) radiative corrections are only available for a few S-wave quarkonium electromagnetic decay processes, exemplified by Υ(J/ψ) → e + e − [19,20], η b,c → γγ [21,22], and B c → ℓν [23,24], as well as the γγ * → η c,b transition form factor [22]. It has been found that the NNLO radiative corrections in aforementioned processes are often negative and substantial. The goal of this work is to address the complete NNLO corrections to P -wave quarkonium annihilation into two photons. The partial widths for χ c0,2 → γγ can be expressed as Γ γγ (χ 0 ) = 1 16π 2|A χ0 1,1 | 2 ,(3a) Γ γγ (χ 2 ) = 1 5 1 16π 2|A χ2 1,1 | 2 + 2|A χ2 1,−1 | 2 , (3b) where A χJ λ1,λ2 signifies the helicity amplitude for χ cJ → γ(λ 1 )γ(λ 2 ). We have employed parity invariance to only enumerate the independent helicity amplitudes in (3). NRQCD factorization approach, which exploits the nonrelativistic nature of heavy quarkonium, provides a systematic and model-independent framework to tackle quarkonium decay [11]. At the lowest order in v, the helicity amplitudes in (3) can be written in a factorized form: A χ0,2 λ1,λ2 = C χ0,2 λ (m, µ R , µ Λ ) 0|χ † K3 P0,2 ψ(µ Λ )|χ c0,2 m 3/2 +O(v 2 ),(4) where K3 P0 = 1 √ 3 (− i 2 ↔ D · σ),(5a)K3 P2 = − i 2 ↔ D (i σ j) ǫ * ij ,(5b) with ǫ ij representing the polarization tensor of χ c2 . C χ0,2 λ (m, µ R , µ Λ ) in (4) signifies the NRQCD shortdistance coefficient (SDC), where m, µ R , µ Λ denote the charm quark mass, renormalization scale, and NRQCD factorization scale, respectively. In phenomenological analysis, these nonpertubative matrix elements are often substituted as the derivative of P -wave radial Schrödinger wave functions at the origin: (6) where N c = 3 is the number of color. In literature, it is often assumed that R ′ χc0 ≈ R ′ χc2 by invoking the approximate heavy quark spin symmetry (HQSS). We stress that, in NRQCD these wave functions at the origin are promoted as scale-dependent quantities. 0|χ † K3 P0,2 ψ(µ Λ )|χ c0,2 = 3N c 2π R ′ χc0,2 (µ Λ ), Thanks to the asymptotic freedom, the SDCs can be computed order by order in α s . Through NNLO in α s , the SDC affiliated with the only helicity channel of χ c0 is C χ0 0 = 4 √ 3πe 2 Q α √ m 1 + C F α s (µ R ) π π 2 8 − 7 6 + α 2 s π 2 C F β 0 4 π 2 8 − 7 6 ln µ 2 R m 2 + ∆ χ0 0 ,(7) and two independent SDCs C χ2 0,2 are C χ2 0 = 4 √ 6παe 2 Q 3 √ m C F α s (µ R ) π 3π 2 8 − 6 ln 2 + 1 (8a) + α 2 s π 2 C F β 0 4 3π 2 8 − 6 ln 2 + 1 ln µ 2 R m 2 + ∆ χ2 0 , C χ2 2 = − 8παe 2 Q √ m 1 − 2C F α s (µ R ) π + α 2 s π 2 − 2C F β 0 4 ln µ 2 R m 2 + ∆ χ2 2 .(8b)J ) → γγ through order α 2 s . β 0 = 11 3 C A − 2 3 (n L + n H )(1) is the one-loop coefficient of the QCD β-function, where n H = 1, and n L signifies the number of light quark flavors (n L = 3 for χ c , 4 for χ b ). The occurrence of the β 0 ln µ R term in (7) and (8) is demanded by renormalization group invariance. To the best of our knowledge, the NLO perturbative correction to the λ = 0 amplitude in (8a) is new. Interestingly, this helicity amplitude turns out to vanish at Born level. Thus, NRQCD framework appears to offer a natural explanation for the tiny value of f 0/2 in (2b) observed by BESIII. The nontrivial task is then to decipher ∆ χ0,2 0,2 . Rather than follow the literal matching doctrine, we employ the standard shortcut of directly extracting the SDCs [19,20]. We compute the on-shell quark amplitude for cc( 3 P (1) J ) → γγ through order α 2 s . In contrast to the S-wave quarkonium decay, we expand the corresponding amplitude to the first order in q, the relative momentum between c andc, to identify the P -wave component, and compose the cc( 3 P (1) 0,2 ) state via the standard procedure [12]. In the end we project out the respective helicity amplitudes. A key simplification originates from the fact that, when conducting the loop integration, q has already been set to zero. We briefly describe the calculation. The package Fey-nArts [25] is employed to generate corresponding Feynman diagrams and amplitudes through O(α 2 s ) in Feynman gauge. There are 108 regular 2-loop diagrams and 12 "light-by-light" (LBL) scattering diagrams, some of which are sketched in Fig. 1. The latter gauge-invariant subsets are UV-and IR-finite. Dimensional regularization (DR) is employed to regularize both UV and IR divergences. We then use FeynCalc/FormLink [26,27] to carry out the trace over Dirac/color matrices. The packages Apart [28] and FIRE [29] are utilized to conduct partial fraction together with integration-by-parts (IBP) reduction. Finally, we end up with around 80 master integrals (MI). For a dozen of simpler MIs, we employ the α parameters [30] as well as the Mellin-Barnes tools AMBRE [31]/MB [32] to infer the (semi-) analytic expressions; for the multi-leg two-loop MIs, we combine FIESTA/CubPack [33,34] to carry out sector decomposition and subsequent numerical integrations with quadru-ple precision. The order-α 2 s expressions for the heavy quark wave function and mass renormalization constants are taken from [35,36]. The strong coupling constant is renormalized to one-loop order under MS scheme. All the UV divergences are eliminated by the renormalization procedure. However, at this stage, the NNLO amplitudes still contain single IR poles, with coefficients differing from the 3 P 0 to the 3 P 2 channel. These single IR poles are intimately connected to the anomalous dimensions of the NRQCD color-singlet currents associated with 3 P 0,2 , as specified in (5). In fact, from the lower-energy effective field theory of NRQCD, Hoang and Ruiz-Femenia are able to predict the anomalous dimensions for NRQCD bilinear carrying general 2S+1 L J quantum number [37]. Particulary, the anomalous dimensions of the operators carrying quantum number 3 P J are predicted to be γ3 P0 = −C F C F 6 + C A 24 α 2 s + O(α 3 s ), (9a) γ3 P2 = −C F 13C F 240 + C A 24 α 2 s + O(α 3 s ). (9b) Their difference signals the violation of HQSS due to spin-dependent interactions. It is reassuring that the coefficients of the uncancelled IR poles in our NNLO amplitudes turn out to exactly match the UV poles as implied in (9). In our opinion, this is a highly nontrivial verification of the correctness of our calculation. We thereby factorize these IR poles into the corresponding χ c0,2 -to-vacuum NRQCD matrix elements in (4) under MS prescription, with ln µ Λ now manifested in the respective SDCs. The ∆ χJ λ receives contributions from both regular and LBL diagrams, where the former is real valued, and the latter complex valued. It is convenient to decompose ∆ χJ λ into two parts: ∆ χJ λ = ∆ χJ reg, λ + ∆ χJ lbl, λ .(10) The regular part can be organized according to their color structure: ∆ χJ reg, λ = C 2 F s χJ A;λ + C F C A s χJ N A,λ + n L C F T F s χJ L,λ + n H C F T F s χJ H,λ ,(11) where C F = 4 3 , C A = 3, T F = 1 2 are SU (3) color factors. The regular pieces of the only helicity component for χ 0 are s χ0 A,0 = − 2π 2 3 ln µ Λ m − 9.14751077(6), s χ0 N A,0 = − π 2 6 ln µ Λ m − 1.69821088(5), s χ0 L,0 = 1 432 − 126ζ(3) − 45π 2 + 244 , s χ0 H,0 = 0.09292479(2).(12) The regular pieces affiliated with the two helicity components of χ 2 are and s χ2 A,2 = − 13π 2 60 ln µ Λ m − 5.93023533(7), s χ2 N A,2 = − π 2 6 ln µ Λ m − 5.78204922(4), Note that the absence of ln µ Λ in (13) originates from the vanishing of LO amplitude for the helicity configuration χ 2 → γ(±1)γ(±1). In contrast to regular part, it is rather challenging for FIESTA to acquire high-precision results for the complex-valued MIs associated with the LBL diagrams. Fortunately, some of them can be worked out analytically. Employing the α-parameters [30] or Mellin-Barnes tools [31,32], it is always feasible to reduce the remaining MIs into one or two-dimensional integrals, which can then be readily computed with high numerical precision. The LBL part for the χ 0 → γ(±1)γ(±1) is ∆ χ0 lbl,0 = (−0.120326 + 0.398547i)n H C F T F + 0.953741 + iπ 6 C F T F nL i e 2 i e 2 Q ,(15) where e i represents the electric charge of the i-th light flavor. The LBL pieces associated with the two helicity com- ponents of χ 2 are ∆ χ2 lbl,0 = − 0.019772 + 0.011196 i n H C F T F + 0.359850 + iπ 7π 2 6 − 23 2 C F T F nL i e 2 i e 2 Q ,(16a)∆ χ2 lbl,2 = − 0.088227 + 0.187239 i n H C F T F + − 0.669873 + π 27 (91 − 12π 2 + 24 ln 2)i ×C F T F nL i e 2 i e 2 Q .(16b) In passing, we recall that the rare decay process χ c2 → e + e − contains uncancelled IR divergences [38]. Since the occurring one-loop box diagrams just comprise subdiagrams of our two-loop LBL diagrams, it is intriguing that our LBL contributions turn out to be completely IR finite. With all the order-α 2 s terms in (10) available in a semianalytic form, we can assemble them together to deduce the corresponding SDCs in (7), (8), and substitute them into (4) to deduce the respective helicity amplitudes, finally obtain the desired two-photon widths for χ c0,2 according to (3). First we would like to predict f 0/2 and R and compare with BESIII experiments. Following the analysis conducted for the ratio of the decay rates of J/ψ → e + e − to η c → γγ [21,39], we also expand these two ratios strictly to the second order in α s : R = 4 15 Ω 1 − π 2 3 + 20 9 α s π (17a) − 5.855 + 22.967 ln µ R m + 15.791 ln m µ Λ α s π 2 , f 0/2 = α 2 s 216π 2 8 + 3π 2 − 48 ln 2 2 ,(17b) where n L = 3 has been taken, and Ω is defined by Ω = R ′ χc2 (µ Λ ) R ′ χc0 (µ Λ ) 2 ,(18) which characterizes the extent of the violation of HQSS. With the nonperturbative matrix elements cancelled, the helicity ratio f 0/2 is entirely determined by the orderα s (leading) contribution of the λ = 0 component from χ 2 decay in (8a). In the following phenomenological analysis, we will use the two-loop quark pole masses as m c = 1.68 GeV and m b = 4.78 GeV [22]. Running strong coupling at a given scale is evaluated by the package RunDec [40]. We first present the NRQCD predictions accurate to NLO in α s : where the uncertainty comes from varying the renormalization scale in the range 1 GeV < µ R < 2m, with central value at µ R = m. Assuming Ω = 1, the predicted R then becomes considerably smaller than the BESIII data in (2a). R = (0.124 +0.032 −0.028 ) Ω, f 0/2 = 0,(19) From (17), we further give our predictions at NNLO accuracy: R = (0.075 +0.044 −0.051 ) Ω, f 0/2 = 0.0009 +0.0009 −0.0004 ,(20) with the central values obtained by setting µ Λ = 1 GeV and µ R = m. Two kinds of uncertainties are included by sliding the µ Λ , µ R in the range m 2 < µ Λ < m and 1 GeV < µ R < 2m, respectively. While the very tiny f 0/2 predicted in (20) fully agrees with the BESIII measurement within errors, the NNLO prediction of R deviates further from the data relative to the NLO prediction, in the HQSS limit. If the HQSS-violating effects would lead to Ω > 1, our NNLO predictions in (20) would still have chance to agree with the data. The spin-dependent interactions such as spin-orbital force and tensor force have been incorporated to study the fine splitting among χ cJ [41]. In order to elucidate the role played by the HQSS violation, we have implemented these spin-dependent forces within the Cornell potential model [χ c0(2) acquires a repulsive (attractive) 1 r 3 potential, respectively], and found that the curvatures of the radial wave functions of χ c0 and χ c2 are changed towards the opposite direction such that Ω < 1. Therefore, the discrepancy between (20) and the BESIII measurement of R even further deteriorates! To closely examine the impact of NNLO corrections, we can also extract the nonperturbative factors R ′ χc0,2 (µ Λ ) from the measured two-photon widths of χ c0,2 in (1). In Table I we tabulated these fitted parameters at various levels of accuracy in α s . Although the NNLO corrections to χ c0 → γγ are sizable, one is still able to obtain a reasonable value for the matrix element; however, for the χ c2 → γγ, both NLO and NNLO corrections are negative yet substantial, such that the partial width turns negative in some parameter space, and we refrain from listing the fitted value of R ′ χc2 in Table I. In Fig. 2, we show the values of |R ′ χc0,2 (µ Λ )| 2 fitted to account for the BESIII data following the NNLO for- FIG. 2: The dependence of |R ′ χ cJ (µΛ)| 2 , which are fitted from the BESIII data using the NNLO formula, as a function of µR. The blue and green bands are obtained by varying µΛ from 1 GeV to m, and the two horizontal lines correspond to the respective values given by B-T and Cornell potential models [42]. mula, as a function of µ R . For χ c0 → γγ, within reasonable choice of µ R and µ Λ , the fitted |R ′ χc0 | 2 agrees with those predicted from the famous Cornell and Buchmüller-Tye (B-T) potential models [42]. However, for small µ R , the fitted |R ′ χc2 | 2 becomes negative, hence unphysical; for large µ R , |R ′ χc2 | 2 > |R ′ χc0 | 2 so that Ω > 1, in contradiction to what is implied by the spin-dependent force. While the NNLO corrections to χ c0 → γγ are under theoretical control, it appears rather challenging to account for the χ c2 → γγ data from our results. It is straightforward to adapt (17) to analyze P -wave bottomonia decays to two photons, by taking n L = 4. The NLO perturbative predictions are R b = (0.169 +0.015 −0.073 ) Ω b , f b 0/2 = 0,(21) where Ω b is the bottomonium counterpart of (18). After incorporating the NNLO corrections, we then predict R b = (0.126 +0.025 −0.144 ) Ω, f b 0/2 = 0.0004 +0.0014 −0.0001 ,(22) where the central values are obtained by setting µ Λ = m b 2 and µ R = m b . The uncertainty is estimated by varying µ Λ , µ R in the range 1 GeV < µ Λ < m b and 1 GeV < µ R < 2m b . Notably, the convergence of perturbative expansion for χ bJ → γγ has been considerably improved with respect to χ cJ decay, and we also expect here the HQSS-violation has smaller impact. To summarize, in this work we have computed, for the first time, the complete order-α 2 s corrections to χ c,b → γγ in NRQCD framework, deducing the corresponding SDCs for each helicity amplitude. The NNLO corrections to χ c0,2 → γγ are found to be substantial, and we find it rather difficult to account for the ratio of their decay rates recently measured by BESIII. This discrepancy even deteriorates after incorporating the spin-dependent inter-quark interaction. To resolve this puzzle, it is worth computing higher-order radiative corrections, as well as including the relativistic corrections. However, our poor knowledge of the higher-order P -wave NRQCD matrix elements renders a sharp order-v 2 prediction unrealistic [43,44] . In contrast, we believe our O(α 2 s ) predictions to χ b0,2 → γγ are trustworthy. Hopefully, the forthcoming Belle II experiments, and the next-generation highenergy colliders, will have a bright prospect to measure these two-photon decay channels, thereby test our predictions. FIG. 1 : 1Representative Feynman diagrams for cc( 3 P TABLE I : IDetermination of |R ′ χ cJ (µΛ)| 2 (GeV 5 ) from BESIII data at various level of perturbative accuracy. The uncer- tainty is estimated by combining the experimental error with that by varying µR from 1 GeV to 2m. Acknowledgment. We thank Estia Eichten for enlightening discussion on the spin-dependent inter-quark force. . T Appelquist, H D Politzer, Phys. Rev. Lett. 3443T. Appelquist and H. D. Politzer, Phys. Rev. Lett. 34, 43 (1975). . A De Rujula, S L Glashow, Phys. Rev. Lett. 3446A. De Rujula and S. L. Glashow, Phys. Rev. Lett. 34, 46 (1975). . W Kwong, P B Mackenzie, R Rosenfeld, J L Rosner, Phys. Rev. D. 373210W. Kwong, P. B. Mackenzie, R. Rosenfeld and J. L. Ros- ner, Phys. Rev. D 37, 3210 (1988). . M B Voloshin, arXiv:0711.4556Prog. Part. Nucl. Phys. 61455hep-phM. B. Voloshin, Prog. Part. Nucl. Phys. 61, 455 (2008) [arXiv:0711.4556 [hep-ph]]. . Z P Li, F E Close, T Barnes, Phys. Rev. D. 432161Z. P. Li, F. E. Close and T. Barnes, Phys. Rev. D 43, 2161 (1991). . S N Gupta, J M Johnson, W W Repko, hep-ph/9606349Phys. Rev. D. 542075S. N. Gupta, J. M. Johnson and W. W. Repko, Phys. Rev. D 54, 2075 (1996) [hep-ph/9606349]. . S Godfrey, N Isgur, Phys. Rev. D. 32189S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985). . C R Munz, hep-ph/9601206Nucl. Phys. A. 609C. R. Munz, Nucl. Phys. A 609, 364 (1996) [hep-ph/9601206]. . D Ebert, R N Faustov, V O Galkin, hep-ph/0302044Mod. Phys. Lett. A. 18601D. Ebert, R. N. Faustov and V. O. Galkin, Mod. Phys. Lett. A 18, 601 (2003) [hep-ph/0302044]. . H W Huang, C F Qiao, K T Chao, hep-ph/9601380Phys. Rev. D. 542123H. W. Huang, C. F. Qiao and K. T. Chao, Phys. Rev. D 54, 2123 (1996) [hep-ph/9601380]. . G T Bodwin, E Braaten, G P Lepage, hep-ph/9407339Phys. Rev. D. 515853Phys. Rev. DG. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Phys. Rev. D 55, 5853 (1997)] [hep-ph/9407339]. . A Petrelli, M Cacciari, M Greco, F Maltoni, M L Mangano, hep-ph/9707223Nucl. Phys. B. 514245A. Petrelli, M. Cacciari, M. Greco, F. Maltoni and M. L. Mangano, Nucl. Phys. B 514, 245 (1998) [hep-ph/9707223]. . J J Dudek, R G Edwards, hep-ph/0607140Phys. Rev. Lett. 97172001J. J. Dudek and R. G. Edwards, Phys. Rev. Lett. 97, 172001 (2006) [hep-ph/0607140]. . K M Ecklund, CLEO CollaborationarXiv:0803.2869Phys. Rev. D. 7891501hep-exK. M. Ecklund et al. [CLEO Collaboration], Phys. Rev. D 78, 091501 (2008) [arXiv:0803.2869 [hep-ex]]. . M Ablikim, BESIII CollaborationarXiv:1205.4284Phys. Rev. D. 85112008hep-exM. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 85, 112008 (2012) [arXiv:1205.4284 [hep-ex]]. . R Barbieri, R Gatto, R Kogerler, Phys. Lett. B. 60183R. Barbieri, R. Gatto and R. Kogerler, Phys. Lett. B 60, 183 (1976). . R Barbieri, M Caffo, R Gatto, E Remiddi, Phys. Lett. B. 9593R. Barbieri, M. Caffo, R. Gatto and E. Remiddi, Phys. Lett. B 95, 93 (1980). . R Barbieri, M Caffo, R Gatto, E Remiddi, Nucl. Phys. B. 19261R. Barbieri, M. Caffo, R. Gatto and E. Remiddi, Nucl. Phys. B 192, 61 (1981). . A Czarnecki, K Melnikov, hep-ph/9712222Phys. Rev. Lett. 802531A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 80 (1998) 2531 [hep-ph/9712222]. . M Beneke, A Signer, V A Smirnov, hep-ph/9712302Phys. Rev. Lett. 802535M. Beneke, A. Signer and V. A. Smirnov, Phys. Rev. Lett. 80 (1998) 2535 [hep-ph/9712302]. . A Czarnecki, K Melnikov, hep-ph/0109054Phys. Lett. B. 519212A. Czarnecki and K. Melnikov, Phys. Lett. B 519 (2001) 212 [hep-ph/0109054]. . F Feng, Y Jia, W L Sang, arXiv:1505.02665hepphF. Feng, Y. Jia and W. L. Sang, arXiv:1505.02665 [hep- ph]. . A I Onishchenko, O L Veretin, hep-ph/0302132Eur. Phys. J. C. 50801A. I. Onishchenko and O. L. Veretin, Eur. Phys. J. C 50, 801 (2007) [hep-ph/0302132]. . L B Chen, C F Qiao, arXiv:1503.05122hep-phL. B. Chen and C. F. Qiao, arXiv:1503.05122 [hep-ph]. . T Hahn, hep-ph/0012260Comput. Phys. Commun. 140T. Hahn, Comput. Phys. Commun. 140, 418 (2001) [hep-ph/0012260]. . R Mertig, M Bohm, A Denner, Comput. Phys. Commun. 64345R. Mertig, M. Bohm and A. Denner, Comput. Phys. Commun. 64, 345 (1991). . F Feng, R Mertig, arXiv:1212.3522F. Feng and R. Mertig, arXiv:1212.3522. . F Feng, arXiv:1204.2314Comput. Phys. Commun. 1832158hep-phF. Feng, Comput. Phys. Commun. 183, 2158 (2012) [arXiv:1204.2314 [hep-ph]]. . A V Smirnov, arXiv:1408.2372Comput. Phys. Commun. 189hep-phA. V. Smirnov, Comput. Phys. Commun. 189, 182 (2014) [arXiv:1408.2372 [hep-ph]]. Analytic tools for Feynman integrals. V A Smirnov, Springer Tracts Mod. Phys. 2501V. A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250, 1 (2012). . J Gluza, K Kajda, T Riemann, arXiv:0704.2423Comput. Phys. Commun. 177hep-phJ. Gluza, K. Kajda and T. Riemann, Comput. Phys. Commun. 177, 879 (2007) [arXiv:0704.2423 [hep-ph]]. . M Czakon, hep-ph/0511200Comput. Phys. Commun. 175M. Czakon, Comput. Phys. Commun. 175, 559 (2006) [hep-ph/0511200]. . A V Smirnov, arXiv:1312.3186Comput. Phys. Commun. 1852090hep-phA. V. Smirnov, Comput. Phys. Commun. 185, 2090 (2014) [arXiv:1312.3186 [hep-ph]]. . R Cools, A Haegemans, ACM Trans. Math. Softw. 293296R. Cools and A. Haegemans, ACM Trans. Math. Softw. 29 (2003), no. 3 287 C296. . D J Broadhurst, N Gray, K Schilcher, Z. Phys. C. 52111D. J. Broadhurst, N. Gray and K. Schilcher, Z. Phys. C 52, 111 (1991). . K Melnikov, T Van Ritbergen, hep-ph/0005131Nucl. Phys. B. 591515K. Melnikov and T. van Ritbergen, Nucl. Phys. B 591, 515 (2000) [hep-ph/0005131]. . A H Hoang, P Ruiz-Femenia, hep-ph/0609151Phys. Rev. D. 74114016A. H. Hoang and P. Ruiz-Femenia, Phys. Rev. D 74, 114016 (2006) [hep-ph/0609151]. . J H Kuhn, J Kaplan, E G O Safiani, Nucl. Phys. B. 157125J. H. Kuhn, J. Kaplan and E. G. O. Safiani, Nucl. Phys. B 157, 125 (1979). . Y Kiyo, A Pineda, A Signer, arXiv:1006.2685Nucl. Phys. B. 841231hep-phY. Kiyo, A. Pineda and A. Signer, Nucl. Phys. B 841, 231 (2010) [arXiv:1006.2685 [hep-ph]]. . K G Chetyrkin, J H Kuhn, M Steinhauser, hep-ph/0004189Comput. Phys. Commun. 133K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, Com- put. Phys. Commun. 133, 43 (2000) [hep-ph/0004189]. . N Brambilla, A Vairo, hep-ph/0411156Phys. Rev. D. 7134020N. Brambilla and A. Vairo, Phys. Rev. D 71, 034020 (2005) [hep-ph/0411156]. . E J Eichten, C Quigg, hep-ph/9503356Phys. Rev. D. 521726E. J. Eichten and C. Quigg, Phys. Rev. D 52, 1726 (1995) [hep-ph/9503356]. . J P Ma, Q Wang, hep-ph/0203082Phys. Lett. B. 537233J. P. Ma and Q. Wang, Phys. Lett. B 537, 233 (2002) [hep-ph/0203082]. . N Brambilla, E Mereghetti, A Vairo, hep-ph/0604190JHEP. 060839Erratum-ibid. 1104, 058 (2011)N. Brambilla, E. Mereghetti and A. Vairo, JHEP 0608, 039 (2006) [Erratum-ibid. 1104, 058 (2011)] [hep-ph/0604190].
[]
[ "Functional Renormalization Group Approach for Inhomogeneous One-Dimensional Fermi Systems with Finite-Ranged Interactions", "Functional Renormalization Group Approach for Inhomogeneous One-Dimensional Fermi Systems with Finite-Ranged Interactions" ]
[ "Lukas Weidinger \nArnold Sommerfeld Center for Theoretical Physics\nCenter for NanoScience\nLudwig-Maximilians-Universität München\nTheresienstrasse 37D-80333MünchenGermany\n", "Florian Bauer \nArnold Sommerfeld Center for Theoretical Physics\nCenter for NanoScience\nLudwig-Maximilians-Universität München\nTheresienstrasse 37D-80333MünchenGermany\n", "Jan Von Delft \nArnold Sommerfeld Center for Theoretical Physics\nCenter for NanoScience\nLudwig-Maximilians-Universität München\nTheresienstrasse 37D-80333MünchenGermany\n" ]
[ "Arnold Sommerfeld Center for Theoretical Physics\nCenter for NanoScience\nLudwig-Maximilians-Universität München\nTheresienstrasse 37D-80333MünchenGermany", "Arnold Sommerfeld Center for Theoretical Physics\nCenter for NanoScience\nLudwig-Maximilians-Universität München\nTheresienstrasse 37D-80333MünchenGermany", "Arnold Sommerfeld Center for Theoretical Physics\nCenter for NanoScience\nLudwig-Maximilians-Universität München\nTheresienstrasse 37D-80333MünchenGermany" ]
[]
We introduce an equilibrium formulation of the functional renormalization group (fRG) for inhomogeneous systems capable of dealing with spatially finite-ranged interactions. In the general third order truncated form of fRG, the dependence of the two-particle vertex is described by O(N 4 ) independent variables, where N is the dimension of the single-particle system. In a previous paper [Phys. Rev. B 89, 045128 (2014)], the so-called coupled-ladder approximation (CLA) was introduced and shown to admit a consistent treatment for models with a purely onsite interaction, reducing the vertex to O(N 2 ) independent variables. In this work, we introduce an extended version of this scheme, called the extended coupled ladder approximation (eCLA), which includes a spatially extended feedback between the individual channels, measured by a feedback length L, using O(N 2 L 2 ) independent variables for the vertex. We apply the eCLA to three types of one-dimensional model systems: First, we study a model of a quantum point contact (QPC) with a parabolic barrier top and short-ranged interactions and show that eCLA achieves convergence for L ≈ lx, where lx is the characteristic length of the QPC. It also turns out that the additional feedback stabilizes the fRG-flow. This enables us, second, to study the geometric crossover between a QPC and a quantum dot, again for a one-dimensional model with short-ranged interactions. Third, the enlarged feedback also enables the treatment of a finite-ranged interaction extending over up to L sites. Using a simple estimate for the form of such a finite-ranged interaction in a QPC with parabolic barrier top, we study its effects on the conductance and the density. We find that for low densities and sufficiently large interaction ranges the conductance develops additional oscillatory features, and the corresponding density shows some fluctuations that can be interpreted as Friedel oscillations arising from a renormalized barrier shape with a rather flat top and steep flanks. arXiv:1609.07423v1 [cond-mat.str-el]
10.1103/physrevb.95.035122
[ "https://arxiv.org/pdf/1609.07423v2.pdf" ]
119,181,706
1609.07423
6e01a85ebcadb79eb32bdc388acebed74a27ca96
Functional Renormalization Group Approach for Inhomogeneous One-Dimensional Fermi Systems with Finite-Ranged Interactions Lukas Weidinger Arnold Sommerfeld Center for Theoretical Physics Center for NanoScience Ludwig-Maximilians-Universität München Theresienstrasse 37D-80333MünchenGermany Florian Bauer Arnold Sommerfeld Center for Theoretical Physics Center for NanoScience Ludwig-Maximilians-Universität München Theresienstrasse 37D-80333MünchenGermany Jan Von Delft Arnold Sommerfeld Center for Theoretical Physics Center for NanoScience Ludwig-Maximilians-Universität München Theresienstrasse 37D-80333MünchenGermany Functional Renormalization Group Approach for Inhomogeneous One-Dimensional Fermi Systems with Finite-Ranged Interactions (Dated: September 26, 2016) We introduce an equilibrium formulation of the functional renormalization group (fRG) for inhomogeneous systems capable of dealing with spatially finite-ranged interactions. In the general third order truncated form of fRG, the dependence of the two-particle vertex is described by O(N 4 ) independent variables, where N is the dimension of the single-particle system. In a previous paper [Phys. Rev. B 89, 045128 (2014)], the so-called coupled-ladder approximation (CLA) was introduced and shown to admit a consistent treatment for models with a purely onsite interaction, reducing the vertex to O(N 2 ) independent variables. In this work, we introduce an extended version of this scheme, called the extended coupled ladder approximation (eCLA), which includes a spatially extended feedback between the individual channels, measured by a feedback length L, using O(N 2 L 2 ) independent variables for the vertex. We apply the eCLA to three types of one-dimensional model systems: First, we study a model of a quantum point contact (QPC) with a parabolic barrier top and short-ranged interactions and show that eCLA achieves convergence for L ≈ lx, where lx is the characteristic length of the QPC. It also turns out that the additional feedback stabilizes the fRG-flow. This enables us, second, to study the geometric crossover between a QPC and a quantum dot, again for a one-dimensional model with short-ranged interactions. Third, the enlarged feedback also enables the treatment of a finite-ranged interaction extending over up to L sites. Using a simple estimate for the form of such a finite-ranged interaction in a QPC with parabolic barrier top, we study its effects on the conductance and the density. We find that for low densities and sufficiently large interaction ranges the conductance develops additional oscillatory features, and the corresponding density shows some fluctuations that can be interpreted as Friedel oscillations arising from a renormalized barrier shape with a rather flat top and steep flanks. arXiv:1609.07423v1 [cond-mat.str-el] I. INTRODUCTION The functional renormalization group (fRG) is a wellestablished tool for studying interacting many-body systems [1][2][3][4][5][6]. This technique treats interactions using an RG-enhanced perturbation theory and is known to provide an efficient way to treat correlations. In particular, fRG can be used to treat spatially inhomogeneous systems, represented by a discretized model with N sites. For example, about N ∼ 10 2 sites are required to represent the electrostatic potential of a quasi-one-dimensional point contact in a manner that is sufficiently smooth to avoid finite-sitze effects [7]. The corresponding twoparticle vertex has O(N 4 ) ∼ 10 8 independent spatial components. To make numerical computations feasible, simplifying approximations have to be made to reduce the number of components used to describe the vertex. Such a scheme, called the coupled-ladder approximation (CLA), was proposed in Ref. 7 for the case of onsite interactions. Bauer, Heyder and von Delft (BHD) [8] supplied a detailed description of the CLA which is in principle applicable to systems of arbitrary dimensionality. The CLA is implemented within the context of generic, third-ordertruncated fRG, meaning that all vertices with three and higher particle number are set to zero throughout the whole flow. In this paper we generalize this scheme to be able to treat a finite-ranged interaction. Since the central aim of our scheme is to extend the spatial range over which information is fed back into the RG flow, we call our scheme the extended coupled ladder approximation (eCLA). The basic idea of the CLA, and by extension the eCLA, lies in reducing the number independent components of the vertex by decomposing it into several interaction channels and then establishing a consistent approximation by controlling the amount of feedback between the individual channels. This strategy follows that used in Refs. 4 and 9 in the context of the single-impurity Anderson model. For a model with short-ranged interactions, this approach reduces the number of independent quantities in the vertex to order ∼ O(N 2 ). From a perturbative point of view, this treatment is exact in second order in the interaction and amounts to summing up a large class of parquet-type diagrams, including mutual feedback between the different interaction channels. The eCLA generalizes the CLA by extending spatial feedback between the channels. As a control parameter for this extended feedback we introduce a feedback length L, where L = 0 corresponds to the previous approximation scheme used by BHD, while L = N − 1 includes the full fRG flow in second order. L thus serves as a control parameter for the number of independent spatial components of the vertex, which scales as ∼ O(N 2 L 2 ). Moreover, the longer-ranged feedback allows us also to treat interactions with finite range up to L U sites (with L U ≤ L) in a manner that is exact to second order in the interaction. In this paper, we present a detailed account of the eCLA, and apply it to two one-dimensional (1D) fermionic systems, modeled to describe the lowest 1D subband of a quantum point contact (QPC) or a quantum dot (QD), respectively. We develop the eCLA for systems described by a Hamiltonian of the form H = ij,σ h σ ij d † iσ d jσ + 1 2 ij,σσ U ijniσnjσ (1 − δ ij δ σσ ) ,(1) where h σ and U are real, symmetric matrices, d † jσ creates an electron in single particle state j with spin σ (=↑, ↓ or +, −, withσ = −σ), and n jσ = d † jσ d jσ . In the context of the applications presented here, we refer to the quantum number j as the "site index". Our eCLA scheme requires the interaction to have a finite range L U ≤ L, such that U ij = 0 if |i − j| > L U .(2) Models of this form, but with onsite interactions (U ij = U δ ij ), have been used to study both QPC and QD systems [7]. To describe a QPC, h σ ij is taken to represent a one-dimensional tight-binding chain, with a potential barrier with parabolic top, whereas for a QD it is chosen to represent a double-barrier potential. The noninteracting physics of both models is well known, whereas the effect of interactions, especially for the QPC, are still a topic of ongoing discussions [10][11][12]. For the QPC the conductance is quantized [13][14][15] in units of the conductance quantum G Q = 2e 2 /h, but shows an additional shoulder at approximately 0.7G Q . This regime, in which other observables show anomalous behavior too [16][17][18], is commonly known as the "0.7-anomaly". The latter has been studied in [7] using a model of the above form, with purely onsite interactions. However, to examine the effect of gate induced screening in a QPC, one needs to consider finite ranged interactions. This goal serves as the main motivation for developing the eCLA put forth in this paper. The numerical results presented here were all obtained using the eCLA in a static approximation, which neglects the frequency-dependence of the two-particle vertex (after which the approach no longer is exact to second order). Nevertheless, BHD have shown that for a QPC model with onsite interactions, the CLA with a static approximation leads to reasonable results for the conductance step shape, though it does produce some artifacts regarding the pinch-off gate voltage when the interaction strength is increased. We find the same to be true for the static eCLA, with the artifacts becoming more pronounced with increasing interaction range, but the step shape behaving in a physically reasonable manner. We use the eCLA for three studies of increasing complexity. (i) We present static eCLA results for a QPC model with short-ranged interaction and successively increase the feedback length L. This systematically improves the treatment of RG-feedback between the various fRG channels, and for sufficiently large L converges to the full solution of the generic, third-order-truncated static fRG. We find that convergence is achieved once L becomes larger than the length l x characterizing the parabolic QPC potential barrier, which for our model with N = 81 is as small as L = 5. For such a system, the eCLA scheme thus speeds up the calculation relative to the full generic, third-order-truncated static fRG by a factor of 10 3 , without any loss of accuracy. (ii) Furthermore, it turns out that the eCLA's enhanced feedback leads to a more stable fRG flow than the CLA scheme, since each interaction channel acts more strongly to limit the tendencies other channels might have to diverge during the fRG flow. This enables us to study the geometric crossover between a QPC and a QD where the barrier top stays close to the chemical potential. This setup features a high local density of states (LDOS) at the chemical potential, and as a result turns out to be intractable when using the CLA without enhanced feedback [19]. In contrast, the eCLA is able to treat this challenging crossover very nicely. (iii) Finally, we illustrate the potential of the eCLA to deal with finite-ranged interactions in a setting where the physics of screening comes into play, namely a QPC model with an interaction whose range extends over up to N sites. The purpose of this study is mainly methodological, i.e. we do not aim here to achieve a fully realistic treatment of screening in a QPC. Nevertheless, the results are interesting: for a sufficiently long ranged interaction and sufficiently low density, there exists a parameter regime where we find oscillatory features in the conductance plateau and corresponding 2k F density fluctuations within the QPC. The paper has three main parts. The first part (section II) develops our improved eCLA feedback scheme. The second part (section III) studies its consequences for QPC and QD models with onsite interaction, focusing on the effects of increasing the feedback length L. Finally, the third part (section IV) is devoted to finiteranged interactions. We estimate the approximate form and strength of the interaction to be used for a 1D depiction of a QPC and show some preliminary results for the conductance and density profile of such a system depending on the screening properties. A detailed study of the physics of long-ranged interactions in QPCs is beyond the scope of this work and left as a topic of future investigation. II. FRG FLOW EQUATIONS Before we introduce our new eCLA scheme, we give a short overview over the general idea and the usual approximations made in fRG. Since numerous detailed treatments of fRG are available, and since our work builds on that of BHD, the discussion below is very brief and structured similarly to that in Ref. 8. The basic idea of fRG is to introduce a flow parameter Λ in the bare propagator of the theory in such a way that for Λ = Λ i = ∞, the structure of the resulting vertex functions are very simple. With our choice for Λ (described later) all but the two-particle vertex will vanish, γ Λi 2 = v γ Λi n = 0 (n = 2),(3) where v is the bare vertex. For the final value of the flow parameter Λ = Λ f = 0, one recovers the full bare propagator and hence the full theory: G Λ 0 → G 0 , with G Λi 0 = 0, G Λ f 0 = G 0 .(4) The RG flow is described by a hierarchy of coupled differential equations for the one particle irreducible (1PI) n-particle vertex functions γ n , d dΛ γ Λ n = F Λ, G Λ 0 , γ Λ 1 , . . . , γ Λ n+1 .(5) Integrating this system from Λ = Λ i to Λ = 0 yields in principle a full description of all interaction vertices. In practice, one can of course not treat an infinite hierarchy of flow equations and has to truncate it at some point. In our form of third-order truncated fRG, we incorporate the one-and two-particle vertex into the flow, but set all vertices with three or more particles to zero d dΛ γ n = 0 (n ≥ 3) .(6) We thus retain only the flow of the self-energy, Σ = −γ 1 , and the flow of the two-particle vertex γ 2 . This differential equation can then be solved numerically, using a standard Runge-Kutta method. As we will see shortly, the flow of the vertex consists of three different parquet-like channels which are coupled to the flow of the self-energy and also directly to each other. This simultaneous treatment moderates competing instabilities in an unbiased way. In principle, the form of the fRG flow equations depends on the choice of the flow parameter, even if in most cases they take the form stated below. In our work, we choose the Λ-dependence of the bare propagator to take the form of an infrared cutoff G Λ 0 (ω n ) = Θ T (|ω n | − Λ)G 0 (ω n ) , Λ i = ∞, Λ f = 0 . (7) We use the Matsubara formalism with the frequencies ω n defined to be purely imaginary, ω n = iT π(2n + 1), and Θ T is a step function broadened on the scale of temperature. Using this cutoff, one can derive the fRG equations in the standard way, see e.g. Refs. 5 and 20 or Ref. 21 for a diagrammatic derivation. The resulting equation for the one-particle vertex is given by d dΛ γ Λ 1 (q 1 , q 1 ) = T q 2 ,q 2 S Λ q 2 ,q 2 γ Λ 2 (q 2 , q 1 ; q 2 , q 1 ) ,(8) where q i is a shorthand for all quantum numbers and the fermionic Matsubara frequency associated with the legs of a vertex, and the full-and single-scale propagators are defined via G Λ = G Λ 0 −1 − Σ Λ −1 ,(9a)S Λ = G Λ ∂ Λ G Λ 0 −1 G Λ ,(9b) respectively. The structure of the vertex consists naturally of three different parquet-like channels γ Λ 2 = v + γ Λ p + γ Λ x + γ Λ d ,(10) where v is the bare vertex and we refer to γ Λ p , γ Λ x , and γ Λ d as the particle-particle channel (P ), and the exchange (X) and direct (D) part of the particle-hole channel. These quantities are defined via their flow equations d dΛ γ Λ 2 = d dΛ (γ Λ p + γ Λ x + γ Λ d ) ,(11) and the initial conditions γ Λi p = γ Λi x = γ Λi d = 0. The explicit form of the flow equations is d dΛ γ Λ p (q 1 , q 2 ; q 1 , q 2 ) = T q 3 ,q 3 ,q 4 ,q 4 γ Λ 2 (q 1 , q 2 ; q 3 , q 4 )S Λ q 3 ,q 3 G Λ q 4 ,q 4 γ Λ 2 (q 3 , q 4 ; q 1 , q 2 ), (12a) d dΛ γ Λ x (q 1 , q 2 ; q 1 , q 2 ) = T q 3 ,q 3 ,q 4 ,q 4 γ Λ 2 (q 1 , q 4 ; q 3 , q 2 ) S Λ q 3 ,q 3 G Λ q 4 ,q 4 + G Λ q 3 ,q 3 S Λ q 4 ,q 4 γ Λ 2 (q 3 , q 2 ; q 1 , q 4 ) ,(12b)d dΛ γ Λ d (q 1 , q 2 ; q 1 , q 2 ) = −T q 3 ,q 3 ,q 4 ,q 4 γ Λ 2 (q 1 , q 3 ; q 1 , q 4 ) S Λ q 4 ,q 4 G Λ q 3 ,q 3 + G Λ q 4 ,q 4 S Λ q 3 ,q 3 γ Λ 2 (q 4 , q 2 ; q 3 , q 2 ) .(12c) At this point, the channels have a full feedback between them. Later on, however, we will control the amount of feedback between channels by the feedback length L. A. Frequency Parametrisation Since we have energy conservation at each vertex, γ 1 (q 1 , q 1 ) ∝ δ n 1 n 1 , γ 2 (q 1 , q 2 ; q 1 , q 2 ) ∝ δ n 1 +n 2 n 1 +n 2 , we can parametrize the frequency dependence of the selfenergy with one frequency, and of the vertex with three frequencies. A detailed discussion of the frequency structure is given in Refs. 4, 8, and 9, and since we proceed analogously, we will be very brief here. A convenient choice for the parametrization of the vertex frequency structure is given in terms of the three bosonic frequencies [7] Π = ω n 1 + ω n 2 = ω n 1 + ω n 2 , (14a) X = ω n 1 − ω n 2 = ω n 1 − ω n 2 ,(14b)∆ = ω n 1 − ω n 1 = ω n 2 − ω n 2 .(14c) In order to keep notation short, the frequency information is separated from the site and spin quantum numbers: γ 2 (j 1 σ 1 ω n 1 , j 2 σ 2 ω n 2 ; j 1 σ 1 ω n 1 , j 1 σ 2 ω n 2 ) = δ n 1 +n 2 n 1 +n 2 γ 2 (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 1 σ 2 ; Π, X, ∆) (15) For convenience, we have here also listed the fermionic frequencies in terms of the bosonic ones: ω n 1 = 1 2 (Π + X + ∆) , ω n 2 = 1 2 (Π − X − ∆) , (16a) ω n 1 = 1 2 (Π + X − ∆) , ω n 2 = 1 2 (Π − X + ∆) . (16b) B. Coupled-Ladder Approximation The basic idea of the CLA scheme was introduced in Refs. 4 and 9 for the frequency parametrization of the single-impurity Anderson model and was further developed for inhomogeneous Fermi systems with onsite interaction in Ref. 7. Here we will go one step further and extend this scheme to treat interacting models with two-particle interactions of finite range, using an idea similar to the singular mode fRG approach introduced in 22. There, the vertex structure in momentum space was decomposed into fermion bilinears that interact via exchange bosons and it was shown that this decomposition admits a systematic approximation by an expansion using form factors. Here, we will proceed similar in position space, introducing "short indices" k, l that will control the extent of our approximation and act similar to the mentioned form factor expansion. In the case of third-order truncated fRG, BHD introduced two different approximation schemes. The simpler "static second order fRG" (sfRG2) neglects the frequency dependence of the vertex; the more elaborate "dynamic second order fRG" (dfRG2) includes the frequency dependence of the vertex within a channel approximation, reducing this dependence from the generic O(N 3 f ) to O(N f ) where N f is the number of used frequencies. In the case of the onsite model, it turned out that static compared to dynamic fRG produces some artifacts concerning the pinch-off point of the conductance of a QPC, but yields essentially the same shape for the conductance steps as dynamic fRG. For this reason and since it is a factor of N f cheaper, we will only compute the static fRG flow in our numerical work. Nevertheless, we will derive here the full dynamic flow equations, and in principle, it should be no problem to implement these too. The dfRG2 scheme exploits the fact that the bare vertex consists of a density-density interaction v(j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 ) = δ L U j 1 j 2 U j 1 j 2 1 − δ j 1 j 2 δ σ 1 σ 2 + δ σ 1σ2 × δ j 1 j 1 δ j 2 j 2 δ σ 1 σ 1 δ σ 2 σ 2 − δ j 1 j 2 δ j 2 j 1 δ σ 1 σ 2 δ σ 2 σ 1 ,(17) and parametrizes the vertex in terms of O(N 2 L 2 U N f ) independent variables. Here δ L U j1j2 =1 if |j 1 − j 2 | ≤ L U and is otherwise set to zero. Using this vertex, we can now consider a simplified version of the vertex flow equation (12), where the feedback of the vertex flow is neglected: on the r.h.s. we replace γ Λ 2 → v. If the feedback of the self-energy were also neglected, this would be equivalent to calculating the vertex in second order perturbation theory. As a consequence, all generated vertex contributions have one of the following structures: P kl jiσσ (Π) := γ Λ p (jσ, j +k σ ; iσ, i+l σ ; Π) O(v 2 ) Π − ω n jσ j + k σ ω n Π − ω n σ σ ω n iσ i + l σ ω n Π − ω n , (18a) P kl jiσσ (Π) := γ Λ p (jσ, j +k σ ; iσ , i+l σ, Π) O(v 2 ) Π − ω n jσ j + k σ ω n Π − ω n σ σ ω n iσ i + l σ ω n Π − ω n , (18b) X kl jiσσ (X) := γ Λ x (jσ, i+l σ ; iσ, j +k σ ; X) O(v 2 ) X + ω n jσ j + k σ ω n X + ω n σ σ ω n iσ i + l σ ω n X + ω n , (18c) X kl jiσσ (X) := γ Λ x (jσ, i+l σ ; iσ , j +k σ; X) O(v 2 ) X + ω n jσ j + k σ ω n X + ω n µ µ ω n iσ i + l σ ω n X + ω n , (18d) D kl jiσσ (∆) := γ Λ d (jσ, i+l σ ; j +k σ, iσ ; ∆) O(v 2 ) jσ j + k σ iσ i + l σ ω n ω n ω n + ∆ ∆ + ω n µ ∆ + ω n µ ω n , (18e) D kl jiσσ (∆) := γ Λ d (jσ, i+l σ ; j +k σ , iσ; ∆) O(v 2 ) jσ j + k σ iσ i + l σ ω n ω n ω n + ∆ ∆ + ω n σ ∆ + ω n σ ω n ,(18f) These terms depend only on a single bosonic frequency. The upper indices kl , are taken to run over the range − L ≤ k, l ≤ L ,(19) where the control parameter L sets the "spatial feedback range". The bounds on the lower indices depend on the upper indices: if one of the site indices of γ 2 lies outside the region [−N , N ] where N is defined by N = 2N +1, γ 2 is zero. Therefore, i, j run between max(−N , −N − l) ≤ i ≤ min(N , N − l),(20)max(−N , −N − k) ≤ j ≤ min(N , N − k).(21) Note that for L = 0,X σσ and D σσ are at least of third order and P σσ , as well asP σσ of fourth order in the interaction. Thus, they where neglected in Ref. 8. Analogously to BHD, we now feed back all those terms on the r.h.s. of the flow equation (12) which conserve this site and spin structure indicated in Eq. (18). As a first consequence, each vertex quantity is fully fed back into its own flow equation. Secondly, the feedback between different quantities is restricted to those site indices which have the appropriate structure. Furthermore, to avoid frequency mixing, the feedback to a given channel from the other two channels is restricted to using only the static, i.e. zero frequency component of the latter. This scheme can be expressed by the replacement γ 2 →γ a(22) on the r.h.s. of channel a = p, x, d in Eq. (12) whereγ a is defined as: γ p (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 , Π) = δ L j 1 j 2 δ L j 1 j 2 γ 2 (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 ; Π, 0, 0) (23a) γ x (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 , X) = δ L j 1 j 2 δ L j 2 j 1 γ 2 (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 ; 0, X, 0) (23b)γ d (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 , ∆) = δ L j 1 j 1 δ L j 2 j 2 γ 2 (j 1 σ 1 , j 2 σ 2 ; j 1 σ 1 , j 2 σ 2 ; 0, 0, ∆) (23c) C. Symmetries As can readily be checked, these flow equations respect the following symmetry relations: G σΛ ij (ω n ) = G σΛ ji (ω n ) = G σΛ ij (−ω n ) * ,(24a)Σ σΛ ij (ω n ) = Σ σΛ ji (ω n ) = Σ σΛ ij (−ω n ) * ,(24b)P kl jiσσ (Π) = P lk ijσσ (Π) = P (−k)(−l) (j+k)(i+l)σ σ (Π) , P kl jiσσ (Π) =P lk ijσ σ (Π) =P (−k)(−l) (j+k)(i+l)σ σ (Π) , P kl jiσσ (Π) = −P −kl j+kiσ σ (Π) = −P k(−l) j(i+l)σσ (Π) , P σσ =P σσ ,(25a)X kl jiσσ (X) = X lk ijσσ (X) = X (−k)(−l) (j+k)(i+l)σ σ (X) * , X kl jiσσ (X) =X lk ijσ σ (X) = [X (−k)(−l) (j+k)(i+l)σσ (X)] * , X σσ =X σσ ,(25b)X = −D ,X = −D ,(25c)P kl jiσσ (Π) = P kl jiσσ (−Π) * , X kl jiσσ (X) = X kl jiσσ (−X) * , X kl jiσσ (∆) = X kl jiσσ (−∆) * . (25d) As a result, all relevant information is contained in a small number of independent frequency-dependent blockmatrices, which we define as follows: P Λ = P Λ ↑↓ , P Λ σ = P Λ σσ , X Λ = X Λ ↑↓ ,(26)D Λ = D Λ ↑↓ , D Λ σ = D Λ σσ , where the superscript Λ signifies a dependence on the flow parameter. The flow equations for these matrices can be derived starting from Eqs. (12). The replacement (22) restricts the internal quantum numbers on the r.h.s. of the flow equation q 3 , q 4 , q 3 , and q 4 according to the definitions (18): P klΛ ji (Π) =γ Λ p (j ↑, j +k ↓; i ↑, i+l ↓; Π) (27a) = T j i k l ,n γ Λ p (j ↑, j +k ↓; i ↑, i +l ↓; Π) S ↑Λ i j (ω n )G ↓Λ i +l j +k (Π−ω n )γ Λ p (j ↑, j +k ↓; i ↑, i+l ↓; Π) +γ Λ p (j ↑, j +k ↓; i ↓, i +l ↑; Π) S ↓Λ i j (ω n )G ↑Λ i +l j +k (Π−ω n )γ Λ p (j ↓, j +k ↑; i ↑, i+l ↓; Π) , P klΛ jiσ (Π) =γ Λ p (jσ, j +kσ; iσ, i+lσ; Π) (27b) = T j i k l ,nγ Λ p (jσ, j +k σ; i σ, i +l σ; Π) S σΛ i j (ω n )G σΛ i +l j +k (Π−ω n )γ Λ p (j σ, j +k σ; iσ, i+lσ; Π) X klΛ ji (X) =γ Λ x (j ↑, i+l ↓; i ↑, j +k ↓; X) (27c) = T i j l k ,nγ Λ x (j ↑, i +l ↓; i ↑, j +k ↓; X) S ↑Λ i j (ω n + X)G ↓Λ j +k i +l (ω n ) + S ↓Λ j +k i +l (ω n )G ↑Λ i j (ω n + X) γ Λ x (j ↑, i+l ↓; i ↑, j +k ↓; X) D klΛ jiσσ (X) =γ Λ d (jσ, i+l σ ; j +k σ, iσ ; ∆) = −T i j l k n,σ γ Λ d (jσ, i +l σ ; j +k σ, i σ ; ∆) S σ Λ i +l j +k (ω n )G σ Λ i j (ω n +∆) + G σ Λ i +l j +k (ω n )S σ Λ i j (ω n +∆) γ Λ d (j σ , i+l σ ; j +k σ , iσ ; ∆) (27d) The initial conditions are P Λi = P Λi σ = X Λi = D Λi σσ = 0 .(28) These equations can be compactly written in blockmatrix form d dΛ P Λ (Π) =P Λ (Π) · W pΛ (Π) ·P Λ (Π) , (29a) d dΛ P Λ σ (Π) =P Λ σ (Π) · W pΛ σ (Π) ·P Λ σ (Π) , (29b) d dΛ X Λ (X) =X Λ (X) · W xΛ (X) ·X Λ (X) , (29c) d dΛ D Λ σσ (∆) = − σ D Λ σσ (∆) · W dΛ σ (∆) ·D Λ σ σ (∆) ,(29d) where '·' denotes a block-matrix multiplication: [A · B] kl ji = j k A kk jj B k l j i(30) and we have introduced the definitions P klΛ ji (Π) =γ Λ p (j ↑, j +k ↓; i ↑, i+l ↓; Π) = δ ji δ kl U jj+k + P klΛ ji (Π) + δ L ji+l δ L ij+k X (i+l−j)(j+k−i)Λ ji (0) + δ L ij δ L j+ki+l D (i−j)(j+k−i−l)Λ j(i+l)↑↓ (0) ,(31a)P klΛ jiσ (Π) =γ Λ p (jσ, j + kσ; iσ, i + lσ; Π) = δ ji δ kl U jj+k − δ k,−l δ (j+k)i U ji + P klΛ jiσ (Π) − δ L i+lj δ L j+ki D (i+l−j)(j+k−i)Λ jiσ (0) + δ L ij δ L j+ki+l D (i−j)(j+k−i−l)Λ j(i+l)σ (0) , (31b) X klΛ ji (X) =γ Λ x (j ↑, i + l ↓; i ↑, j + k ↓; X) = δ ji δ kl U jj+k + X klΛ ji (X) + δ L i+lj δ L j+ki P (i+l−j)(j+k−i)Λ ji (0) + δ L ij δ L j+ki+l D (i−j)(i+l−j−k)Λ j(j+k)↑↓ (0) ,(31c)D klΛ jiσσ (∆) =γ Λ d (jσ, i + lσ ; j + kσ, iσ ; ∆) = δ 0k δ 0l U ji − δ σσ δ ji δ kl U jj+k + D klΛ jiσσ (∆) + δ L i+lj δ L j+ki P (i+l−j)(i−j−k)Λ j(j+k)σσ (0) + δ L ij δ L j+ki+l X (i−j)(i+l−j−k)Λ j(j+k)σσ (0) ,(31d) which account for the inter-channel feedback contained in equation (22). Note that Eq. (31d) is not fully expressed in terms of the definitions (26). This can only been done once σ and σ are specified explicitly and then leads to three independent equations. W p , W x and W d each represent a specific bubble, i.e. a product of two propagators summed over an internal frequency: Figure 1. The linear conductance g = G/GQ of a QPC as function of gate voltage, plotted for the cases with and without feedback of D ↑↓ in an intermediate parameter regime for four equidistant magnetic fields. Note that the difference between the two cases is suppressed with increasing the magnetic field. W kl,pΛ ij (Π) = T n S ↑Λ ij (ω n )G ↓Λ i+lj+k (Π−ω n ) +S ↓Λ i+lj+k (ω n )G ↑Λ ij (Π−ω n ) (32a) with D without D L = 0, U/ √ Ωxτ = 3 B/Ωx = 0 B/Ωx ≃ 0.5 0 0.5 1 −2 −1 0 1 Vg/Ωx gW kl,pΛ ijσ (Π) = T n S σΛ ij (ω n )G σΛ i+lj+k (Π−ω n ) (32b) W kl,xΛ ij (X) = T n S ↓Λ i+lj+k (ω n )G ↑Λ ij (ω n +X) + G ↓Λ j+ki+l (ω n )S ↑Λ ij (ω n +X) , (32c) W kl,dΛ ijσ (∆) = T n S σΛ i+lj+k (ω n )G σΛ ij (ω n +∆) + G σΛ i+lj+k (ω n )S σΛ ij (ω n +∆) . (32d) D. The Role of D ↑↓ We have already mentioned above that for the special case L = 0, D ↑↓ is only generated in the third order of the interaction. Thus, if one considers purely onsite interactions and strictly limits the flow to have only the second order index structure, the quantity D ↑↓ would not be present. In the general case, however, where L > 0, D ↑↓ is already generated in second order and therefore has to be included in the flow. Since our actual implementation of the fRG flow is, of course, written for the general case, we always take D ↑↓ into account, even if L = 0. This is the reason why our results will slightly differ from the results obtained in Refs. 7 and 8, which considered only onsite models and thus never took the contribution from D ↑↓ into account. In Fig. 1, we compare the dependence of the QPC conductance on the magnetic field for a model with purely onsite interactions (defined in Sec. III below) for both cases and see that the difference is noticeable but not very big (of course this holds only in intermediate parameter regimes, i.e. in regimes where both the eCLA and the CLA are convergent). E. The flow equation of the self-energy Using the above definitions, the flow equation of the self-energy, (8), can be written explicitly as d dΛ Σ σΛ ji (ω n ) = −T k,σ ,n l S σ Λ i+l,j+k (ω n ) U i(i+l) δ lk δ ji − U ij δ k,−l δ j(i+l) δ σσ + P klΛ jiσσ (ω n + ω n ) + X klΛ jiσσ (ω n − ω n ) + i2 S σ Λ i2,i2+k (ω n )D (i−j)kΛ ji2σσ (0) ,(33) where the l, k-summation is restricted to |l|, |k| ≤ L, whereas the sum over i 2 runs over the whole interacting region. To summarize, dfRG2 is defined by the flow equations (29) and (33), together with the definitions (9), (18), (26), (31) and (32). F. Restrictions for actual computations In our actual computations, we restrict ourself to the case of zero temperature and use so called static fRG, meaning that we treat the vertices as frequency independent. The zero-temperature limit enables us to transform the summation over discrete Matsubara frequencies into continuous integrals along the imaginary axis, and the Θ T in Eq. (7) is a sharp step function. Using this, we are able to apply Morris' lemma [23], which enables us to simplify the integral expressions containing the singlescale propagator S in the flow equations (27): under integration over ω, the following relations hold: S Λ (iω) T =0 = δ(|ω| − Λ) G Λ (iω), (34a) G Λ (iω) = [G 0 (iω)] −1 − Σ Λ (iω) −1 , (34b) S Λ i,j (iω 1 )G Λ k,l (iω 2 ) T =0 = δ(|ω 1 | − Λ)Θ(|ω 2 | − Λ) G Λ i,j (iω 1 ) G Λ k,l (iω 2 ) . (34c) The static fRG approximation treats the vertex quantities γ Λ p , γ Λ x , γ Λ d as frequency independent, setting the bosonic frequencies Π, X, ∆ to zero. Via Eq. (8), this automatically implies that the self-energy is frequency independent, too. In the case of QPC models with onsite interaction this approximation was compared with results of the frequency dependent fRG-scheme, so called "dynamic fRG" and was seen to yield reasonable results for the zero-frequency Green's function at zero temperature. However, for models with finite-ranged interactions we find more pronounced static fRG artifacts (described in section IV) which might be improved by the use of the dynamical method. This is a topic for future research. We stress here that it should in principle be straightforward to implement the dynamical method. The main restriction is simply the effort in computation time, which scales like the number of used frequencies, N f , which in Ref. 7 is typically of the order 10 2 . G. Numerical implementation In a numerical implementation, the flow will start at a value Λ i which is usually chosen as large, but is not infinite. For Λ i large enough, one can show [5] that the flow of the self-energy from Λ = ∞ to Λ = Λ i results in a value of γ Λi 1 given by γ Λi 1 (q 1 , q 1 ) = − 1 2 q v(q, q 1 ; q, q 1 ) .(35) This is then used as the initial condition for γ 1 in the numeric fRG flow. The initial condition for the vertex γ 2 , given by Eq. (3), stays the same. In the case of sfRG2 the vertices and the self-energy only depend on Λ. In order to carry out the resulting integration, we mapped the domain of the flow parameter Λ ∈ [0, ∞) onto the finite domain x ∈ [0, 1) by using the substitution Λ = x 1−x , c.f. Ref. 8. To integrate the resulting flow, we followed Dormand-Prince [24], using a 4-th order Runge-Kutta method with adaptive step-size control. The computationally most expensive step is the blockmatrix multiplication of Eq. (29), which scales as O(N 3 L 3 ). In some fRG schemes, e.g. Keldysh fRG with hybridisation flow [4], for intermediate N 10 2 , most of the calculation time is spent on the Keldysh version of the bubble integrals Eq. (32), whose calculation time scales as O(N 2 L 2 N f ). Since this is less costly than the blockmatrix multiplication of Eq. (29), it might be possible to implement the eCLA within the Keldysh formalism, too. III. RESULTS: ONSITE-INTERACTIONS Having derived our eCLA scheme in the last section, we are now able to apply it to the two models of primary interest here, namely the QPC and the QD. In the present section, we study purely onsite models U ij = δ ij U ,(36) where we treat the strength U of the interaction as a tunable and space-independent parameter, which is suppressed smoothly to zero at the ends of the interacting region. The focus of this section lies on comparing our results to the ones obtained previously by BHD to explore the consequences of the improved feedback for a wellstudied example. All plots in this section are calculated with µ = 0, i.e. with half-filled leads. A. Models for QPC and QD Our interest lies in the low energy physics of a QPC or a QD. For this reason, we consider only the lowest subband of a QPC, or a QD coupled to one-dimensional leads. We use a one-dimensional model Hamiltonian of the same form as used in Refs. [7,8,19]: − 2 −1.5 V j /τ −40 0 40 j ǫ F Vg N ′ QPC (a)H = jσ [E σ jnjσ − τ (d † jσ d j+1σ + h.c.)] + j U jnj↑nj↓ .(37) It describes an infinite tight-binding chain with constant lattice spacing a, constant hopping amplitude τ , onsite interaction U j , and on-site potential energy E σ j = V j − σB 2 . Here V j will be used to model the smooth electrostatic QPC or QD potential defined by gates (as described below and illustrated in Fig. 2), and the Zeeman energy B accounts for a uniform external magnetic field parallel to the 2DEG. We take U j and V j to be nonzero only within a (single or double) "barrier region" of N = 2N + 1 sites centered around j = 0, containing the QPC or QD. The rest of the chain represents two noninteracting leads with bandwidth 4τ , chemical potential µ, bulk Fermi energy ε F = 2τ + µ and effective mass m * = 2 /(2τ a 2 ) (defined as the curvature of the dispersion at the band bottom in the bulk). Adopting the convention in Ref. 8, we choose the center of the bulk band as energy origin. Throughout the present section, we choose µ = 0, implying half-filled leads. In order to arrive at a discrete QPC potential V j , we start with a continuous QPC potential V (x) = (V g + ε F ) exp −γ 2 (x/L bar ) 2 1−(x/L bar ) 2 , |x| ≤ L bar 0 |x| > L bar(38) where 2L bar is the whole barrier length and V g controls the barrier height, measured w.r.t. ε F . Near the barrier top, the potential (38) can be expanded as V (x) = V g + ε F − 1 2 m * 2 Ω 2 x x 2 + O(x 4 ) ,(39) where the curvature parameter Ω x is given by Ω x = γ L bar 2(V g + ε F ) m * .(40) It has units of energy and serves as characteristic energy scale for the QPC. It also defines a characteristic length scale for the QPC barrier top, l x = / √ 2m * Ω x . The dimensionless parameter γ in the exponent of Eq. (38) can be used to vary the barrier curvature [Eq. (40)] without changing the barrier height. Throughout section III, we will keep γ = 1 constant and consider only gate-voltages small compared to ε F , such that the curvature can be assumed to be independent of V g . However, when dealing with longer-ranged interactions in Sec. IV, we will need to choose γ = 1. We discretize the QPC potential (38) by choosing a number of sites N and setting the lattice spacing a = 2L bar /N , to arrive at V j = V (j · a) = (V g + ε F )e −γ 2 (j/N ) 2 1−(j/N ) 2 , |j| ≤ N , 0, |j| > N . (41) The resulting barrier shape given by Eq. (41) is plotted in Fig. 2(a). The leading behavior around the maximum at j = 0 is quadratic and the same as in Ref. 8: V j = V g + ε F − Ω 2 x 4τ j 2 + O(j 4 ),(42) and the curvature can be expressed through the discrete quantities as Ω x = γ 2 √ τ (ε F +Vg) N . For our onsite studies, where V g is only varied in a small region around V g = 0, we use the approximation Ω x = γ 2 √ τ ε F N . In order to avoid discretization artifacts, the discretization length a should be chosen significantly smaller than l x . In our actual computations for the QPC with onsite interactions we use a ratio l x /a 5. To model a QD, we use a potential that can be tuned smoothly from the QPC shape described above to a double-barrier structure, as shown in Fig. 2(b). The discretization procedure is analogous to the QPC and we state here only the resulting discrete dot potential, which is the same as used in Refs. 7 and 19: V j =                    0 , ∀ |j| ≥ N , (V s + ε F ) 2 |j|−N js−N 2 − |j|−N js−N 4 , ∀ j 0 ≤ |j| ≤ N , V g + ε F +Ω 2 x j 2 4τ sgn(V s − V g ), ∀ 0 ≤ |j| < j 0 .(43) We can vary the dot width via j s , and the depth of the quadratic well in the middle via V s and V g . These choices determine the values of j 0 andΩ x in order to make the potential continuously differentiable. Of course, this is just one convenient way to model the dot structure, and the qualitative behavior of the physical results does not depend on the specific implementation. For the onsite interaction we use both for the QPC and the QD the form used by BHD [7]: U j = U e −(j/N ) 6 /[1−(j/N ) 2 ] .(44) It is almost constant and equal to U in the center of the QPC and drops smoothly to zero at the flanks of the barrier region. B. Physical behavior of the models We now briefly summarize the physics of these models, which was already discussed in great detail by BHD in Refs. 7 and 19. Our main handle for tuning the QPC potential is the gate voltage V g , which controls the height of the barrier. If the barrier top lies well above the chemical potential, the QPC is closed. Lowering the barrier, the QPC opens up and the linear conductance g increases smoothly from 0 to 1 in the region of gate voltages 0 V g Ω x , where Ω x is the curvature of the QPC introduced above. Additionally, the width of the conductance step, i.e. the gate-voltage interval in which the conductance increases from zero to one, is also set by Ω x . The general shape of the conductance curve for a parabolic barrier in the absence of interactions is a step described by a Fermi-function, as was shown by Büttiker in Ref. 15. If one switches on onsite interactions, the conductance curve becomes asymmetric and flattens increasingly at the top. This effect can be traced back to the fact that when the barrier top drops below the chemical potential as the QPC is being opened up, the maximum in the LDOS just above the barrier top (called van Hove ridge in Ref. [7]) is aligned with the chemical potential, thereby strongly enhancing interaction effects. It turns out that the effective onsite interaction strength is in fact given by U eff j = U · A j (µ),(45) where A j (ω) = − 1 π ImG 0 jj (ω + i0 + )(46) is the non-interacting local density of states per site. Near the barrier center, the resulting U eff scales like U/ √ Ω x τ . In the QD case, we can vary the width and depth of the middle well, [c.f. Fig. 5 (d,e) below]. Typically, we want to study the crossover between QPC and QD, thus we start out with a QPC setup and lower the potential of the central region to change the geometry to a QD model. The characteristic physics of the quantum dot is determined by the structure of the discrete levels of the bound states in the well. This quantization leads to a conductance peak whenever such a level crosses the chemical potential and the dot gets filled by one electron more. In the interacting case, the degenerate levels split on a scale of the interaction strength U . However, there is a further effect: the odd valleys, i.e. the regions between the peaks where the dot contains an odd number of electrons, become conductance plateaus with G Q ≈ 1. This behavior reflects the occurrence of the Kondo [25] effect since the singly-occupied dot level behaves like a localized spin coupled to a fermionic bath. In this work, we will apply our eCLA first to the same type of onsite models of QPCs as used by BHD [7,8,19] and analyse the resulting effects. Importantly, we find that in comparison to the CLA used previously, the eCLA yields an improved stability of the fRG flow in the case of large bare LDOS at the chemical potential. This improvement allows us to additionally study the QPC-QD crossover, which involves a very high LDOS due to the flat barrier top that occurs in this transition. Using the CLA, it had not been possible to study this transition when the barrier top lies close to the chemical potential µ, since the CLA equations did not converge. Due to this problem, in the real-space approach chosen by Heyder et al. [19] it was not possible to study dots which contain just a few electrons. Since our new feedback scheme significantly ameliorates the convergence problem, we are now able to study the crossover from a QPC to a QD which is just occupied by a single electron. This will be shown in section III D. C. Increasing the feedback length Let us now study the influence of the feedback length L on the zero-temperature linear conductance [26], g = 1 2 σ 2πρ σ (µ + i0 + )G σ −N N (µ + i0 + ) 2 .(47) Here ρ(ω) is the density of states at the boundary of a semi-infinite tight-binding chain; two such chains represent the two one-dimensional non-interacting leads, coupled to the central interacting region. Let us first look at the QPC case. We are interested in the shape of the conductance trace as a function of applied gate voltage and how this shape changes with external parameters, such as applied magnetic field. For pure onsite interactions, it is natural to choose the feedback length L = 0. This is what has been done in Refs. 7,8,19, and 27, and the results have been discussed therein in detail. Here, we will allow a nonzero L, although the actual interaction is purely onsite. This implies that a certain class of additional third order terms will be generated during the RG flow which introduce a better coupling between the channels in the sense of the feedback in Eq. (23). For L → N the third-order truncated static fRG scheme is recovered fully regarding the spatial structure of the two-particle vertex (but not for its frequency structure, since we are using the static approximation). Figs. 3(a) to (c) show the conductance G as a function of gate voltage V g for different values of magnetic field B, calculated at fixed U and different values of feedback parameters L. Increasing the latter from L = 0 to L = 3, c.f. Fig. 3(b), leads to quantitative but not qualitative changes in the shape of the conductance curves -the main effect is that the width of the B-induced subplateau decreases. In this regard, increasing L has a qualitatively similar effect to decreasing U (at L = 0), c.f. Fig. 3(d) to (f). Note, though, that increasing L hardly affects the V g position of the conductance step, whereas decreasing U does shift the step slightly towards higher V g values, as expected physically due to the lowering of the Hartree barrier. Increasing the feedback beyond L = 5 does not lead to any significant quantitative changes, as can be seen in Fig. 3(c) where L = 5 (black line) is directly compared with L = 8 (red dashed line). Hence, for the present model convergence is reached for L = 5. In general this value depends on the strength of interaction U , and more importantly on the actual shape of the barrier. For the given case, L = 5 corresponds to the geometric length-scale l x /a = τ /Ω x = 5, which is the width of the region where the LDOS is enhanced. The extended feedback between the channels becomes increasingly important with increasing interaction strength. For L = 5 the eCLA yields meaningful, converged results for interaction values for which the L = 0 flow obtained by CLA is divergent. This is the case for U 4 √ Ω x τ . Figs. 3(d) to (f) show the conductance for such large values of interaction and L = 5. The qualitative behaviour is unchanged w.r.t. smaller values of the interaction, and the quantitative strength of the impact of the interaction increases continuously, in that the width of the spin-split subplateau increases with U . To shed light on the effect of the enhanced coupling between the channels, we now analyse the resulting twoparticle vertex quantitatively, by studying its extremal value γ ext 2 = max q 1 q 2 q 1 q 2 |γ 2 (q 1 , q 2 ; q 1 , q 2 )|,(48) where the q's stand here both for site and spin indices. Furthermore, we identify the two most contributing parts to these value as γ ext x = max j 1 j 2 j 1 j 2 γ x (j 1 ↑, j 2 ↓; j 1 ↑, j 2 ↓) γ ext p = min j 1 j 2 j 1 j 2 γ p (j 1 ↑, j 2 ↓; j 1 ↑, j 2 ↓) .(49) Note that we used the minimum in the definition of γ ext p , since the γ p contribution is mainly negative, whereas γ x is dominated by its positive part. Fig. 4 shows these quantities and the conductance as a function of V g for L = 0 and L = 5. The main message of this figure is that for intermediate interaction strength (solid black curves) the flow converges for both L = 0 (left column) and L = 5 (right column) and yields qualitatively the same results for the conductance in Fig. 4 (a,b). If, however, one increases the interaction strength further (red solid curves) the flow for L = 0 starts to diverge [ Fig. 4 (c,e)] and the values of physical observables computed from it become wrong, reflected for example in the kink of the red conductance curve in Fig. 4 (a). A good measure for the behavior of the flow is the maximum value of the two-particle vertex, plotted in Fig. 4 (c,d). We see that the kink in the conductance curve corresponds to a very large value of γ ext 2 /U = 58.2 [lying outside of the range of Fig. 4(c)]. In contrast, for L = 5, γ ext shed light on this stabilizing effect of the enhanced feedback, we show in Fig. 4 (e,f) the P ↑↓ and X ↑↓ part of the channels, which constitute the contributions to γ ext 2 with the largest moduli. In the case of intermediate interac-tion (black curves) the X and P contributions are of the same order of magnitude but differ in their relative sign. If one looks at the completely uncoupled channels, i.e. the pure ladder contributions (c.f. the study in Ref. 8) and increases the interaction strength, the X-channel is the first one to diverge. Our interpretation of the stabilizing effect is now as follows. Since the channels are coupled, a slight increase in the modulus of the X-channel leads via the feedback to a slight increase of the modulus of the P-channel, and due to their relative sign difference they partially cancel, so that the resulting additional contribution to γ 2 is small. If the effective interaction becomes too strong, this ameliorating effect eventually breaks down and the flow diverges. In the L = 5 case, we take much more feedback between the individual channels into account than for L = 0 and it is therefore reasonable that the divergence point of the flow is shifted toward larger effective interactions. D. Crossover between QPC and QD As we have seen above, the increase of the feedback length L leads to a more stable fRG flow in regions for high LDOS, corresponding to a large effective interaction strength. This stabilization effect enables us to study parameter regimes that have been inaccessible to previous fRG schemes. We illustrate this below for a model known to suffer from fRG divergence problems, namely the crossover from a QPC to a QD. The fRG flow for this transition suffers from divergences if the flat barrier top is too close to the chemical potential, c.f. Ref. 19. For this reason, it was not possible for fRG to smoothly describe how the dot filling increases with decreasing V g , and the region where no or only a few electrons occupy the dot remained inaccessible. eCLA enables us to study precisely this region. In Fig. 5 (a) we show the conductance curve for a QPC-QD crossover, in which the first two bound state levels cross the chemical potential as the dot is made deeper. As one expects, we can see conductance plateaus arising in the V g regions where the occupation of the dot is odd. These Kondo plateaus, caused by Kondo screening of the dot spin, get suppressed with increasing magnetic field since the spin degeneracy is broken. This suppression happens in the first and second Kondo plateau for magnetic fields on the scale ∼ 1 · 10 −4 τ (solid black lines), and ∼ 3 · 10 −4 τ (dashed black lines), respectively. A quantitative extraction and analysis of the Kondo scales of the setup is beyond the scope of this paper. Our main purpose here is to illustrate that the finite-ranged feedback of eCLA enables us to treat a parameter regime which was not accesible with previous fRG schemes and produces qualitatively correct Kondo physics. To outline this, we have indicated in Fig. 5 (a) how the range of convergence increases with increasing L from 0 to 30. We see that the L = 0 method is only convergent in the parameter regimes where the occupancy of the dot is even and hence the conductance is small. By increasing L from 0 over 5 to 10, we see that also the conductance plateaus become more and more visible. At L = 20 the whole Kondo plateau is accesible. Upon further increasing the feedback up to L = 30 (not shown here), we find that the conductance results for L = 20 are already properly converged. IV. FINITE-RANGED INTERACTIONS In this section we consider a model of a QPC with an interaction whose range extends over up to N sites, in contrast to the purely onsite interaction studied in Sec. III. The purpose of this study is to illustrate the potential of the eCLA to deal with finite-ranged interactions in a setting where the screening of a longer-ranged interaction comes into play, and to take a first step towards exploring the physical consequences of screening. We should emphasize, though, that we do not aim here to achieve a fully realistic treatment of screening in a QPC. That would require including higher-lying transport modes (we consider just the lowest-lying one), which would go well beyond the scope of the present paper. Our model is described by the following Hamiltonian: H = ijσ [E σ jnjσ −τ (d † jσ d j+1σ + h.c.)] + 1 2 i,j,σ,σ U ijniσnjσ (1 − δ ij δ σσ ).(50) Here E σ j is chosen as described in section III, and U ij can differ from zero for all sites with separation |i − j| < L U , where L U determines the bare interaction range. Note that we now also have a bare interaction between electrons with the same spin, which was absent in the onsite case. In the previous section, the interaction strength was controlled by a single value U [c.f. eq. (44)] and treated as a tunable parameter, whose strength was varied by hand. However, now U ij is a matrix with N 2 parameters, and we need to specify its form explicitly. For this we start with a continuous 3D model of a QPC, and for the Hilbert spaces associated with transverse motion in the y-and zdirections we reduce the dimensionality down to one, by taking into account only the ground states of the respective confining potentials, c.f. Ref. 28. In this way, we arrive at a continuous effective theory in 1D for the xdirection, which in a last step is discretized using a finite difference method, already applied by BHD in Ref. 8. We use the resulting model to compute the conductance and the density profile of a QPC, and study their dependence on the screening effects of the long-ranged interaction and the geometric dimensions of the QPC. A. Derivation of a 1-dimensional Hamiltonian We start from the HamiltonianĤ =Ĥ 0 +Ĥ 1 witĥ H 0 = σ d 3 rΨ † σ (r) V QPC (r) − 2 2m ∇ 2 Ψ σ (r) H 1 = 1 2 σ1,σ2 d 3 r 1 d 3 r 2 U (r 1 − r 2 ) ×Ψ † σ1 (r 1 )Ψ † σ2 (r 2 )Ψ σ2 (r 2 )Ψ σ1 (r 1 ),(51) where the fermionic fieldΨ † σ (r) creates an electron with spin σ at the continuous position variable r. The interaction is of screened Coulomb form with screening length l s and relative dielectric constant κ, which is given in ESU-CGS units by: U (r 1 − r 2 ) = e 2 κ 1 |r 1 − r 2 | − 1 |r 1 − r 2 | 2 + l 2 s ,(52) c.f. Hirose et al. [29]. This interaction form results from taking image charges on the top gate into account, which is positioned at a distance of l s /2 above the 2DEG. We use a QPC potential given by V QPC (x, y, z) = αV (x) + m * Ω y (x) 2 2 y 2 2 ∆(z),(53) with Ω y (x) = 2βV (x), and m * = 0.067m e is the effective mass of GaAs. The function ∆(z) ensures the confinement to the 2DEG and the one-dimensional potential V (x) which enters here is the same as that used in our onsite-model studies, Eq. (38). The QPC potential V QPC has a saddle-like form: it defines a quadratic confinement in y-direction with a positiv curvature Ω y (x) that decreases with increasing |x|, whereas the curvature in x-direction is negative, with magnitude Ω x . The confinement in y-direction disappears for |x| → ∞, where V (x) = 0. For the coefficients α and β, we impose the condition α + β = 1, which turns out to ensure that the effective one-dimensional potential resulting from eliminating the y-and z-direction is precisely V (x). We specify the transverse curvature at the center of the QPC to be Ω y = Ω y (0), thereby fixing the parameter β = Ωy 2V (0) . We now project onto the ground state subspace for the transverse directions. With this step, taken for the sake of simplicity, we ignore all transport modes except the one contributing to the first conductance step. For a truly realistic description of screening, the higher-lying modes would have to be taken into account, too. This would lead to stronger screening and an effective interaction of shorter range than that obtained below. Concretely, we thus represent our quantized fields aŝ Ψ σ (r) = φ x (y)ϕ(z)ψ σ (x).(54) Here φ x (y) and ϕ(z) are the normalized ground state wave-functions of the confining potentials in the y and z directions, respectively , ϕ(z) = δ (z),(55)φ x (y) = 1 (2π) 1/4 l y (x) e −y 2 /(4l 2 y (x)) ,(56) (57) and the operatorψ σ (x) creates an electron in a state with wavefunction δ(x)φ x (y)ϕ(z). In our 2DEG setup,δ(z) is a peak of weight one, very narrow compared to the scales in x-and y-direction, whereas φ x (y) is the ground state of a harmonic oscillator with characteristic length l y (x) = 2m * Ω y (x) .(58) With this, we arrive at an effective 1D continuous theory described by the effective 1D Hamiltonian H eff = σ dxψ † σ (x) 2m ∂ 2 x + (α + β)V (x) ψ σ (x) + σ1,σ2 dx 1 dx 2 U (x 1 , x 2 ) 2ψ † σ1 (x 1 )ψ † σ2 (x 2 )ψ σ2 (x 2 )ψ σ1 (x 1 ).(59) We now choose α + β = 1 as stated above, thus ensuring that the resulting effective one-dimensional potential is indeed given by V (x). The matrix elements of the interaction are given by U (x 1 , x 2 ) = e 2 κ 1 2π(l 2 y (x 1 ) + l 2 y (x 2 )) × exp (x 1 − x 2 ) 2 4(l 2 y (x 1 ) + l 2 y (x 2 )) · K 0 (x 1 − x 2 ) 2 4(l 2 y (x 1 ) + l 2 y (x 2 )) − exp (x 1 − x 2 ) 2 + l 2 s 4(l 2 y (x 1 ) + l 2 y (x 2 )) · K 0 (x 1 − x 2 ) 2 + l 2 s 4(l 2 y (x 1 ) + l 2 y (x 2 )) .(60) For a typical 2DEG of GaAs-AlGaAs the relative dielectric constant has the value κ ≈ 12.9. K 0 is the modified Bessel function of second kind in zeroth order. It diverges logarithmically when its argument approaches zero. In order to discretize our 1D continuous theory along the x-direction, we set x := a · j and replace the continuous fieldψ σ (x) by the discrete set of operators d jσ , where a is the lattice spacing and j the site index. This results in a Hamiltonian of the form (50). Treating the second derivative in the kinetic term using a finite difference method, the single-particle part of the Hamiltonian takes the form H 0 = ijσ h σ ij , with h σ ij = (V i − σB 2 )δ ij − τ (δ i,i+1 + δ i,i−1 ),(61) where V i is just the discretized version of the effective 1D potential, B is the magnetic field, and τ = 2 2m * a 2 is the hopping matrix element. We define a discretized form of the interaction by U ij :=U (ai, aj), if i = j (62) U ii := 1 a 2 a(i+1/2) a(i−1/2) dx 1 a(i+1/2) a(i−1/2) dx 2 U (x 1 , x 2 ),(63) where we treat the on-site case separately, since U (x 1 , x 2 ) has an integrable singularity as x 1 approaches x 2 . The above treatment presupposes that the transverse wavefunctions do not change significantly on a scale set by a. If a is much smaller than the characteristic length of the electrostatic potential, the above discretization scheme correctly captures the physical behavior of the continuous theory while regularizing the short-distance of the interaction, with U ii = − e 2 κ √ πly(ai) · log[a/l y (ai)] + O(1) for a → 0. Having arrived at the discretized Hamiltonian (50), let us take a final look at the parameters that characterize our system. From the dimensionful constants , e 2 /κ and m * one can construct an intrinsic length scale 2 m * e 2 κ ≈ 10nm and intrinsic energy scale m * e 4 2 2 κ 2 ≈ 5.5meV. It is possible to express all our model's length and energy scales in terms of these two dimensionful constants. However, it is often convenient to be able to relate quantities like the gate-voltage dependence of the conductance or the spatial resolution of the density directly to the geometry of the QPC. For this reason, we introduce in our studies below for each QPC a characteristic energy scalē Ω x , and a corresponding length scalel x = / 2m * Ω x , which we measure in absolute units and which characterize the mean geometry of the QPC barrier. Concretely, we will take forΩ x the curvature of the bare barrier at the renormalized conductance pinchoff gate voltage V po g , where the conductance just begins to increase from zero (and the barrier height is ε F + V po g ). All the other geometric quantities are then specified relative toΩ x . To be specific, we will characterize our QPC by the following rescaled dimensionless quantities (denoted by tildes): (i) Ω x =Ω x meV , (ii) V g = V ḡ Ω x , (iii) Ω y = Ω y (0) Ω x , (iv) l s = l s l x , (v) L bar = L bar l x , (vi) x = x l x , (vii) Ω y =l 2 x Ω x ∂ ∂ 2 x Ω y (x) x=0 .(64) Ω x describes the longitudinal barrier curvature in units of meV, V g the normalized gate voltage, Ω y the transverse curvature at the barrier center, l s the screening length, L bar the total barrier length which controls the behavior of the flanks, x the longitudinal coordinate, and Ω y the x-dependence of the transverse curvature at the barrier center. Note that if one chooses to specify Ω x , Ω y , Ω y , l s , and L bar , this implicitly also fixes ε F : its values has to be chosen in such a way that the resulting curvature at pinchoff has the specified value Ω x . It is instructive to express the interaction U (x 1 , x 2 ) of Eq. (60) in terms of the rescaled dimensionless parameters. If we define U b = e 2 /(κl x ), the dimensionless ratiõ U ( x 1 , x 2 ) = U (x 1 , x 2 )/U b depends only on the dimensionless parameters (64)(ii)-(vii), but not onΩ x . Thus, the dependence of the interaction strength (in absolute units) on the longitudinal curvatureΩ x of the QPC is fully encapsulated in U b . The corresponding dimensionless parameter U b = U b /Ω x = √ 2m * e 2 κ 1 Ω x(65) characterizes the effective onsite interaction strength at the barrier center for the present long-ranged interaction model, and plays a role analogous to the parameter U eff 0 = U · A 0 (µ) of Eq. (45) (which likewise scales as 1/ Ω x ) for the onsite interaction model of section III. Evidently, U b increases with decreasingΩ x , implying that interactions become ever more important the smaller the curvature of the barrier top. Typical values for U b for the plots below range between 4.2 and 4.9. B. Discretization dependence We begin our treatment of long-ranged interactions by investigating to what extent our results depend on the number of discretization points, N , with all other parameters held fixed. Fig. 6 shows this dependence for two QPCs whose parameters were chosen to yield somewhat different ranges of Ω x curvatures. The first point to notice involves the V g value of the conductance pinchoff: whereas in the absence of interactions it occurs near V g = 0 , turning on our long-ranged interactions shifts it towards the left, i.e. towards a larger gate voltage. This behavior is unphysical, since for any fixed V g at which the density is nonzero, turning on interactions should generate a Hartree barrier that causes the conductance to decrease, not increase. We suspect that this unphysical behavior is an fRG artefact, possibly due to our use of the static approximation. We leave the issue of exploring what will happen when using a dynamic version of our eCLA as a topic for future study. We remark, however, that similar unphysical shift artifacts where encountered in Ref. 8 when comparing various different fRG methods that treated the details of the vertex flow in somewhat different ways. Nevertheless, although the V po g values of the conductance curves in Ref. 8 depended on methological details, the overall shape of the conductance steps were essentially the same, i.e. when plotted as functions of V g − V po g , they coincided. We find a similar trend here: if we increase N , V po g increases, because changing N slightly changes the strength and shape of the interaction function U ij , causing corresponding changes in V po g andΩ x ; however, the shape of the conductance steps in Figs. 6 (a),(b) seems at least qualitatively convergent when N increases [c.f. insets in (a) and (b)], despite the N dependence of the step's position. For the remainder of this paper we will therefore only address the overall shape of the conductance step. In Fig. 6(a),(b) we expressed all parameters in terms of absolute units. In all the remaining plots where physical properties are discussed [ Fig. 8 to Fig. 10], we use instead the more convenient dimensionless quantities introduced in Eq. (64) (and denoted by tildes). We have also extracted these dimensionless parameters for Figs. 6(a),(b) and summarized them for further use in the parameter sets A and B given in the caption of Fig. 6. In Fig. 6 we used the maximal feedback length L = N − 1 to fully take interactions over the whole QPC into account. However, due to numerical costs, this limited the number of sites that could be treated to N ≤ 71. For this reason, we have also explored using a cutoff length L U for the interaction range, setting U ij = 0 for |i − j| > L U . When using such a cutoff, we found rapid convergence when increasing L beyond L U for a fixed N : for example, Fig. 7 contains two curves for L U = 10, one computed with L = 60 (solid), the other with L = 15 (dashed), which essentially coincide. However, the shape of the conductance step becomes independent of L U only for rather large values of L U , implying that the tail of the long-ranged interaction actually matters significantly. Therefore, we did not pursue using L U < N Fig. 6(b). Note that while convergence in L is rapid, the conductance becomes independent of the cutoff length only for LU > 40. any further and for the remainder of this work show only data obtained without interaction cutoff and with full feedback length, L = N − 1. C. Effects of long-ranged interactions on QPC properties After these technical considerations, let us now study how the fact that the interaction range is not zero affects the QPC properties. For this, we first briefly discuss the dependence of our finite-ranged interaction on the given physical parameters and then study the resulting consequences on the conductance and the density. As pointed out earlier, this study does not aim to achieve a fully realistic description of screening in a QPC, but rather serves as a first illustration of the potential of the eCLA for treating a model with reasonably long-ranged interactions. The structure of the long-ranged interaction for typical choices for the physical parameters is shown in Figs. 8(a),(b). In (a), we plotted the dimensionless ratio U (0,x) = U (0, x ·l x )/U b for three values of the rescaled screening length l s , as a function of positive x = x/l x . This ratio is independent ofΩ x itself, but increases significantly with increasing screening length. In (b), we again showŨ (0, x) (central peak) and for comparison alsõ U ( x s , x) = U ( x s ·l x , x ·l x )/Ũ b for fixed x s = 4.5 as function of x, where the x range contains now the whole QPC. Due to the reflection symmetry of our system about the QPC center, U (0, x) is a symmetric function of x. In contrast, U ( x s , x) is an asymmetric function of x around the point x = x s , decreasing more quickly when x − x s becomes large positive than large negative, because the transverse potential is wider in the former case. This widening of the transverse potential is also the reason whyŨ ( x s , x) as a function of x s − x with fixed x s is in general smaller thanŨ (0, x) as a function of x. Figs. 8(c) and (d) show, for two different values of the curvature Ω x , respectively, three conductance curves corresponding to the three choices of l s used in Fig. 8(a). For both choices of Ω x , we obtain an onsite-like conductance step shape when l s is small. When l s is increased, i.e. when the amount of screening is reduced, the step shape acquires some additional features, such as the emergence of a "preplateau" at a value of g slightly lower than 1, followed by a much slower increase towards 1 in Fig. 8(c). These features are more pronounced for the longer QPC (i.e. smaller curvature) of Fig. 8(d), where the conductance quickly reaches a preplateau around g 0.8 and thereafter increases much more slowly. U (0, x) U ( x s , x) g 0 1 (c) Ω x = 1.2 U b = 4.2 −2 0 2 4 6 V g V g −2 0 2 4 6 g 0 1 (d) Ω x = 1.0 U b = 4.6 In order to explore the origin of this behavior, we show in Fig. 9(a) and 9(b) two density profiles (thin lines), calculated, respectively, for two fixed parameter choices from Fig. 8(d), indicated in the latter by the right (red) marker for l s = 0.86, V g = −1.43 and the left (blue) marker for l s = 2.15, V g = 3.73. In Fig. 9(b), for which the rescaled screening length l s is larger, we observe three qualitative changes relative to Fig. 9(a). First, the flanks of the density profile are somewhat steeper. Second, the Fig. 8(d), respectively. For comparison, the thick lines depict (a vertically rescaled version of) the imaginary part of the interacting single-particle propagator at the chemical potential, A 0, x = − 1 πlx Im G R 0, x (ω = 0) . Horizontal dashed lines indicate where A 0, x = 0. In (b), the distance between the two density maxima (marked by the dashed vertical lines) is λ = 3.62lx. This agrees well with two estimates of λF /2, either from the distance between the two central zeros of A 0, x finding λF /2 = 3.82lx or from the mean densityn in the center of the QPC (shaded region) finding λF /2 = 3.55lx. spatial region in which the density is low has become wider. And third, in this low-density region the density shows some weak density oscillations that are absent in Fig. 9(a). The first two features suggest that the long-range interactions have generated a renormalized barrier whose shape has a flatter top and steeper flanks than the bare parabolic barrier. This flattening occurs because the bare density is larger in the flanks than near the center, hence the upward Hartree-type shift of the barrier potential, which is proportional to the bare density, is larger in the flanks than near the center. The upward renormalization in the flanks becomes stronger the larger the interaction range, because then the upward Hartree-type shift at a given site is determined by a weighted average of the density over a range of nearby sites (whose extent is set by the screening length), and since the bare density profile is convex, the sites in the flanks contribute more strongly. To shed further light on the third feature, namely the weak density oscillations in the low-density region, we compare their oscillation period with estimates for the "local Fermi wavelength" λ F at the QPC center, which can be extracted from either the interacting Green's function or the mean density in the center of the QPC. To illustrate the first method, the thick lines in Figs. 9(a) and (b) indicate the oscillatory behavior of A 0, x = − 1 πlx Im G R 0, x (ω = 0). For a homogeneous system the Green's function oscillates with period λ F , and likewise we can here define an effective λ F /2 in the middle of the QPC by taking the distance between the two central zeros of the thick line. For Fig. 9(b), the position of these zeros is in good agreement with the position of the density maxima of the QPC (indicated by the two dashed vertical lines), whereas the density in Fig. 9(a) shows no features on the scale of λ F . An alternative way to extract an effective λ F is to calculate the mean densityn Fig. 6]. (a,b) Conductance as function of gate voltage, and (c-h) density as function of position and gate voltage. While the conductance changes its shape for both QPCs, the shorter one (b) shows stronger features, preeminently a shoulder in the conductance step. In the density, both QPCs show the development of oscillations with approximate wavelength λF /2, which is determined by the Green's function as in Fig. 9 and indicated by the distance between the black lines. In the last plots (g) and (h) the density oscillations transition at smaller gate voltages from two to three maxima. The cut along the dashed white line in (f) is precisely the density profile plotted in Fig. 9(b). in the center of the QPC between the two density maxima (shaded region in Fig. 9), and use λ F = 2π/k F = 4/n. For Fig. 9(b), the first method yields λ F /2 = 3.82l x , and the second λ F /2 = 3.55l x , which are both in reasonable agreement with each other and the distance λ = 3.62l x between the two density maxima. Thus, we conclude that the period of the density oscillations observed here can be associated with λ F /2, or equivalently wavenumber 2k F . In Fig. 10 we examine this behavior more systematically, using two QPCs having a comparatively long screening length of l s = 2.15, but which differ slightly in L bar , i.e. in their total barrier length. For both QPCs the conductance step [Figs. 10(a),(b)] changes its shape with decreasing curvature Ω x and for the right QPC with smaller L bar develops additional pronounced features in the plateau region. In Figs. 10(c)-(h) we show the corresponding densities (color scale) as functions of gate voltage and longitudinal position, and find that with decreasing curvature Ω x the density develops oscillations. The period of these oscillations is again set by λ F /2, which is indicated in Figs. 10(c)-(h) by the distance between the black lines. While for the right QPC the two density maxima follow very accuratly the black lines, in the left QPC they lie slightly further apart than λ F /2. The reason for this might be that the left QPC is slightly longer ( L bar is larger), giving the electrons in the center more space to form the two repelling density maxima, but not enough space to fit a third density maximum into the available region. In summary, we find that when increasing the geometric proportions of the QPC compared to the scale set by the interactions, i.e. when decreasing Ω x , the conductance develops additional features in the plateau region, and simultaneously density oscillations arise on a scale set by λ F /2. We interpret the 2k F density oscillations seen in Fig. 9(b) as Friedel oscillations generated by the inhomogeneity induced by the renormalized QPC potential. A similar interpretation was envoked in Iqbal et al. [30] where they also found a wavelength λ F /2, or equivalently a wavenumber of 2k F , for their spin polarized, emergent localized states (ELS) obtained from SDFT calculations in long QPCs. To support this interpretation, we show in Figs. 11 and 11(f) are reminiscent, respectively, of those seen in Figs. 9(b) and 10(c-h) for QPCs with interactions whose range is longer than the characteristic QPC length (i.e. with l s > 1). This supports the interpretation offered above that such QPCs indeed have renormalized barriers with rather flat tops and steep flanks. However, for higher gate voltages where the QPC is beginning to close off and the density in the center becomes very low, we see a qualitative difference between the density profiles shown in Fig. 11(f) and those of Figs. 10(c-h): the former shows a weak density maximum, whereas the latter do not, because in the regime of very low densities, the Hartree-type renormalization of the barrier shape is not yet strong enough to generate a flattish barrier top. In the light of the above discussion, it would be interesting to study the geometric crossover from a QPC to a homogeneous wire obtained by making the QPC lengthl x very long, or by using flat-topped bare barriers of increasing width. In a paper by Schulz [31], concerning Wigner crystal physics in 1D, it was predicted that in a homogeneous 1D model with long-ranged Coulomb interactions in the low-density limit, the density-density correlator ρ(x)ρ(0) contains both 2k F and 4k F oscillations. The latter decay more slowly with x, and are argued by Schulz to lead to a Wigner crystal in a homogeneous system. During the aforementioned geometric crossover from a QPC to a long wire, well-developed 4k F density oscillations can be expected to emerge, which could be regarded as precursors for the formation of a Wigner crystal. A systematic study of this behavior would be extremely interesting, but falls beyond the scope of this paper and is left for future study. In particular, future work would have to incorporate screening also due to higher transport channels, leading to a shorter-ranged interaction, so that the effects discussed above would likely turn out to be somewhat less pronounced than found here. V. CONCLUSION AND OUTLOOK Building on previous works [7,8], we have introduced an improved approximation scheme for 3rd-order truncated fRG. We use an extended coupled ladder approximation (eCLA), splitting the fRG-flow into three channels depending on the internal index structure. When treated independently, each of these channels behaves as in the random phase approximation. The complexity of the eCLA scheme depends on the amount of feedback admitted between the individual channels. For the frequency dependence, we only used static feedback between the channels. In order to control the amount of feedback in the spatial structure, we have introduced the feedback length L. In the case L = 0 we get the minimal feedback between the channels, corresponding to the CLA of previous works [8], whereas for L → N − 1 we recover the full spatial vertex flow in 2nd order. For actual computations, we restricted ourselves to static fRG, i.e. in addition to using only a static feedback between the channels we also neglected the frequency dependence of the vertices altogether. In this additional approximation, we calculated the zero-temperature Green's function at the chemical potential, which is the relevant quantity in order to compute the linear conductance of the system. We first applied our new method to a QPC model with onsite interactions, which has extensively been studied in the past. Here, we observed that the longer-ranged feedback leads to a quantitative but not qualitative change as long as both methods are convergent for the respective parameters. In particular, we observed convergence with increasing L once L l x , where l x is the characteristic length of the QPC. Additionally, we observed that the enhanced feedback stabilizes the fRG-flow and therefore leads also to convergence in parameter regimes which could not be studied with the L = 0 method. To illustrate this increased stability, we studied QPC-QD crossovers analogous to those discussed by Heyder et al. in Ref. [19] using the CLA. There, the convergence of the fRG flow suffers especially from the high LDOS at the chemical potential that occurs during the crossover when the barrier top becomes flat in an extended region close to the chemical potential. Our stabilized flow, however, enabled us to study this type of transition. In particular, we succeeded to study regimes of very shallow dots, containing only a few electrons, and observed the Kondo plateau in the conductance expected for such dots. Finally, in order to test the full potential of our improved feedback, we applied it to a QPC with finiteranged interactions. The most striking observation was that for a relatively flat QPC in the regime of low density and sufficiently long-ranged interactions, the conductance reaches a preplateau somewhat below g = 1 (before slowly climbing towards g = 1), accompanied by the onset of oscillations in the density. The wavelength of these density oscillations was determined to be approximately λ F /2, admitting an interpretation as Friedel oscillations arising from a renormalized barrier shape with a rather flat top and steep flanks. This behavior is consistent with that observed by Iqbal et al. [30] in SDFT calculations for their emergent localized states (ELS) in a spin-polarized QPC. It would be of great interest to explore these type of effects more systematically in the future, within a more realistic model that incorporates the effects of higher transport modes when deriving the effective screened interaction for the lowest-lying transport mode. In particular, the geometric crossover between a QPC potential and a homogeneous quantum wire, expected to show Wigner crystallization, could be explored in this fashion. However, it remains to be seen whether fRG will be able to cope with the truly homogeneous limit; such a study will presumably also have to employ tools more powerful than fRG, such as the density matrix renormalization group. By way of an outlook to future technical fRG developments, let us remark that it would be desirable to find ways of avoiding an fRG artifact that is present in our results: upon turning on a long-ranged interaction, the position of the conductance step shifts not to smaller gate voltages, as physically expected, but to larger ones. We suspect that this is artefact results from our use of static fRG. A next possible step to remedy this problem could be to change from static to dynamic fRG, i.e. to implement the frequency dependence of the vertices. Moreover, it would also be possible to use our enhanced feedback scheme in the context of Keldysh fRG, which is additionally able to treat the temperature dependence and non-equilibrium behavior of QPCs. This would be numerically challenging since the Keldysh scheme in the L = 0 implementation is already very costly by itself. However, one might profit from the fact that the most expensive part of the Keldysh calculation scales with O(L 2 ), and not with O(L 3 ) as in our case. Work in that direction is currently in progress. Figure 2 . 2(color online). QPC and QD barrier shapes controlled via the parameters, εF , Vg, N and additionally for the QD Vs and js. Figure 3 . 3(color online). Linear conductance g calculated using the static eCLA for five equidistantly chosen magnetic fields B between 0 and Ωx/2. (a-c) Conductance at fixed U = 3.0 √ Ωxτ , and four values of L. (d-f) Conductance at fixed L = 5, for three values of U . Figure 4 . 4(color online). Conductance and vertex quantities calculated for the two feedback lengths L = 0 (left column) and L = 5 (right column) with three different effective interaction strengths U/ √ Ωxτ , at zero magnetic field. Figure 5 . 5(color online). The crossover from a QPC to a QD. (a) The conductance as function of gate-voltage Vg, calculated for several magnetic fields (black solid lines: B = 0, 1, 2, 3 · 10 −4 , black dashed lines: B = 6, 9, 12 · 10 −4 ) with feedback-length L = 20. Colored symbols indicate the conductance values obtained with smaller feedback lengths. (b), (c) The electron density per site, nj, computed for the two gate voltages marked by the left and right vertical arrows in (a), respectively, which both lie within Kondo plateaus. Summing nj over all sites between the two density minima yields n dot = 1.01 and n dot = 2.98, respectively. (d), (e) Non-interacting LDOS (color scale) and barrier shape (solid white curve) for the two gate voltages used in (b) and (c), respectively. Horizontal white dashed lines indicate the chemical potential µ. The LDOS maximum near ω = µ causes convergence problems if the feedback length L is small, but not if L is chosen sufficiently large. Figure 6 . 6(color online). QPC conductance step shape for three choices of the number of discretization points N (with maximal feedback length L = N − 1), for two QPCs with different curvatures. We used the following parameters, in absolut units [c.f. Eqs. (38) and (60)]: In (a), γ = 0.85, εF = 13.89meV, Ωy = 2.35meV, L bar = 146.11nm, ls = 46.17nm; and in (b), γ = 0.85, εF = 11.00meV, Ωy = 2.00meV, L bar = 158.24nm, ls = 50.00nm. The insets zoom into the range g ∈ [0.8, 1.05] and plot g as function of Vg − V po g to align the pinchoffs. When expressed in terms of the dimensionless parameters of Eq. (64), the parameter choices in (a) and (b) differ only in Ωx. For example, for the middle N = 61 curves (green) we obtain for panel (a) A = { Ωx = 1.23, Ωy = 1.91, L bar = 6.79, Ω y = −0.060, ls = 2.15}, and for panel (b), B = { Ωx = 1.05, Ωy = 1.91, L bar = 6.79, Ω y = −0.060, ls = 2.15}. Figure 7 . 7(color online). QPC conductance curves at fixed N , calculated with feedback length L = N − 1 for several values of the interaction cutoff LU (solid lines), and with L = 15 for LU = 10 (dashed line). The QPC parameters were chosen as in Figure 8 . 8(color online). (a) Distance dependence of the bare interaction U (0, x) between an electron located at the QPC center and one at x, plotted on a logarithmic scale, for three values of ls. The other physical parameters aside from ls are the same as in B [c.f. caption of Fig. 6]. The dashed black line shows the limit of ls → ∞ and the dots on the lowest curve (red) illustrate the chosen discretization points for the case N = 61. (b) U (0, x) (central peak) and U ( xs = 4.5, x) (side peak), plotted for ls = 2.15 on a linear scale for both negative and positive x values. (c) and (d) The conductance curves corresponding to the interactions depicted in (a), for two different QPC mean curvatures Ωx, respectively. The arrows at the right (red) ls = 0.86 and the left (blue) ls = 2.15 curve in (d) indicate the gate voltages Vg = −1.43 and Vg = 3.73 at which the density profiles in Figs. 9(a) and (b) were calculated, respectively. Figure 9 . 9(color online). Density profiles (thin lines) calculated for two fixed parameter choices fromFig. 8, indicated for panels (a) and (b) by the right and left arrows in Figure 10 . 10(color online). Study of two QPCs with different L bar , for three choices of Ωx. The other dimensionless parameters were chosen the same as in B [c.f. caption of Figure 11 . 11(color online). (a-c) Barrier shapes (dashed lines) and corresponding noninteracting densities (solid lines) for almost open QPCs with (a) a parabolic barrier top, (b) a flat barrier top with wide flanks, (c) and a flat barrier top with steep flanks. (d-f) Density profiles corresponding to these three barrier shapes, plotted as functions of position and gate voltage. In these plots, λF /2 is again indicated by the distance between the black lines. The flat barrier top with steep flanks of panel (c) yields pronounced Friedel oscillations in the density profile shown in panel (f), which resemble the density oscillations caused by the long-range interaction in the open regime of the QPCs ofFig. 10(e-h). This suggests that for the latter, the renormalized barriers have a rather flat tops with steep flanks. (a-c) some density profiles (solid lines) obtained for a QPC model of noninteracting electrons traversing a QPC, comparing three different barrier shapes (dashed lines): (a) a parabolic top, (b) a flat top with a slow transition to broad flanks, and (c) a flat top with a rather quick transition to steep flanks. For a given gate voltage, the overall shape of the density profile mirrors that of the barrier top for all three cases. Moreover, pronounced additional density oscillations arise for case (c). Panels (d) to (f) show the corresponding evolution of such density profiles with gate voltage. For gate voltages where the QPC is sufficiently open that the density in the center is not very low, the density oscillations seen inFigs. 11(c) as well as the conductance stay well behaved and, in fact, the flow converges without problems [Fig. 4 (b,d)]. In order to ACKNOWLEDGMENTSWe thank Jan Heyder, Volker Meden, Yigal Meir and Dennis Schimmel for very helpful discussions. We acknowledge support from the DFG via SFB-631, SFB-TR12, De730/4-3, and the Cluster of Excellence Nanosystems Initiative Munich. . W Metzner, M Salmhofer, C Honerkamp, V Meden, K Schönhammer, 10.1103/RevModPhys.84.299Rev. Mod. Phys. 84299W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Schönhammer, Rev. Mod. Phys. 84, 299 (2012). S Andergassen, T Enss, C Karrasch, V Meden, 10.1007/978-1-4020-8512-3_1cond-mat/0612229Lecture Notes in Physics. B. Barbara, Y. Imry, G. Sawatzky, and P. C. E. StampBerlin Springer Verlag645Lecture Notes in PhysicsS. Andergassen, T. Enss, C. Karrasch, and V. Meden, in Lecture Notes in Physics, Berlin Springer Verlag, Lec- ture Notes in Physics, Berlin Springer Verlag, Vol. 645, edited by B. Barbara, Y. Imry, G. Sawatzky, and P. C. E. Stamp (2008) p. 1, cond-mat/0612229. . C Karrasch, T Enss, V Meden, 10.1103/PhysRevB.73.235337Physical Review B (Condensed Matter and Materials Physics). 73235337C. Karrasch, T. Enss, and V. Meden, Physical Review B (Condensed Matter and Materials Physics) 73, 235337 (2006). . S G Jakobs, M Pletyukhov, H Schoeller, 10.1103/PhysRevB.81.195109Phys. Rev. B. 81195109S. G. Jakobs, M. Pletyukhov, and H. Schoeller, Phys. Rev. B 81, 195109 (2010). C Karrasch, arXiv:cond-mat/0612329v1Transport Through Correlated Quantum Dots -A Functional Renormalization Group Approach, Master's thesis. C. Karrasch, Transport Through Correlated Quantum Dots -A Functional Renormalization Group Approach, Master's thesis, Georg-August Universität Göttingen (2006), arXiv:cond-mat/0612329v1. The Functional Renormalization Group for Zero-Dimensional Quantum Systems in and out of Equilibrium. C Karrasch, AachenPh.D. thesis, RWTHC. Karrasch, The Functional Renormalization Group for Zero-Dimensional Quantum Systems in and out of Equi- librium, Ph.D. thesis, RWTH Aachen (2010). . F Bauer, J Heyder, E Schubert, D Borowsky, D Taubert, B Bruognolo, D Schuh, W Wegscheider, J Delft, S Ludwig, 10.1038/nature12421Nature. 50173F. Bauer, J. Heyder, E. Schubert, D. Borowsky, D. Taubert, B. Bruognolo, D. Schuh, W. Wegscheider, J. von Delft, and S. Ludwig, Nature 501, 73 (2013). . F Bauer, J Heyder, J Von Delft, 10.1103/PhysRevB.89.045128Phys. Rev. B. 8945128F. Bauer, J. Heyder, and J. von Delft, Phys. Rev. B 89, 045128 (2014). . C Karrasch, R Hedden, R Peters, T Pruschke, K Schnhammer, V Meden, Journal of Physics: Condensed Matter. 20345205C. Karrasch, R. Hedden, R. Peters, T. Pruschke, K. Schn- hammer, and V. Meden, Journal of Physics: Condensed Matter 20, 345205 (2008). . Y Meir, Journal of Physics: Condensed Matter. 20164208Y. Meir, Journal of Physics: Condensed Matter 20, 164208 (2008). . C Sloggett, A I Milstein, O P Sushkov, 10.1140/epjb/e2008-00092-2The European Physical Journal B. 61427C. Sloggett, A. I. Milstein, and O. P. Sushkov, The Eu- ropean Physical Journal B 61, 427 (2008). . K Aryanpour, J E Han, 10.1103/PhysRevLett.102.056805Phys. Rev. Lett. 10256805K. Aryanpour and J. E. Han, Phys. Rev. Lett. 102, 056805 (2009). . B J Van Wees, H Van Houten, C W J Beenakker, J G Williamson, L P Kouwenhoven, D Van Der Marel, C T Foxon, 10.1103/PhysRevLett.60.848Phys. Rev. Lett. 60848B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988). . D A Wharam, T J Thornton, R Newbury, M Pepper, H Ahmed, J E F Frost, D G Hasko, D C Peacock, D A Ritchie, G A C Jones, Journal of Physics C: Solid State Physics. 21209D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, Journal of Physics C: Solid State Physics 21, L209 (1988). . M Büttiker, 10.1103/PhysRevB.41.7906Phys. Rev. B. 417906M. Büttiker, Phys. Rev. B 41, 7906 (1990). . K J Thomas, J T Nicholls, M Y Simmons, M Pepper, D R Mace, D A Ritchie, 10.1103/PhysRevLett.77.135Phys. Rev. Lett. 77135K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace, and D. A. Ritchie, Phys. Rev. Lett. 77, 135 (1996). . N J Appleyard, J T Nicholls, M Pepper, W R Tribe, M Y Simmons, D A Ritchie, 10.1103/PhysRevB.62.R16275Phys. Rev. B. 6216275N. J. Appleyard, J. T. Nicholls, M. Pepper, W. R. Tribe, M. Y. Simmons, and D. A. Ritchie, Phys. Rev. B 62, R16275 (2000). . S M Cronenwett, H J Lynch, D Goldhaber-Gordon, L P Kouwenhoven, C M Marcus, K Hirose, N S Wingreen, V Umansky, 10.1103/PhysRevLett.88.226805Phys. Rev. Lett. 88226805S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon, L. P. Kouwenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen, and V. Umansky, Phys. Rev. Lett. 88, 226805 (2002). . J Heyder, F Bauer, E Schubert, D Borowsky, D Schuh, W Wegscheider, J Delft, S Ludwig, 10.1103/PhysRevB.92.195401arXiv:1409.3415Phys. Rev. B. 92195401cond-mat.str-elJ. Heyder, F. Bauer, E. Schubert, D. Borowsky, D. Schuh, W. Wegscheider, J. von Delft, and S. Ludwig, Phys. Rev. B 92, 195401 (2015), arXiv:1409.3415 [cond-mat.str-el]. Microscopic Origin of the 0.7 Anomaly in Quantum Point Contacts. F Bauer, LMU-MünchenPh.D. thesisF. Bauer, Microscopic Origin of the 0.7 Anomaly in Quantum Point Contacts, Ph.D. thesis, LMU-München (2014). . S G Jakobs, V Meden, H Schoeller, 10.1103/PhysRevLett.99.150603Phys. Rev. Lett. 99150603S. G. Jakobs, V. Meden, and H. Schoeller, Phys. Rev. Lett. 99, 150603 (2007). . C Husemann, M Salmhofer, 10.1103/PhysRevB.79.195125Phys. Rev. B. 79195125C. Husemann and M. Salmhofer, Phys. Rev. B 79, 195125 (2009). . T R Morris, 10.1142/S0217751X94000972International Journal of Modern Physics A. 092411T. R. Morris, International Journal of Modern Physics A 09, 2411 (1994). . J Dormand, P Prince, 10.1016/0771-050X(80)90013-3Journal of Computational and Applied Mathematics. 619J. Dormand and P. Prince, Journal of Computational and Applied Mathematics 6, 19 (1980). . J Kondo, Prog. Theor. Phys. 3237J. Kondo, Prog. Theor. Phys. 32, 37 (1964). S Datta, Electronic Transport in Mesoscopic Systems, Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. Cambridge University PressS. Datta, Electronic Transport in Mesoscopic Systems, Cambridge Studies in Semiconductor Physics and Mi- croelectronic Engineering (Cambridge University Press, 1997). . O Goulko, F Bauer, J Heyder, J Von Delft, 10.1103/PhysRevLett.113.266402arXiv:1408.0746v1Phys. Rev. Lett. 113266402condmat.mes-hallO. Goulko, F. Bauer, J. Heyder, and J. von Delft, Phys. Rev. Lett. 113, 266402 (2014), arXiv:1408.0746v1 [cond- mat.mes-hall]. . A M Lunde, A D Martino, A Schulz, R Egger, K Flensberg, New Journal of Physics. 1123031A. M. Lunde, A. D. Martino, A. Schulz, R. Egger, and K. Flensberg, New Journal of Physics 11, 023031 (2009). . K Hirose, Y Meir, N S Wingreen, 10.1103/PhysRevLett.90.026804Phys. Rev. Lett. 9026804K. Hirose, Y. Meir, and N. S. Wingreen, Phys. Rev. Lett. 90, 026804 (2003). . M J Iqbal, R Levy, E J Koop, J B Dekker, J P Jong, J H M Van Der Velde, D Reuter, A D Wieck, R Aguado, Y Meir, C H Van Der Wal, 10.1038/nature12491Nature. 50179M. J. Iqbal, R. Levy, E. J. Koop, J. B. Dekker, J. P. de Jong, J. H. M. van der Velde, D. Reuter, A. D. Wieck, R. Aguado, Y. Meir, and C. H. van der Wal, Nature 501, 79 (2013). . H J Schulz, 10.1103/PhysRevLett.71.1864Phys. Rev. Lett. 711864H. J. Schulz, Phys. Rev. Lett. 71, 1864 (1993).
[]
[ "On generalized Witt algebras in one variable", "On generalized Witt algebras in one variable" ]
[ "Ki-Bong Nam ", "Jonathan Pakianathan " ]
[]
[]
We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples.We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra which does not contain any abelian Lie subalgebras of dimension greater than one.We develop an invariant of these generalized Witt algebras called the spectrum, and use it to show that there exist infinite families of nonisomorphic, simple, generalized Witt algebras and infinite families of nonisomorphic, nonsimple, generalized Witt algebras.We develop a machinery that can be used to study the endomorphisms of a generalized Witt algebra in the case that the spectrum is "discrete". We use this to show, that among other things, every nonzero Lie algebra endomorphism of the classical Witt algebra is an automorphism and every endomorphism of the centerless Virasoro algebra fixes a canonical element up to scalar multiplication.However, not every injective Lie algebra endomorphism of the centerless Virasoro algebra is an automorphism.
10.3906/mat-1003-201
[ "https://arxiv.org/pdf/1007.3247v1.pdf" ]
16,352,333
1007.3247
504da0432632aac277e6f86029aec1fc2040eae0
On generalized Witt algebras in one variable 19 Jul 2010 July 20, 2010 Ki-Bong Nam Jonathan Pakianathan On generalized Witt algebras in one variable 19 Jul 2010 July 20, 2010Infinite dimensional Lie algebraVirasoro algebra 1991 Mathematics Subject Classification Primary: 17B6517C20; Sec- ondary: 17B40 We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples.We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra which does not contain any abelian Lie subalgebras of dimension greater than one.We develop an invariant of these generalized Witt algebras called the spectrum, and use it to show that there exist infinite families of nonisomorphic, simple, generalized Witt algebras and infinite families of nonisomorphic, nonsimple, generalized Witt algebras.We develop a machinery that can be used to study the endomorphisms of a generalized Witt algebra in the case that the spectrum is "discrete". We use this to show, that among other things, every nonzero Lie algebra endomorphism of the classical Witt algebra is an automorphism and every endomorphism of the centerless Virasoro algebra fixes a canonical element up to scalar multiplication.However, not every injective Lie algebra endomorphism of the centerless Virasoro algebra is an automorphism. Introduction Throughout this paper, we will work over a field k of characteristic zero. Also note that there will be no finiteness constraints on the dimension of the Lie algebras in this paper -in fact, most of the Lie algebras that we will consider will be infinite dimensional. We now sketch the basic results and ideas of this paper in this introductory section. Precise definitions of the concepts can be found within the paper. Let R be the field of fractions of the power series algebra k [[x]]. Following [6], we define a stable algebra to be a subalgebra of R which is closed under formal differentiation ∂. Notice that we confine ourselves to the one variable case throughout this paper. Important examples of stable algebras are the polynomial algebra k[x], the power series algebra k [[x]] and the Laurent polynomial algebra k[x, x −1 ]. Following [8] and [10], to every stable algebra A, we associate a Lie algebra W itt(A). We refer to W itt(A) as a generalized Witt algebra. (The reader is warned, that there are different definitions of what a generalized Witt algebra is in the literature. Please look at Definition 3.1 for ours.) W itt(k[x]) is the classical Witt algebra, (See [2]) and W itt(k[x, x −1 ]) is called the centerless Virasoro algebra in the literature. (See [7].) A Lie algebra is called self-centralizing if it contains no abelian Lie subalgebras of dimension greater than one. We prove: Theorem 1.1 (Theorem 3.8 and Proposition 3.11). Every generalized Witt algebra is self-centralizing. Furthermore, if it is infinite dimensional (which is the case for all but one trivial example where A = k), then a generalized Witt algebra must be semisimple and indecomposable. To contrast, over an algebraically closed field, it is shown that the only finite dimensional Lie algebra which is self-centralizing, semisimple and indecomposable is sl 2 , the Lie algebra of 2 × 2 matrices of trace zero. However a generalized Witt algebra need not be simple, some are and some are not. If a generalized Witt algebra has a nonzero ad-diagonal element, i.e., nonzero α such that ad(α) is diagonal in some basis, we show that the set of eigenvalues of ad(α) possesses the algebraic structure of a pseudomonoid. We call this pseudomonoid, the spectrum of α. We then show in Proposition 7.10 that any other nonzero ad-diagonal element of this Lie algebra, has to have an equivalent spectrum. This allows us to define the spectrum of L to be the spectrum of any nonzero ad-diagonal element. It is then shown that this is indeed an invariant for these kinds of Lie algebras, i.e., isomorphic Lie algebras have equivalent spectra. The constraint that the Lie algebra possesses a nonzero ad-diagonal element, is not so bad as all the classical examples possess this property. In these pseudomonoids, one can define the notion of an ideal subset. We show: Proposition 1.2 (Proposition 6.5). Let L be a generalized Witt algebra with nonzero ad-diagonal element and let G be its spectrum. Then there is a one-toone correspondence between the ideal subsets of G and the ideals of L. If G is actually an abelian group then it is simple as a psuedomonoid and hence L is simple. Since the classical Witt algebra and centerless Virasoro algebra have nonzero ad-diagonal elements, and their spectra are simple pseudomonoids, we recover the well-known fact, that they are simple, as a corollary. Using this spectrum invariant, we can distinguish between nonisomorphic generalized Witt algebras and show that there is a rich variety of such algebras (with nonzero ad-diagonal element): In fact for every submonoid of (k, +), there is a generalized Witt algebra with that monoid as its spectrum. Thus, in particular since every torsion-free abelian group embeds into the additive group of some rational vector space, we may get any torsion-free abelian group as the spectrum of a generalized Witt algebra in one variable by suitable choice of the base field k. A machinery is obtained to find the set of eigenvalues of any element in a generalized Witt algebra. It uses formal calculus and in particular, the logarithmic derivative. It is stated in Theorem 5.11. Finally, motivated by [12], we discuss injective Lie algebra endomorphisms of generalized Witt algebras. In the case where the generalized Witt algebra possesses a "discrete" spectrum, one can show that such an endomorphism must essentially fix a nonzero ad-diagonal element. (See Theorem 8.7.) As corollaries of this fact we can easily obtain information about endomorphisms of these Lie algebras and prove things such as: Theorem 1.4 (Corollaries 8. 8 and 8.9). Any nonzero Lie algebra endomorphism f of the classical Witt algebra is actually an automorphism and furthermore, f (x∂) = (x + b)∂ for some b ∈ k. If f is a nonzero Lie algebra endomorphism of the centerless Virasoro algebra, then f is injective and f (x∂) = 1 a x∂ for some nonzero integer a. However f need not be onto. More precisely, the centerless Virasoro algebra possesses injective Lie algebra endomorphisms which are not automorphisms. g = ∞ i=N α i x i for suitable α i ∈ k and N ∈ Z. Notice that R acts on itself by left multiplication and this gives us a monomorphism of k vector spaces: τ : R → End k (R). Furthermore, there also exists ∂ ∈ End k (R) which corresponds to formal differentiation with respect to x, i.e., ∂( ∞ i=N α i x i ) = ∞ i=N iα i x i−1 . It is easy to verify that ∂(g) = 0 if and only if g is a constant. Definition 2.3. Given a stable algebra A, we define W eyl(A) to be the subalgebra of End k (R) generated by τ (A) and ∂. Thus, W eyl(A) is an associative algebra with identity element equal to the identity endomorphism of R. We will identify A with its image τ (A) ⊆ End k (R) from now on. Lemma 2.4. Let A be a stable algebra. For any f ∈ A, one has ∂f − f ∂ = f ′ in W eyl(A). Thus for any α ∈ W eyl(A), one has α = N i=0 α i ∂ i for suitable N ∈ N and α i ∈ A. Definition 2.1. A stable algebra A is a subalgebra of R with the property that ∂(A) ⊆ A. Furthermore, if {e i |i ∈ I} is a k-basis for A, then {e i ∂ j |i ∈ I, j ∈ N} is a k-basis for W eyl(A). Proof. The proof is standard and is left to the reader. Remark 2.5. W eyl(k[x] ) is the classical Weyl algebra. It is a simple algebra which has no zero divisors, (see [2]). In general, one can define an order on W eyl(R) such that the order of a nonzero element is equal to the highest exponent of ∂ in its canonical expression and is defined to be −∞ for the zero element. Then one shows that ord(αβ) = ord(α) + ord(β) for any α, β ∈ W eyl(R) (see [2]) and it easily follows that W eyl(R) has no zero divisors. Hence, W eyl(A), which is a subalgebra of W eyl(R), has no zero divisors in general. Note however, that in general, W eyl(A) need not be simple. Generalized Witt algebras Definition 3.1. Let W itt(A) be the subspace of W eyl(A) consisting of the order 1 elements together with zero. Thus α ∈ W itt(A) if α can be written as f ∂ for some f ∈ A. It is easy to check that W itt(A) is a Lie subalgebra of W eyl(A). (Note, it is not a subalgebra of W eyl(A).) If {e i } i∈I is a k-basis for A then {e i ∂} i∈I is a k-basis for W itt(A). Proposition 5.12 shows how our definition is related to the one found in [3]. Remark 3.2. W itt(k[x] ) is the classical Witt algebra. It is the Lie algebra of derivations of the classical Weyl algebra (see [2]), and is a simple Lie algebra. However, in general, W itt(A) is not neccessarily simple. W itt(k[x, x −1 ]) is called the centerless Virasoro algebra in the literature. (See [7].) In general, we cannot claim that W itt(A) is simple, but these generalized Witt algebras do share one important common property -they are selfcentralizing. Definition 3.3. Given a Lie algebra L and an element l ∈ L, we define the centralizer of l, C(l) = {x ∈ L|[l, x] = 0}. Notice, by the Jacobi identity, C(l) is always a Lie subalgebra of L containing l. Proof. The proof is easy and left to the reader. Definition 3.5. A Lie algebra L is said to be self-centralizing if it satisfies any of the equivalent conditions of Proposition 3.4. Remark 3.6. Thus a self-centralizing Lie algebra is one where the centralizers have as small a dimension as possible. Notice that a self-centralizing Lie algebra of dimension strictly greater than one must have trivial center. Furthermore, a Lie algebra isomorphic to a self-centralizing one, is itself self-centralizing. Remark 3.7. It is easy to check that the nonabelian Lie algebra of dimension two is self-centralizing but is not simple. Similarly sl n , the Lie algebra of n × n, trace zero matrices is simple but contains an abelian Lie subalgebra of dimension greater than one for n ≥ 3 and hence is not self-centralizing. We now make a useful observation: Theorem 3.8. For any stable algebra A, W itt(A) is a self-centralizing Lie algebra. Proof. Let f ∂ be a nonzero element of W itt (A). Suppose [f ∂, g∂] = 0. Then as [f ∂, g∂] = (f g ′ − gf ′ )∂, we conclude that f g ′ − gf ′ = 0 in A ⊆ R. Then we can rewrite f g ′ − gf ′ = 0 as (g/f ) ′ f 2 = 0 in R which is possible since f is not the zero element. Since the only elements in R which have zero derivative, are the constants, we conclude that g/f is a constant or that g is a multiple of f . Thus we conclude C(f ∂) is one dimensional. This concludes the proof. Remark 3.9. It follows immediately from Theorem 3.8, that the classical Witt algebra and the centerless Virasoro algebra are self-centralizing. Definition 3.10. Recall that a Lie algebra is called semisimple if it does not possess any nontrivial solvable ideals. It is a standard fact that a Lie algebra is semisimple if it does not possess any nontrivial abelian ideals. (See [5].) Let us record some consequences of the self-centralizing property in the following proposition. Proposition 3.11. Let L be a self-centralizing Lie algebra, then: (a) Any Lie subalgebra is also self-centralizing. (b) If L possesses a finite dimensional ideal I of dimension n > 1, then dim(L) ≤ n 2 . If L possesses an ideal of dimension 1, then dim(L) ≤ 2. (c) If L is infinite dimensional, then L does not possess any finite dimensional, nontrivial ideals. (d) If α, β are two linearly independent elements of L and x is a common eigenvector of ad(α) and ad(β) then x is a multiple of [α, β]. (e) If α, β are two linearly independent elements of L, then there is no basis for L, in which both α and β are ad-diagonal. (f ) L is indecomposable i.e., L cannot be written as a direct sum of two nonzero Lie algebras. (g) If dim(L) > 2 then L is semisimple. (h) If L is finite dimensional and k is algebraically closed, then L is either isomorphic to the nonabelian Lie algebra of dimension two, sl 2 , or a Lie algebra of dimension less than or equal to one. Proof. (a) follows at once from the definition of a self-centralizing Lie algebra. To prove (b), suppose I is a nontrivial, finite dimensional ideal of dimension n. Then define θ : L → End k (I) by θ(x) = ad(x)| I . Note that End k (I) is finite dimensional of dimension n 2 . If n > 1, then θ is injective by the self-centralizing property of L. This is because if z were a nonzero element in Ker(θ), then I ⊆ C(z). However, C(z) has dimension 1 as L is self-centralizing, while I is assumed to have dimension bigger than 1 giving a contradiction. It follows easily from the injectivity of θ that dim(L) ≤ dim(End k (I)) = n 2 . If n = 1 and x is a generator of I, then Ker(θ) is codimension at most one in L. However, Ker(θ) = C(x) = I since L is self-centralizing. Thus dim(L) ≤ 2. (c) follows quickly from (b). (d) and (e) follow from quick calculations and the self-centralizing property. (f) is a trivial verification. For (g), note that if dim(L) > 2, then by (b), L does not possess any nontrivial ideals of dimension one. On the other hand, because L is self-centralizing, it cannot possess any abelian ideals of dimension greater than one and so we conclude that L does not possess any nontrivial abelian ideals and hence is semisimple. For (h), note that if dim(L) ≤ 2, the result is easy. So we can assume 2 < dim(L) < ∞, and so by (g), L is semisimple. From standard results (see [5] or [4]), since we are over a field of characteristic zero, L is the direct sum of simple Lie algebras. However by (f), we see that in fact L must be simple. If we assume k to be algebraically closed, then the Cartan subalgebra of L is abelian, and since L is self-centralizing, it must have rank one. From the classification of simple finite dimensional Lie algebras over an algebraically closed field, we see that L is isomorphic to sl 2 . Remark 3.12. By Proposition 3.11, we see that there aren't very many finite dimensional self-centralizing Lie algebras. Thus it is somewhat striking that all of the generalized Witt algebras are self-centralizing. We will see later that we can find infinitely many nonisomorphic generalized Witt algebras so that the class of self-centralizing Lie algebras is pretty rich. In the class of infinite dimensional Lie algebras, Proposition 3.11 shows that being self-centralizing is a stronger condition than being semisimple and yet is usually easier to verify than simplicity. Since stable algebras A are infinite dimensional in all but some trivial cases, W itt(A) is usually infinite dimensional and since it is self-centralizing by Theorem 3.8, it follows by Proposition 3.11, that W itt(A) is both semisimple and indecomposable. However there are examples where W itt(A) is simple and there are examples where it is not. We will discuss this more later on. Eigenvalues and eigenspaces We have seen that all generalized Witt algebras are self-centralizing. Given a Lie algebra L, and α ∈ L, let E a (α) ⊆ L be the eigenspace of ad(α) corresponding to the eigenvalue a ∈ k. In this language, a self-centralizing Lie algebra L is one such that dim(E 0 (α)) = 1 for all nonzero α ∈ L. We have seen that a generalized Witt algebra is selfcentralizing and hence satisfies this condition on the eigenspaces. We will now extend this result by studying further constraints on these eigenspaces in a generalized Witt algebra. Before we can do this, we need to recall the concept of the logarithmic derivative on R, and some of its basic properties. LD(f ) = f ′ f . where f ′ is the formal derivative of f . It is easy to check that LD is a group homomorphism from (R ♯ , ×) to (R, +). It is also routine to see that Ker(LD) is exactly the constant functions. Thus if u, v ∈ R ♯ have LD(u) = LD(v), then u is a scalar multiple of v. Now we are ready to prove an important lemma which generalizes Theorem 3.8. Lemma 4.2. If f ∂ ∈ W itt(R) is a nonzero element, then dim(E a (f ∂)) ≤ 1 for all a ∈ k. Furthermore, if g∂ is a nonzero element in E a (f ∂), then g = f u where LD(u) = a/f . Proof. Suppose g∂ is a nonzero element in E a (f ∂). Then [f ∂, g∂] = ag∂ (f g ′ − gf ′ )∂ = ag∂ (g/f ) ′ f 2 = ag Thus we conclude (g/f ) ′ f = a(g/f ). If we let u = g/f , this becomes u ′ f = au or LD(u) = a/f . Thus we conclude g = f u where LD(u) = a/f . If h∂ is another nonzero element in E a (f ∂), then similarly we would conclude h = f v where LD(v) = a/f . However LD(u) = LD(v) = a/f so v is a scalar multiple of u and hence h is a scalar multiple of g. Thus we see dim(E a (f ∂)) ≤ 1 as we sought to show. Lemma 4.2 shows that for any nonzero f ∂ ∈ W itt(R), and a ∈ k, the eigenspace of ad(f ∂) corresponding to a is at most one dimensional. It remains to decide when this eigenspace is one dimensional and when it is zero dimensional. To do this, it turns out we need to find the image of LD : R ♯ → R. We will now introduce a few more concepts in formal calculus that will let us do this. Definition 4.3. Given a nonzero f ∈ R, we can write f = ∞ i=N α i x i where α i ∈ k for all i ≥ N and α N = 0. N is called the Weierstrass degree (see [9]) of f and will be denoted by W (f ). α −1 is called the residue of f and will be denoted res(f ). We also define W (0) = ∞ and res(0) = 0. We now collect some elementary properties of the Weierstrass degree in the next lemma. The proof is simple and will be left to the reader. Lemma 4.5. Given nonzero f ∈ R, we can write f = x W (f ) u with u ∈ U . Furthermore such an expression for f is unique. Given f, g ∈ R, W (f g) = W (f ) + W (g). We now define formal integration: Definition 4.6. Recall (x) is the unique maximal ideal of k[[x]]. We define formal integration : k[[x]] → (x) by ( ∞ i=0 α i x i ) = ∞ i=0 α i x i+1 i + 1 = ∞ i=1 α i−1 x i i It follows easily that ∈ End k (k[[x]]) and that if f ∈ k[[x]] is nonzero, W ( f ) = W (f ) + 1. Furthermore, we have of course ∂( f ) = f for all f ∈ k[[x]]. We will also need to compose two power series. Recall that given g ∈ k[[x]] and f ∈ (x), we have a well-defined composition power series g • f ∈ k[[x]] given in the following manner: If g = ∞ i=0 α i x i then g • f ∈ k[[x]] is given formally by ∞ i=0 α i f i . We collect well-known results on this composition in the following proposition: Proposition 4.7. If g ∈ k[[x]] and f ∈ (x). Then there exists a series g • f ∈ k[[x]] such that (g • f ) ′ = (g ′ • f )f ′ . Furthermore, (g • f )(0) = g(0) and g • x = g. We are now ready to study the image of the logarithmic derivative LD : R ♯ → R. Lemma 4.8. Let u ∈ R ♯ , (a) If W (u) = 0 then W (LD(u)) = −1 and res(LD(u)) is equal to W (u) which is of course an integer. (b) If W (u) = 0 then W (LD(u)) ≥ 0. (c) If W (g) < −1 or if W (g) = −1 and res(g) is not an integer, then g is not in the image of LD : R ♯ → R. Proof. The proof will be left to the reader. It follows from writing u as a Laurent series and explicitly calculating LD(u). We have seen in Lemma 4.8, conditions that ensure an element g ∈ R is not in the image of LD : R ♯ → R. We now show that in the remaining situations, the element g is in the image. First recall e x ∈ k[[x]] is the power series given by e x = ∞ i=0 x i i! . It is easy to verify that ∂e x = e x and that e x evaluated at x = 0 is 1. Given g ∈ k[[x]], g lies in (x), the maximal ideal of k[[x]]. Thus by Proposition 4.7 we can form the power series e x • ( g) which we will denote by e g . It follows from the same proposition that ∂e g = e g ∂( g) = ge g . Furthermore since e g (0) = e x (0) = 1, we see that e g ∈ U for all g ∈ k[[x]]. We will use these facts in the next theorem. Theorem 4.9. Let g ∈ R. Then either: (a) W (g) ≥ 0 and g = LD(e g ). (b) W (g) = −1 and res(g) is an integer then g = res(g) x + u for some unique u ∈ k[[x]] and we have g = LD(x res(g) e u ). (c) W (g) < −1 or W (g) = −1 and res(g) is not an integer in which case g is not in the image of LD : R ♯ → R. Proof. (c) follows from Lemma 4.8. For (a), assume g has W (g) ≥ 0 so that e g ∈ U . Then we calculate LD(e g ) = ∂e g e g = ge g e g = g and so (a) is proven. Assume g as in the statement of (b). Then it is obvious that we may write g = res(g) x + u with u ∈ k[[x] ] determined uniquely. Since res(g) is an integer x res(g) e u certainly defines an element in R ♯ . We compute LD(x res(g) e u ) = res(g)LD(x) + LD(e u ), since LD is a homomorphism = res(g) 1 x + u, using the calculation in (a) = g. Thus we are done. We are now ready to complete the analysis of the eigenspaces of elements in ad(W itt(R)) which was started in Lemma 4.2. Theorem 4.10 (Spectral theorem for R). Let f ∂ be a nonzero element in W itt(R). Then: (a) If W (f ) > 1, then dim(E a (f ∂)) = 0 for all nonzero a ∈ k and dim(E 0 (f ∂)) = 1. (b) If W (f ) ≤ 0, then dim(E a (f ∂)) = 1 for all a ∈ k. Furthermore, f e a f ∂ ∈ E a (f ∂). (c) If W (f ) = 1 then dim(E a (f ∂)) = 0 if a = N f ′ (0) for some integer N . dim(E N f ′ (0) (f ∂)) = 1 for all N ∈ Z. Furthermore f x N e ( N (f ′ (0)x−f ) f x ) ∂ ∈ E N f ′ (0) (f ∂) for all N ∈ Z. Proof. Let f ∂ ∈ W itt(R) be nonzero and let a ∈ k. Then by Lemma 4.2, we see that dim(E a (f ∂)) is either zero or one and it is one if and only if a f = LD(u) for some u ∈ R ♯ . Furthermore, in this case, f u∂ is a nonzero element of E a (f ∂). Since we know dim(E 0 (f ∂)) = 1 we can assume a = 0 for the rest of the proof. It follows that W ( a f ) = −W (f ). If W (f ) > 1 then W ( a f ) < −1 and so by Theorem 4.9, a f is not in the image of the logarithmic derivative and hence we have proven (a). If If W (f ) ≤ 0 then W ( a f ) ≥ 0W (f ) = 1 then we can write f = xf ′ (0)v where v ∈ U has v(0) = 1. Then W ( a f ) = −1 and res( a f ) = a f ′ (0) . Again by Theorem 4.9, a f is in the image of the logarithmic derivative if and only if this residue is an integer which happens if and only if a is an integral multiple of f ′ (0). If this is the case, then a = N f ′ (0) and we can write N f ′ (0) f = N x + w where w ∈ k[[x]]. Theorem 4.9 then shows that N f ′ (0) f = LD(x N e w ) . Now it remains only to note that w = N f ′ (0) f − N x = N (f ′ (0)x − f ) f x and we are done. Spectra We now discuss the concept of a spectrum which we will find to be very useful in the remainder of this paper. Definition 5.1. Given a Lie algebra L, and α ∈ L, we define the L-spectrum of α to be spec L (α) = {a ∈ k| dim(E a (α)) = 0}. We write spec(α) for spec L (α) when there is no danger of confusion. Thus the spectrum of α is the set of eigenvalues of ad(α) ∈ End k (L). Notice that in a nonzero Lie algebra L, spec(0) = {0} and 0 ∈ spec(α) for all α ∈ L. In general the spectrum possesses no significant algebraic structure. However, we will soon see that if L is self-centralizing, spec(α) possesses the structure of a pseudomonoid (which we will define shortly) for all α ∈ L. Definition 5.2. A subset P of k is a pseudomonoid if it satisfies the following conditions: (a) 0 ∈ P . (b) If a, b ∈ P and a = b then a + b ∈ P where + is addition in k. Remark 5.3. Notice that a pseudomonoid differs from a monoid because in a monoid we may also add an element to itself, i.e., if a ∈ P and P is a monoid under + then a + a ∈ P . This need not hold for a pseudomonoid as can be seen by the following example: Let A = {−1, 0, 1, . . . } be the set of integers greater than or equal to negative one. This set is a pseudomonoid under addition but is not a monoid as (−1) + (−1) = −2 / ∈ A. The concept of a pseudomonoid turns out to be important for us because of the following lemma: Lemma 5.4. Let L be a Lie algebra, and α ∈ L be a nonzero element. Then for any a, b ∈ k, we have [E a (α), E b (α)] ⊆ E a+b (α). Thus if L is self-centralizing, then spec(α) is a pseudomonoid for all α ∈ L. Proof. For a proof of the first statement, take α ∈ L and a, b ∈ k. Then for e a ∈ E a (α) and e b ∈ E b (α) we have by the Jacobi identity: [α, [e a , e b ]] = [[α, e a ], e b ] + [e a , [α, e b ]] = [ae a , e b ] + [e a , be b ] = (a + b)[e a , e b ]. Thus we see that [e a , e b ] ∈ E a+b (α) which proves the first statement. Now suppose that L is self-centralizing. spec(0) = {0} is a pseudomonoid so assume α = 0. Let a, b ∈ spec(α) with a = b. Then if we take nonzero e a ∈ E a (α) and e b ∈ E b (α), since a = b it follows that e a , e b are linearly independent. Thus since L is self-centralizing, it follows that [e a , e b ] = 0 which shows that E a+b (α) = 0. Thus a + b ∈ spec(α) and so spec(α) is a pseudomonoid. Definition 5.5. Let α ∈ L. Then we define M L (α) = ⊕ a∈k E a (α). Thus M L (α) is the subspace of L spanned by the eigenspaces of α. It is the maximal subspace on which ad(α) is diagonal (with respect to some basis). It is easy to argue that we can also write M L (α) = ⊕ a∈spec(α) E a (α). It follows from Lemma 5.4 that M L (α) is a Lie subalgebra of L. We will write M (α) for M L (α) when there is no danger of confusion. We will now look at a few examples before proceeding any further. To do this, it is useful to introduce the concept of a differential spanning set. Definition 5.6. S ⊆ R is called a differential spanning set if it satisfies the following conditions: (a) 1 ∈ S. (b) If f, g ∈ S, then f g ∈ S. (c) If f ∈ S then ∂f is a linear combination of elements in S. Given a differential spanning set S, the vector space A spanned by S in R is easily seen to be a stable algebra. Example 5.7. Let S = {x n |n ∈ N}, then it is easy to check that S is a differential spanning set which spans the polynomial stable algebra k[x] and is in fact a basis for this algebra. In W itt(k[x]), one calculates: [x n ∂, x m ∂] = (x n (x m ) ′ − x m (x n ) ′ )∂ = (m − n)x m+n−1 ∂. Thus we see easily that M (x∂) = W itt(k[x]) and that spec(x∂) = {−1, 0, 1, . . . }. Example 5.8. Let S = {x n |n ∈ Z}, then S is a differential spanning set which forms a basis for the Laurent polynomial stable algebra k[x, x −1 ]. Exactly as in Example 5.7, one can show that M (x∂) = W itt(k[x, x −1 ]) and that spec(x∂) = {. . . , −2, −1, 0, 1, 2, . . . } = Z. Note that the spectrum of x∂ depends on which Lie algebra we are in and so we stress that the reader should keep in mind the surpressed subscript L in the notation for spec. Example 5.9. Let G be a submonoid of (k, +), then S = {e ax |a ∈ G}, is a differential spanning set (since e (a+b)x = e ax e bx as the reader can verify). Let A(G) be the stable algebra that this spanning set spans. In W itt(A(G)), we calculate: [e ax ∂, e bx ∂] = (e ax be bx − e bx ae ax )∂ = (b − a)e (a+b)x . ¿From this, it follows that M (1∂) = W itt(A(G)) and that spec(1∂) = G. Since e bx ∂ ∈ E b (1∂) for all b ∈ G, it also follows that S is a basis for A(G). Remark 5.10. It is a standard fact that every torsion-free abelian group embeds into a torsion-free divisible group and that a torsion-free divisible group is isomorphic to the additive group of a rational vector space (see [11]). Any rational vector space is isomorphic to a subgroup of (k, +) for suitable choice of k. (Need the dimension of k over its characteristic subfield Q to be big enough.) Thus by Example 5.9, we conclude that every torsion-free abelian group is the spectrum of some ad-diagonal element in some generalized Witt algebra in one variable. Finally we state a general spectral theorem for generalized Witt algebras. It is based on Theorem 4.10. Theorem 5.11 (Spectral theorem). Let W itt(A) be a generalized Witt algebra and f ∂ be a nonzero element in W itt(A). Then for all a ∈ k, dim(E a (f ∂)) ≤ 1 and: (a) If W (f ) > 1, then spec(f ∂) = {0}. (b) If W (f ) ≤ 0 then for all a ∈ k, a ∈ spec(f ∂) ⇐⇒ f e a f ∈ A. (c) If W (f ) = 1 then spec(f ∂) ⊆ Zf ′ (0), where Zf ′ (0) stands for the set of integral multiples of f ′ (0) ∈ k. Furthermore, for all N ∈ Z, N f ′ (0) ∈ spec(f ∂) ⇐⇒ f x N e N (f ′ (0)x−f ) f x ∈ A. Proof. First note that A ⊆ R so W itt(A) is a Lie subalgebra of W itt(R). Then if f ∂ ∈ W itt(A), and a ∈ k, the a-eigenspace of ad(f ∂) for W itt(A) lies inside the one for W itt(R). Thus spec W itt(A) (f ∂) ⊆ spec W itt(R) (f ∂) and a ∈ spec W itt(R) (f ∂) lies in spec W itt(A) (f ∂) if and only if one of the eigenvectors in R corresponding to a actually lies in A. With these comments, the rest now follows from Theorem 4.10. Theorem 5.11, will show that our definition of generalized Witt algebras is related to the definition in papers such as [3]. We do this in the next proposition. Similarly, in the remaining case where W (f ) = 1, we set e N f ′ (0) = f x N e N (f ′ (0)x−f ) f x ∂ for all N f ′ (0) ∈ spec(f ∂), and again compute that [e N f ′ (0) , e Mf ′ (0) ] = (M − N )f ′ (0)e (N +M)f ′ (0) for all M f ′ (0), N f ′ (0) ∈ spec(f ∂). We are now ready to study the issue of simplicity of a generalized Witt algebra. We will do this in the next section. Note that this means that spec(e 0 ) = G. Remark 6.2. Of course, by Theorem 5.11 and Proposition 5.12, if W itt(A) is a generalized Witt algebra, and α ∈ W itt(A) is nonzero, then M (α) is a strongly graded Lie algebra, graded by the pseudomonoid spec(α) where α plays the role of e 0 . Remark 6.3. It is obvious that two strongly graded Lie algebras, graded by the same pseudomonoid G ⊆ k, are isomorphic as Lie algebras. Now we set out to get a complete correspondence between the ideals of a strongly graded Lie algebra and the ideals of the pseudomonoid which grades it. Definition 6.4. Let G be a pseudomonoid. Then S ⊆ G is called a closed subset if for all distinct a, b ∈ S, we have a + b ∈ S. Note that a closed subset S need not be a subpseudomonoid of G since we do not require that 0 ∈ S. In fact the empty set ∅ is always a closed subset. I ⊆ G is called an ideal subset if for all a ∈ I and all b ∈ G such that b = a, we have a+ b ∈ I. Again the empty set is always an ideal subset and G is always an ideal subset of G. These are called the trivial ideal subsets. A pseudomonoid G which has no nontrivial ideal subsets is called a simple psuedomonoid. A nonzero element x ∈ G is called invertible if −x ∈ G. (Recall, all pseudomonoids are by definition in k and hence −x exists in k and is distinct from x.) Fix a strongly graded Lie algebra L, graded by a pseudomonoid G, then L = ⊕ g∈G E g such that there is e 0 ∈ E 0 , with spec(e 0 ) = G and E g equal to the eigenspace of ad(e 0 ) corresponding to g. Then for any S ⊂ G, we define: Θ(S) = ⊕ a∈S E a . (We use the convention that Θ(∅) = 0.) Thus Θ is a map from the subsets of G to the subspaces of L which is obviously injective. Notice that if S is a closed subset of G, then Θ(S) is a Lie subalgebra of L. Similarly, if I is an ideal subset of G, then Θ(I) is an ideal of L. Proposition 6.5. Let L be a strongly graded Lie algebra, graded by the pseudomonoid G. The map Θ defined above takes closed subsets of G to Lie subalgebras of L and this correspondence is injective. The map Θ takes closed subsets of G containing 0, to Lie subalgebras of L containing e 0 and this correspondence is bijective. The map Θ takes ideal subsets of G to ideals of L and this correspondence is bijective. Proof. All but the surjectivity of the last two correspondences has been proven. So assume J is an ideal of L (or a Lie subalgebra containing e 0 ). First, let us show that there is a subset I of G such that Θ(I) = J. We can of course assume J = 0 as Θ(∅) = 0. Define I ⊂ G as follows. Recall that by the grading, for any x ∈ L, we can write x uniquely as x = a∈G x a with x a ∈ E a and only finitely many x a nonzero. We call x a the a-th component of x. Then set: I = {a ∈ G such that there exists y ∈ J whose a-th component is nonzero}. It is clear that J ⊆ Θ(I). So it remains only to show Θ(I) ⊆ J. We do this by showing that E a ⊆ J for any a ∈ I. This follows immediately from the following fact: Fact: If y ∈ J, then all of the components of y are also in J. We will prove this fact by induction on n, the number of nonzero components of y. If n = 0, 1, it follows trivially. So assume n > 1 and we have proven the fact for all smaller n. So let y ∈ J and assume we can write (1 − a i a 1 )y ai is in J. However by induction, it follows that the components of y − 1 a1 [e 0 , y] lie in J and hence that y ai lie in J for all 2 ≤ i ≤ n. However y = y a1 + n i=2 y ai , so it also follows that y a1 is in J. Thus by induction, we have proven the fact and hence that J = Θ(I). All that remains, is to show that I is an ideal subset if J is an ideal or that I is a closed subset containing zero if J is a Lie subalgebra containing e 0 . We prove only the former, the proof of the latter being similar. If a ∈ I then by definition, there is y ∈ J such that y = g∈G y g with y g ∈ E g and y a = 0. If b ∈ G and b = a, take nonzero z b ∈ E b . Then [z b , y] = g∈G [z b , y g ] ∈ J as J is an ideal. Notice that since our pseudomonoids are defined to be subpseudomonoids of (k, +), the only term in the sum that can lie in E a+b is [z b , y a ] which is nonzero as z b , y a are nonzero and since we are in a strongly graded Lie algebra. All the other terms, live in other eigenspaces and so we conclude [z b , y a ] has nonzero (a + b)-component and hence a + b ∈ I showing that I is an ideal subset of G. Corollary 6.6. If L is a strongly graded Lie algebra, graded by a pseudomonoid G. Then L is simple if and only if G is simple. Let W itt(A) be a generalized Witt algebra and α ∈ W itt(A) be nonzero, then M (α) is a simple Lie algebra if and only if spec(α) is a simple pseudomonoid. (Note it is easy to see that spec M(α) (α) = spec W itt(A) (α).) Proof. Follows immediately from previous remarks and Proposition 6.5. So we see that it would be useful to have some conditions that ensure the simplicity of a pseudomonoid. This is the purpose of the next lemma. Proof. Let I be an ideal subset with 0 ∈ I. Then for any nonzero a ∈ G, we have 0 + a = a ∈ I since I is an ideal subset. Thus I = G. This proves (a). Suppose I contained an invertible element x. Then as x = −x, and −x ∈ G, we have x + (−x) = 0 ∈ I as I is an ideal subset. Thus I = G by (a). So this proves (b). If G is an (abelian) group and I a nonempty ideal subset. Then take a ∈ I. If a = 0 then I = G by (a) and if a is nonzero then a is invertible as G is a group, and so I = G by (b). Thus we conclude G is a simple pseudomonoid. Corollary 6.8. If L is a strongly graded Lie algebra, graded by an abelian group A ⊆ k, then L is simple. In Example 5.7 we saw that the classical Witt algebra, W itt(k[x]) is strongly graded, graded by the pseudomonoid G = {−1, 0, 1, . . . }. If I is a nonempty ideal subset of this pseudomonoid, by adding −1 repeatedly to an element in I if necessary, we see −1 ∈ I. Since −1 is invertible in G, we conclude by Lemma 6.7, that I = G. So G is a simple pseudomonoid and so the classical Witt algebra is a simple Lie algebra. In Example 5.8 we saw that the centerless Virasoro algebra, W itt(k[x, x −1 ]) is strongly graded, graded by the pseudomonoid Z. Since this is a group, it is simple as a pseudomonoid and we have proven: Corollary 6.9. The classical Witt algebra and the centerless Virasoro algebra are simple. Example 6.10. The natural numbers N = {0, 1, 2, . . . } is a monoid which is not simple as a pseudomonoid. In fact if we define I k = {k, k + 1, . . . } for all k ∈ N, then the reader can easily verify that I k is an ideal subset of N. (There is exactly one more nonempty ideal subset not covered by these which we leave the reader to find if they wish.) So from Example 5.9, W itt(A(N)) gives us an example of a generalized Witt algebra which is not simple. Definition 6.11. Two subsets S 1 , S 2 of k are said to be equivalent if there exists nonzero k ∈ k such that S 1 = kS 2 ≡ {kx|x ∈ S 2 }. It is easy to see that this defines an equivalence relation on the subsets of k. We write [[S]] for the equivalence class of the set S under this equivalence relation. For any Lie algebra L, nonzero α ∈ L, and nonzero k ∈ k, it is easy to see that M (α) = M (kα) and spec(kα) = k spec(α). Thus we have [[spec(kα)]] = [[spec(α)]]. It is also easy to see that two strongly graded Lie algebras, graded by equivalent pseudomonoids, are isomorphic as Lie algebras. Given a strongly graded Lie algebra L, graded by the pseudomonoid G, we have L = M (e 0 ) with spec(e 0 ) = G ⊆ k where e 0 is obtained from the definition of a strongly graded Lie algebra. We would like to define [[spec(e 0 )]] as an invariant of L. However, it turns out that this is not apriori, intrinsic enough to be useful, i.e., it is not obvious that we might not find another nonzero element f such that L = M (f ) and [[spec(f )]] = [[spec(e 0 )]]. In the next section, we show that this in fact cannot occur, and hence define an invariant which helps us find infinite families of nonisomorphic generalized Witt algebras! 7 Invariance of the spectrum Before we proceed any further, we need to develop a somewhat technical tool. We need to weakly order any field (of characteristic zero). We define this notion now. Definition 7.1. A weak order on k is a linear order on k such that if x y then x + z y + z for all z ∈ k. (Recall a linear order is a partial order with the property that for any two elements e, f either e f or f e (or both).) Note the field of real numbers R has a weak order (the usual one) and so any subfield of R has a weak order. A weak order on an abelian group is defined in exactly the same way. As is common, we will write x ≺ y if x y and x = y. There is also a stronger notion of ordered field in the literature (see [9]). However for example C, the field of complex numbers, cannot be made into an ordered field. However, we show in the next proposition, that any field (of characteristic zero) has a weak order. Proof. We identify the characteristic subfield of k with the rational numbers Q as is usual. Then of course, k is a vector space over Q. Define the set S as follows: S = {(A, )|A is a Q-subspace of k and is a weak order on A.}. We make S into a partially ordered set (S, ≤) as follows: (A 1 , 1 ) ≤ (A 2 , 2 ) ⇐⇒ A 1 ⊆ A 2 and 2 | A1 = 1 . The characteristic subfield Q of k can be viewed as the characteristic subfield of the real numbers and so we can put the standard order on it. Thus S is not empty. It is easy to verify that any chain {(A i , i ) i∈I } in (S, ≤) has an upper bound (∪ i∈I A i , ) and thus Zorn's lemma gives us a maximal element (M, ) of (S, ≤). Suppose M = k, then we can find a ∈ k \ M and thus M ′ = M ⊕ Qa is a Q-subspace of k. We define ′ on M ′ as follows: m 1 + q 1 a ≺ ′ m 2 + q 2 a ⇐⇒ m 1 ≺ m 2 or m 1 = m 2 and q 1 < q 2 . It is easy to verify that ′ is a weak order on M ′ which restricts to on M . Thus (M, ) < (M ′ , ′ ) which is a contradiction as (M, ) is maximal. Thus we conclude M = k and hence that we can weakly order k. We now use Proposition 7.2 to weakly order any pseudomonoid. We say that x ∈ (G, ) is positive if 0 ≺ x and we say x is negative if x ≺ 0. If we set P to be the set of positive elements in (G, ) and N to be the set of negative elements in (G, ), then it is easy to see that {P, N, {0}} is a partition of G. A maximum element M of (G, ) is an element such that x M for all x ∈ G. Similarly a minimum element m of (G, ) is an element such that m x for all x ∈ G. Notice that if there is a maximum element, it is unique as is a linear order and similarly for a minimum element. An extreme element of (G, ) is either a maximum or a minimum element. We collect in the next lemma some basic but useful facts about ordered psuedomonoids. Remark 7.6. From Proposition 3.11, we have a complete list of finite dimensional, self-centralizing Lie algebras (in the case that k is algebraically closed). The reader can easily verify, that each of these is strongly graded, graded by a finite pseudomonoid of size one, two or three. Definition 7.7. Let L = ⊕ g∈G E g be a strongly graded Lie algebra and suppose we have a weak order on the pseudomonoid G. Then if α ∈ L is nonzero we can uniquely write α = n i=1 e gi where g 1 ≺ g 2 ≺ · · · ≺ g n ∈ G and e gi ∈ E gi is nonzero for all 1 ≤ i ≤ n. We call g 1 ∈ (G, ) the initial index of α and write g 1 = Init(α). We call g n ∈ (G, ) the terminal index of α and write g n = Term(α). Proof. L = ⊕ g∈G E g so we can take a basis {e g } g∈G of L with e g ∈ E g for all g ∈ G. First we expand x in the basis {e g } g∈G . Thus x = n i=1 x gi e gi with g 1 ≺ g 2 ≺ · · · ≺ g n and x gi = 0 for all 1 ≤ i ≤ n. Thus Init(x) = g 1 and Term(x) = g n . We can expand y in a similar manner. Hence, if g n = h m then 0 = [e gn , e hm ] ∈ E gn+hm and g n + h m is easily seen to be the terminal index of [x, y], and similarly, if g 1 = h 1 then g 1 + h 1 is the initial index of [x, y]. Thus Init(α) = 0 which contradicts our hypothesis. Thus we conclude every eigenvector of α must have the same initial index as α. The proof of (b) is similar and is left to the reader. For (c), note that if both Init(α) and Term(α) are zero, then α is a nonzero scalar multiple of e 0 and the result is clear. So we can assume one of Init(α) or Term(α) is nonzero. For concreteness, let us assume Init(α) = 0, the proof for the case where Term(α) = 0 being similar and left to the reader. Then if dim(E a (α)) ≥ 2 for some a ∈ k. We can find linearly independent x, y ∈ E a (α). By (a), we have Init(x) = Init(y) = Init(α). Then it is clear we can form a nonzero linear combination of x and y whose Init(α)-component is zero. Call this element z then this means that Init(z) is not Init(α). This is a contradiction as z is nonzero and in E a (α) and so, by (a) again, must have Init(z) = Init(α). We are now ready to prove an important proposition. This proposition will enable us to define the spectrum of a strongly graded Lie algebra and use it as a tool to distinguish between two such Lie algebras. Proposition 7.10. Let L be an infinite dimensional, strongly graded Lie algebra, graded by a pseudomonoid G. Choose a weak order on G and let {e g } g∈G be the usual basis of L. Suppose we have nonzero α ∈ L such that M (α) = L, then: (a) If (G, ) has no nonzero extreme elements, α = ke 0 for some nonzero k ∈ k. Thus spec(α) = k spec(e 0 ) and Proof. Assume the setup as in the statement of the proposition. First note that if Init(α) = 0 then Corollary 7.9 shows that all the eigenvectors of α have initial index equal to Init(α). However these eigenvectors span L as M (α) = L and so it follows easily that Init(α) is a nonzero minimal element of (G, ). Similarly if Term(α) = 0 then Term(α) is a nonzero maximal element of (G, ). For (a), note that our previous arguments show that if (G, ) has no nonzero extreme elements, that Init(α) = 0 = Term(α) and hence that α = ke 0 for some nonzero k ∈ k from which the rest of the conclusion in (a), is obvious. For (b), note that we can assume that at least one of Term(α), Init(α) is a nonzero extreme element of (G, ) or else the conclusion would follow from our argument for (a). Since L is infinite dimensional, G is infinite and hence (G, ) can possess at most one extreme element by Lemma 7.5, part (d). Thus for (b), we can assume (G, ) has exactly one extreme element m and that it is a minimum. (If it was a maximum, reorder G by setting x ≺ ′ y ⇐⇒ y ≺ x. This reordering switches Init(α) and Term(α) but does not change the conclusions of this proposition.) Thus we have that without loss of generality, Init(α) = m ≺ 0 is the minimum of (G, ) and that Term(α) = 0 (Recall if Term(α) = 0, we showed before that it would be a nonzero maximum which is a contradiction to our assumption). Thus we have α = k ′ e m + T + ke 0 where k, k ′ ∈ k are nonzero and T consists of terms which have components corresponding to elements in g ∈ G which have m ≺ g ≺ 0. By Lemma 7.5, part (a), there are no such elements g, and so we conclude that α = k ′ e m + ke 0 . It remains to show that [[spec(α)]] = [[G]]. Since [[spec(α)]] does not change if we scale α, we will assume from now on that k = 1. So α = k ′ e m + e 0 . Suppose x is an eigenvector of ad(α) corresponding to eigenvalue µ ∈ k with 0 ≺ Term(x). Then x = ae Term(x) + D where D has nonzero components only in indices g ∈ G with g ≺ Term(x), and a ∈ k is nonzero. Then [α, x] = [k ′ e m + e 0 , ae Term(x) + D] = a Term(x)e Term(x) + D ′ . where D ′ has nonzero components only in indices g ∈ G with g ≺ Term(x). However [α, x] = µx and so we have a Term(x)e Term(x) + D ′ = µae Term(x) + µD from which it follows that µ = Term(x) ∈ G. Now if x is an eigenvector of ad(α) with Term(x) 0 then x = ce m +de 0 and it is easy to check that x must be a scalar multiple of e m or of α corresponding to the eigenvalues m and 0 respectively. In any case, we have [α, x] = Term(x)x. Thus we see that if x is any eigenvector of ad(α), then x corresponds to the eigenvalue Term(x) ∈ G. So spec(α) ⊆ spec(e 0 ) = G. Furthermore, m, 0 ∈ spec(α), with m a minimum element of spec(α) under the ordering inherited from G. However, we also see that if x, y are eigenvectors of ad(α) corresponding to different eigenvalues, then Term(x) = Term(y) and we must have [x, y] = 0 by Lemma 7.8. Since M (α) = L and dim(E a (α)) ≤ 1 for all a ∈ k by Corollary 7.9, we conclude that L is strongly graded with respect to the eigenspaces of α. Thus reversing the roles of α and e 0 in the part of the proof where we showed spec(α) ⊆ spec(e 0 ), and noting that e 0 = α − k ′ e m , we conclude that spec(e 0 ) ⊆ spec(α) and hence that spec(e 0 ) = spec(α) and thus we are done. Definition 7.11. Let L be a strongly graded Lie algebra. We define spec(L) = [[spec(α)]] where α is a nonzero element in L with M (α) = L. Note that spec(L) is well-defined if L is infinite dimensional, by Proposition 7.10. If L is finite dimensional, then Corollary 7.9, part (c), shows that dim(E a (α)) ≤ 1 for all a ∈ k and so we must have the order of spec(α) is equal to the dimension of L for any nonzero α with M (α) = L. Since there is exactly one pseudomonoid of order spec(α) up to equivalence by Lemma 7.5, spec(L) is well-defined in this case also. We now show that spec(L) is truly an invariant of L. Proposition 7.12. Let L, L ′ be two Lie algebras and f : L → L ′ be a Lie algebra homomorphism. Then: (a) For every α ∈ L and a ∈ k, we have f (E a (α)) ⊆ E a (f (α) ). Hence f (M (α)) ⊆ M (f (α)). (b) If f is injective, then spec(α) ⊆ spec(f (α)). (c) If f is bijective, then spec(α) = spec(f (α)) and furthermore f (M (α)) = M (f (α)). ∈ E a (f (α)). Also M (α) = ⊕ a∈k E a (α) and so f (M (α)) = ⊕ a∈k f (E a (α)) ⊆ ⊕ a∈k E a (f (α)) = M (f (α)). This gives us (a). For (b), notice that if f is injective, and we had nonzero x ∈ E a (α), then f (x) would be nonzero, and by (a), it would lie in E a (f (α)). This proves (b). For (c), notice that since f is bijective, f −1 exists and is in fact a Lie algebra homomorphism. Thus from (a) and (b) applied to (f, α) and (f −1 , f (α)) we get f (M (α)) ⊆ M (f (α)) and f −1 (M (f (α))) ⊆ M (f −1 (f (α))) giving us f (M (α)) = M (f (α)). We also get spec(α) ⊆ spec(f (α)) and spec(f (α)) ⊆ spec(f −1 (f (α))) giving us spec(α) = spec(f (α)). For ( f : G → G ′ such that (a) f (0) = 0 and (b) f (x + y) = f (x) + f (y) for all distinct x, y ∈ G. It is easy to see that if [[G]] = [[G ′ ]], then G is isomorphic to G ′ . Example 7.14. The field k is a vector space over its characteristic subfield Q. If dim Q (k) = ∞ then we can find Q-vector subspaces V n of k of dimension n for every n ∈ N. Certainly the {V n } n∈N are a family of nonisomorphic pseudomonoids which are simple pseudomonoids by Lemma 6.7 as they are abelian groups. Thus the construction of Example 5.9 gives us a family W itt(A(V n )) of simple, strongly graded Lie algebras by Corollary 6.6. Furthermore since spec(W itt(A(V n ))) = [[V n ]] we see that In contrast, over an algebraically closed field, the only finite dimensional Lie algebra which is indecomposable, semisimple and has no abelian Lie subalgebras of dimension greater than one is sl 2 . Thus the following is a list of nonisomorphic generalized Witt algebras: the classical Witt algebra, the centerless Virasoro algebra, W itt(A(M m,n )) for relatively prime m, n > 1 and W itt(A(V n )) for Q-vector subspaces V n of k, where dim Q (V n ) = n for all n ∈ N. Thus, we hope we have conveyed the rich variety of generalized Witt algebras available! In the final section, we verify the Jacobian conjecture for a class of generalized Witt algebras. That is, we show that under suitable hypothesis, any nonzero Lie algebra endomorphism of a generalized Witt algebra is actually an automorphism. The Jacobian conjecture A polynomial map f : C n → C n is a map with the property that each of its components is a complex polynomial in n-variables. Such a map is called invertible if it is bijective, and if its inverse is a polynomial map also. It is easily seen that an invertible polynomial map has the property that the determinant of its Jacobian matrix is a nonzero constant as a function on C n . (See [2]). The classical Jacobian conjecture is that the converse is true and remains open for all n ≥ 2. One can ask the following question about the classical Weyl algebra in nvariables. (Defined similarly as we did in the beginning of the paper but using n-variables instead of one.) Is every nonzero algebra endomorphism of a classical Weyl algebra actually an automorphism? The answer to this question is unknown for all n ≥ 1. If the statement is true for some n, then it implies the classical Jacobian conjecture in dimension n. (See [2]). One can generalize to: Definition 8.1. Given a Lie algebra L, one says that the Jacobian conjecture holds for L, if every nonzero Lie algebra endomorphism is actually an automorphism. Certainly the Jacobian conjecture does not hold for all Lie algebras but does hold for finite dimensional, simple Lie algebras. We will show, among other things that the Jacobian conjecture holds for the classical Witt algebra which is the Lie algebra of derivations of the classical Weyl algebra where the corresponding conjecture remains open. One can see immediately, the spectral theory machinery developed earlier has a lot to say about this. For example one has: Corollary 8.2. If W itt(A) is a generalized Lie algebra and f ∂ is a nonzero element such that spec(f ∂) = {0}. Then for every injective Lie algebra endomorphism F of W itt(A), one has F (f ∂) = g∂ with W (g) ≤ 1. Proof. This follows immediately from Theorem 5.11 and Proposition 7.12. Corollary 8.2 shows that the image of an element under an injective endomorphism, is reasonably constrained by its spectrum. Of course, Corollary 8.2 is a rough application of these ideas and we will have to refine them a bit to get our desired result. To this end, we define: Definition 8.3. A pseudomonoid G ⊆ k is called self-containing if there is nonzero a ∈ k such that aG ⊂ G and aG = G. Notice in this case that aG is a subpseudomonoid of G which is equivalent to G so we could also define a pseudomonoid to be self-containing if it possesses a proper subpseudomonoid equivalent to itself. The integers Z = {. . . , −1, 0, 1, . . . } is an example of a self-containing pseudomonoid since nZ is a proper subpseudomonoid equivalent to Z for all natural numbers n ≥ 2. The reader can verify that this is in fact a complete list of all such proper subpseudomonoids. We next give examples of pseudomonoids which are not self-containing. Proof. Suppose aE ⊆ E for some nonzero a ∈ k. Since 1 ∈ E, it follows that a ∈ E. Then given x ∈ E, xa −1 ∈ E and x = a(xa −1 ). Thus aE = E. So E is not self-containing. Give G = {−1, 0, 1, . . . } the weak order inherited by viewing it as the usual subset of the real numbers. If aG ⊆ G for some nonzero a ∈ k, it again follows as 1 ∈ G, that a ∈ G. Clearly a = −1 so a > 0. Then we must have a(−1) = −a ∈ G and hence −a = −1 and a = 1. Thus aG = G and so G is not a self-containing pseudomonoid either. Definition 8.5. Let L be a strongly graded Lie algebra, graded by G. Then we can write L = ⊕ g∈G E g as usual. For nonzero x ∈ L, we let x g be the g-component of x. We define the support of x as Supp(x) = {g ∈ G|x g = 0}. We also define Supp(0) = ∅. Definition 8.6. A weak order on a pseudomonoid G is called discrete if for every a, b ∈ G, the order of the set {g ∈ G|a g b} is finite. A pseudomonoid which possesses a discrete order is called discrete. Every subpseudomonoid of the integers is discrete by restricting the standard weak order. We are now ready to prove: Theorem 8.7. Let L be an infinite dimensional, strongly graded Lie algebra, graded by a pseudomonoid G. Suppose G possesses a discrete order . Write L = ⊕ g∈G E g as usual and let {e g } g∈G be a basis of L with the usual properties. Let Θ be the correspondence map of Proposition 6.5. Then for every injective Lie algebra endomorphism f of L, we have one of the following two possibilities: (a) f (e 0 ) = 1 a e 0 for some nonzero a ∈ k such that aG ⊆ G. In this case f (L) = Θ(aG). Hence if G is not self-containing, then f is onto. (b) f (e 0 ) = 1 a e 0 + D for some nonzero a ∈ k such that aG ⊆ G and Supp(D) consists of elements ≺ ′ 0. (Here ′ is either equal to , or is reversed.) Furthermore there is ′ -minimal I ∈ Supp(D) such that I ′ ag for all g ∈ G. In the situation of (b), if G is not self-containing, then I is actually a minimum element of (G, ′ ), and f (e 0 ) = 1 a e 0 + k ′ e I . Furthermore f is onto. Proof. Let f : L → L be an injective endomorphism of Lie algebras. Then f (L) is an infinite dimensional Lie subalgebra of L. Write L = ⊕ g∈G E g as in the statement of the theorem and let be a discrete order on G. Let us assume both I and T are nonzero to derive a contradiction. Let S = {g ∈ G|I g T }. Since (G, ) is discrete, S is finite. Since I, T are nonzero, we have seen that every eigenvector of ad(f (e 0 )) will lie in Θ(S), and hence f (L) ⊆ Θ(S) which is a contradiction as f (L) is infinite dimensional. So at least one of I, T is zero. By reordering G if necessary, we can assume T = 0. (Notice, if you reverse a discrete order by setting x ≺ ′ y ⇐⇒ y ≺ x, you get a discrete order where T and I interchange. Also notice that this reordering will not affect the conclusion of the theorem.) Now if I = 0 also then f (e 0 ) = ke 0 for nonzero k ∈ k. Now by Proposition 7.12, G = spec(e 0 ) ⊆ spec(f (e 0 )) = spec(ke 0 ) = kG. Thus 1 k G ⊆ G. Then notice that f (E b (e 0 )) ⊆ E b (ke 0 ) = E b k (e 0 ) for all b ∈ G by Proposition 7.12. Since E b k is one dimensional, we conclude that f (E b ) = E b k for all b ∈ G and hence that f (L) = f (⊕ g∈G E g ) = ⊕ g∈G E g k = Θ( 1 k G). So in this case, we get the situation described in (a) of the theorem if we set a = 1 k . So we may now assume I = 0, and hence that I ≺ 0. Thus f (e 0 ) = ke 0 + D where every element of Supp(D) is negative with minimum element I. Now if x is an eigenvector of ad(f (e 0 )) corresponding to µ ∈ spec(f (e 0 )), we may write: x = n i=1 x gi where g 1 ≺ · · · ≺ g n ∈ G and x gi ∈ E gi is nonzero for all 1 ≤ i ≤ n. Then a simple calculation shows that [f (e 0 ), x] = kg n x gn + D ′ where Supp(D ′ ) ⊆ {g ∈ G|g ≺ g n }. Since this must equal µx, we conclude that kg n = µ or in other words k Term(x) = µ. Thus we conclude that spec(f (e 0 )) ⊆ k spec(e 0 ). However, by Proposition 7.12, it follows that spec(e 0 ) ⊆ spec(f (e 0 )). Thus G = spec(e 0 ) ⊆ spec(f (e 0 )) ⊆ k spec(e 0 ). Hence 1 k G ⊆ G in this case also. Now since I = 0, every eigenvector x corresponding to µ of f (e 0 ) has Init(x) = I. Thus I = Init(x) Term(x) = µ/k and we conclude that I g k for all g ∈ G since G ⊆ spec(f (e 0 )). Now if G is not self-containing, we must have 1 k G = G and hence I is a mimimum element of G. Since I ≺ 0, it is the unique such element. Thus since we had f (e 0 ) = ke 0 + D where Supp(D) ⊆ {g ∈ G|g ≺ 0}, we conclude that f (e 0 ) = ke 0 + k ′ e I . Now kI ∈ G as 1 k G = G. Then by Proposition 7.12, we have 0 = f (e kI ) ∈ E kI (f (e 0 )). By our previous analysis, k Term(f (e kI )) = kI and so Term(f (e kI )) = I. Since I is a minimum of (G, ), we conclude f (e kI ) is a nonzero multiple of e I . Hence e I ∈ f (L). Since f (e 0 ) = ke 0 + k ′ e I in f (L), we conclude that f (L) contains e 0 . Now by Proposition 6.5, it follows that f (L) = Θ(S) where S consists of the union of the supports of the elements in f (L). However for every g ∈ G, kg ∈ G and Term(f (e kg )) = g by an analysis similar to the one done previously. Hence S = G and f is onto. Thus we are done. Corollary 8.8. Let L be a strongly graded Lie algebra, graded by a discrete pseudomonoid which is not self-containing. Then every injective Lie algebra endomorphism of L is an automorphism. If f is any nonzero Lie algebra endomorphism of the classical Witt algebra, then f is an automorphism, and furthermore f (x∂) = (x + b)∂ for some b ∈ k. Thus the Jacobian conjecture holds for the classical Witt algebra. Proof. The first part follows immediately from Theorem 8.7. By Example 5.7, the classical Witt algebra is a strongly graded Lie algebra graded by the pseudomonoid G = {−1, 0, 1, . . . } which is obviously discrete and is not self-containing by Lemma 8.4. We have already seen that this Lie algebra is simple, hence any nonzero Lie algebra endomorphism f is injective and hence an automorphism by Theorem 8.7. Furthermore, in the strong grading of the classical Witt algebra, we can take x∂ = e 0 and x n ∂ ∈ E n−1 for all n ∈ N. Notice further that if aG ⊆ G, in fact a = 1 as we saw in the proof of Lemma 8.4. Thus applying Theorem 8.7 again and noting that we must have I = −1 if we are in situation (b), we conclude furthermore that f (x∂) = (x + b)∂ for some b ∈ k. Corollary 8.9. If f is a nonzero Lie algebra endomorphism of the centerless Virasoro algebra then f is injective and f (x∂) = 1 a x∂ for some nonzero integer a. However, the Jacobian conjecture is false for this Lie algebra. Thus there exist injective Lie algebra endomorphisms of the centerless Virasoro algebra which are not automorphisms. Proof. By Example 5.8, the centerless Virasoro algebra is strongly graded by the pseudomonoid G = Z = {. . . , −1, 0, 1, . . . }, with basis e n = x n+1 ∂ ∈ E n for all n ∈ Z. G is obviously discrete. Let f be a nonzero Lie algebra endomorphism. Since the centerless Virasoro algebra is simple, f is injective. It is easy to see that aZ ⊆ Z if and only if a is an integer. Also if we use the standard order of Z, then there is no I as in situation (b) of Theorem 8.7, and so we immediately conclude from the same theorem that: f (x∂) = 1 a x∂ for some nonzero integer a and Image(f ) = Θ(aZ). We will now construct such a Lie algebra endomorphism for every nonzero intger a. Thus for a = ±1, we obtain injective Lie algebra endomorphisms which are not onto. Define f a (e n ) = a −(n+1) e an for all n ∈ Z. Certainly this defines a vector space endomorphism which is not onto if a = ±1. We calculate Hence f is a homomorphism of Lie algebras and we are done. This concludes our initial study of generalized Witt algebras. One sees that for this family of self-centralizing Lie algebras, spectral analysis provides a powerful tool to answer basic questions locally. (On M (α) for nonzero α ∈ L.) We found this extremely useful in the case where L = M (α) for some nonzero α, but it should be possible to push these results to the more general case by patching together the local spectra to get some sort of global scheme. There exist infinite families of nonisomorphic, simple, generalized Witt algebras and there exist infinite familes of nonisomorphic, nonsimple, generalized Witt algebras. Remark 2 . 2 . 22Three important examples of stable algebras are the polynomial algebra k[x], the power series algebra k[[x]], and the Laurent polynomial algebra k[x, x −1 ]. (Recall a Laurent polynomial is an element of the form M i=N α i x i for suitable N, M ∈ Z and α i ∈ k.) Proposition 3 . 4 . 34Given a Lie algebra L, the following conditions are equivalent. (a) For any nonzero l ∈ L, [l, x] = 0 implies x = βl for some β ∈ k. (b) C(l) is one dimensional for all nonzero l ∈ L. (c) L does not contain any abelian Lie algebras of dimension greater than one. (d) If α, β ∈ L are linearly independent, then [α, β] = 0. Definition 4. 1 . 1Let R ♯ denote the group of nonzero elements in the field R under multiplication. (Recall R is the field of fractions of k[[x]].) The logarithmic derivative LD : R ♯ → R is defined by Definition 4 . 4 . 44Let U = {f ∈ R|W (f ) = 0}. Then f ∈ U if and only if f ∈ k[[x]] and f (0) = 0 and this happens if and only if f is a unit of k[[x]]. Thus U is the group of units of k[[x]] under multiplication. Proposition 5 . 12 . 512Let W itt(A) be a generalized Witt algebra and let f ∂ be a nonzero element of W itt(A). Then there exists a basis {e a } a∈spec(f ∂) of M (f ∂) such that [e a , e b ] = (b − a)e a+b for all a, b ∈ spec(f ∂). (Here (b − a)e a+b is considered to be zero for a = b even though a + b might not be in spec(f ∂).) Furthermore, we can take e a ∈ E a (f ∂) for all a ∈ spec(f ∂) and e 0 = f ∂. Proof. By Theorem 5.11, if W (f ) > 1, then spec(f ∂) = {0} and the result is obvious. If W (f ) ≤ 0, then set e a = f e a f ∂ for all a ∈ spec(f ∂). Then by Theorem 5.11, {e a } a∈spec f ∂ is a basis for M (f ∂). One computes using [g∂, h∂] = (gh ′ − hg ′ )∂, that indeed [e a , e b ] = (b − a)e a+b . . A Lie algebra L is said to be strongly graded if there exists a pseudomonoid G and a vector space decomposition:L = ⊕ a∈G E awith the following properties: (a) dim(E a ) = 1 for all a ∈ G. (b) There is a basis {e a } a∈G of L such that e a ∈ E a for all a ∈ G and [e a , e b ] = (b − a)e a+b . (Because [E a , E a ] = 0 for all a ∈ G.) Furthermore, if 0 ∈ S, then e 0 ∈ Θ(S). with y ai ∈ E ai nonzero and {a i } n i=1 a set of distinct elements in I. Also without loss of generality, a 1 = 0 Lemma 6 . 7 . 67Let G be a pseudomonoid. Then: (a) If I is an ideal subset, and 0 ∈ I then I = G. (b) If I is an ideal subset, and there is an invertible element x ∈ I then I = G. (c) A pseudomonoid which is a group is a simple pseudomonoid. Proposition 7 . 2 . 72Any field k (of characteristic zero) possesses a weak order. Definition 7. 3 . 3Let G ⊆ k be a psuedomonoid. A weak order on G is the restriction of some weak order on k.Proposition 7.2 shows all pseudomonoids possess a weak order. (Since we require our pseudomonoids to be in k, by definition.) Definition 7.4. Let G be a pseudomonoid with weak order . Lemma 7 . 5 . 75Let (G, ) be a pseudomonoid with a weak order. Then: (a) If G possesses a minimum element m then either m = 0 or m is the unique negative element in (G, ). (b) If G possesses a maximum element M then either M = 0 or M is the unique positive element in (G, ).(c) If G possesses a minimum and a maximum element then the order of G is less than or equal to 3. (d) If the order of G is infinite, then G possesses at most one extreme element. (e) If G is a finite pseudomonoid, then the order of G is either one, two or three. Furthermore, for each of these orders, there is a unique pseudomonoid up to equivalence.Proof. For (a), let m be a minimum element and assume m is not zero. Then we must have m ≺ 0 as m is a minimum element.Suppose there were x ≺ 0 with x = m, then x + m ≺ 0 + m = m with x + m ∈ G as G is a pseudomonoid. This contradicts the minimality of m and thus we conclude there is no such x, i.e., m is the unique negative element.The proof of (b) is similar to (a) and is left to the reader. For (c), note that if G has a minimum element m and a maximum element M then it follows that the set of nonpositive elements is{0, m} by (a) and the set of nonnegative elements is {0, M } by (b). Thus G = {0, m, M } and hence G has order less than or equal to 3. (Exact order depends on whether or not the elements {0, m, M } are distinct or not.) (d) follows immediately from (c). The first part of (e) also follows immediately from (c) since any weak order on a finite pseudomonoid has a maximum and a minimum element. Note that if the order of G is three and G = {0, m, M }, then we must have M = −m since M +m ∈ G. Thus it is easy to see that [[G]] = [[{−1, 0, 1}]]. If G has order two, obviously [[G]] = [[{0, 1}]] and if G has order one, then G = {0}. So we are done. Lemma 7. 8 . 8Let L be a strongly graded Lie algebra, graded by a weakly ordered pseudomonoid (G, ). Then if x, y are nonzero elements of L, we have: (a) If Term(x) = Term(y) then [x, y] = 0 and Term([x, y]) = Term(x) + Term(y).(b) If Init(x) = Init(y) then [x, y] = 0 and Init([x, y]) = Init(x) + Init(y). yx hj e hj with h 1 ≺ · · · ≺ h m and y hj = 0 all 1 ≤ j ≤ m. Thus Init(y) = h 1 and Term(y) = h m . gi y hj [e gi , e hj ]. Corollary 7 . 9 . 79Let L be as inLemma 7.8. Suppose α ∈ L is nonzero. Then: (a) If Init(α) = 0 then every eigenvector x of ad(α) has Init(x) = Init(α).(b) If Term(α) = 0 then every eigenvector x of ad(α) has Term(x) = Term(α). (c) dim(E a (α)) ≤ 1 for all a ∈ k.Proof. For (a), let α have Init(α) = 0 and assume x is an eigenvector of ad(α) with Init(x) = Init(α). Then by Lemma 7.8 we have [α, x] is nonzero andInit([α, x]) = Init(x) + Init(α).However, as x is an eigenvector, we also have [α, x] = µx for some µ ∈ k. Since [α, x] = 0 we conclude µ = 0 and hence that Init(α) + Init(x) = Init([α, x]) = Init(µx) = Init(x). [ [spec(α)]] = [[spec(e 0 )]] = [[G]]. (b) If (G, ) has a nonzero extreme element m, then m is unique and α = ke 0 + k ′ e m for some k, k ′ ∈ k with k = 0. Furthermore we still have [[spec(α)]] = [[spec(e 0 )]] = [[G]]. ( d ) dIf L, L ′ are two strongly graded Lie algebras, and f is an isomorphism, then spec(L) = spec(L ′ ). Proof. For (a), notice that if x ∈ E a (α), then [α, x] = ax and hence f ([α, x]) = af (x). Since f is a Lie algebra homomorphism, we have f ([α, x]) = [f (α), f (x)] and so we conclude [f (α), f (x)] = af (x) and thus f (x) d), note that spec(L) = [[spec(α)]] for some nonzero α ∈ L with M (α) = L. Since f is an isomorphism, we have f (α) is nonzero with M (f (α)) = f (M (α)) = f (L) = L ′ . Hence by Proposition 7.10, we have spec(L ′ ) = [[spec(f (α))]] = [[spec(α)]] = spec(L). Thus we are done. Definition 7.13. Two pseudomonoids G and G ′ are isomorphic if there is a bijection {W itt(A(V n ))} n∈N is an infinite family of nonisomorphic, simple, generalized Witt algebras.Example 7.15. Let N be the monoid of natural numbers. For every pair of relatively prime integers n, m > 1, we define M n,m to be the submonoid of N generated by n and m. It is easy to see that M n,m is never simple as a pseudomonoid as one can find nontrivial restrictions of ideal subsets from N.(See Example 6.10.) Furthermore M n,m is isomorphic to M n ′ ,m ′ if and only if {n, m} = {n ′ , m ′ }. Thus again using the construction of Example 5.9, we get an infinite family W itt(A(M n,m )) 1<n<m,gcd(n,m)=1 of nonisomorphic, nonsimple, generalized Witt algebras. By Proposition 3.11, all of these Lie algebras are semisimple and indecomposable and have no abelian Lie subalgebras of dimension greater than one. W itt(k[x])) = [[{−1, 0, 1, . . . }]] and spec(W itt(k[x, x −1 ])) = Z by examples 5.7 and 5.8. These spectra are easily seen not to be isomorphic to those discussed in examples 7.14 and 7.15, and not isomorphic to each other of course. Lemma 8. 4 . 4Any subfield E of k is not a self-containing pseudomonoid. {−1, 0, 1, . . . } ⊆ k is not a self-containing psuedomonoid. Now f (L) = f (M (e 0 )) ⊆ M (f (e 0 )) and spec(e 0 ) ⊆ spec(f (e 0 )) by Proposition 7.12. Thus f (e 0 ) ∈ f (L) is ad-diagonalizable on f (L). (In other words, there is a basis for f (L) consisting of eigenvectors of ad(f (e 0 )).) Using the chosen order on G, we can speak of I = Init(f (e 0 )) and T = Term(f (e 0 )) which both lie in G. By Corollary 7.9, we conclude that if I = 0 then every eigenvector x of ad(f (e 0 )) has Init(x) = I. Similarly, if T = 0, then every eigenvector x of ad(f (e 0 )) has Term(x) = T . [ f a (e n ), f a (e m )] = a −(n+m+2) [e an , e am ] = (am − an)a −(n+m+2) e a(n+m) = (m − n)a −(n+m+1) e a(n+m) = f a ((m − n)e n+m ) = f a ([e n , e m ]). Infinite-dimensional Lie Algebras. R Amayo, I Stewart, Noordhoff Int. Publishing115Amayo, R., Stewart, I.: Infinite-dimensional Lie Algebras, 115-120, No- ordhoff Int. Publishing (1974). A Primer of Algebraic D-modules. S Coutinho, London Math. Soc. Student Texts. 33Cambridge Univ. PressCoutinho, S.: A Primer of Algebraic D-modules, London Math. Soc. Student Texts 33, Cambridge Univ. Press, 1995. Derivations, isomorphisms and second cohomology of a generalized Witt algebra. D Doković, K Zhao, Trans. A.M.S. 350Doković, D., Zhao, K.: Derivations, isomorphisms and second cohomol- ogy of a generalized Witt algebra, Trans. A.M.S. 350, 2-7 (1998). Introduction to Lie Algebras and Representation Theory. J Humphreys, Springer-Verlag9Humphreys, J.: Introduction to Lie Algebras and Representation The- ory, Springer-Verlag, G.T.M. 9, 1-21, (1987). Lie Algebras. N Jacobson, Dover PublicationsJacobson, N.: Lie Algebras, Dover Publications, 1979. Description of Filtered Lie Algebra with which Graded Lie algebras of Cartan type are Associated. V Kac, Izv. Akad. Nauk SSSR, Ser. Mat. Tom. 38Kac, V.: Description of Filtered Lie Algebra with which Graded Lie algebras of Cartan type are Associated, Izv. Akad. Nauk SSSR, Ser. Mat. Tom 38, 832-834 (1974). The Virasoro algebra. I Kaplansky, Comm. in Mathematical Physics. 86Kaplansky, I.: The Virasoro algebra, Comm. in Mathematical Physics, 86, 49-52 (1982). Generalizations of Witt algebras over a field of characteristic zero. N Kawamoto, Hiroshima Math. J. 16Kawamoto, N.: Generalizations of Witt algebras over a field of charac- teristic zero, Hiroshima Math. J.,16, 417-426 (1986). Algebra. S Lang, Addison-Wesley Pub. Co3rd ed.Lang, S.: Algebra, 3rd ed., Addison-Wesley Pub. Co., 1993. Generalized W and H type Lie algebras, Algebra Colloquium. K Nam, Springer Verlag6Nam, K.: Generalized W and H type Lie algebras, Algebra Colloquium, Springer Verlag, 6:3, 329-340 (1999). A Course in the Theory of Groups. D Robinson, Springer-Verlag80Robinson, D.: A Course in the Theory of Groups, Springer-Verlag, G.T.M. 80, 95-98 (1996). Groups of Automorphisms of Infinite-Dimensional Simple Lie Algebras. A Rudakov, Math. USSR-Izvestija. 3Rudakov, A.: Groups of Automorphisms of Infinite-Dimensional Simple Lie Algebras, Math. USSR-Izvestija, 3, 836-837 (1969).
[]
[ "A MCKAY CORRESPONDENCE FOR THE POINCARÉ SERIES OF SOME FINITE SUBGROUPS OF SL 3 (C) Dedicated to the memory of Egbert Brieskorn with great admiration", "A MCKAY CORRESPONDENCE FOR THE POINCARÉ SERIES OF SOME FINITE SUBGROUPS OF SL 3 (C) Dedicated to the memory of Egbert Brieskorn with great admiration" ]
[ "Wolfgang Ebeling " ]
[]
[]
A finite subgroup of SL 2 (C) defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity.Here we consider some non-abelian finite subgroups G of SL 3 (C). They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that we obtain a similar relation between the group G and the corresponding surface singularity.
10.5427/jsing.2018.18t
null
119,145,755
1712.07985
203e21a70b771e8a77b8c76a4213b027858a048d
A MCKAY CORRESPONDENCE FOR THE POINCARÉ SERIES OF SOME FINITE SUBGROUPS OF SL 3 (C) Dedicated to the memory of Egbert Brieskorn with great admiration 5 Jun 2018 Wolfgang Ebeling A MCKAY CORRESPONDENCE FOR THE POINCARÉ SERIES OF SOME FINITE SUBGROUPS OF SL 3 (C) Dedicated to the memory of Egbert Brieskorn with great admiration 5 Jun 2018 A finite subgroup of SL 2 (C) defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity.Here we consider some non-abelian finite subgroups G of SL 3 (C). They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that we obtain a similar relation between the group G and the corresponding surface singularity. Introduction In [E4] we showed that the Poincaré series of the coordinate algebra of a twodimensional quasihomogeneous singularity is the quotient of two polynomials one of which is related to the characteristic polynomial of the monodromy of the singularity. There are two special cases of this result. One is the case of a Kleinian singularity not of type A 2n . The Kleinian singularities are defined by quotients of C 2 by finite subgroups of SL 2 (C). In this case, the relation means that the Poincaré series is the quotient of the characteristic polynomials of the Coxeter element and the affine Coxeter element of the corresponding root system of type ADE. We derived this relation from the McKay correspondence. The other case is the case of a Fuchsian singularity. A Fuchsian singularity is defined by the action of a Fuchsian group (of the first kind) Γ ⊂ PSL(2, R) on the tangent bundle T H of the upper half plane H. For a Fuchsian hypersurface singularity (or more generally for a Fuchsian singularity of genus 0 [EP]), we showed that the Poincaré series is the quotient of two characteristic polynomials of Coxeter elements [E5]. Here we consider a similar relation for the Poincaré series of some non-abelian finite subgroups of SL 3 (C). The non-abelian finite subgroups of SL 3 (C) define nonisolated three-dimensional Gorenstein quotient singularities. We consider those groups where the natural three-dimensional representation is irreducible and the corresponding quotient singularity has a certain hyperplane section which is a Kleinian or Fuchsian singularity. We show, that in this way, we again obtain relations between the Poincaré series of the algebra of invariants of the group and the characteristic polynomials of certain Coxeter elements determined by the corresponding Kleinian or Fuchsian singularity. G |G| x, y, z c G R(x, y, z) Sing. α 1 , . . . , α m C 2n+1 2n + 1 2, 2n + 1, 2n + 1 1 x 2n+1 + y 2 + z 2 A 2n 2n C 2n 2n 2, 2n, 2n 2 x 2n + y 2 + z 2 A 2n−1 2n − 1 D n 4n 4, 2n, 2n + 2 2 x n+1 + xy 2 + z 2 D n+2 2, 2, n T 24 6, 8, 12 2 x 4 + y 3 + z 2 E 6 2, 3, 3 O 48 8, 12, 18 2 x 3 y + y 3 + z 2 E 7 2, 3, 4 I 120 12, 20, 30 2 x 5 + y 3 + z 2 E 8 2, 3, 5 Table 1. Subgroups of SL 2 (C) and surface singularities The famous paper [Br] of E. Brieskorn is fundamental for the study of Kleinian singularities. The Kleinian singularities were a central theme in Brieskorn's research and we owe Brieskorn many beautiful and important results about these singularities. Therefore I would like to express my great admiration for him in dedicating this paper to his memory. 1. Finite subgroups of SL 2 (C) and SL 3 (C) and normal surface singularities Let G be a finite subgroup of SL 2 (C). The classification of finite subgroups of SL 2 (C) up to linear equivalence is well-known, see e.g. [Kl]. There are up to conjugacy five classes of such groups: the cyclic groups C ℓ , the binary dihedral groups D n , the binary tetrahedral group T , the binary octahedral group O, and the binary icosahedral group I. The quotients of C 2 by these groups were studied by E. Brieskorn [Br]. Equations for these singularities can be obtained from generators and relations of the algebra of invariant polynomials with respect to G. This algebra has three generators x, y, z in each case which satisfy an equation R(x, y, z) = 0. The degrees of the generators and the equation R(x, y, z) = 0 are indicated in Table 1. (They can be found, e.g., in [Sp].) The equations define isolated hypersurface singularities in C 3 , the so called Kleinian singularities. The finite subgroups of SL 3 (C) were classified up to linear equivalence by H. F. Blichfeldt, G. A. Miller, and L. E. Dickson [Bl, MBD] with two missing cases (see [YY]). There are 12 types of finite subgroups of SL 3 (C): (A)-(L). There are four infinite series (A)- (D). The groups of type (A) are the diagonal abelian groups and the groups of type (B) are isomorphic to transitive finite subgroups of GL 2 (C). Here the natural 3-dimensional representation is not irreducible. Type (C) is the infinite series ∆(3n 2 ) of groups and type (D) the series ∆(6n 2 ) (for the notation see [HH, LNR, EL]). Moreover, we have 8 exceptional subgroups (E)-(L). We consider those subgroups of type (C)-(L) which admit a certain hyperplane section which defines a Kleinian or Fuchsian singularity. Generators and relations for the algebra of invariant polynomials with respect to G have been computed in [YY] (see also [We] for some cases). They correspond to non-isolated Gorenstein quotient singularities C 3 /G. These singularities are either hypersurface singularities in C 4 or complete intersection singularities in C 5 . We denote the coordinates of these spaces by w, x, y, z and w, x, y, z, u respectively. We consider hyperplane sections of these singularities, namely we consider the restrictions of the equations to the hyperplane w = 0. For the series (C) and (D) the hyperplane sections of the corresponding singularities for the first few elements of these series are listed in Table 2. It turns out that the singularities corresponding to the series (C) G |G| w, x, y, z(, u) c G R(0, x, y, z(, u)) Sing. ∆(3 · 2 2 ) 12 2, 3, 4, 6 1 z 2 + 4y 3 + 27x 4 E 6 ∆(3 · 3 2 ) 27 3, 3, 6, 9 3 z 2 + 4y 3 + 27x 6 E 8 ∆(3 · 4 2 ) 48 4, 3, 8, 12 1 z 2 + 4y 3 + 27x 8 E 14 ∆(3 · 5 2 ) 75 5, 3, 10, 15; 30 1 z 2 + 4y 3 + 27x 10 E 18 ∆(6 · 2 2 ) 24 2, 4, 6, 9 1 z 2 + 4xy 3 + 27x 3 E 7 ∆(6 · 3 2 ) 54 6, 6, 6, 6, 9 3 z 2 − xy u 2 + 4xyz + 27x 3 δ1 ∆(6 · 4 2 ) 96 4, 6, 8, 15 1 z 2 + 4xy 3 + 27x 5 Z 11 ∆(6 · 5 2 ) 150 10, 6, 8, 10, 15 1 z 2 − xy u 2 + 4x 2 yz + 27x 5 no name ∆(6 · 6 2 ) 216 6, 6, 12, 21 3 z 2 + 4xy 3 + 27x 7 Z 1,0 ∆(6 · 7 2 ) 294 14, 6, 10, 14, 21 1 z 2 − xy u 2 + 4x 3 yz + 27x 7 no name ∆(6 · 8 2 ) 384 8, 6, 16, 27; 54 1 z 2 + 4xy 3 + 27x 9 Z 19 Table 2. The first subgroups of types (C) and (D) and surface singularities (∆(3n 2 )) belong to Arnold's E-series whereas those of type (D) (∆(6n 2 )) belong to Arnold's Z-series (n even) or are complete intersection singularities (n odd) (for the definition of these series see [Arn]). The subgroups which correspond to Kleinian singularities are the tetrahedral group T = ∆(3 · 2 2 ) and the octahedral group O = ∆(6 · 2 2 ) which correspond to the Kleinian singularities E 6 and E 7 respectively. Those which correspond to Fuchsian singularities are ∆(3 · 4 2 ) (E 14 ), ∆(6 · 4 2 ) (Z 11 ), ∆(6 · 6 2 ), ∆(6 · 6 2 ) (Z 1,0 ), and ∆(6 · 3 2 ) which corresponds to the elliptic complete intersection singularity δ1 in C. T. C. Wall's notation [Wa2]. (For a list of Fuchsian hypersurface and complete intersection singularities see [E5].) These are 6 cases. The remaining 8 exceptional subgroups of types (E)-(L) all correspond to Fuchsian singularities except in the case (H) which is the icosahedral group I corresponding to the Kleinian singularity E 8 . Altogether we have 14 cases which we will consider in this paper. They are listed in Table 3. These singularities are surface singularities and they are isolated except in the three cases ∆(6 · 3 2 ), (E) and (J). They correspond to Kleinian singularities in the cases T , O and (H) (the icosahedral group I) and to Fuchsian singularities in the other cases. They correspond to simple (T , O, I), unimodal (∆(3·4 2 ), ∆(6·4 2 ), (F), (I), (K), (L)) and bimodal (∆(6 · 6 2 ), (G), (J)) hypersurface singularities, to the unimodal complete intersection singularity of type K ′ 1,0 (type (E)) in Wall's notation [Wa1], and to the elliptic complete intersection singularity δ1. The names of the hypersurface singularities according to V. I. Arnold's classification [Arn] are indicated in the last column of Table 3. Poincaré series and Coxeter elements We now consider the isolated singularities corresponding to these singularities. They are quasihomogeneous. This means the following. A complex polynomial f (x 1 , . . . , x n ) is called quasihomogeneous, if there are positive integers w 1 , . . . , w n (called weights) and d (called degree) such that f (λ w1 x 1 , . . . , λ wn x n ) = λ d f (x 1 , . . . , x n ) for λ ∈ C * . A complete intersection singularity given as the zero G |G| w, x, y, z(, u) c G R(0, x, y, z(, u)) Sing. (C): T 12 2, 3, 4, 6 1 z 2 + 4y 3 + 27x 4 E 6 ∆(3 · 4 2 ) 48 4, 3, 8, 12 1 z 2 + 4y 3 + 27x 8 E 14 (D): O 24 2, 4, 6, 9 1 z 2 + 4xy 3 + 27x 3 E 7 ∆(6 · 3 2 ) 54 6, 6, 6, 6, 9 3 z 2 − xy u 2 + 4xyz + 27x 3 δ1 ∆(6 · 4 2 ) 96 4, 6, 8, 15 1 z 2 + 4xy 3 + 27x 5 Z 11 ∆(6 · 6 2 ) 216 6, 6, 12, 21 3 z 2 + 4xy 3 + 27x 7 Z 1,0 (E) 108 6, 6, 9, 12, 12 3 9u 2 − 12z 2 432y 2 − x 3 − 36xz K ′ 1,0 (F) 216 6, 9, 12, 12 3 4z 3 − 144yz 2 +1728y 2 z − 186624x 4 U 12 (G) 648 9, 12, 18, 18 6 4z 3 − 9yz 2 + 6y 2 z − y 3 + 6912x 3 y U 1,0 (H)=I 60 2, 6, 10, 15 1 Table 3. Subgroups of SL 3 (C) and surface singularities z 2 − y 3 + 1728x 5 E 8 (I) 168 4, 6, 14, 21 1 z 2 − y 3 − 1728x 7 E 12 (J) 180 6, 6, 12, 15 3 y 3 − xz 2 + 64x 2 y 2 Q 2,0 (K) 504 6, 12, 18, 21 3 y 3 − xz 2 − 256x 3 y Q 11 (L) 1080 6, 12, 30, 45 3 459165024z 2 − 25509168y 3 −(7558272 − 2519424 √ 15i)x 5 y E 13set of polynomials f 1 (x 1 , . . . , x n ), . . . , f k (x 1 , . . . , x n ) is called quasihomogeneous, if f 1 , . . . , f k are quasihomogeneous with respect to the same weights w 1 , . . . , w n but degrees d 1 , . . . , d k respectively. We call the system W := (w 1 , . . . , w n ; d 1 , . . . , d k ) the weight system corresponding to the set of polynomials. Let c W be the greatest common divisor of w 1 , . . . , w n , d 1 , . . . , d k . The weight system is called reduced if c W = 1. We assume that f 1 (0) = · · · = f k (0) = 0 and the system of equations f 1 = · · · = f k = 0 has an isolated singularity at the origin. The coordinate algebra A f := C[x 1 , . . . , x n ]/(f 1 , . . . , f k ) is a Z-graded algebra with respect to the system of weights (w 1 , . . . , w n ; d 1 , . . . , d k ). Therefore we can consider the decomposition of A f as a Z-graded C-vector space: A f := ∞ k=0 A f,k , A f,k := g ∈ A f g(λ w1 x 1 , . . . , λ wn x n ) = λ k g(x 1 , . . . , x n ) . The formal power series p f (t) := ∞ k=0 (dim C A f.k )t k is called the Poincaré series of A f . It is given by p f (t) = k i=1 (1 − t di ) n j=1 (1 − t wj ) . Let (X, x) be a Kleinian singularity. Then the minimal resolution of the singularity x has an exceptional divisor with the dual graph depicted in Fig. 1 with m = 1 in the case of the A n -singularities and m = 3 in the other cases (see, e.g., [Br]). Here all vertices correspond to rational curves of self-intersection number −2, the mutual intersection numbers are either 0 or 1, and two vertices are joined by an edge if and only if the intersection number of the corresponding rational curves is equal to 1. The values of the numbers α 1 , . . . , α m are indicated in Table 1. They are Figure 1. The graph T − α1,...,αm the Dolgachev numbers of the singularity, see [ET]. It turns out that these graphs are precisely the ordinary Coxeter-Dynkin diagrams of type ADE. (Note that the corresponding intersection matrix is the Cartan matrix multiplied by −1.) Now let (X, x) be a Fuchsian hypersurface or complete intersection singularity. A natural compactification of X is given by X := Proj(A f [t]), where t has degree 1 for the grading of A f [t] (see [P]). This is a normal projective surface with a C * -action. The surface X may acquire additional singularities on the boundary X ∞ := X \ X = Proj(A f ). Let g = g(X ∞ ) be the genus of the boundary. We assume g = 0. Let π : S → X be the minimal normal crossing resolution of all singularities of X. The preimage X ∞ of X ∞ under π : S → X consists of the strict transform δ 0 of X ∞ and m chains δ i 1 , . . . , δ i αi−1 , i = 1, . . . , m, of rational curves of self-intersection −2 which intersect again according to the dual graph shown in Figure 1 (see, e.g., [D, E5]). By the adjunction formula and g = 0, the selfintersection number of the rational curve δ 0 is also −2. The numbers α 1 , . . . , α m of the Fuchsian singularities corresponding to finite subgroups of SL 3 (C) are indicated in Table 4. They are again the Dolgachev numbers of the singularity, see [ET,E3]. · · · • δ 2 1 · · · • δ 2 α 2 −1 • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ δ0 • δ m−1 α m−1 −1 · · · • δ m−1 1 • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ δ 1 α 1 −1 • ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ δ m αm −1 · · · ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ · · · ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ • δ 1 1 • δ r 1 We call the graph T − α1,...,αm a Coxeter-Dynkin diagram. Let V − be the free Z-module with the basis δ 1 1 , . . . , δ 1 α1−1 , δ 2 1 , . . . , δ 2 α2−1 , . . . , δ m 1 , . . . , δ m αm−1 , δ 0 equipped with the symmetric bilinear form −, − given by the intersection matrix corresponding to Fig. 1. This defines a lattice (V − , −, − ). We consider two extensions of this lattice. Let V 0 = V − ⊕Zδ 1 with the symmetric bilinear form defined by Fig. 2. Here the double dashed line between δ 0 and δ 1 means δ 0 , δ 1 = −2. Let V + = V 0 ⊕ Zδ 2 with the symmetric bilinear form defined by Fig. 3. If (V, −, − ) is an arbitrary lattice and e ∈ V is a root, i.e. e, e = −2, then the reflection corresponding to e is defined by s e (x) = x − 2 x, e e, e e = x + x, e e for x ∈ V. G Name Normal form Weights α 1 , . . . , α m (C): T E 6 z 2 + y 3 + x 4 3,4,6;12 2, 3, 3 ∆(3 · 4 2 ) E 14 z 2 + y 3 + x 8 3,8,12;24 3, 3, 4 (D): O E 7 z 2 + y 3 + yx 3 4,6,9;18 2, 3, 4 ∆(6 · 3 2 ) δ1 xy + z 2 x 3 + y 3 + z 3 + w 2 2,2,2,3;4,6 2, 2, 2, 2, 2, 2 ∆(6 · 4 2 ) Z 11 z 2 + xy 3 + x 5 6,8,15;30 2, 3, 8 ∆(6 · 6 2 ) Z 1,0 z 2 + xy 3 + x 7 2,4,7;14 2, 2, 2, 4 (E) K ′ 1,0    xu + y 2 ax 4 + xy 2 + z 2 + u 2 , a = 0, 1 4    2,3,4,4;6,8 2, 2, 4, 4 (F) U 12 z 3 + y 3 + x 4 3,4,4;12 4, 4, 4 (G) U 1,0 z 3 + yz 2 + x 3 y 2,3,3;9 2, 3, 3, 3 (H)=I E 8 z 2 + y 3 + x 5 6,10,15;30 2, 3, 5 (I) E 12 z 2 + y 3 + x 7 6,14,21;42 2, 3, 7 (J) Q 2,0 xz 2 + y 3 + x 4 y 2,4,5;12 2, 2, 2, 5 (K) Q 11 xz 2 + y 3 + yx 3 4,6,7;18 2, 4, 7 (L) E 13 z 2 + y 3 + x 5 y 4,10,15;30 2, 4, 5 Table 4. Normal forms, reduced weight systems, and Dolgachev numbers Figure 2. The graph T α1,...,αm If B = (e 1 , . . . , e n ) is an ordered basis consisting of roots, then the Coxeter element τ corresponding to B is defined by τ = s e1 s e2 · · · s en . · · · • ✤ ✤ ✤ ✤ ✤ ✤ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ δ1 • δ 2 1 · · · • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ δ 2 α 2 −1 • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ δ0 • δ m−1 α m−1 −1 · · · • δ m−1 1 • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ δ 1 α 1 −1 • ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ δ m αm −1 · · · ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ · · · ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ • δ 1 1 • δ r 1 For a Coxeter element τ , let ∆(t) = det(1 − τ −1 t) be its characteristic polynomial, using a suitable normalization. Figure 3. The graph T + α1,...,αm If D is a Coxeter-Dynkin diagram, then we denote by ∆ D (t) the characteristic polynomial of the Coxeter element corresponding to the graph D. These polynomials can be computed as in [E1] and one gets · · · • δ2 • ✤ ✤ ✤ ✤ ✤ ✤ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✌ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ δ1 • δ 2 1 · · · • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ δ 2 α 2 −1 • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ δ0 • δ m−1 α m−1 −1 · · · • δ m−1 1 • ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ δ 1 α 1 −1 • ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ δ m αm −1 · · · ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ · · · ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ • δ 1 1 • δ r 1∆ T − α 1 ,...,αm (t) = (1 + t) m i=1 1 − t αi 1 − t − t m i=1 1 − t αi−1 1 − t m j=1 j =i 1 − t αj 1 − t , ∆ Tα 1 ,...,αm (t) = (1 − t) 2−m (1 − t α1 ) · · · (1 − t αm ), ∆ T + α 1 ,...,αm (t) = (1 − 2t − 2t 2 + t 3 ) m i=1 1 − t αi 1 − t + t 2 m i=1 1 − t αi−1 1 − t m j=1 j =i 1 − t αj 1 − t . (The last two formulas can also be found in [E2, p. 98], but note that, unfortunately, there is a misprint in [E2, p. 98].) Now we can state the main result of [EP]. Theorem 1. (i) For a Kleinian singularity not of type A 2n we have p f (t) = ∆ T − α 1 ,...,αm (t) ∆ Tα 1 ,...,αm (t) . (ii) For a Fuchsian singularity with g = 0 we have p f (t) = ∆ T + α 1 ,...,αm (t) ∆ Tα 1 ,...,αm (t) . Remark 2. Unfortunately, the exclusion of the case A 2n is only implicit in [EP] and was forgotten in the statement of [EP,Theorem 1]. Remark 3. Note that we have T 2,3,3 ∼ T − 3,3,3 , T 2,3,4 ∼ T − 2,4,4 , T 2,3,5 ∼ T − 2,3,6 , where ∼ means equivalence under the braid group action, see [E6]. Similarly, one can show that the graphs T 2n−1 , n ≥ 1, and T 2,2,n , n ≥ 2, are equivalent under the braid group action to the extended Coxeter-Dynkin diagrams of type A 2n−1 and D n+2 respectively. 3. Poincaré series of subgroups of SL 2 (C) and SL 3 (C) Let G be a finite subgroup of SL n (C) for n = 2, 3. Consider the algebra of complex polynomials C[x 1 , . . . , x n ] graded by the degree for homogeneous ones. It is isomorphic to the symmetric algebra S := S(C n ) = ∞ k=0 S k (C n ), where S k (C n ) denotes the k-th symmetric power of C n . Let S G be the algebra of invariant polynomials with respect to G. For n = 2, it is generated by 3 elements x, y, z which satisfy a relation R(x, y, z) = 0. The elements x, y, z correspond to invariant polynomials and their degrees correspond to the weights of these variables. Let c G denote the greatest common divisor of these weights. The weights of the variables x, y, z, the number c G , and the polynomial R(x, y, z) are indicated in Table 1. Now let G be one of the finite subgroups of SL 3 (C) of Table 3. Except in the cases (E) and ∆(6 · 3 2 ), the algebra S G is generated by 4 elements w, x, y, z which satisfy a relation R(w, x, y, z). In the cases (E) and ∆(6 · 3 2 ), S G is generated by 5 elements w, x, y, z, u which satisfy two relations R 1 (w, x, y, z, u) = 0 and R 2 (w, x, y, z, u) = 0. The degrees of the invariants and the polynomials R(w, x, y, z) and R 1 (w, x, y, z, u), R 2 (w, x, y, z, u) respectively can be found in [YY]. The degrees of the invariant polynomials and the restriction to the hyperplane w = 0 of the polynomials R(w, x, y, z) and R 1 (w, x, y, z, u), R 2 (w, x, y, z, u) respectively are indicated in Table 3. Let c G be the greatest common divisor of the weights of the remaining variables x, y, z(, u) (with the weight of w excluded). The number c G is also indicated in Table 3. Note that, except in the case (G), c G also divides the weight of w. For n = 2, the algebra A G := S G = C[x, y, z]/R(x, y, z) coincides with the coordinate algebra A f of the corresponding singularity indicated in the last column of Table 1 up to the grading. The grading is shifted by c G . For n = 3 and G one of the cases of Table 3 except the cases (E) and ∆(6 · 3 2 ), the algebra A G := C[x, y, z]/R(0, x, y, z) coincides with the coordinate algebra A f of the corresponding singularity indicated in the last column of Table 3 with the grading shifted by c G . In the cases (E) and ∆(6 · 3 2 ), the algebra A G := C[x, y, z, u]/(R 1 (0, x, y, z, u), R 2 (0, x, y, z, u)) coincides with the coordinate algebra A f of the complete intersection singularity K ′ 1,0 and δ1 respectively, again with the grading shifted by c G . Let p G (t) be the Poincaré series of the algebra of A G . Then we have p G (t) = p f (t cG ) for G ⊂ SL 2 (C), p G (t) = p f (t cG ) (1 − t deg w ) for G ⊂ SL 3 (C). The finite subgroups G ⊂ SL n (C) for n = 2, 3 under consideration have a natural n-dimensional representation γ which is irreducible (except in the cases G = C l ). Let γ * be its contragredient representation. Let γ 0 , . . . , γ l be the irreducible representations of G, where γ 0 is the trivial representation. Let B = (b ij ) and B * = (b * ij ) be the (l + 1) × (l + 1)-matrices defined by decomposing the tensor products γ j ⊗ γ = i b ij γ i and γ j ⊗ γ * = i b * ij γ i respectively into irreducible components. For each integer k ≥ 0, let ρ k be the representation of G on S k (C n ) induced by its natural action on C n . We have a decomposition ρ k = l i=0 v ki γ i with v ki ∈ Z. We associate to ρ k the vector v m = (v m0 , . . . , v ml ) t ∈ Z l+1 . As in [K, p. 211] we define P G (t) := ∞ m=0 v m t m . This is a formal power series with coefficients in Z l+1 . We also put P G (t) i := ∞ m=0 v mi t m . Note that P G (t) 0 is the usual Poincaré series p G (t) of the group G. Let V denote the set of all formal power series x = ∞ m=0 x m t m with x m ∈ Z l+1 . This is a free module of rank l + 1 over the ring R of formal power series with integer coefficients. Now let n = 2 and G ⊂ SL 2 (C) be a finite subgroup not of type C 2n+1 . Then c G = 2 and we have p f (t 2 ) = P G (t) 0 . Moreover, we have γ * = γ and therefore B * = B. The irreducible representations of SL 2 (C) are of the form ρ m , m a non-negative integer. The Clebsch-Gordon formula reads in this case ρ m ⊗ γ = ρ m+1 ⊕ ρ m−1 setting ρ −1 = 0 (cf., e.g., [FH,Exercise 11.11]). This yields the equation Bv m = v m+1 + v m−1 . Following [K, p. 222], one can easily derive from this equation that x = P G (t) is a solution of the following linear equation in V : ((1 + t 2 )I − tB)x = v 0 . Let M (t) be the matrix (1+t 2 )I −tB and M 0 (t) be the matrix obtained by replacing the first column of M (t) by v 0 = (1, 0, . . . , 0) t . By Cramer's rule we can derive the following theorem [E4,Sect. 3] (see also [St]). Theorem 4. For a finite subgroup G ⊂ SL 2 (C) not of type C 2n+1 we have p f (t 2 ) = P G (t) 0 = det M 0 (t) det M (t) = det(t 2 I − τ ) det(t 2 I − τ a ) , where τ is the Coxeter element and τ a the affine Coxeter element of the corresponding root system of type ADE associated to the singularity defined by the equation f = 0 with the same name. Now let n = 3 and G ⊂ SL 3 (C) be a finite subgroup. For a pair a, b of nonnegative integers, let Γ a,b be the unique irreducible, finite-dimensional representation of SL 3 (C) of [FH,Theorem 13.1]. By [FH,Proposition 15.25] and [FH,(13.5)], we have for a non-negative integer m (setting Γ −1,b = 0) the following Clebsch-Gordon formulas: Γ m,0 ⊗ γ = Γ m+1,0 ⊕ Γ m−1,1 , Γ m,0 ⊗ γ * = Γ m−1,0 ⊕ Γ m,1 . Since Γ m,0 = ρ m , we can derive from these formulas v m+2 = Bv m+1 − B * v m + v m−1 . Therefore x = P G (t) is a solution of the following linear equation in V (see also [BI, BP]): ((1 − t 3 )I − tB + t 2 B * )x = v 0 . Let M (t) be the matrix (1 − t 3 )I − tB + t 2 B * and M 0 (t) be the matrix obtained by replacing the first column of M (t) by v 0 = (1, 0, . . . , 0) t . Again Cramer's rule yields P G (t) 0 = det M 0 (t) det M (t) . We have the following theorem: Theorem 5. Let G ⊂ SL 3 (C) be one of the groups T , ∆(3 · 4 2 ), O, ∆(6 · 3 2 ), ∆(6·4 2 ), ∆(6·6 2 ), (E), (F), (G), (H)=I, (I), (J), (K), or (L), let c G be the greatest common divisor of the weights of the variables x, y, z(, u), and let α 1 , . . . , α m be the Dolgachev numbers of the singularity corresponding to G. Moreover, let q where (a, b) = (3, 0), (3, 1), (4, 0) respectively. (ii) For G=(I), ∆(3 · 4 2 ), ∆(6 · 4 2 ) (E 12 , E 14 , Z 11 ) we have c G = 1 and det M 0 (t) = q (4) a,b (t)∆ T + α 1 ,α 2 ,α 3 (t), det M (t) = (1 − t)q (4) a,b (t)∆ T4,α 1 ,α 2 ,α 3 (t). where (a, b) = (3, 0), (3, 2), (2, 3) respectively. (iii) For G= (K), (L), (F), ∆(6 · 6 2 ), (J), (E) (Q 11 , E 13 , U 12 , Z 1,0 , Q 2,0 , K ′ 1,0 ) we have c G = 3 and det M 0 (t) = q where (a, b) = (6, −1), (7, −1), (7, −2), (7, 1), (8, −2), (8, −3) respectively. (iv) For G = ∆(6 · 3 2 ) (δ1) we have c G = 3, m = 6, α i = 2 for i = 1, . . . , m, and det M 0 (t) = (1 − t 3 )q (2) 9,−3 (t 3 )∆ T + 2,α 2 ,...,αm (t 3 ), det M (t) = q (2) 9,−3 (t 3 )∆ Tα 2 ,...,αm (t 3 ). (v) For G=(G) (U 1,0 ) we have c G = 6 and det M 0 (t) = q(t 3 )∆ T + α 1 ,α 2 ,α 3 ,α 4 (t 6 ), det M (t) = (1 − t 9 )q(t 3 )∆ Tα 1 ,α 2 ,α 3 ,α 4 (t 6 ), where q(t) = (1−t) 4 (1−t 2 )(1−t 3 ) (1−t 6 ) . G det M 0 (t) det M (t) T (1 − t) 3 ∆ T − 2,3,3 (t) (1 − t) 4 ∆ T2,2,3,3 (t) ∆(3 · 4 2 ) (1 − t) 3 (1 − t 4 ) 2 ∆ T + 3,3,4 (1 − t) 4 (1 − t 4 ) 2 ∆ T3,3,4,4 (t) O (1 − t) 3 (1 − t 2 )∆ T − 2,3,4 (t) (1 − t) 4 (1 − t 2 )∆ T2,2,3,4 (t) ∆(6 · 3 2 ) (1−t 3 ) 10 (1−t 6 ) 3 ∆ T + 2,2,2,2,2,2 (t 3 ) (1−t 3 ) 9 (1−t 6 ) 3 ∆ T2,2,2,2,2 (t 3 ) ∆(6 · 4 2 ) (1 − t) 2 (1 − t 4 ) 3 ∆ T + 2,3,8 (1 − t) 3 (1 − t 4 ) 3 ∆ T2,3,4,8 (t) ∆(6 · 6 2 ) (1 − t 3 ) 7 (1 − t 6 )∆ T + 2,2,2,4 (t 3 ) (1 − t 3 ) 8 (1 − t 6 )∆ T2,2,2,2,4 (t 3 ) (E) (1−t 3 ) 8 (1−t 6 ) 3 ∆ T + t) a (1 − t e ) b for a, b, e ∈ Z. ( i ) iFor G = T, O, I (E 6 , E 7 , E 8 ) we have c G = 1 and det M 0 (t) = q b (t)∆ T2,α 1 ,α 2 ,α 3 (t), b (t 3 )∆ T + α 1 ,...,αm (t 3 ), det M (t) = (1 − t 3 )q (2) a,b (t 3 )∆ T2,α 1 ,...,αm (t 3 ), Mathematics Subject Classification. 32S25, 14E16, 13A50, 20G05. Partially supported by DFG.1 ,2,4,4 (t 3 )(1−t 3 ) 9(1−t 6 ) 3 ∆ T2,2,2,4,4 (t 3 )(F) (1−t 3 ) 7 (1−t 6 ) 2 ∆ T + 4,4,4 (t 3 ) (1−t 3 ) 8 (1−t 6 ) 2 ∆ T2,4,4,4 (t 3 ) (G) (1−t 3 ) 4 (1−t 6 )(1−t 9 ) (1−t 18 ) ∆ T + 2,3,3,3 (t 6 ) (1−t 3 ) 4 (1−t 6 )(1−t 9 ) 2 (1−t 18 ) ∆ T2,3,3,3 (t 6 ) (H)=I (1 − t) 4 ∆ T − 2,3,5 (t) (1 − t) 5 ∆ T2,2,3,5 (t) (I) (1 − t) 3 ∆ T + 2,3,7 (t) (1 − t) 4 ∆ T2,3,4,7 (t) (J) (1−t 3 ) 8 (1−t 6 ) 2 ∆ T + 2,2,2,5 (t 3 ) (1−t 3 ) 9(1−t 6 ) 2 ∆ T2,2,2,2,5 (t 3 )(K) (1−t 3 ) 6 (1−t 6 ) ∆ T + 2,4,7 (t 3 ) (1−t 3 ) 7 (1−t 6 ) ∆ T2,2,4,7 (t 3 ) (L) (1−t 3 ) 7 (1−t 6 ) ∆ T + 2,4,5 (t 3 ) (1−t 3 ) 8(1−t 6 ) ∆ T2,2,4,5 (t 3 )Table 5. Determinants of the matrices M 0 (t) and M (t) AcknowledgementsThis paper grew out of a common research project with A. Takahashi. The author is grateful to him for useful discussions. He also would like to thank the anonymous referee for useful comments which helped to improve the paper.Proof. The character tables of the tetrahedral and icosahedral group are given in[Art]. The character table of the octahedral group can be found, e.g., in[HH]. From these tables, one can calculate the matrices B. The matrices B for the remaining cases are given in[BP]. For the case(D), only one example is treated. More complete results for the cases ∆(3n 2 ) and ∆(6n 2 ) can be found in[LNR]and[EL]respectively. From these results, one can derive the corresponding matrices B. The proof of the theorem is then obtained by a direct calculation from these matrices using the computer algebra system Singular[DGPS].The results are summarized inTable 5.Remark 6. Let G be one of the groups T, O, I. In this case, the matrix B is symmetric and we have B * = B. Therefore M (t) = (1 − t 3 )I − tB + t 2 B * = (1 − t)((1 + t + t 2 )I − tB). Critical points of smooth functions and their normal forms. V I Arnold, Engl. translation in Russ. Math. Surv. 305Usp. Math. Nauk.V. I. Arnold: Critical points of smooth functions and their normal forms. Usp. Math. Nauk. 30:5 (1975), 3-65 (Engl. translation in Russ. Math. Surv. 30:5 (1975), 1-75). . M Artin, Prentice Hall, IncEnglewood Cliffs, NJM. Artin: Algebra. Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. Itzykson: A case study in finite groups: PSL(2, F7). M Bauer, C , Recent Developments in Conformal Field Theories, Trieste Conference. M. Bauer, C. Itzykson: A case study in finite groups: PSL(2, F7). In: Recent De- velopments in Conformal Field Theories, Trieste Conference 1989, World Scient., 1990, pp. 10-38. H F Blichfeldt, Finite Collineation Groups. The Univ. ChicagoChicago PressH. F. Blichfeldt: Finite Collineation Groups. The Univ. Chicago Press, Chicago, 1917. E Brieskorn, Rationale Singularitäten komplexer Flächen. Inventiones Math. 4E. Brieskorn: Rationale Singularitäten komplexer Flächen. Inventiones Math. 4 (1968), 336-358. Branching law for finite subgroups of SL3(C) and McKay correspondence. F Butin, G S Perets, J. Group Theory. 172F. Butin, G. S. Perets: Branching law for finite subgroups of SL3(C) and McKay correspondence. J. Group Theory 17 (2014), no. 2, 191-251. McKay's correspondence for cocompact discrete subgroups of SU(1, 1). I V Dolgachev, CRM Proc. Lecture Notes. Providence, RIAmer. Math. Soc47In: Groups and symmetriesI. V. Dolgachev: McKay's correspondence for cocompact discrete subgroups of SU(1, 1). In: Groups and symmetries, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009, pp.111-133. Singular 4-1-0 -A computer algebra system for polynomial computations. W Decker, G.-M Greuel, G Pfister, H Schönemann, W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann: Singular 4-1-0 -A computer algebra system for polynomial computations. http://www.singular.uni- kl.de (2016). Milnor lattices and geometric bases of some special singularities. W Ebeling, Noeuds, tresses et singularités. Genève31Enseign. Math.W. Ebeling: Milnor lattices and geometric bases of some special singularities. In: Noeuds, tresses et singularités (Ed. C.Weber), Monographie Enseign. Math. 31, Genève, 1983, 129-146 and Enseign. Math. (2) 29 (1983), 263-280. The Monodromy Groups of Isolated Singularities of Complete Intersections. W Ebeling, Lect. Notes in Math. 1293Springer-VerlagBerlin etc.W. Ebeling: The Monodromy Groups of Isolated Singularities of Complete Inter- sections. Lect. Notes in Math., Vol. 1293, Springer-Verlag, Berlin etc., 1987. Strange duality, mirror symmetry, and the Leech lattice. W Ebeling, Singularity theory. Liverpool; CambridgeCambridge Univ. Press263W. Ebeling: Strange duality, mirror symmetry, and the Leech lattice. In: Singular- ity theory (Liverpool, 1996), London Math. Soc. Lecture Note Ser. 263, Cambridge Univ. Press, Cambridge, 1999, pp. 55-77. Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity. W Ebeling, Manuscripta math. 107W. Ebeling: Poincaré series and monodromy of a two-dimensional quasihomoge- neous hypersurface singularity. Manuscripta math. 107 (2002), 271-282. The Poincaré series of some special quasihomogeneous surface singularities. W Ebeling, Publ. Res. Inst. Math. Sci. 39W. Ebeling: The Poincaré series of some special quasihomogeneous surface singu- larities. Publ. Res. Inst. Math. Sci. 39 (2003), 393-413. A note on distinguished bases of vanishing cycles. In: Special Issue -Real and Complex Singularities and their Applications in Geometry and Topology. W Ebeling, Topology Appl. 234W. Ebeling: A note on distinguished bases of vanishing cycles. In: Special Issue - Real and Complex Singularities and their Applications in Geometry and Topology, Topology Appl. 234 (2018), 259-268. Ploog: McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities. W Ebeling, D , Math. Ann. 347W. Ebeling, D. Ploog: McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities. Math. Ann. 347 (2010), 689-702. Strange duality of weighted homogeneous polynomials. W Ebeling, A Takahashi, Compos. Math. 147W. Ebeling, A. Takahashi: Strange duality of weighted homogeneous polynomials. Compos. Math. 147 (2011), 1413-1433. The flavor group ∆(6n 2 ). J A Escobar, Ch Luhn, J. Math. Phys. 501ppJ. A. Escobar, Ch. Luhn: The flavor group ∆(6n 2 ). J. Math. Phys. 50 (2009), no. 1, 013524, 22 pp. Representation Theory. A First Course. W Fulton, J Harris, Graduate Texts in Math. 129Springer-VerlagW. Fulton, J. Harris: Representation Theory. A First Course. Graduate Texts in Math., Vol. 129, Springer-Verlag, New York etc., 1991. He: Non-abelian finite gauge theories. A Hanany, Y.-H , J. High Energy Phys. 2Paper 13A. Hanany, Y.-H. He: Non-abelian finite gauge theories. J. High Energy Phys. 1999, no. 2, Paper 13 Vorlesungenüber das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Herausgegeben mit einer Einführung und mit Kommentaren von Peter Slodowy. F Klein, Birkhäuser, Basel Boston BerlinF. Klein: Vorlesungenüber das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Herausgegeben mit einer Einführung und mit Kommentaren von Peter Slodowy, Birkhäuser, Basel Boston Berlin, 1993. The McKay correspondence, the Coxeter element and representation theory. B Kostant, Élie Cartan et les mathématiques d'aujourd'hui. LyonAstérisque, Numéro hors sérieB. Kostant: The McKay correspondence, the Coxeter element and representa- tion theory. In:Élie Cartan et les mathématiques d'aujourd'hui (Lyon, 1984), Astérisque, Numéro hors série, 1985, pp. 209-255. Flavor group ∆(3n 2 ). Ch, S Luhn, P Nasri, Ramond, J. Math. Phys. 487ppCh. Luhn, S. Nasri, P. Ramond: Flavor group ∆(3n 2 ). J. Math. Phys. 48 (2007), no. 7, 073501, 21 pp. G A Miller, H F Blichfeldt, L E Dickson, Theory and Applications of Finite Groups. New YorkDoverG. A. Miller, H. F. Blichfeldt, L. E. Dickson: Theory and Applications of Finite Groups. Dover, New York, 1916. Normal surface singularities with C * action. H Pinkham, Math. Ann. 227H. Pinkham: Normal surface singularities with C * action. Math. Ann. 227 (1977), 183-193. Invariant Theory. T A Springer, Lecture Notes in Mathematics. 585Springer-VerlagT. A. Springer: Invariant Theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin-New York, 1977. R Stekolshchik, Notes on Coxeter Transformations and the McKay Correspondence. Springer Monographs in Mathematics. BerlinSpringer-VerlagR. Stekolshchik, Notes on Coxeter Transformations and the McKay Correspon- dence. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2008. Classification of unimodal isolated singularities of complete intersections. C T C Wall, Singularities, Part. Arcata, Calif; Providence, RIAmer. Math. Soc40Proc. SympC. T. C. Wall: Classification of unimodal isolated singularities of complete inter- sections. In: Singularities, Part 2 (Arcata, Calif., 1981), Proc. Symp. Pure Math. Vol. 40, Part 2, Amer. Math. Soc., Providence, RI, 1983, pp. 625-640. Elliptic complete intersection singularities. C T C Wall, Singularity theory and its applications. D. Mond, J. MontaldiPart I, WarwickSpringer1462C. T. C. Wall: Elliptic complete intersection singularities. In: Singularity theory and its applications, Part I, Warwick 1989 (D. Mond, J. Montaldi, eds.), Lecture Notes in Math., Vol. 1462, Springer, Berlin etc., 1991, pp. 340-372. Lehrbuch der Algebra. H Weber, Friedrich Vieweg und SohnII1899BraunschweigSecond editionH. Weber: Lehrbuch der Algebra, Volume II. Second edition. Friedrich Vieweg und Sohn, Braunschweig, 1899. Gorenstein quotient singularities in dimension three. S St, Y Yau, Yu, Mem. Amer. Math. Soc. 105505St. S.-T. Yau, Y. Yu: Gorenstein quotient singularities in dimension three. Mem. Amer. Math. Soc. 105 (1993), no. 505. D-30060 Hannover, Germany E-mail address: [email protected]. Postfach. 6009deInstitut für Algebraische Geometrie, Leibniz Universität HannoverInstitut für Algebraische Geometrie, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany E-mail address: [email protected]
[]
[ "Pair-Linking for Collective Entity Disambiguation: Two Could Be Better Than All", "Pair-Linking for Collective Entity Disambiguation: Two Could Be Better Than All" ]
[ "Minh C Phan ", "Aixin Sun ", "Yi Tay ", "Jialong Han ", "Chenliang Li " ]
[]
[]
Collective entity disambiguation, or collective entity linking aims to jointly resolve multiple mentions by linking them to their associated entities in a knowledge base. Previous works largely based on the underlying assumption that entities within the same document are highly related. However, the extend to which these mentioned entities are actually connected in reality is rarely studied and therefore raises interesting research questions. For the first time, this paper shows that the semantic relationships between mentioned entities within documents are in fact less dense than expected. This could be attributed to several reasons such as noise, data sparsity and knowledge base incompleteness. As a remedy, we introduces MINTREE, a new tree-based objective for the problem of entity disambiguation. The key intuition behind MINTREE is the concept of coherence relaxation which utilizes the weight of a minimum spanning tree to measure the coherence between entities. Based on this new objective, we devise a novel iterative solution for MINTREE optimization problem which we call Pair-Linking. The idea of Pair-Linking is simple: instead of considering all the given mentions, Pair-Linking iteratively selects the pair with highest confidence at each step for decision making. Via extensive experiments on 8 publicly available benchmark datasets, we show that our approach is not only more accurate but also surprisingly faster than many state-of-the-art collective linking algorithms.
10.1109/tkde.2018.2857493
[ "https://arxiv.org/pdf/1802.01074v1.pdf" ]
3,616,748
1802.01074
e7cda53e4710557ec6dfb1424dd481a025b1beda
Pair-Linking for Collective Entity Disambiguation: Two Could Be Better Than All Minh C Phan Aixin Sun Yi Tay Jialong Han Chenliang Li Pair-Linking for Collective Entity Disambiguation: Two Could Be Better Than All Index Terms Collective Entity DisambiguationMINTREEPair-Linking Collective entity disambiguation, or collective entity linking aims to jointly resolve multiple mentions by linking them to their associated entities in a knowledge base. Previous works largely based on the underlying assumption that entities within the same document are highly related. However, the extend to which these mentioned entities are actually connected in reality is rarely studied and therefore raises interesting research questions. For the first time, this paper shows that the semantic relationships between mentioned entities within documents are in fact less dense than expected. This could be attributed to several reasons such as noise, data sparsity and knowledge base incompleteness. As a remedy, we introduces MINTREE, a new tree-based objective for the problem of entity disambiguation. The key intuition behind MINTREE is the concept of coherence relaxation which utilizes the weight of a minimum spanning tree to measure the coherence between entities. Based on this new objective, we devise a novel iterative solution for MINTREE optimization problem which we call Pair-Linking. The idea of Pair-Linking is simple: instead of considering all the given mentions, Pair-Linking iteratively selects the pair with highest confidence at each step for decision making. Via extensive experiments on 8 publicly available benchmark datasets, we show that our approach is not only more accurate but also surprisingly faster than many state-of-the-art collective linking algorithms. 1) "The Sun and The Times reported that Greece will have to leave the Euro soon". 2) "Wood played at 2006 Master held in Augusta, Georgia" where entity mentions are underlined. In the first example, only two entities are closely related, which is shown in Figure 1a. On the other hand, the entities in the second example are coherent in a chain-like fashion as illustrated in Figure 1b. Both examples illustrate the sparse coherence (between mentioned entities) which is commonplace in generic documents. This qualitatively shows that the fundamental assumption and objective of ALL-Link leaves much to be desired. In lieu of the apparent weakness of ALL-Link, this paper proposes a novel and simple paradigm. Our approach relaxes the pairwise coherence assumption and affirms the narrative that maintaining pairwise coherence between all entities is unnecessary. Furthermore, relaxation of this assumption allows us to improve not only the accuracy but also the runtime of collective entity disambiguation significantly. Overall, the prime contributions of this work are as follows: • For the first time, we study the form of coherence between mentioned entities (i.e., whether it is sparse or dense). We show that not all entities (in a general document) are highly related to each other. This insight leads us to develop a new objective that relaxes the coherence condition, aiming towards effective and significantly faster solution for entity disambiguation. • We propose a tree-based model that utilizes the weight of spanning tree as the objective. We provide detailed analysis showing that our proposed tree-based objective is highly correlated with conventional objective and can be used to effectively measure the disambiguation quality. • We introduce Pair-Linking, an approximate solution for the tree-based model. Pair-Linking achieves state-of-the-art performance while being extremely fast (approximately up to 100 times) in comparison to other collective linking algorithms. RELATED WORK Generally, techniques for collective entity disambiguation can be characteristically dichotomized into two families, optimizationbased approaches and graph-based approaches. The optimization-based approach models the entity disambiguation problem and solves the objective function by optimization techniques. On the other hand, graph-based approach directly approximates the solution by doing influence propagation on the mention-entity graph built from mentions and candidate entities. We will describe the two approaches in this section. Optimization-based Approach The common approach for finding the optimal matching, denoted by Γ * , is to maximize the local confidence of each assignment φ(mi, ei), while enforcing the pairwise coherence among all linked entities ψ(ei, ej). The objective is referred to as ALL-Link and is expressed as follows: Γ * = arg max Γ   N i=1 φ(mi, ei) + N i=1 N j=1,j =i ψ(ei, ej)  (1) Local confidence or local score φ(mi, ei) reflects the likelihood of mapping mi → ei solely based on mi's textual context and ei's profile, without considering any other mappings in the same document. It is computed through the textual similarity between the entity mention and the candidate entity's name, and/or the semantic similarity between the context around the entity mention and the document associated with the candidate entity [5]. On the other hand, the pairwise coherence ψ(ei, ej) represents the semantic relatedness between entities and it is often computed by utilizing the linkage structure in knowledge base (e.g., Wikipedia) or entity embedding. Detailed computation of the two components will be described in Section 3. Solving the optimization expressed in Equation 1 is NP-Hard therefore approximation is needed. For example, Shen et al. [6] use iterative substitution (i.e., hill climbing technique): the optimal assignment is obtained by substituting an assignment mi → ei with another mapping mi → ej as long as it improves the objective score. Furthermore, in the works [3], [7], Loopy Belief Propagation (LBP) [8] is utilized to solve the inference problem. Both approaches have complexity of O I × N 2 k 2 where I is the number of iterations required for convergence, N and k are the number of mentions and candidates per mention respectively. Other methods follow the idea proposed by Ratinov et al. [9]. First, they identify a set of unambiguous mentions and entities identified by the local confidence score φ(mi, ei). The set of unambiguous entities will be used as a disambiguation context Γ . Then the global optimization task is decomposed into the optimization of individual coherence, described by the formula: Fig. 2. An example of mention-entity graph. The weight between mention and entity represents the local confidence while the weight between entities represents the semantic relatedness. Γ * = arg max Γ N i=1   φ(mi, ei) + e j ∈Γ ψ(ei, ej)  (2) The challenge with the Ratinov's approach is that the unambiguous set of mention is not always obtainable beforehand. It happens commonly that all mentions within a document can be ambiguous due to noisy and unclear context. Therefore, authors in [7], [10] consider not only the unambiguous mentions but also the ambiguous ones. The supporting evidence to disambiguate a mention is collected from all other mentions. In details, suppose Sij(ei) is the support for label ei from mention mi, Sij(ei) is calculated as follows: Sij(ei) = max e j [φ(mj, ej) + ψ(ei, ej)](3) The disambiguated entity ei for mention mi is defined as follows: ei = arg max e i   φ(mi, ei) + N j=1,j =i Sij(ei)  (4) Interestingly, the work in [7] reveals that the best performance is obtained by considering evidence not from all mentions but only top-k supporting mentions. Furthermore, the authors also study the SINGLE-Link, which considers only the the most related entity into the coherence objective, expressed as follows: Γ * = arg max Γ N i=1 φ(mi, ei) + N max j=1 ψ(ei, ej)(5) In another work [11], fast collective linking is achieved by looking at only the neighbouring connections i.e., the previous and subsequent mentions. The associated objective function can be written as follows: Γ * = arg max Γ N i=1 φ(mi, ei) + N −1 i=1 ψ(ei, ei+1)(6) Dynamic programming, specifically Forward-Backward algorithm [12] (FwBw) is utilized to solve the optimization. Although the approach has shown its effectiveness for short text (i.e., query) [11], the coherence component is currently restricted to a local region. Therefore, it is challenging to simulate more complicated situations that is commonplace in long texts. Graph-based Approach Graph-based approaches solve the problem by performing collective linking on mention-entity graph. The graph is constructed with edges between mentions and their candidate entities weighted by score of local context matching φ(mi, ei), while edges between candidate entities reflect the semantic coherence ψ(ei, ej). An example of such graph is illustrated in Figure 2. Hoffart et al. [13] cast the joint mapping into the problem of identifying dense subgraph that contains exactly one mentionentity edge for each mention. Many other works are based on the Random Walk and PageRank algorithm [14], [15], [16], [17], [18], [19]. Specifically, authors in [20] introduce a new 'pseudo' topic node into the mention-entity graph to enforces the disambiguated entities to be consistent with the topic node's context. The node represents all unambiguous and already-disambiguated mentions. By DoSeR [20], Personalized PageRank is iteratively performed on the mention-entity graph with the newly introduced topic node. At each step, candidate entities with high stabilized scores will be assigned to its associated mentions and the entities are added into the pseudo topic node. The algorithm stops when all mentions are disambiguated (i.e., after few iterations). Although graphbased approaches are shown to produce robust and competitive performance, they are computationally expensive because the mention-entity graph may contain hundreds of vertices, in case of long documents. m i . Γ A mapping M → W representing a disambiguation outcome. φ(m i , e i ) Local confidence of linking m i to e i . ψ(e i , e j ) Pairwise coherence or semantic relatedness between two entities e i to e j . d(e i , e j ) Semantic distance between two entities e i to e j in the MINTREE coherence graph. Dicussion. Exiting studies on collective linking problem either propose an objective and its solution (e.g., the optimization based approaches) or directly approximate the problem (e.g., PageRank). There is no work to study the coherence structure of mentioned entities. Specifically, the research question is "to what extent the mentioned entities are related with each other? (by a specific relatedness measure)". To the best of our knowledge, we are the first to address this research problem. We also study a new tree-based objective to model the coherence between entities. PRELIMINARIES In this section, we give an overview of concepts and core components in disambiguation system. For ease of presentation we summarize primary notations used throughout this paper in Table 1. Given a document with a set of mentions to be linked, candidate entities for each mention are identified based on the mention's surface form. Collective linking works on the sets of candidates and selects for each memtion a candidate entity that optimizes the objective consists of local confidence φ(mi, ei) and the pairwise coherence ψ(ei, ej) (described in the section above). It is worth mentioning that the work in this paper does not focus on improving the local confidence or the semantic relatedness which is used to measure the pairwise coherence. In contrast, our work mainly focus on the study and evaluation of different collective linking models and solutions. In this section, we will detail the methods they are the state-of-the-art approaches for computing the local confidence and the semantic relatedness. First, we will describe the way we obtain word and entity embeddings being used in the two components. Word and Entity Embeddings. Embedding models aim to generate a continuous representation for every word, such that two words that are close in meaning are also close in the embedding vector space. It assumes that words are similar if they co-occur often with the same words [21]. Correspondingly, we can assume two entities to be semantically related if they are found in analogous contexts. The context is defined by the surrounding words or surrounding entities. Jointly modeling words and entities in the same continuous space have been shown to improve the quality of both word and entity embeddings [22] and benefit entity disambiguation task [4], [23]. We use the word2vec with skip-gram model [21] to jointly learn the distributional representation of words and entities. Let T denote the set of tokens. Token τ ∈ T can be either a word (e.g., Tiger, Wood) or an entityID (e.g., [Tiger Wood]). Suppose τ1, ..., τN is a given sequence, the model tries to maximize the following average log probability: L = 1 N N i=1 −c≤j≤c,j =0 log P (τi+j|τi)(7) where c is the size of context window, τi denotes the target token, and τi+j is a context token. The conditional probability P (τi+j|τi) is defined by the softmax function: P (τO|τI ) = exp(v τ O T vτ I ) τ ∈T exp(v τ T vτ I )(8) where vτ and v τ are the 'input' and 'output' vector representations of τ , respectively. After training, we use the 'output' v τ as the embedding for word or entity. To co-train word and entity embeddings, we create a 'token corpus' by exploiting the existing hyperlinks in Wikipedia. Specifically, for each sentence in Wikipedia which contains at least one hyperlink to another Wikipedia entry, we create an additional sentence by replacing each anchor text with its associated entityID. Then, for each Wikipedia page, we also create a 'pseudo sentence' which is the sequence of entityIDs linked from this page, in the order of their appearances. For Local confidence score φ(mi, ei). We adopt the approach proposed in [24] for estimating the matching score of the given mention (with its local context) and an candidate entity. A leaning to rank model by Gradient Boosting Tree is trained based on textual and embedding features to estimate the probability that a mention mi will be mapped to a candidate entity ei. The features to be used include the prior probability that an entity is linked given the mention's surface form P (e|m), several string similarity features between the mention's surface form and the entity's title, and finally the semantic similarity between the candidate entity and the mention surrounding context. The output obtained from the ranking model for each (mi, ei) will be used as the local confidence score. It is worth mentioning that there are more effective ways to model the local confidence with the use of deep neural networks [1], [25], [26]. However, since the modelling is not the focus of this work, we will use the most straightforward and efficient way for the estimation as described above. Pairwise coherence score (or relatedness measure) ψ(ei, ej). We study wide range of semantic similarity measures (ψ(ei, ej)) including the Wikipedia Links-based measure and the Entity Embedding similarity. The Wikipedia Link-based measure (WLM) [27] is widely used to estimate the coherence under the assumptions that two entities are related if there are many Wikipedia pages that link to both. The WLM score for two entities e1, e2 is calculated as follows: W LM (e1, e2) = 1 − log(max(|U1|, |U2|) + 1) − log(|U1 ∩ U2| + 1) log(|W | + 1) − log(min(|U1|, |U2|) + 1)(9) where |U1| and |U2| are the set of Wikipedia articles that have hyperlinks to e1 and e2 respectively, and W is the set of all Wikipedia articles. We also exploit Jaccard-like similarity. Different with the original formula in Guo et al. [28], here we take logarithm scale as it yields better results. The Normalized Jaccard Similarity (NJS) is then defined as follows: N JS(e1, e2) = log(|U1 ∩ U2| + 1) log(|U1 ∪ U2| + 1)(10) Furthermore, we study the entity embedding similarity (EES) which is the cosine similarity of the two representations: EES(e1, e2) = cos(embeding(e1), embeding(e2))(11) The embedding of entity is trained jointly with word's embedding taken from the Wikipedia corpus (details in [1]). Using the entity embedding to estimate the semantic relatedness has been shown to be effective for entity disambiguation in recent works [2], [4], [20]. ENTITATIVE COHERENCE IN DOCUMENT For the first time, we study the form of coherence between entities appear within a document, i.e., whether entities are densely or sparsely connected. As illustrated by the two examples in the introduction section, documents (in general) may contain non-salient entities or entities that do not have adequate connections in KB. Therefore, the basic assumption used by conventional collective linking approach that all the entities mentioned should be densely related leaves much to be desired. For a quantitative study, we aim to measure the denseness of coherence for 8 testing datasets (details about the datasets will be presented in Section 7.2). First, we define the measures for denseness of coherence given the entity relatedness graph G(V, E) which consists of all ground-truth entities mentioned in a document. The edges between every pair of entities are weighted by the semantic relatedness. To be comprehensive, we will analyse and report the result with all three relatedness measures: the Wikipedia link-based measures (WLM), the normalized Jaccard similarity (NJS) and the entity embedding (cosine) similarity (EES). Note that our intent is to measure the denseness (or sparseness) of the connections, not the degree of coherence. The degree of coherence can be estimated through the averaged weight of the relatedness graph. However, we are more interested in knowing the denseness of connections i.e., whether entities are densely or sparsely connected regardless the degree of coherence. Figure 3 illustrates four standard forms of coherence between mentioned entities. By focusing on the denseness, if all pairs of entities are coherent at the same degree (can be at high or low pairwise coherence scores), we would say the entities are densely connected (Figure 3a). On the other hand, if there are only few pairs dominate the pairwise coherence, we will view it as sparse (Figures 3d, 3b, 3c). The illustration hints that the denseness of coherence graph can be estimated through the averaged degree of a filtered graph G θ (V, E θ ) which only consists of edges between entities with highest pairwise relatedness scores (i.e., E θ = {e|e ∈ E ∧weight(e) ≥ θ}). To this end, we determine a dynamic threshold θ for each document as follows. The θ is chosen as a largest value such that every vertex (or entity) in V is incident to at least one edge in E θ . In other words, the associated filtered edges E θ is a valid edge cover 2 of the graph G. Finally, we calculate the average degree of G θ (V, E θ ) and refer it as the denseness of coherence for the entity set V. Denseness(V ) = Avg deg(G θ ) = 2 × |E θ | |V |(12) Note that the filtered graph G θ only contains highly related connections between entities. The average degree of G θ will reflect the density of the connections. Higher value means the entity set V is more densely connected while lower value indicates a sparse coherence within the entities. As illustrated in Figure 3d, if G θ is sparse (i.e., every entity is strongly related to only one other entity) its theoretical average degree is equal to 1. On the other hand, if entities in G θ are connected by tree-like or chain-like fashion (see Figures 3b, 3c), the theoretical coherence sparseness is 2 * (n − 1)/n. Furthermore, the expected value for densely connected case ( Figure 3a) is (n − 1) where n is the number of entities (or vertices). We report the coherence sparseness for 7 benchmark datasets in Table 2. We consider only the documents having at least 4 mentions because the ones with 3 or less mentions will lead to a fixed sparseness score by the calculation described above. As a result, since all RSS500 documents contain fewer than 4 linkable mentions, the dataset is ignored in the report. Table 2 shows, in general, the calculated values lie closer to tree (or chain) form's expected values rather than that of the dense form. The result is observable for three different relatedness measures (WLM, NJS and EES). Especially, for long documents like MSNBC and AQUAINT, each mentioned entity is highly related to only 3-5 other entities (with the NJS measure) although the number of entities in each document in the two datasets is more than 13 (on average). The result reveals that not all entities mentioned in a document are densely related to each other; therefore, considering all the pairwise connections are not necessary in an entity disambiguation model. Next, we define a new graph-based model that relaxes the ALL-Link coherence objective (Equation 1), allowing us to propose an fast and effective linking algorithm. MINIMUM SPANNING TREE REPRESENTATIVE We introduce MINTREE, a new tree-based objective to effectively model the entity disambiguation problem. Firstly, we define a new coherence measure for a set of entities. MINTREE coherence measure. Given an entity relatedness graph G(V, E) for a set of entity V, edges between every pair of entity are weighted by a specific semantic distance. The coherence of the graph G is the weight of the minimum-spanning tree (MST) that can be formed from G. The MINTREE coherence measure defined in this way relaxes the conventional ALL-Link-like measure which is the sum of all edge's weights in G. The measure will be used as an objective for the MINTREE model described as follows. MINTREE problem statement. Given N mentions and N subsets C1, ..., CN ⊂ W where each Ci represents the candidate entity set for mention mi, an undirected entity coherence graph G(V, E) is defined with the set of vertices V containing all the candidate entities in C1, ..., CN . Undirected edges between two candidates ei ∈ Ci and ej ∈ Cj (with i = j) are weighted by the semantic distance between the two associated entities. The distance is computed from their local confidence and pairwise coherence scores, defined as follows: d(ei, ej) = 1 − φ(mi, ei) + ψ(ei, ej) + φ(mj, ej) 3(13) The edge weight defined in this manner not only reflects the semantic relatedness distance between the two candidate entities but also encodes the local confidence for making a pair of assignments mi → ei and mj → ej. We aim to find in each subset Ci an entity ei such that the MINTREE coherence score of the selected entities Γ = {e1, ..., eN } is minimized. In other words, the MINTREE problem can be viewed as finding the minimum spanning tree on an N-partite graph G such that each of N subsets has one representative in the tree. Fig. 4. Entity coherence graph for a document with 4 mentions, each has 2 candidate entities. The weight of minimum-spanning tree obtained from the set selected entities will be used to measure the coherence of the entity set. ALL-Link - 0.986 -0.983 - 0.995 -0.994 - 0.989 -0.990 SINGLE-Link - -0.985 - -0.992 - -0.986 MINTREE - - - An illustration of MINTREE output is shown in Figure 4. In this example, the document contains 4 mentions and 4 associated sets of candidate entities. The disambiguated entity for each mention is highlighted (in red) and a sample of the spanning tree is illustrated by the solid edges. The weight of the spanning tree is used to measure the coherence of the selected entities. Using the MINTREE coherence measure has advantage of flexibility. It is capable of modelling complicated context such as sparse-content documents or social text whose documents may contain non-salient entities or entities that are not densely related in knowledge base. In the following section, we will present a quatitative study of MINTREE and show that it is as good as other conventional models in the disambiguation task. Quantitative study of MINTREE. It is undoubted that objective score of a coherence model should be correlated to the disambiguation quality. Specifically, for a set of disambiguated entities within a document, the MINTREE objective score has to be lowered as the number of correct disambiguation decisions increases. We simulate the disambiguation quality by considering N+1 disambiguation results having the number of correct assignment increasing from 0 to N. The details is described as follows: • The first disambiguation result has all mentions linking to wrong entities. • The second disambiguation result differs with the first result by having the first mention linking to its correct entity. • The k th (2 < k ≤ N + 1) result differs with the (k − 1) th result by having the (k − 1) th mention linking to its correct entity. We calculate the MINTREE objective score associated with each of the N+1 results. Finally, the Spearman's correlation is calculated from the list of objective scores and the numbers of correct decisions made in N+1 disambiguations. In the ideal case, the rank-based correlation should be equal to -1 because the MINTREE score should be inversely correlated with the disambiguation quality. We also analyse the Spearman's correlation with ALL-Link objective (Equation 1) as well as SINGLE-Link objective (Equation 5) in the same manner. Furthermore, to show that MINTREE is correlated with other objective models, we study the correlation between each pair of objectives. The results are reported in Table 3. It shows that the Spearman's correlation score between MINTREE and the disambiguation result is as high as other objectives. The score is about 0.92 for WLM measure and more than 0.94 for NJS and EES measures. Moreover, MINTREE is highly correlated to ALL-Link and SINGLE-Link with the pairwise correlation scores are more than 0.98 across different relatedness measure. Therefore, MINTREE is reasonably as good as other objective when being used to model the disambiguation quality. To end this, we want to note that the correlations between the objective score and the disambiguation quality by WLM measure are lower than ones by NJS and EES measures. As a result, we would expect NJS and EES to be more effective when being used as a relatedness measure for a collective linking algorithm. We will be back to this discussion in experiment section. Next, we will present Pair-Linking, an heuristic solution for the MINTREE problem. PAIR-LINKING Idea. Two well-known algorithms for finding minimum spanning tree (MST) in a general graph is Kruskal's [29] and Prim's [30]. However, the special setting of MINTREE problem makes the direct application of Kruskal's or Prim's becoming infeasible. In this section, we introduce Pair-Linking, a heuristic for finding the minimum spanning tree representative in the MINTREE problem. Similar to the Kruskal's algorithm, the main idea of Pair-Linking is iteratively taking an edge with smallest distance into consideration. Specifically, Pair-Linking works on the entity coherence graph G (see the problem statement, Section 5). It iteratively takes an edge of the least possible distance that connects two entities e x i , e y j (in two candidate sets Ci and Cj respectively) to form the tree. The difference with the original Kruskal's algorithm is that after e x i is selected, Pair-Linking removes other vertex ex i from G such that ex i = e x i ∧ ex i ∈ Ci. Similar removal is done with e y j . The removing steps will ensure there will be no other entities within the same candidate set being selected. The algorithm stops when every candidate set has one entity being selected. Intuitively, each step of Pair-linking aims to find and resolve the most confident pair of mentions (represented by the least weighted edge on the entity coherence graph G). Furthermore, once the the edge (e x i , e y j ) is selected, it implies that the mentions mi and mj are disambiguated to the entities e x i and e y j respectively. Note that our Pair-Linking algorithm approximates MINTREE solution by simulating the Kruskal's but not the Prim's algorithms. The reason is twofold. First, instead of building the MST by merging smaller trees (like Kruskal's algorithm), Prim's grows the tree from a root. However, the strategy is less effective than Kruskal's in the entity disambiguation task because (Kruskal-like) Pair-linking performs disambiguation by the order of confidence score, enforcing the subsequent and less confident decisions to be consistent with previously made and more confident assignments. The strategy has also been used in other works [9], [20], [31] and been shown to improve the disambiguation performance noticeably. Another advantage of Kruskal-like over Prim-like approach is that in the case of not well-connected (sparse) coherence graph, the output from Kruskal-like Pair-Linking will contain multiple coherent trees (see Figure 3d). Therefore, it is capable of modelling the sparse and noisy context. Pair-Linking example. An example of Pair-Linking outcome is illustrated in Figure 5. In the example, the given document consists of 5 mentions, each has 2 candidate entities. Edges between entities are weighted by the semantic distance. For the ease of presentation, only the ones with lowest distance (i.e., highest pairwise confidence) are shown. Pair-Linking will traverse through the list of edges by the sorted order of their weights and its disambiguation step is described as follows. In the first step, Pair-Linking considers the edge with the lowest semantic distance (e 2 1 , e 2 2 ) and make a pair of linkings with highest confidence m1 → e 2 1 and m2 → e 2 2 . The edge with second lowest semantic distance is (e 1 2 , e 1 3 ). However, since m2 is already disambiguated (to e 2 2 ), any entity other than e 2 2 is removed from m2's candidates, including its edge. Therefore, the next edge to be considered becomes (e 1 4 , e 1 5 ). As a result, m4 and m5 are disambiguated to e 1 4 and e 1 5 respectively. Lastly, (e 1 3 , e 1 4 ) is taken into account and one additional linking is made i.e., m3 → e 1 3 . Pair-Linking stops at this step because all the 5 mentions are already disambiguated to its associated entities (highlighted in red in Figure 5). Note that in this example, the selected entities are separated into two well-connected components. Therefore, the input document may be context-sparse. While it is challenging to apply the traditional ALL-Link objective in this situation, by relaxing the pairwise coherence objective using the tree-based model, Pair-Linking can be used to derive the disambiguation effectively. Pair-Linking procedure. We detail Pair-Linking procedure in Algorithm 1. Specifically, Pair-Linking maintains a priority queue Q and each element Qm i ,m j tracks the most confident linking pairs involving mentions mi and mj. Qm i ,m j is initialized by calling function top pair(mi, Ci, mj, Cj), where Ci is the set of candidate entities that mention mi can link to. The function returns a pair assignment mi → e x i and mj → e y j , such that e x i ∈ Ci, e y j ∈ Cj, and the confidence score of the pair assignment is the highest among Ci ×Cj (i.e., the edge distance is the smallest according to Equation 13). After initialization, Pair-Linking iteratively retrieves the most confident pair assignment from Q (Line 7) and links the pair of mentions to the associated entities (Lines 8-9). Then, Pair-Linking updates Q, more precisely, Qm k ,m i and Qm k ,m j (Lines 10-13). For Qm k ,m i , the possible pairs of assignments between m k and mi are now conditioned by mi → e x i , and the same applies to Qm k ,m j . Algorithm 1: Pair-Linking algorithm input : N mentions (m1, ..., mN ). Mention mi has candidate set {Ci ⊂ W } output: Γ = (e1, ..., eN ) 1 ei ← null, ∀ei ∈ Γ 2 for each pair (mi, mj) ∧ mi = mj do 3 Qm i ,m j ← top_pair(mi, Ci, mj, Cj) Speed up with early stop. The most expensive part of the algorithm is the initialization of Q which requires to compute top pair between every two mentions. A straightforward implementation of the function top pair(mi, Ci, mj, Cj) will scan through all possible candidate pairs between the two mentions which requires O k 2 where k is the number of candidates per mention. This leads to an overall complexity of O N 2 k 2 for the Q's initialization (Lines 2-5), where N is number of mentions. However, since only the pair of candidates with the highest confidence score is recorded for a pair of mentions mi and mj, Pair-Linking uses early stop to prevent checking all possible candidate pairs in Ci × Cj. Specifically, it sorts each of N candidate set by the local scores (O (N k log k)) and traverses the sorted list in descending order. Early stop is applied if the current score is worse than the highest score by a specific margin, i.e., the largest possible value of ψ(ei, ej), see Equation 13. In the best case, if early stop is applied right after getting the first score, the complexity of top pair(mi, Ci, mj, Cj) is O (1) and the overall time complexity becomes O N 2 + N k log k . Indeed, early stop significantly reduces the running time of Pair-Linking in practice while still maintaining the correctness of the algorithm. EXPERIMENT We use Wikipedia dump on 01-Jul-2016 as the target knowledge base which consists of 5,187,458 entities. In the following subsections, we will describe the experiment setting protocol, datasets and methods in comparison. Lastly, we present and discuss the experiment results. Experimental Setting Candidate Generation and Filtering. As a common approach [2], [5], [20], our candidate generation is purely based on the textual similarity between a mention's surface form and an entity's title including all its variants. We used the name dictionary based techniques for candidate retrieval [5]. The dictionary is built by exploiting entity titles, anchor texts, redirect pages, and disambiguation pages in Wikipedia. If a given mention does not present in the dictionary, we use its n-grams to retrieve the candidates. We further improve the recall of candidate generation by correcting the mention's boundary. In several situations, a given mention may contain trivial words (e.g., the, Mr., CEO, president) that are not indexed by the dictionary. We use an off-the-shelf Named Entity Recognizer (NER) 3 to refine the mention's boundary in these cases. As in [32], we also utilize the NER output to expand the mention's surface form. Specifically, if mention m1 appears before m2 and m1 contains m2 as a substring, we consider m1 as an expanded form of m2, and candidates of m1 will be included to the candidate set of m2. We train a Gradient Boosted Regression Trees model [33] as the candidate ranker to reduce the size of candidate set. For each pair of (mention, candidate) i.e., (m, e), we use the following statistical and lexical features for ranking. • Prior probability P (e|m). P (e|m) is the likelihood that the mention with surface form m being mapped to entity e. P (e|m) is pre-calculated based on the hyperlinks in Wikipedia. • String similarity. We use several string similarity measures including: (i) edit distance, (ii) whether mention m exactly matches entity e's name, (iii) whether m is a prefix or suffix of the entity name, and (iv) whether m is an abbreviation of the entity name. Note that the string similarity features are calculated for the original mention as well as the boundary-corrected mention and the expanded mention described earlier. We use the IITB labeled dataset [34] to train the ranker and take the top 20 scored entities as the final candidate set for each mention. Taking fewer candidates per mention will lead to low recall while using more candidates degrades disambiguation accuracy in later step. Similar observations are also reported in [3], [20]. Note that the candidate ranker described above is different from the model used to estimate the local confidence score presented in Section 3. The former aims to maximize the recall of top-k ranked candidates while the latter targets on the accuracy of prediction, i.e., the top-1 ranked candidate. 3. We used the Standford NER tool in this work. Local confidence score and pairwise coherence score. We use the the local score which is the output of a learning to rank model (see Section 3). Furthermore, for pairwise coherence, we study and report the results with three kinds of measures: Wikipedia link-based measure (WLM), normalized Jaccard similarity (SNS) and entity embedding similarity (EES). In addition, we use a hyper parameter β to control the contribution between local confidence and pairwise coherence components in the final objective. For example, the refined objective for Equation 1 can be written as follows: Γ * = arg max Γ   (1 − β) N i=1 φ(mi, ei) + β N i=1 N j=1,j =i ψ(ei, ej)   (14) Cross validation. We use 5-fold cross validation to evaluate and report the results. For each dataset, the learning to rank model GBT and the β parameter are learnt on 4 training partitions and the best setting are then used to perform disambiguation on the remaining test partition. The final disambiguation result on a dataset is the aggregation of the predictions on 5 test partitions. Datasets and Methods in Comparison Datasets. We evaluate on 8 benchmark datasets from different domains, including short and long text, formal and informal text (details in Table 4). Note that, we only consider the mentions whose linked entities present in the Wikipedia dump; the same setting has been used in [2], [3], [4], [20]. We describe each dataset as follows: • Reuters128 [35] contains 128 economic news articles taken from the Reuters-21587 corpus. There are 111 documents containing linkable mentions (based on the KG from the Wikipedia 01-Jul-2016 dump). • ACE2004 [9] is a subset of ACE2004 co-reference documents annotated by Amazon Mechanical Turk. It has 35 documents, each has 7 mentions on average. • MSNBC [36] is created from MSNBC news articles. It contains 20 documents, each has 33 mentions on average. The dataset includes many entities that can be linked via direct relation in DBpedia. Therefore, many disambiguation systems can easily achieve high accuracy on this dataset. • DBpedia Spotlight (DBpedia) is a news corpus and contains many non-named entity mentions such as parents, car, dance. It is an average-size dataset in which each document contains 5 to 6 mentions on average. • RSS500 [37] is RSS feeds -a short formal text collection covers a wide range of topics e.g., world, business, science, etc. The dataset is one of N3 datasets [35] which are carefully created as a benchmark for named entity disambiguation system. • KORE50 [38] contains 50 short sentences on various topics e.g., music, celebrities, and business. Most mentions are first names referring to persons with high level of ambiguity. It is considered as a challenging dataset for any disambiguation system. • Microposts2014 (Micro2014) [39] is a collection of tweets, introduced in the 'Making Sense of Microposts 2014' challenge. The textual context for a document is very limited and noisy due to the nature of tweet. The dataset has train/test partitions. We use the test partition here so that our results can be compared with others. • AQUAINT [27] contains 50 news documents from Xinhua News Service, the New York Times and Associated Press news corpus. Collective linking methods. We compare our Pair-Linking algorithm with the following state-of-the-art collective linking (CL) algorithms. • Iterative Substitution (Itr Sub (AL)) [6] is an approximate solution for the ALL-Link objective (Equation 1). The mentions are first assigned to the candidates with highest local scores. Then, the algorithm iteratively substitutes an assignment mi → e x i with another mapping mi → e y j as long as it improves the objective score. We also study the performance of Iterative Substitution with the Sing-Link objective (Equation 5) and refer it as IterSub (SL). • Loopy Belief Propagation. (LBP(AL)) [3], [7] solves the inference problem (Equation 1) through loopy belief propagation technique [8]. Similar to the Iterative Substitution algorithm, we also study another setting with SINGLE-Link objective and refer it as LBP(SL). • Forward-Backward (FwBw) algorithm [12] limits the topical coherence to consider only the adjacent linkings of a mention and uses dynamic programming to derive the optimal assignments. The work in [11] shows that the approach is effective and efficient with the task of entity extraction in query. • Personalized PageRank (PageRank) used by DoSeR [20], it performs personalized PageRank on the mention-candidate graph and uses the stabilized scores for disambiguation. By introducing the 'pseudo' topic node, DoSeR enforces the coherence between disambiguated entities and the main topic's context therefore improves the disambiguation accuracy. We acknowledge a relevant work in [7] also addresses the issue of mentioned entities that are not salient or not well-connected in KB. The authors propose a model that considers only top-k most related connections for each entity to perform collective linking. However, the model is trained in end-to-end fashion together with the parameters for local confidence and coherence scores. In contrast, our work only focuses on the collective linking component and uses existing local similarity and pairwise coherence measures. Therefore a comparison to their work is not included in our study. Evaluation measures. To evaluate performance of different collective linking methods, we use the Gerbil benchmarking framework [40] (Version 1.2.4) with a fixed local setting for fair comparison. We evaluate the disambiguation results by the widely used measures: Precision, Recall and F1. Specifically, let Γg be the set of groundtruth assignments mi → e k i and Γ * be the linkings produced by a disambiguation system, The precision is computed as the fraction of correctly linked entity mentions that are generated by the system: P recision P = |Γ * ∩ Γg| |Γ * | Precision considers all entity mentions that are produced by the system and determines how correct entity mentions linked by the entity linking system are. On the other hand, Recall takes into account all entity mentions that should be linked and determines how correct linked entity mentions are with regard to total entity mentions that should be linked. Recall therefore is expressed as follows: Recall R = |Γ * ∩ Γg| |Γg| Precision and Recall are usually used together in F1-measure to provide a single measurement for a system. F1-measure is defined as the harmonic mean of precision and recall: F 1 = 2 × P × R P + R For all the measures, we report the micro-averaged score (i.e., aggregated across mentions not documents), and refer the micro-averaged F 1 as the main metric for evaluation. Result and Discussion Collective linking performance. We study the performance of different collective linking algorithms with different coherence measures. The results are listed in Table 5. Note that in this experiment, we use a Gradient Boosting model to estimate the local confidence score (see Section 3). This is different from our previous work [1] where we utilize a deep neural network model for the estimation. Therefore the results in this table are slightly different from the former one. As illustrated in Table 5, the coherence measure significantly affects the performance of all collective linking algorithms. The Normalized Jaccard Similarity (NJS) and entity embedding similarity (EES) are shown to be more effective than the Wikipedia Link-based Measure (WLM). We further combine the two former measures (i.e., NJD and EES) by taking the average of their coherence scores and the combined schema works best among other individual coherence measures (see Table 6d). The approximation algorithm Loopy Belief Propagation (LBP) is consistently better than the Iterative Substitution in both two objective settings ALL-Link (AL) and SINGLE-Link (SL). Furthermore, considering the two objective settings ALL-Link and SINGLE-Link, the Iterative Substitution optimizing toward the two objectives give comparable results across different coherence measures. Similar observation is obtained with the performance of LBP (AL) and LBP (SL). Graph based algorithm like PageRank is sensitive to the coherence measure. It only produces good results when working with the NJS coherence measure, i.e., 0.825 F1 score versus 0.744 and 0.789 when working with WLM and EES measure, respectively. On the other hand, Pair-Linking is quite robust to all three measures. It outperforms other methods on more challenging and short text datasets such as Reuters128, RSS500 and KORE50. Forward-Backward algorithm (FwBw) is shown to perform better with short text datasets (RSS and Micro2014) than long text datasets (Reuters and AQUAINT). The reason is that for long documents, the useful evidence may not be presented in neighbouring mentions but in the distant ones. Therefore considering only the adjacent mentions does not always provide sufficient context to disambiguate correctly. Table 7. FwBw has the lowest time complexity in worst case since it only considers adjacent mentions. By using dynamic programming [12], suppose the last mention mi is disambiguated to entity ei, FwBw calculates the associated objective score of the assignment by considering all possible states of the previous decision (i.e., mi−1 → ei−1), resulting in the complexity of O (k) where k is the number of candidate entities per mention. Therefore, the overall time complexity of FwBw is O N k 2 where N is the number of mentions. Not surprisingly, Itr Sub, LBP and PageRank have highest time complexity. While Itr Sub and LBP require multiple iterations to refine assignments, PageRank iteratively operates on the mention-entity matrix for convergence leading to the complexity of O I × N 2 k 2 with I is the number of iterations required. On the other hand, Pair-Linking only needs to traverse all possible pairs of linking assignment (i.e., (mi, ti), (mj, tj)) at most once which results in the complexity of O N 2 k 2 . Furthermore, the worst case of Pair-Linking is the prerequisite of any graph-based algorithm (e.g., PageRank) because building the mention-entity graph for N mentions, each has k candidate entities will require N k vertices and N 2 k 2 edges. It is also worth to mention that Pair-Linking is interested in only pairs of linking assignments with highest confident scores. Therefore, by using a priority queue to keep track of the top confident pairs, it can avoid traversing through every pair at each step. Empirical results show that Pair-Linking is indeed fast, partially due to "early stop" in implementation described in Section 6. Since only few pairs of assignments dominate the Pair-Linking scores, a large number of pairs are ignored by early stop. Table 6 shows that the running time of Pair-Linking is even faster than FwBw on 6 out of 8 datasets, making Pair-Linking the most effective and efficient linking algorithm. LBP O(N 2 k 2 ) O(I×N 2 k 2 ) FwBw O(N k 2 ) O(N k 2 ) PageRank O(N 2 k 2 ) O(I×N 2 k 2 ) Pair-Linking O(N 2 +N k log k) O(N 2 k 2 ) Considering the long text, the most time consuming dataset is MSNBC, where on average there are 32.9 mentions per document. Among all algorithms that consider all mentions in the same document for collective linking, Pair-Linking is nearly 50-100 times faster than the next efficient algorithm LBP(AL), as shown in Table 6. FwBw is faster than Pair-Linking but it does not consider all mentions in the same document for collective linking, and its linking accuracy is worse than Pair-Linking (see Table 6c). Comparison with other disambiguation systems. We compare the disambiguation performance of the best setting of Pair-Linking (the one employs the combined NJS&EES coherence measure) with other state-of-the-art disambiguation systems described as follows: • PBoH [3] is a light-weight disambiguator which is based on probabilistic graphical model and loopy belief propagation to perform collective linking. The model utilizes Wikipedia statistics about the co-occurrence of words and entities to compute the local matching and pairwise coherence scores. • DoSeR [20] carefully designs the collective disambiguation algorithm using Personalized PageRank on the mention-candidate graph. The edge weight is the cosine similarity between context and entity embeddings. DoSeR heavily relies on the collective linking algorithm to produce good results. Additionally, we report the results of two simple baselines. One is the prior probability model P (e|m). It simply disambiguates a mention bases on the statistics from Wikipedia hyperlinks. The other baseline is the learning to rank Gradient Boosting Tree (GBT) model which uses only the local confidence score for ranking and selecting candidates. In both baselines, each mention is disambiguated in isolation with other mentions. Therefore, the two can be viewed as local (non-collective) disambiguation models . Pair-Linking's performance is detailed in Table 8 and the comparison with other systems are shown in Table 9. Note that some results of DoSeR and PBoH are slightly different from the ones reported in their original papers [3], [20]. The reason is that Gerbil (Version 1.2.4) has improved the entity matching and entity validation procedures to adapt to the knowledge base's changes over time. 4 Pair-Linking performs quite well on short text, i.e., RSS500, KORE50, Micro2014. For the most challenging dataset KORE50, Pair-Linking improves the disambiguation performance by 0.30 F1 compared to the local approach P (e|m) which disambiguates based on only the local context. Furthermore, Pair-Linking also outperforms PBoH by 0.14 F1 score on the same dataset. Overall, Pair-Linking outperforms the second best disambiguator DoSeR by a large margin (0.045 F1 score) making Pair-Linking a new state-of-the-art entity disambiguation system. 4. http://svn.aksw.org/papers/2016/ISWC Gerbil Update/public.pdf In this work, we do not consider the case where a mention refers to a not-in-link (NIL) entity (i.e., the entity that does not present in the given knowledge base). One possible solution to detect the NIL mention is to base on the local confidence score. Specifically, a mention is considered to map to NIL entity if the highest local confidence score among its candidates is less than a predefined threshold. However, since the performance of the threshold-based approach relies on the local confidence modelling which is not the focus of our study, we do not involve the NIL detection in this paper. Instead, we will address a more interesting research question: "How robust is Pair-Linking if NIL mentions are presenting in a document?". Specifically, for each document, we randomly sample few mentions and remove the ground-truth entities from their candidate sets. We report the disambiguation performance of Pair-Linking with the new setting. Note that in this experiment, we only consider medium-to-long text document which contains sufficient number of mentions and the performance is measured only with the linkable mentions. As reported in Table 10, the presence of NIL mentions does not degrade the performance of Pair-Linking for other linkable mentions, even in the case that 60% mentions are NIL. The robust disambiguation performance of our algorithm can be explained as follows. Since the local confidence of a NIL-mention and its candidate is usually low, any pair of linking assignment involving the NIL-mention will have low confidence score. As a result, the pair will be selected at the latest in the procedure of Pair-Linking (see Section 6). Therefore, assignment of the NIL-mention is not likely to affect the results of linking other mentions. CONCLUSIONS In this work, we study the collective entity disambiguation problem. While conventional approaches assume that all entities mentioned in a document should be densely related, our study reveals low degree of coherence is not occasional in general text (news, tweet, RSS). We propose MINTREE, a new tree-based collective linking model that utilizes the weight of minimum spanning tree to measure the coherence in entity graph. Using the tree-based objective allows us to model sparse and noisy context effectively. Furthermore, we also show that MINTREE is highly correlated to previously introduced collective linking models therefore it can be used as an replacement. Finally, we introduce Pair-Linking, an approximate solution for the MINTREE optimization problem. Despite being simple, Pair-Linking performs notably fast and achieves comparable (even better in some cases) accuracy in comparison to other algorithms. The superior performance of Pair-Link on 8 benchmark datasets makes it a new state-of-the-art collective linking algorithm. Fig. 1 . 1Illustration of coherence between linked entities in two examples. The edge represents strong semantic relatedness between the two associated entities. 's rank-order correlations between the disambiguation quality (represented by the number of correct linking decisions) and three objective scores. The correlations are averaged across datasets. The results are reported with three relatedness measures: Wikipedia Link-based Measure (WLM), Normalized Jaccard Similarity (NJS) and Entity Embedding Similarity (EES). Spearman's Correlation WLM NJS EES ALL-L SINGLE-L MINTREE ALL-L SINGLE-L MINTREE ALL-L SINGLE- Fig. 5 . 5An example of entity coherence graph for 5 mentions, each have 2 candidates. There are edges between candidate entities that are weighted by the semantic distance. However, only edges with lowest semantic distances are illustrated. The solid edges are the ones selected by Pair-Linking. ej (Disambiguate mj to e x i ) 10 for k := 1 → N ∧ e k = null do11 Qm k ,m i ← top_pair(m k , C k , mi, {ei})12 Qm k ,m j ← top_pair(m k , C k , mj, {ej}) linking running time. The theoretical time complexities of different collective linking methods are listed in TABLE 1 1Frequently used notations.The k th entity candidate of mentionNotation Definition and description M List of mentions to be linked in a document. m i The i th mention in M. W Set of all entity in knowledge base (we use Wikipedia in this work). e i An entities in W being assigned to mention m i . N Number of mentions given in a document. k Number of candidate entities each mention has. C i List of candidate entities for mention m i . e k i example, assume that the Wikipedia page about Tiger Wood contains only 2 sentences: "Woods [Tiger Woods] was born in Cypress [Cypress, California] . He has a niece, Cheyenne Woods [Cheyenne Woods] .", the following sentences are in our 'token corpus'.(a) Dense (b) Tree-like (c) Chain-like (d) Forest-like Fig. 3. Illustration of four different levels of denseness in entity coherence graph. • Wood was born in Cypress. He has a niece, Cheyenne Woods. • [Tiger Woods] was born in [Cypress, California]. He has a niece, [Cheyenne Woods]. • [Tiger Woods] [Cypress, California] [Cheyenne Woods]. TABLE 2 2Average coherence sparseness calculated for each dataset. Only documents whose number of mentions greater than 3 are considered. The results are reported with three relatedness measures: Wikipedia Link-based Measure (WLM), Normalized Jaccard Similarity (NJS) and Entity Embedding Similarity (EES).Dataset |D| Coh deg (theoretical) Coh deg (calculated) Forest Tree Dense WLM NJS EES Reuters128 30 1.00 1.64 5.93 3.21 2.13 2.68 ACE2004 25 1.00 1.69 7.20 3.23 2.83 2.75 MSNBC 19 1.00 1.83 14.89 6.35 4.48 7.08 Dbpedia 35 1.00 1.71 6.60 3.08 2.55 2.92 KORE50 9 1.00 1.54 3.44 1.36 1.58 1.36 Micro2014 80 1.00 1.53 3.33 1.81 1.72 1.82 AQUAINT 50 1.00 1.84 12.82 5.78 3.39 4.53 TABLE 4 4Statistics of the 8 test datasets used in experiments. |D|, |M |, Avgm, and Length are number of documents, number of mentions, average number of mentions per document, and document length in number of words, respectively.Dataset Type |D| |M | Avgm Length Reuters128 news 111 637 5.74 136 ACE2004 news 35 257 7.34 375 MSNBC news 20 658 32.90 544 DBpedia news 57 331 5.81 29 RSS500 RSS-feeds 343 518 1.51 30 KORE50 short sentences 50 144 2.88 12 Micro2014 tweets 696 1457 2.09 18 AQUAINT news 50 726 14.52 220 TABLE 5 5Micro-averaged F1 of different collective linking algorithms with different coherence measures. The best scores are in boldface and the second-best are underlined. The number of win and runner-up each method performs across different datasets are also illustrated.(d) Combination of NJS&EES as coherence measure.CL Method Reuters128 ACE2004 MSNBC Dbpedia RSS500 KORE50 Micro2014 AQUAINT Average #1st #2nd Iter Sub(AL) 0.795 0.873 0.809 0.821 0.775 0.506 0.798 0.857 0.779 0 1 Iter Sub(SL) 0.778 0.849 0.874 0.827 0.758 0.484 0.794 0.849 0.777 1 0 LBP(AL) 0.800 0.867 0.847 0.837 0.776 0.487 0.798 0.855 0.783 0 2 LBP(SL) 0.793 0.865 0.850 0.828 0.772 0.496 0.805 0.868 0.785 2 0 FwBw 0.788 0.876 0.850 0.844 0.772 0.526 0.799 0.859 0.789 2 2 PageRank 0.767 0.832 0.791 0.722 0.769 0.490 0.772 0.812 0.744 0 0 Pair-Linking 0.802 0.871 0.864 0.842 0.785 0.535 0.796 0.862 0.795 3 3 (a) WLM as coherence measure. CL Method Reuters128 ACE2004 MSNBC Dbpedia RSS500 KORE50 Micro2014 AQUAINT Average #1st #2nd Iter Sub(AL) 0.840 0.877 0.882 0.810 0.783 0.689 0.814 0.869 0.821 0 1 Iter Sub(SL) 0.821 0.876 0.878 0.812 0.795 0.671 0.812 0.859 0.815 0 0 LBP(AL) 0.839 0.883 0.883 0.825 0.790 0.728 0.812 0.871 0.829 0 2 LBP(SL) 0.813 0.886 0.886 0.833 0.788 0.726 0.818 0.868 0.827 1 3 FwBw 0.813 0.883 0.870 0.849 0.792 0.728 0.815 0.869 0.827 1 2 PageRank 0.835 0.897 0.864 0.833 0.783 0.707 0.808 0.875 0.825 2 1 Pair-Linking 0.846 0.876 0.892 0.831 0.797 0.764 0.814 0.870 0.836 4 0 (b) NJS as coherence measure. CL Method Reuters128 ACE2004 MSNBC Dbpedia RSS500 KORE50 Micro2014 AQUAINT Average #1st #2nd Iter Sub(AL) 0.852 0.905 0.875 0.837 0.795 0.556 0.806 0.872 0.812 1 1 Iter Sub(SL) 0.807 0.871 0.864 0.820 0.801 0.565 0.809 0.860 0.800 0 1 LBP(AL) 0.852 0.884 0.897 0.851 0.801 0.581 0.809 0.877 0.819 2 3 LBP(SL) 0.846 0.889 0.882 0.836 0.802 0.631 0.817 0.872 0.822 1 2 FwBw 0.834 0.885 0.891 0.850 0.805 0.587 0.809 0.870 0.816 0 3 PageRank 0.817 0.874 0.877 0.827 0.768 0.503 0.790 0.860 0.789 0 0 Pair-Linking 0.856 0.879 0.894 0.846 0.806 0.637 0.817 0.885 0.827 5 1 (c) Entity Embedding Similarity (EES) as coherence measure. CL Method Reuters128 ACE2004 MSNBC Dbpedia RSS500 KORE50 Micro2014 AQUAINT Average #1st #2nd Iter Sub(AL) 0.856 0.894 0.879 0.839 0.793 0.682 0.811 0.876 0.829 0 1 Iter Sub(SL) 0.807 0.883 0.870 0.835 0.809 0.653 0.808 0.850 0.814 0 0 LBP(AL) 0.864 0.861 0.895 0.833 0.777 0.715 0.822 0.877 0.831 1 2 LBP(SL) 0.823 0.875 0.900 0.843 0.814 0.762 0.824 0.872 0.839 1 3 FwBw 0.830 0.895 0.905 0.832 0.802 0.749 0.818 0.866 0.837 1 1 PageRank 0.837 0.882 0.888 0.822 0.785 0.512 0.797 0.872 0.799 0 0 Pair-Linking 0.859 0.883 0.910 0.845 0.823 0.787 0.813 0.879 0.850 5 1 TABLE 6 6Average time to disambiguate mentions in one document (in milisecond) for each dataset (lower is better). The number of win and runner-up each method performs across different datasets are also illustrated.CL method Reuters128 ACE2004 MSNBC Dbpedia RSS500 KORE50 Micro2014 AQUAINT #1st #2nd Iter Sub(AL) 97.515 21.369 3010.214 12.922 0.127 2.235 0.682 293.271 0 0 Iter Sub(SL) 67.772 20.183 3211.341 11.603 0.108 2.284 0.684 107.640 0 0 LBP(AL) 40.049 41.911 1584.504 42.673 0.331 11.515 3.667 269.854 0 0 LBP(SL) 92.625 43.173 4421.172 44.263 0.289 8.627 3.170 403.140 0 0 FwBw 0.940 1.975 8.880 2.034 0.103 1.190 0.367 4.959 2 6 PageRank 110.572 77.398 4293.670 132.009 5.436 64.982 15.796 375.239 0 0 Pair-Linking 1.721 0.590 28.699 0.491 0.025 0.951 0.117 3.105 6 2 TABLE 7 7Time complexity of linking algorithms, where N is number of mentions, k is average number of candidates per mention, and I is number of iterations for convergence.Collective Linking Best case Worst case ItrSub O(N 3 k) O(I×N 3 k) TABLE 8 8Micro-averaged Precision, Recall, F 1 score of Pair-Linking with the combined NJS&EES as coherence measure.Data set Precision Recall F1 Reuters128 0.866 0.853 0.859 ACE2004 0.888 0.877 0.883 MSNBC 0.910 0.910 0.910 Dbpedia 0.847 0.842 0.845 RSS500 0.823 0.823 0.823 KORE50 0.787 0.787 0.787 Micro2014 0.820 0.806 0.813 AQUAINT 0.882 0.875 0.879 TABLE 9 9Micro-averaged F 1 of Pair-Linking with the combined NJS&EES coherence measure versus other disambiguation systems. The best results are in boldface and the second-best are underlined.TABLE 10 Micro-averaged F 1 performance of Pair-Linking (with NJS&EES as coherence measure) with four different percentage of NIL-mention settings. The F 1 score is calculated only with the linkable mentions. 7.3.4 Discussion about the NIL mention.System Reuters128 ACE2004 MSNBC Dbpedia RSS500 KORE50 Micro2014 AQUAINT Average PBoH [3] 0.759 0.876 0.897 0.791 0.711 0.646 0.725 0.841 0.781 DoSeR [20] 0.873 0.921 0.912 0.816 0.762 0.550 0.756 0.847 0.805 P (e|m) (local) 0.697 0.861 0.781 0.752 0.702 0.354 0.650 0.835 0.704 Xgb (local) 0.776 0.872 0.834 0.818 0.756 0.496 0.789 0.855 0.775 Pair-Linking (NJS&EES) 0.859 0.883 0.910 0.845 0.823 0.787 0.813 0.879 0.850 Dataset 0% 20% 40% 60% Reuters128 0.859 0.842 0.850 0.848 ACE2004 0.883 0.879 0.900 0.869 MSNBC 0.910 0.890 0.887 0.893 AQUAINT 0.879 0.873 0.875 0.863 Neupl: Attention-based semantic matching and pair-linking for entity disambiguation. M C Phan, A Sun, Y Tay, J Han, C Li, CIKM. M. C. Phan, A. Sun, Y. Tay, J. Han, and C. Li, "Neupl: Attention-based semantic matching and pair-linking for entity disambiguation," in CIKM, 2017. Lightweight multilingual entity extraction and linking. A Pappu, R Blanco, Y Mehdad, A Stent, K Thadani, WSDM. A. Pappu, R. Blanco, Y. Mehdad, A. Stent, and K. Thadani, "Lightweight multilingual entity extraction and linking," in WSDM, 2017. Probabilistic bag-of-hyperlinks model for entity linking. O Ganea, M Ganea, A Lucchi, C Eickhoff, T Hofmann, WWWO. Ganea, M. Ganea, A. Lucchi, C. Eickhoff, and T. Hofmann, "Probabilistic bag-of-hyperlinks model for entity linking," in WWW, 2016, pp. 927-938. Joint learning of the embedding of words and entities for named entity disambiguation. I Yamada, H Shindo, H Takeda, Y Takefuji, CoNLLI. Yamada, H. Shindo, H. Takeda, and Y. Takefuji, "Joint learning of the embedding of words and entities for named entity disambiguation," in CoNLL, 2016. Entity linking with a knowledge base: Issues, techniques, and solutions. W Shen, J Wang, J Han, IEEE TKDE. 272W. Shen, J. Wang, and J. Han, "Entity linking with a knowledge base: Issues, techniques, and solutions," IEEE TKDE, vol. 27, no. 2, pp. 443-460, 2015. LIEGE: : link entities in web lists with knowledge base. W Shen, J Wang, P Luo, M Wang, SIGKDD. W. Shen, J. Wang, P. Luo, and M. Wang, "LIEGE: : link entities in web lists with knowledge base," in SIGKDD, 2012, pp. 1424-1432. Collective entity resolution with multi-focal attention. A Globerson, N Lazic, S Chakrabarti, A Subramanya, M Ringgaard, F Pereira, ACL. A. Globerson, N. Lazic, S. Chakrabarti, A. Subramanya, M. Ringgaard, and F. Pereira, "Collective entity resolution with multi-focal attention," in ACL, 2016. Loopy belief propagation for approximate inference: An empirical study. K P Murphy, Y Weiss, M I Jordan, UAI. K. P. Murphy, Y. Weiss, and M. I. Jordan, "Loopy belief propagation for approximate inference: An empirical study," in UAI, 1999, pp. 467-475. Local and global algorithms for disambiguation to wikipedia. L Ratinov, D Roth, D Downey, M Anderson, ACL. L. Ratinov, D. Roth, D. Downey, and M. Anderson, "Local and global algorithms for disambiguation to wikipedia," in ACL, 2011, pp. 1375-1384. TAGME: on-the-fly annotation of short text fragments (by wikipedia entities). P Ferragina, U Scaiella, CIKM. P. Ferragina and U. Scaiella, "TAGME: on-the-fly annotation of short text fragments (by wikipedia entities)," in CIKM, 2010, pp. 1625-1628. Lightweight multilingual entity extraction and linking. A Pappu, R Blanco, Y Mehdad, A Stent, K Thadani, WSDM. A. Pappu, R. Blanco, Y. Mehdad, A. Stent, and K. Thadani, "Lightweight multilingual entity extraction and linking," in WSDM, 2017, pp. 365-374. The forward-backward search algorithm. S Austin, R Schwartz, P Placeway, IEEE ICASSP. S. Austin, R. Schwartz, and P. Placeway, "The forward-backward search algorithm," in IEEE ICASSP, 1991, pp. 697-700. Robust disambiguation of named entities in text. J Hoffart, M A Yosef, I Bordino, H Fürstenau, M Pinkal, M Spaniol, B Taneva, S Thater, G Weikum, EMNLP. J. Hoffart, M. A. Yosef, I. Bordino, H. Fürstenau, M. Pinkal, M. Spaniol, B. Taneva, S. Thater, and G. Weikum, "Robust disambiguation of named entities in text," in EMNLP, 2011, pp. 782-792. Collective entity linking in web text: a graph-based method. X Han, L Sun, J Zhao, SIGIR. X. Han, L. Sun, and J. Zhao, "Collective entity linking in web text: a graph-based method," in SIGIR, 2011, pp. 765-774. Graph-based named entity linking with wikipedia. B Hachey, W Radford, J R Curran, WISEB. Hachey, W. Radford, and J. R. Curran, "Graph-based named entity linking with wikipedia," in WISE, 2011, pp. 213-226. Robust entity linking via random walks. Z Guo, D Barbosa, CIKM. Z. Guo and D. Barbosa, "Robust entity linking via random walks," in CIKM, 2014, pp. 499-508. From tagme to WAT: a new entity annotator. F Piccinno, P Ferragina, ACM Workshop on Entity Recognition & Disambiguation. F. Piccinno and P. Ferragina, "From tagme to WAT: a new entity annotator," in ACM Workshop on Entity Recognition & Disambiguation, 2014, pp. 55-62. Graph ranking for collective named entity disambiguation. A Alhelbawy, R J Gaizauskas, ACL. Short Papers2A. Alhelbawy and R. J. Gaizauskas, "Graph ranking for collective named entity disambiguation," in ACL Volume 2: Short Papers, 2014, pp. 75-80. Entity linking meets word sense disambiguation: a unified approach. A Moro, A Raganato, R Navigli, TACL. 2A. Moro, A. Raganato, and R. Navigli, "Entity linking meets word sense disambiguation: a unified approach," TACL, vol. 2, pp. 231-244, 2014. Robust and collective entity disambiguation through semantic embeddings. S Zwicklbauer, C Seifert, M Granitzer, SIGIR. S. Zwicklbauer, C. Seifert, and M. Granitzer, "Robust and collective entity disambiguation through semantic embeddings," in SIGIR, 2016, pp. 425-434. Distributed representations of words and phrases and their compositionality. T Mikolov, I Sutskever, K Chen, G S Corrado, J Dean, NIPS. T. Mikolov, I. Sutskever, K. Chen, G. S. Corrado, and J. Dean, "Distributed representations of words and phrases and their compositionality," in NIPS, 2013, pp. 3111-3119. Knowledge graph and text jointly embedding. Z Wang, J Zhang, J Feng, Z Chen, EMNLP. Z. Wang, J. Zhang, J. Feng, and Z. Chen, "Knowledge graph and text jointly embedding." in EMNLP, 2014, pp. 1591-1601. Entity disambiguation by knowledge and text jointly embedding. W Fang, J Zhang, D Wang, Z Chen, M Li, CoNLLW. Fang, J. Zhang, D. Wang, Z. Chen, and M. Li, "Entity disambiguation by knowledge and text jointly embedding," in CoNLL, 2016. Joint learning of the embedding of words and entities for named entity disambiguation. I Yamada, H Shindo, H Takeda, Y Takefuji, SIGNLL. I. Yamada, H. Shindo, H. Takeda, and Y. Takefuji, "Joint learning of the embedding of words and entities for named entity disambiguation," in SIGNLL, 2016, pp. 250-259. Modeling mention, context and entity with neural networks for entity disambiguation. Y Sun, L Lin, D Tang, N Yang, Z Ji, X Wang, IJCAI. Y. Sun, L. Lin, D. Tang, N. Yang, Z. Ji, and X. Wang, "Modeling mention, context and entity with neural networks for entity disambiguation," in IJCAI, 2015, pp. 1333-1339. Capturing semantic similarity for entity linking with convolutional neural networks. M Francis-Landau, G Durrett, D Klein, NAACL HLT. M. Francis-Landau, G. Durrett, and D. Klein, "Capturing semantic similarity for entity linking with convolutional neural networks," in NAACL HLT, 2016, pp. 1256-1261. Learning to link with wikipedia. D N Milne, I H Witten, CIKM. D. N. Milne and I. H. Witten, "Learning to link with wikipedia," in CIKM, 2008, pp. 509-518. To link or not to link? A study on end-to-end tweet entity linking. S Guo, M Chang, E Kiciman, HLT-NAACL. S. Guo, M. Chang, and E. Kiciman, "To link or not to link? A study on end-to-end tweet entity linking," in HLT-NAACL, 2013, pp. 1020-1030. On the shortest spanning subtree of a graph and the traveling salesman problem. J B , Proceedings of the American Mathematical society. 71J. B. Kruskal, "On the shortest spanning subtree of a graph and the traveling salesman problem," Proceedings of the American Mathematical society, vol. 7, no. 1, pp. 48-50, 1956. Shortest connection networks and some generalizations. R C Prim, Bell Labs Technical Journal. 366R. C. Prim, "Shortest connection networks and some generalizations," Bell Labs Technical Journal, vol. 36, no. 6, pp. 1389-1401, 1957. TSDW: two-stage word sense disambiguation using wikipedia. C Li, A Sun, A Datta, JASIST. C. Li, A. Sun, and A. Datta, "TSDW: two-stage word sense disambiguation using wikipedia," JASIST, pp. 1203-1223, 2013. Linking entities to a knowledge base with query expansion. S Gottipati, J Jiang, EMNLP. S. Gottipati and J. Jiang, "Linking entities to a knowledge base with query expansion," in EMNLP, 2011, pp. 804-813. Greedy function approximation: a gradient boosting machine. J H Friedman, Annals of statisticsJ. H. Friedman, "Greedy function approximation: a gradient boosting machine," Annals of statistics, pp. 1189-1232, 2001. Collective annotation of wikipedia entities in web text. S Kulkarni, A Singh, G Ramakrishnan, S Chakrabarti, SIGKDD. S. Kulkarni, A. Singh, G. Ramakrishnan, and S. Chakrabarti, "Collective annotation of wikipedia entities in web text," in SIGKDD, 2009, pp. 457-466. N 3 -a collection of datasets for named entity recognition and disambiguation in the nlp interchange format. M Röder, R Usbeck, S Hellmann, D Gerber, A Both, LREC. M. Röder, R. Usbeck, S. Hellmann, D. Gerber, and A. Both, "N 3 -a collection of datasets for named entity recognition and disambiguation in the nlp interchange format." in LREC, 2014, pp. 3529-3533. Large-scale named entity disambiguation based on wikipedia data. S Cucerzan, EMNLP-CoNLL. S. Cucerzan, "Large-scale named entity disambiguation based on wikipedia data," in EMNLP-CoNLL, 2007, pp. 708-716. Real-time RDF extraction from unstructured data streams. D Gerber, S Hellmann, L Bühmann, T Soru, R Usbeck, A N Ngomo, ISWC. D. Gerber, S. Hellmann, L. Bühmann, T. Soru, R. Usbeck, and A. N. Ngomo, "Real-time RDF extraction from unstructured data streams," in ISWC, 2013, pp. 135-150. KORE: keyphrase overlap relatedness for entity disambiguation. J Hoffart, S Seufert, D B Nguyen, M Theobald, G Weikum, CIKM. J. Hoffart, S. Seufert, D. B. Nguyen, M. Theobald, and G. Weikum, "KORE: keyphrase overlap relatedness for entity disambiguation," in CIKM, 2012, pp. 545-554. Making sense of microposts (#microposts2014) named entity extraction & linking challenge. A E C Basave, G Rizzo, A Varga, M Rowe, M Stankovic, A Dadzie, WWWA. E. C. Basave, G. Rizzo, A. Varga, M. Rowe, M. Stankovic, and A. Dadzie, "Making sense of microposts (#microposts2014) named entity extraction & linking challenge," in WWW, 2014, pp. 54-60. GERBIL: general entity annotator benchmarking framework. R Usbeck, M Röder, A N Ngomo, C Baron, A Both, M Brümmer, D Ceccarelli, M Cornolti, D Cherix, B Eickmann, P Ferragina, C Lemke, A Moro, R Navigli, F Piccinno, G Rizzo, H Sack, R Speck, R Troncy, J Waitelonis, L Wesemann, WWWR. Usbeck, M. Röder, A. N. Ngomo, C. Baron, A. Both, M. Brümmer, D. Ceccarelli, M. Cornolti, D. Cherix, B. Eickmann, P. Ferragina, C. Lemke, A. Moro, R. Navigli, F. Piccinno, G. Rizzo, H. Sack, R. Speck, R. Troncy, J. Waitelonis, and L. Wesemann, "GERBIL: general entity annotator benchmarking framework," in WWW, 2015, pp. 1133-1143.
[]
[ "FUNCTIONAL CURRENTS : A NEW MATHEMATICAL TOOL TO MODEL AND ANALYSE FUNCTIONAL SHAPES", "FUNCTIONAL CURRENTS : A NEW MATHEMATICAL TOOL TO MODEL AND ANALYSE FUNCTIONAL SHAPES" ]
[ "Nicolas Charon ", "Alain Trouvé " ]
[]
[]
This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometricofunctional data, that is a data structure that involves a deformable shape (roughly a finite dimensional sub manifold) together with a function defined on this shape taking value in another manifold.Indeed, if mathematical currents have already proved to be very efficient theoretically and numerically to model and process shapes as curves or surfaces ([6] and [11]), they are limited to the manipulation of purely geometrical objects. We show that the introduction of the concept of functional currents offers a genuine solution to the simultaneous processing of the geometric and signal information of any functional shape. We explain how functional currents can be equipped with a Hilbertian norm mixing geometrical and functional content of functional shapes nicely behaving under geometrical and functional perturbations and paving the way to various processing algorithms. We illustrate this potential on two problems: the redundancy reduction of functional shapes representations through matching pursuit schemes on functional currents and the simultaneous geometric and functional registration of functional shapes under diffeomorphic transport.
10.1007/s10851-012-0413-4
[ "https://arxiv.org/pdf/1206.3564v1.pdf" ]
15,798,413
1206.3564
af7d61e70c43d0c35fcdfa4ba1061b5f558cb48c
FUNCTIONAL CURRENTS : A NEW MATHEMATICAL TOOL TO MODEL AND ANALYSE FUNCTIONAL SHAPES Nicolas Charon Alain Trouvé FUNCTIONAL CURRENTS : A NEW MATHEMATICAL TOOL TO MODEL AND ANALYSE FUNCTIONAL SHAPES This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometricofunctional data, that is a data structure that involves a deformable shape (roughly a finite dimensional sub manifold) together with a function defined on this shape taking value in another manifold.Indeed, if mathematical currents have already proved to be very efficient theoretically and numerically to model and process shapes as curves or surfaces ([6] and [11]), they are limited to the manipulation of purely geometrical objects. We show that the introduction of the concept of functional currents offers a genuine solution to the simultaneous processing of the geometric and signal information of any functional shape. We explain how functional currents can be equipped with a Hilbertian norm mixing geometrical and functional content of functional shapes nicely behaving under geometrical and functional perturbations and paving the way to various processing algorithms. We illustrate this potential on two problems: the redundancy reduction of functional shapes representations through matching pursuit schemes on functional currents and the simultaneous geometric and functional registration of functional shapes under diffeomorphic transport. Introduction Shape analysis is certainly one the most challenging problem in pattern recognition and computer vision [4,14,5,3,17]. Moreover, during the last decade, shape analysis has played a major role in medical imaging through the emergence of computational anatomy [13,25,19,1,18,20]. More specifically, the quest of anatomical biomarkers through the analysis of normal and abnormal geometrical variability of anatomical manifolds has fostered the development of innovative mathematical frameworks for the representation and the comparison of a large variety of geometrical objects. Among them, since their very first significant emergence in the field of computational anatomy, mathematical currents have become more and more commonly used framework to represent and analyse shapes of very various natures, from unlabelled landmarks to curves ( [11]), fiber bundles ( [7]) surfaces ( [12]) or 3D volumes. The reasons of this success, which we shall detail in the next section, lie basically in the generality of the framework with respect to a very wide collection of geometrical features as well as in their robustness to change of topology and of parametrization. The crucial step at this point is to define a proper distance between currents that faithfully transcribes variations of geometry itself. This problem has been successfully addressed by embedding current spaces into Reproducing Kernel Hilbert Spaces (RKHS), providing kernel-based norms on currents which are fully geometric (independent of parametrization) and enable practical computations in a very nice setting. Such norms and the resulting distances allow to define attachment terms between the geometrical objects, which are then used for instance to drive registration algorithms on shapes ( [12], [11]) and perform statistical analysis of their variability ( [7], [9]). More recently though, an increasing number of data structures have emerged in computational anatomy that not only involve a geometrical shape but some signal attached on this shape, to which we give the general name of functional shapes. The most basic example is, of course, classical images for which the geometrical support is simply a rectangle on which is given a 'grey level' signal. In many cases however, the support can have a much more complex geometry like, for instance, the activation maps on surfaces of cortex obtained through fMRI scans. Signals can also include structures that are more sophisticated than simple real values : we could think of a vector field on a surface as well as tensor-valued signal that appear in DTI imaging. Such a diversity both in shape and signal makes it a particularly delicate issue to embed all geometrico-functional objects in one common framework. Despite several attempts to model them directly as currents, important limitations of currents were found in such problems, which we will develop in section 2. As a result, recent approaches have been rather investigating methods where shape and signal are treated separately instead of trying to define an attachment distance between geometrico-functional objects. This is the case for instance in [23] where authors propose a registration algorithm for fMRI data in which is performed an anatomic matching followed by a second one based on the values of the signals. However, all these frameworks have two important drawbacks : they are first very specific to a certain type of dataset and they require an exact one to one correspondence between the two shapes in order to further compare functional values, whereas in many applications inexact matchings are far more appropriate. The purpose of this paper is to describe and explore a new analytical setting to work on the most general problem of representation and comparison of geometricofunctional structures (compatible with any change of parametrisation of their geometrical supports) treated as elements of an embedding functional vector space, here a Reproducing Kernel Hilbert Space, on which many desirable operations can be performed. Our new analytical setting shares some common features with the mathematical current setting that will be recalled briefly in section 2 but overcome its main limitations when dealing with functional shapes. The core idea, developed in section 3 is to augment usual currents with an extra component embedding the signal values by a natural tensor product leading to our definition of functional currents. We consider then various actions on functional currents by diffeomorphic transport in section 3 and shows in section 4 that kernel norms can provide a suitable Hilbertian structure on functional currents generalizing greatly what has been done for currents. We also show in what sense this representation and RKHS metric on functional currents is consistent with the idea of comparing functional shapes with respect to deformations between them, which makes it a good approach for defining attachment distances. The two main results on this topic are the control results of propositions 3 and 4. We then illustrate the potential of this new metric setting in section 5 on two different problems. The first illustration is the construction, via a matching pursuit algorithm, of redundancy reduction or compression algorithm of the representation of functional shapes by functional currents with few examples of compression on curves and surfaces with real-valued data. The second illustration is about the potential benefits of functional currents in the field of computational anatomy. In particular, we show a few basic results of diffeomorphic matching between functional shapes with our extension of large deformation diffeomorphic metric mapping (LDDMM) algorithm [2] to functional currents. 2. Currents in the modelling of shapes 2.1. A brief presentation of currents in computational anatomy. Currents were historically introduced as a generalization of distributions by L. Schwartz and then G. De Rham in [21]. The theory was later on considerably developed and connected to geometric measure theory in great part by H. Federer [10]. In the first place, these results found interesting applications in calculus of variations as well as differential equations. However, the use of currents in the field of computational anatomy is fairly more recent since it was considered for the first time in [11]. In the following, we try to outline the minimum background of theory about currents needed to recall the link between shapes and currents. First of all, we fix some notations. Let's call E a generic euclidean space of dimension n. We will denote by Ω p 0 (E) the space of continuous p-differential forms on E that vanish at infinity. Every element ω of Ω p 0 (E) is then a continuous function such that for all x ∈ E, ω(x) ∈ Λ p E * . Since we have the isomorphism Λ p E * ≈ (Λ p E) * , we can see both ω(x) as a p-multilinear and alternated form on E and as a linear form on the n p -dimensional space of p-vectors in E. For all the following, we will use the notation ω x (ξ) as the evaluation of a differential form ω at point x ∈ E and on the p-vector ξ. On Λ p E can be defined an euclidean structure induced by the one of E, which is such that if ξ = ξ 1 ∧ .. ∧ ξ p and η = η 1 ∧ .. ∧ η p are two simple p-vectors, ξ, η = det( ξ i , η j ) i,j . The norm of a simple p-vector is therefore the volume of the element. The space Ω p 0 (E) is then equipped with the infinite norm of bounded functions defined on E. These notations adopted, we define the space of p-currents on E as the topological dual Ω p 0 (E) , i.e. the space of linear and continuous forms on Ω p 0 (E). Note that in the special case where p = 0, the previous definition is exactly the one of usual distributions on E that can be also seen as signed measures on E. Simplest examples of currents are given by generalization of a Dirac mass : if x ∈ E and ξ ∈ Λ p E, δ ξ x is the current that associates to any ω ∈ Ω p 0 (E) its evaluation ω x (ξ). Now, the relationship between shapes and currents lies fundamentally in the fact that every d-dimensional and oriented sub-manifold X of E of finite volume can be represented by an element of Ω p 0 (E) . Indeed, we know from integration theory on manifolds ( [10], [15]) that any d-differential form of Ω p 0 (E) can be integrated along X, which associates to X a d-current C X such that : (1) C X (ω) = X ω for all ω ∈ Ω p 0 (E). The application X → C X is also injective. Equation (1) can be rewritten in a more explicit way if X admits a parametrization given by a certain smooth immersion F : U → E with U an open subset of R d . Then, C X (ω) = (x1,..,x d )∈U ω F (x1,..,x d ) ∂F ∂x 1 ∧ ... ∧ ∂F ∂x d dx 1 ...dx d . It is a straightforward computation to check that the last expression is actually independent of the parametrization (as far as the orientation is conserved). In the general case, there always exists a partition of the unit adapted to the local charts of X, so that C X could be expressed as a combination of such terms. The representation is fully geometric in the sense that it only depends on the manifold structure itself and not on the choice of a parametrization. Currents' approach therefore allows to consider sub-manifolds of given dimension (curves, surfaces,...) as elements of a fixed functional vector space. This also gives a very flexible setting to manipulate shapes since addition, combination or averages become straightforward to define. On the other hand, spaces of currents contain a lot more than sub-manifolds because general currents do not usually derive from sub-manifolds (think for instance of a punctual current δ ξ x ). However, it encompasses in a unified approach a wide variety of geometrical objects as for instance sets of curves and surfaces which can be relevant in some anatomy problems. In registration issues, a fundamental operation is the transport of objects by a diffeomorphism of the ambient space. If C ∈ Ω p 0 (E) and φ ∈ Diff(E), we define the transport of C by φ as the classical push-forward operation denoted φ C : (2) ∀ω ∈ Ω p 0 (E), (φ * C) (ω) = C (φ * ω) where φ * ω is the usual pull-back of a differential form defined for all x ∈ E and ξ = ξ 1 ∧ ... ∧ ξ p ∈ Λ p E by : (3) (φ * ω) x (ξ) = ω φ(x) (d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ p )) d x φ being the notation we use for the differential of the diffeomorphism at point x. With this definition, it's a straightforward proof to check that φ * C X = C φ(X) , which means that the d-current associated to a submanifold transported by φ is the d-current associated to the transported submanifold φ(X). To complete this brief presentation of currents applied to computational anatomy, we still need to explain how the currents' representation can be practically implemented and how computations can be made on them. This step consists mainly in approximating the integral in (1) into a discrete sum of punctual currents C X ≈ k=1..N δ ξ k x k where x k are points in E and d-vectors ξ k encode local elements of volume of the manifold X. A manifold X would be then stored as a list of N momenta δ ξ k x k consisting of points' coordinates and corresponding d-vectors. However, the transition between X and its approximation as a discrete current cannot usually be performed in a standard way. Computationally, a mesh on the sub-manifold is needed. Let's examine the two most frequent cases of curves and surfaces. Let γ : I → E be a continuous curve in E given by a sampling of N points {x k = γ(t k )} k=1..N . Starting from this approximation of γ as a polygonal line, we can associate the 1-current defined by : C γ = N −1 j=1 δ τj cj with c j the center of segment [x j x j+1 ] and τ j the vector x j+1 − x j . It can be proved easily that |C X (ω) −C X (ω)| tends toward zero for all 1-form ω as max k {|t k+1 − t k |} → 0, i.e. as the sampling gets more accurate (cf [11]). Same process can be applied to a triangulated surface S immersed in E = R 3 . We associate to each triangle of the mesh x j x j+1 x j+2 a punctual current δ ξj cj with c j = 1 3 (x j +x j+1 +x j+2 ) and ξ j = 1 2 (x j+1 − x j ) ∧ (x j+2 − x j ). Since we have Λ 2 R 3 ≈ R 3 , the previous formal 2-vector can be identified to the usual wedge product of vectors in R 3 , that is the normal vector to the surface whose norm encodes the area of the triangle. Again, it can be shown that this approximated current gets closer and closer to the actual C S as the mesh is refined. Eventually, the surface is represented as a finite collection of points and normal vectors in the space E. Finally, the question of building a metric on the space of currents should be addressed. There are several norms traditionally defined on Ω p 0 (E) such as the mass norm or the flat norm. However, those are either not easily computable in practice or unfitted to comparison between shapes (see [6] chap 1.5). A particularly nice framework to avoid both problems is to define a Hilbert space structure on currents through reproducing kernel Hilbert space (RKHS) theory. This approach consists in defining a vector kernel on E (K : E × E → L(Λ p E)) and its associated RKHS W . Under some assumptions on the kernel, it can be shown that the space of p-currents is continuously embedded in the dual W which is also a Hilbert space. Therefore, in applications, we generally consider W instead of Ω p 0 (E) as our actual space of currents. For more details on the construction of RKHS on currents, we refer to [6] and [11]. Since, in applications, manifold are represented by sums of punctual currents, it's sufficient to be able to compute inner products between two punctual currents. RKHS framework precisely gives simple closed expressions of such products. Indeed, one can show that δ ξ1 x1 , δ ξ2 x2 W = ξ T 1 K(x 1 , x 2 )ξ 2 . Computation of distances between shapes then reduces to simple kernel calculus which can be performed efficiently for well-suited kernels either through fast Gauss transform schemes as in [11] or through convolutions on linearly spaced grids as explained in [6]. In summary, this succinct presentation was meant to stress two essential advantages of currents in shape representation. The first one being its flexibility due to the vector space structure and the wide range of geometrical objects that are comprehended without ever requiring any parametrization. The second important point is the fact that computations on currents are made very efficient by the use of kernels which makes them appropriate in various applications as simplification, registration or template estimation. All these elements motivate an extension of the framework of currents to incorporate functional shapes, which will be discussed thoroughly in all the following. 2.2. Functional shapes and the limitations of currents. We now consider, as in the previous section, a d-dimensional sub-manifold X of the n-dimensional vector space E but in addition, we assume that functional data is attached to every points of X through a function f defined on X and taking its values in a differentiable manifold M , the signal space. What we call a functional shape is then a couple (X, f ) of such objects. The natural question that arises is this : can we model such functional shapes in the framework of currents like purely geometrical shapes ? In the following, we are discussing two possible methods to address this question directly with usual currents and explain why both of them are not fully satisfying in the perspective of applications to computational anatomy. First attempts to include signals supported geometrically in the currents' representation were investigated in [6] with the idea of colored currents. This relies basically on the fact already mentioned that the set of d-currents contains a wider variety of objects than d-dimensional sub-manifolds like rectifiable sets or flat chains (cf [10]). In particular, weighted sub-manifolds can be considered as currents in the following very natural way : suppose that X is a sub-manifold of E of dimension d and f : X → R is a weight or equivalently a real signal at each point of X such that f is continuous, then we can associate to (X, f ) a d-current in E : T (X,f ) (ω) = X f ω Although this approach seems to be the most straightforward way to apply currents to functional shapes since we are still defining a d-current in E, it's quite obvious that such a representation suffers from several important drawbacks. The first thing is the difficulty to generalize colored currents for signals that are not simply real-valued, particularly if the signal space is not a vector space (think for instance of the case of a signal consisting of directions in the 3D space, where M is therefore the sphere S 2 ). The second point arises when the previous equation is discretized into Dirac currents, which leads to an expression of the form k=1..N f (x k )δ ξ k x k . We notice an ambiguity appearing between the signal and the volume element ξ since for any r = 0, f (x k )δ ξ k x k = rf (x k )δ ξ k /r x k ; separating geometry from signal in the discretized version appears as a fundamental difficulty. In addition, the energy of Dirac terms are proportional to the value of the signal at the corresponding point which induces an asymmetry between low and high-valued signals. In this setting, areas having very small signals become negligible in terms of current, which is both not justified in general and can affect drastically the matching of colored currents. We show a simple illustration of this issue when matching two colored ellipsoids with this approach in figure 2. Finally, we could also mention some additional pitfalls resulting in that colored currents do not separate clearly geometry from signal. Most problematic is the fact that there is no flexibility to treat signals at different scale levels than geometry which can make the approach highly sensitive to noise. Figure 2. An example of matching between two ellipsoids provided by the classical LDDMM algorithm. On the left, the adaptation with the colored currents' representation. Values of the signals are two diffused stains both on the source ellipsoid (inside surface) and the target one (exterior shaded surface). We display in blue trajectories of the points. The points compounding to zerovalued area of the signal in the source shape are not matched to the corresponding points in the target surface. On the right, we show what should be the expected result. It is obtained through the approach of functional currents that shall be presented in the next parts of the paper. Another possible and interesting way to represent a functional shape by a current is to view it as a shape in the product space E × M . Somehow, it generalizes the idea of seeing a 2D image as a 3D surface. However, at our level of generality, it is not a completely straightforward process. If the signal function f is assumed to be C 1 , the set G := {(p, f (p)) | p ∈ X} inherits a structure of d-dimensional manifold of E × M . With M a vector space, it results directly from the previous that G can be represented as a d-current in the product space, that is as an element of Ω d 0 (E × M ). For a general signal manifold though, we would need to extend our definitions of currents to the manifold case, which could be done (cf [21]) but the definition of kernels on such spaces would then become a much more involved issue in general compared to the vector space case. This difficulty set apart, there still are some important elements to point out. The first one is the increase of dimensionality of the approach because, while we are still considering manifold of dimension d, the co-dimension is higher : the space of d-vectors characterizing local geometry Λ d (E × M ) is now of dimension n+dim(M ) d , with significant consequences from a computational point of view. From a more theoretical angle, we see that, in such an approach, geometrical support and signal play a symmetric role. In this representation, the modelled topology is no more the one of the original shape because we also take into account variations within the signal space. Wether this is a strength or a weakness is not obvious a priori and would highly depend on the kind of applications. What we can state is that this representation is not robust to topological changes of the shape : in practice, the connectivity between all points becomes crucial, what we illustrate on the simplest example of a plane curve carrying a real signal in figure 3. In the field of computational anatomy, the processing of data such as fiber bundles, where connections between points of the fibers are not always reliable, this would be a clear drawback. We shall illustrate these consequences from the point of view of diffeomorphic matching in the last section of the paper. To sum up this section, we have investigated two direct ways to see a functional On the left, we show a disconnected 2D curve with signal values 0 in blue and 1 in red as well as the connected curve in dashed line. On the right hand side are the corresponding curves in the 3-dimensional geometry × signal space. What we want to emphasize here is the fact that no RKHS norm on product currents would provide a continuity of this representation with respect to connectivity : the difference between the two curves is the magenta dashed part which represents a pure variation in the signal domain. shape as a current. The colored currents' setting, although being very close to the modelling of purely geometrical shapes, is to be discarded mainly because it mixes geometry and signal in an inconsistent way. As for the second idea of immersing the functional shape in a product space, we have explained its limits both from the difficulty of the practical implementation and from the lack of robustness with respect to topology of the geometrical support. These facts constitute our motivation to redefine a proper class of mathematical objects that would preserve the interest of currents while overcoming the previous drawbacks. Definition and basic properties of functional currents In this section, we propose an extension of the notion of currents to represent functional shapes. The new mathematical objects we introduce, we call 'functional currents', are not usual currents strictly speaking, contrarily to the methods presented in section 2.2. They would rather derive from the very general concept of double current introduced originally by De Rham in [21]. Here, we adapt it in a different way to fit with the applications we aim at in computational anatomy. 3.1. Functional p-forms and functional currents. Like in the previous section, let (X, f ) be a functional shape, with X a d-dimensional sub-manifold of the ndimensional Euclidean space E and f a measurable application from X to a signal space M . In our framework, M can be any Riemannian manifold. Most simple examples are provided by surfaces with real signal data like activation maps on cortex in fMRI imaging but the framework that we present here is made general enough to incorporate signals from very different natures : vector fields, tensor fields, grassmannians. We now define the space of functional currents again as the dual of a space of continuous forms : Definition 1. We call a functional p-form on (E, M ) an element of the space C 0 (E × M, Λ p E * ) which will be denoted by Ω p 0 (E, M ) hereafter. We consider the uniform norm on Ω p 0 (E, M ) defined by : ω ∞ = sup (x,m)∈E×M ω (x,m) . A functional p-current (or fcurrent in short) is defined as a continuous linear form on Ω p 0 (E, M ) for the uniform norm. The space of functional p-current will be therefore denoted Ω p 0 (E, M ) . Just as one can establish a correspondence between shapes and currents, to any functional shape we now associate a fcurrent. Proposition 1. Let (X, f ) be a functional shape, with X an oriented sub-manifold of dimension d and of finite volume and f a measurable function from X to M . For all ω ∈ Ω d 0 (E, M ), x → ω (x,f (x)) can be integrated along X. We set : (4) C (X,f ) (ω) := X ω (x,f (x)) . Then C (X,f ) ∈ Ω d 0 (E, M ) and therefore (X, f ) → C (X,f ) associates, to any functional shape, a functional current. To be more explicit, recall that the integral in (4) is simply defined through local parametrization with a given partition of the unit of sub-manifold X. If F : U → E is a parametrization of X with U an open subset of R d , then C (X,f ) (ω) = (x1,..,x d )∈U ω (F (x1,..,x d ),f •F (x1,..,x d )) ∂F ∂x 1 ∧ ... ∧ ∂F ∂x d dx 1 ...dx d . Note also, although we did not state it explicitly, that the previous proposition could include sub-manifolds with boundary in the exact same way since the boundary is of zero Hausdorff measure on the sub-manifold. Of course, like for regular currents, the previous correspondence between functional shapes and functional currents is not surjective. For instance, a sum of functional currents of the form C (X,f ) do not generally derive from a functional shape. In the functional current framework, Dirac masses are naturally generalized by elementary functional currents or Dirac fcurrents δ ξ (x,m) for x ∈ X, m ∈ M and ξ ∈ Λ p E such that δ ξ (x,m) (ω) = ω (x,m) (ξ). In the same way as explained in the previous part, one can give a discretized version of functional currents associated to (X, f ) when a mesh is defined on X. C (X,f ) is then approximated into a sum of punctual currents : (5) C (X,f ) ≈ k=1..N δ ξ k (x k ,m k ) In the particular case of a triangulated surface, the discretized version of the fcurrent can be simply obtained as explained for classical currents by adding the interpolated value of signal at each center point of triangles. From the previous equation, we can observe that functional currents have a very simple interpretation. It consists in attaching values of the signal f to the usual representation of X as a d-current. At this stage, we could also point out an alternative way to define fcurrents by considering them as tensor products of d-currents in E and 0-current (i.e. measure) in M , following for instance [21]. Diffeomorphic transport of fcurrents. What about diffeomorphic transport of functional shapes and currents ? This question cannot be addressed as simply as for the classical current setting if we want to remain completely general. The reason is that, depending on the nature of the signal defined on the manifold, there is not a unique way a deformation can act on a functional shape. In the most simple case where the signal values are not directly correlated to geometry (for instance an activation map on a cortical surface), the natural way to deform a functional shape (X, f ) by a diffeomorphism φ is to transport the geometry of the shape with the values of the signal unchanged. Therefore, the image of (X, f ) would be (φ(X), f • φ −1 ). But imagine now that f is a tangent vector field on X. A diffeomorphism φ, by transporting the geometrical support also has to act on the signal through its differential in order to have a tangent vector field on the image shape. In this case, the image of ( X, f ) is (φ(X), g) where, for all y ∈ φ(X), g(y) = d φ −1 (y) φ(f • φ −1 (y)) . In other cases, for instance a tensor field defined on a manifold, the expression of the transport would differ again. In all cases though, what we have is a left group action of diffeomorphisms of E on the set of considered functional shapes. Thus, to remain general, suppose that a certain class of functional shapes together with such a group action are fixed, we will note φ.(X, f ) the action of φ ∈ Diff(E) on a functional shape (X, f ). Then, Definition 2. We call a deformation model on the space of functional currents an action of the group of diffeomorphisms of E on Ω d 0 (E, M ) which is such that for any functional shape (X, f ) and any diffeomorphism φ, if φ * stands for the action on fcurrents, the following property holds : (6) [φ * C (X,f ) ](ω) = C φ.(X,f ) (ω) for all ω ∈ Ω d 0 (E, M ). Note the difference with (2) : the action of a diffeomorphism on usual currents is always the simple push forward operation which is automatically compatible with the transport of a shape. Here, it is necessary to adapt the definition of the action on fcurrents to be compatible with a given action on functional shapes by satisfying (6). In practical applications, this is usually not a difficulty. In the first case mentioned above, the action of φ ∈ Diff(E) on a functional current C can be derived in a very similar way to the case of usual currents : (7)            φ * C(ω) . = C(φ * ω), ∀ω ∈ Ω d 0 (E, M ) where for all ∀x ∈ E, m ∈ M, ξ = ξ 1 ∧ ... ∧ ξ p ∈ Λ p E (φ * ω) (x,m) (ξ) . = ω (φ(x),m) (d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ d )) . It can be easily checked from the previous equations that for all functional shape (X, f ), we have φ * C (X,f ) = C (φ(X),f •φ −1 ) as we expected under this model. Since we do not want to focus this paper specifically on deformation, the examples of matching that we will give in the last section are under the hypothesis of this model of transport, which is the simplest and will lead to a convenient generalization of matching algorithms on functional currents. We could go a step further and introduce also a contrast change ψ → ψ • f for ψ ∈ Diff(M ) so that we end up with a new action of Diff(E) × Diff(M ) on Ω d 0 (E, M ) defined by (8) ((φ, ψ) * ω) (x,m) (ξ) . = ω (φ(x),ψ(m)) (d x φ(ξ 1 ) ∧ ... ∧ d x φ(ξ d )) and the corresponding action on fcurrent (φ, ψ) * C(ω) . = C((φ, ψ) * ω) given by duality for which we easily check that (9) (φ, ψ) * (δ ξ x,m ) = δ dxφ(ξ1)∧...∧dxφ(ξ d ) φ(x),ψ(m) . Note that it is not significantly more difficult to express and implement the deformation model on functional currents that corresponds to other types of action, as for instance in the case of tangent vector signal we mentioned earlier. A Hilbert space structure on functional currents In this section, we address the fundamental question of comparing functional currents through an appropriate metric. For this purpose, we adapt the ideas of RKHS presented briefly for currents in the first part of the paper. This approach allows to view functional currents as elements of a Hilbert space of functions, which opens the way to various processing algorithms on functional shapes as will be illustrated in the next section. 4.1. Kernels on fcurrent spaces. As we have seen for currents, the theory of RKHS defines an inner product between currents through a certain kernel function satisfying some regularity and boundary conditions. Following the idea that functional p-currents can be considered as well as tensor product of p-currents on E and 0-currents on M, we can generically define a kernel on E × M . Proposition 2. Let K g : E × E → L(Λ p E) be a positive kernel on the geometrical space E and K f : M × M → R a positive kernel on the signal space M . We assume that both kernels are continuous, bounded and vanishing at infinity. Then K g ⊗ K f defines a positive kernel from E × M on Λ p E whose corresponding reproducing Hilbert space W is continuously embedded into Ω p 0 (E, M ). Consequently, every functional p-current belongs to W . Proof. This relies essentially on classical properties of kernels. From the conditions on both kernels, we know that to K g and K f correspond two RKHS W g and W f that are respectively embedded into Ω p 0 (E) and Ω 0 0 (M ) (cf [11]). It is a classical result in RKHS theory that K := K g ⊗ K f defines a positive kernel. Moreover, since K f is real-valued, we have the following explicit expression of K: K ((x 1 , m 1 ), (x 2 , m 2 )) = K f (m 1 , m 2 ).K g (x 1 , x 2 ) . To the kernel K corresponds a unique RKHS W that is the completion of the vector space spanned by all the functions {K f (., m).K g (., x)ξ} for x ∈ E, m ∈ M, ξ ∈ Λ p E. Since functions K f (., m) and K g (., x) are both continuous and vanishing at infinity from what we have said, it is also the case for K f (., m).K g (., x)ξ so that W is indeed embedded into Ω p 0 (E, M ). There only remains to prove that we have a continuous embedding, which reduces to dominate the uniform norm by . W . Let ω ∈ W . For all (x, m) ∈ E × M and ξ ∈ Λ p E such that |ξ| = 1, we have (10) |ω (x,m) (ξ)| = |δ ξ (x,m) (ω)| . Since W is a RKHS, all δ ξ (x,m) are continuous linear forms on W . In addition, Riesz representation theorem provides an isometry K W : W → W . Then : δ ξ1 (x1,m1) , δ ξ2 (x2,m2) W = K W (δ ξ1 (x1,m1) ), K W (δ ξ2 (x2,m2) ) W = K f (., m 1 )K g (., x 1 )ξ 1 , K f (., m 2 )K g (., x 2 )ξ 1 W = K f (m 1 , m 2 ).ξ T 2 K g (x 1 , x 2 )ξ 1(11) Now, back to equation (10), we have : |ω (x,m) (ξ)| ≤ δ ξ x,m W ω W ≤ K f (m, m).ξ T K g (x, x)ξ ω W Since we assume that m → K f (m, m) and x → K g (x, x) are bounded we deduce that K f (m, m).ξ T K g (x, x)ξ is bounded with respect to x, m and ξ with |ξ| = 1. Hence, by taking the supremum in the previous equation, we finally get ω ∞ ≤ C ω W which precisely means that the embedding is continuous. By duality, we get that every functional current is an element of W . Note that the dual application is not necessarily injective unless W is dense in Ω p 0 (E, M ), which is the case in particular if both W g and W f are respectively dense in Ω p 0 (E) and Ω 0 0 (M ). In other words, a quite natural (but not unique) way to build kernels for functional currents is to make the tensor product of kernels defined separately in the geometrical domain (p-currents in E) and in the signal domain (0-currents in M ). As we see, everything eventually relies on the specification of kernels on E and M . Kernels on vector spaces have been widely studied in the past and obviously do not arise any additional difficulty in our approach compared to usual current settings. Among others, classical examples of kernels on a vector space E taking values in another vector space H are provided by radial scalar kernels defined for x, y ∈ E by K(x, y) = k(|x − y|).Id H where k is a function defined on R + and vanishing at infinity. This family of kernels is the only one that induces a RKHS norm invariant through affine isometries. The most popular is the Gaussian kernel defined by K(x, y) = exp − |x−y| 2 σ 2 Id H , σ being a scale parameter that can be interpreted somehow as a range of interactions between points. The definition of a kernel on a general manifold M is often a more involved issue as we already mentioned in subsection 2.2. Generally, the procedure is reversed : the kernel is defined through a compact operator on differential forms of M , which can be diagonalized and hopefully provide a closed expression of the kernel on M (cf [28]). The case of the two-dimensional sphere for instance is thoroughly treated in [11]. However, it's important to note that, in our setting of functional currents, this issue is drastically simplified because we only need to define a real-valued kernel on M . This is contrasting with the idea of product space currents of subsection 2.2, which requires the definition of kernels living in the exterior product of the fiber bundle of M . For instance, if M is a sub-manifold of a certain vector space, obtaining real-valued kernels on M becomes straightforward by restriction to M of kernels defined on the ambient vector space. 4.2. Convergence and control results on the RKHS norm. We are now going to explore a little more some properties of the RKHS norm on fcurrents and show the theoretical benefits of our approach with respect to the original problem raised by this article. Suppose, under the same hypotheses as the previous section, that two kernels K g and K f are given respectively on space E and manifold M , providing two RKHS W g and W f . By a simple triangular inequality, we get for any x 1 , x 2 ∈ E, any ξ 1 , ξ 2 ∈ Λ p E and any m 1 , m 2 ∈ M (12) δ ξ2 (x2,m2) − δ ξ1 (x1,m1) W ≤ δ m1 W f δ ξ2 x2 − δ ξ1 x1 W g + δ ξ2 x2 W g δ m2 − δ m1 W f . Since both kernels K f and K g are assumed to be bounded as in Proposition 2, δ m1 W f and δ ξ2 x2 W g are uniformly bounded so that eventually δ ξ2 (x2,m2) − δ ξ1 (x1,m1) W ≤ Cst ( δ ξ2 x2 − δ ξ1 x1 W g + δ m2 − δ m1 W f ) . Therefore, the RKHS distance between punctual fcurrents is dominated both with respect to the variation of their geometrical parts and of their functional values. This is the general idea we will formulate in a more precise way with the two following propositions. We denote by d M the geodesic distance induced on M by its Riemannian structure. The next proposition examines the case where the geometrical support is a fixed sub-manifold X and shows that the variation of the W -norm is then dominated by the L 1 norm on X. Proposition 3. Let X be a d-dimensional sub-manifold of E of finite volume and f 1 : X → M and f 2 : X → M two measurable functions defined on the sub-manifold X taking value in M . We assume that W f is continuously embedded into C 1 0 (M, R). Then, there exists a constant β such that : C (X,f1) − C (X,f2) W ≤ β X d M (f 1 (x), f 2 (x))dσ(x) where σ is the uniform measure on X. Proof. We recall the definition C (X,f ) = X ω (x,f (x)) . We will first restrict the proof to the case where X admits a parametrization given by a function G : U → E where U is an open subset of R d . The general result follows by the use of an appropriate partition of the unit on X. Denoting ξ(u) . = ∂G ∂u1 (u) ∧ .. ∧ ∂G ∂u d (u) for u = (u 1 , .., u d ) ∈ U , we get C (X,f ) (ω) = u∈U ω (G(u),f •G(u)) (ξ(u))du Now, for g 1 . = f 1 • G and g 2 . = f 2 • G, we have by triangular inequality on . W : (13) C (X,f1) − C (X,f2) W ≤ U δ ξ(u) (G(u),g1(u)) − δ ξ(u) (G(u),g2(u)) W du . From (12), δ ξ(u) (G(u),g1(u)) − δ ξ(u) (G(u),g2(u)) W ≤ δ ξ(u) G(u) W g δ g2(u) − δ g1(u) W f . Now, for any m 1 , m 2 ∈ M and h ∈ W f we have |(δ m1 − δ m2 )(h)| = |h(m 1 ) − h(m 2 )| ≤ Dh ∞ d M (m 1 , m 2 ) ≤ Cst h W f d M (m 1 , m 2 ) the last inequality resulting from the continuous embedding W f → C 1 0 (M, R). Therefore we get δ g2(u) − δ g1(u) W f ≤ Cst d M (g 1 (u), g 2 (u)) . Moreover, since we assume that the kernel K g is bounded, we also have δ ξ(u) G(u) W g ≤ Cst |ξ(u)|. Back to equation (13), we get from the previous derivations the existence of a constant β > 0 such that : C (X,f1) − C (X,f2) W ≤ β U d M (g 1 (u), g 2 (u))|ξ(u)|du which precisely proves the stated result. A straightforward consequence of Proposition 3 and dominated convergence theorem is that if f n is a sequence of function on X that converges pointwisely to a function f , then C (X,fn) → C (X,f ) . In other words, pointwise convergence of signal implies convergence in terms of fcurrents. Following the same kind of reasoning we eventually give a local bound of the RKHS distance between a functional shape and the same shape deformed through small diffeomorphisms both in geometry and signal. As it is now classical, we consider deformations modelled as flows between 0 and 1 of differential equations given through time varying vector fields. In appendix A, we remind the basic definitions about this modelling and a few useful results for the following. Let u(t, x) (resp. v(t, m)) be a smooth time dependent vector fields on the geometrical space E (resp. on the signal space M ) and let φ (resp. ψ) the solution at time 1 of the flow of the ODE y = u(t, y) (resp. y = v(t, y)). On these spaces of vector fields, we define the norms : Proposition 4. Let X be a sub-manifold of E of finite volume and f : X → M a measurable function. Assume that W g and W f are continuously embedded respectively into C 1 0 (E, Λ d E * ) and C 1 0 (M, R). There exists a universal constant γ > 0 such that, if u χ 1 and v χ 0 are sufficiently small (which means that deformations are 'close' to identity), then : u χ 1 =C (X,f ) − C (φ(X),ψ•f •φ −1 ) W ≤ γVol(X) u χ 1 + v χ 0 Proof. The full proof of proposition 4 relies mostly on a few controls which are summed up in appendix A. Given again a local parametrization of X, G : U → X, then, similarly to the previous proposition and using same notations, we have : (14) C (X,f ) − C (φ(X),ψ•f •φ −1 ) W ≤ U δ ξ(u) (G(u),f •G(u)) − δξ (u) (φ•G(u),ψ•f •G(u)) W du where for the volume element ξ(u) = ξ 1 (u) ∧ ... ∧ ξ d (u),ξ(u) is the transported volume element by φ equal toξ(u) = dφ x (ξ 1 (u)) ∧ ... ∧ dφ x (ξ d (u)). From (12) we get δ ξ(x) (x,f (x)) − δξ (x) (φ(x),ψ•f (x)) W ≤ δ ψ•f (x) W f δξ (x) φ(x) − δ ξ(x) x W g + δ ξ(x) x W g δ ψ•f (x) − δ f (x) W f . and using δ ξ(x) x W g ≤ Cst |ξ(x)| and δ ψ•f (x) − δ f (x) W f ≤ Cst d M (ψ • f (x), f (x)) we get δ ξ(x) x W g δ ψ•f (x) − δ f (x) W f ≤ Cst |ξ(x)|d M (ψ • f (x), f (x)) ≤ Cst |ξ(x)| v χ 0 (15) In a similar way, we know that δ ψ•f (x) W f ≤ Cst. Moreover : δ ξ(x) x − δξ (x) φ(x) W g ≤ Cst (|ξ(x)| Id − φ ∞ + |ξ(x) − ξ(x)|) ≤ Cst |ξ(x)| u χ 1 the last inequality being obtained thanks to theorem 3 and corollary 1 of appendix A with s = 0 and t = 1. This leads to : (16) δ (15) and (16) in (14), we finally get : ψ•f (x) W f δ ξ(x) x − δξ (x) φ(x) W g ≤ Cst |ξ(x)| u 1,∞ . PluggingC (X,f ) − C (φ(X),ψ•f •φ −1 ) W ≤ Cst ( u χ 1 + v χ 0 ) U |ξ(u)|du which concludes the proof since U |ξ(u)|du = Vol(X). This property shows that the RKHS norm is continuous with respect to deformations of the functional shape (both in its geometry and its signal). More specifically, it is not hard to see that C (φ(X),ψ•f •φ −1 ) = (φ, ψ) * C (X,f ) for the action given by (8) and (9) and to extend the proof of the previous proposition to a more general situation of a fcurrent C ∈ W having finite "mass norm" M (C) where M (C) . = sup ω∈W, ω ∞ ≤1 C(ω) is the proper extension of the previous finite volume condition. Then we get (17) (φ, ψ) * C − C W ≤ γM (C) u χ 1 + v χ 0 . where γ is a universal constant. This result also provides an answer to wether there is a resversed domination in Proposition 3 for two functional shapes that have the same geometrical support. Indeed, consider a particular case where ψ = Id and φ is a small deformation that leaves X globally invariant (φ(X) = X). We wish to compare the initial functional shape (X, f ) with the deformed one (φ(X), f • φ −1 ) = (X, f • φ −1 ). By proposition 4, we know that, for any function f , the fcurrent's distance remains small if the deformation φ is small. It is no longer true if we compute instead X |f − f • φ −1 | p , the L p distance on X (0 < p ≤ ∞). This is easily seen if we choose for X the unit circle S 1 and consider crenellated signals as in figure 4. Introducing the operator τ dθ that acts on functional shapes by rotation of an angle dθ, we see indeed that : sup f ∈L p (S 1 ), f L p ≤1 S 1 |f − f • τ −1 dθ | p = 1 whereas, according to Proposition 4 sup f ∈L p (S 1 ), f L p ≤1 C (X,f ) − C (X,f •τ −1 dθ ) W = O(dθ) . In conclusion, this gives an answer to the previous question : W norm and L p norm on a fixed geometrical support are not equivalent in general. Again, such a fact speaks in favor of the use of RKHS norms on fcurrents : somehow, the approach we have presented allows a coherent collaboration between signal and geometry to define a proper attachment term for functional shapes that shall be used in section 5. Processing functional shapes with fcurrents : Two examples We would like to illustrate now how the concept of functional currents introduced before offers a genuine solution to the simultaneous processing of the geometric and signal information of any functional shape. We have explained how functional currents can be equipped with a Hilbertian norm mixing geometrical and functional content of functional shapes and how this norm has nice properties with respect to geometrical and functional perturbations. It is more or less clear that the embedding in this convenient Hilbert setting is opening to way to various processing algorithms that will be developed in a near future. Since the purpose of this paper is to stay focused on the theoretical exposition of the fcurrents, you will not try here to develop a full range of applications but we will present briefly two illustrative applications in order to shed a light on the potential of the developed framework. The first application illustrates the full potential of the Hilbertian structure with the design of redundancy reduction or compression algorithms for functional shapes representations through matching pursuit schemes on functional currents. The second one, closer to the core engine of computational anatomy, is the design of large deformation matching algorithm for the simultaneous geometric and functional registration of functional shapes under diffeomorphic transport. 5.1. A compression algorithm for functional current representations. Let us start with the issue of the redundancy of fcurrent representations. If we consider for instance a segment in the 2D space with constant signal, the discretization in punctual fcurrents given by (5) will provide a representation with a number of elements that corresponds to the initial sampling of the curve. Generally, this representation could be clearly reduced since, for such a simple functional shape, only a few terms should capture most of the shape. However, the quality of the approximation needs to be quantified in a meaningful way, especially when the functional part is also involved, through an appropriate norm for which we have a natural candidate given by the Hilbert structure. This issue of redundancy reduction or compression is important for instance when making means of currents because without further treatment, the number of Dirac currents involved in the representation of the mean would increase dramatically. This is even more important when considering higher order statistics for the estimation of noise or texture models around a mean functional shape possibly coupled with a deformation model learned from a set of inexact geodesic matchings, as provided for instance by the matching algorithm provided in subsection 5.2. In the following, we only provide a general overview of the algorithm and few numerical results to show the functional current behaviours. The details of numerical optimization that may deserve a more in depth exposition are out of the scope of the present paper. As we have said, the problem of redundancy reduction or compression is deeply simplified thanks to the Hilbert space structure that has been defined on functional currents in the previous section. Indeed, classical matching-pursuit algorithms in general Hilbert spaces have already been studied by Mallat and Zhang in [16] and later adapted to currents in [8]. We can proceed in a similar way for functional currents. Consider again a discretized fcurrent C = i=1..N δ ξi (xi,mi) ∈ W . N, the number of momenta, is automatically given by the mesh on the sub manifold (point sampling for curves, triangulation for surfaces,...). This sub manifold might have some very regular regions with low geometrical and functional variations, in which results a very redundant representation by fcurrent due to the fact that many adjacent nodes present the same local geometry and signal. The goal of matchingpursuit is to find a more appropriate and reduced representation of C in elementary functional currents. Given a certain threshold ε > 0, we want to find Π n (C) such that C = Π n (C) + R n (C) and R n (C) W ε. R n (C) will be called the residue of the approximation. Somehow, this is linked to the problem of finding the best projection of C on a subspace of W . This problem is however too much timeconsuming computationally for usual applications. Instead, matching pursuit is a greedy algorithm that constructs a family of approximating vectors step by step. The result is a suboptimal fcurrent that approximates the functional current C with a residue whose energy is below threshold. The algorithm basically proceeds as follows. We need to specify a 'dictionary' D of elements in W . In our case, we typically consider the set of all elementary functional currents {δ ξ (x,m) } with ξ a unit vector in Λ d E. The first step of matching pursuit algorithm is to find δ ξ 1 (x 1 ,m 1 ) ∈ D that is best correlated to C. In other words, we try to maximise, with respect to x, m, ξ, the quantity : (18) C, δ ξ (x,m) W = ξ T N i=1 K((x, m), (x i , m i ))ξ i Since ξ is taken among unit vectors, the problem is strictly equivalent to maximize i=1..N K((x, m), (x i , m i ))ξ i = γ(x, m) with respect to (x, m) and take ξ as the unit vector of same direction. We get a first approximation of C : C = Π 1 (C) + R 1 (C) . The algorithm then applies the same procedure to the residue R 1 (C), which provides a second vector δ ξ 2 (x 2 ,m 2 ) ∈ D, and a residue R 2 (C). The algorithm is stopped when the RKHS norm of the residue decreases below the given threshold ε. In most cases, it appears that the compression is better with the orthogonal version of the previous scheme, in which the family of vectors is orthonormalized at each step, in order to impose the projection and the residue to be orthogonal in W . The classical algorithm is based on a Gram-Schmidt orthonormalization at each step. In our case, it's possible to obtain a similar result in a more optimal way by keeping the values of (x i , m i ) found during previous steps and simply modify the vectors ξ i . This is done by imposing the following orthogonality condition. Let's call (e k ) the canonical basis of the vector space Λ d E. If C = Π n (C) + R n (C) and Π n (C) = i=1..n δ α n i (x i ,m i ) , we will add the orthogonality constraint : δ e k (x i ,m i ) ⊥R n (C) ⇐⇒ C, δ e k (x i ,m i ) W * = Π n (C), δ e k (x i ,m i ) W * for all basis vectors e k and for all i ∈ {1, .., n}. It is then straightforward to check that these conditions are strictly equivalent to the following system of linear equations to find the α n i : (19) ∀i ∈ {1, .., n}, n j=1 K((x i , m i ), (x j , m j ))α n j k = γ(x i , m i ) k We could show that the norm of the residue R n (C) monotonically decreases to zero as n → ∞. Hence the algorithm converges and eventually when the residue goes below the given threshold at a certain step n, we obtain a compressed representation of C with n orthogonal dirac fcurrents (with generally n N , as we shall see on the coming examples). At each step, the time-consuming part of the algorithm is mainly the computation of sums of kernels, which has quadratic complexity with respect to the number of Diracs of the original current but can be speeded up tremendously by making computations on a fixed grid with FFT, as introduced for currents in [6]. The same kind of numerical trick can be performed with fcurrents but we will not elaborate on that in this paper. Here are now a few illustrative examples for real valued data on curves or surfaces. We will always consider kernels on fcurrents that are the tensor product of a Gaussian kernel in R 3 of scale parameter λ g with a real Gaussian kernel in the signal space of scale parameter λ f . In figure 5 and 6, we emphasize the influence of both kernel sizes on the compression factor as well as on the precision of the functional values of the compressed shape. The bigger the parameter λ g , the coarser the scale of representation is and fewer punctual fcurrents are therefore needed to compress shapes but more smaller features are lost. In figure 7, we focus more precisely on the compression's behaviour when computing matching-pursuit on a simulated fiber bundle of 2D curves carrying different signals. The scale λ g is the same for both figures but we show the results of matching-pursuit for two radically different values of λ f . In both cases, matching pursuit provides an accurate approximation of the mean (accordingly to the kernel norm) with a very limited number of Diracs compared to the original sampling. However, note the important influence of λ f . Taking a big value for this parameter means that the matching-pursuit will average values of the signals and provide a representation essentially with Dirac fcurrents having values for their signal parts close to the average (left figure) whereas for a smaller λ f , the algorithm will only average the diracs that have close values of signal (right figure). In conclusion, these first examples of functional shape processing were meant to highlight that the combination of the fcurrent's representation with the use of RKHS metrics provides an easy solution to address the issue of redundancy and compression. The method provides important compression factors and enables scale analysis on geometry and signal through the kernel parameters λ g and λ f . 5.2. A large deformation matching algorithm for functional shapes. As a second illustrative example, we would like to briefly highlight the potentials of fcurrent representations in the context of computational anatomy and more generally in the context of shape spaces. It is clear that many important anatomical manifolds are coming with interesting data lying on it (for instance cortical thickness in anatomical MRI or activation maps in fMRI scans among many possibilities) and are perfect examples of functional shapes as defined in this paper. The statistical analysis of a population of such functional shapes is however a real challenge since the relevant information in a functional shape may be buried in two sources : the pure geometrical shape defined by the manifold itself and the signal information spread on the support. However the geometrical and functional parts are more likely intertwined with each other. When only pure geometrical shapes are considered, the concept of shape space equipped with a Riemannian metric offers proper tools for the local analysis of a population of shapes seen as a distribution of points in a shape space. In particular, the use of Riemannian exponential map around a template conveys an efficient linearization of the shape space to describe the differences between shapes. However, observed shapes are contaminated by many errors coming from various pre-processing pipelines driving the extraction of shapes from raw data and the shape space is not sufficient to accommodate any observed shape. Moreover, and more fundamentally, shapes in a shape space are ideal exemplars of real shapes with controlled complexity to address properly estimation issues from a limited sample. Consequently a discrepancy measure or a noise model is needed to link ideal shapes in shape space with observed shapes. A coherent framework is provided by the current framework : indeed observed shapes can be represented as a vector in a Hilbert space of currents which is also embedding a Riemannian shape space M of ideal shapes : M → W so that a population of observed shapes (S i ) can be represented as a sum S i = m i + r i where m i ∈ M and the residual noise r i ∈ W . Introducing a template m 0 and using the linearization provided around m 0 by the Riemannian exponential map Exp m0 we can write for any observed shape S: (20) S = Exp m0 (u) + r where (u, r) ∈ T m0 M × W . Note that the (u, r) are lying in a vector spaces and t → m t . = Exp m0 (tu) is a geodesic on M. Introducing the metric m0 at m 0 and the metric W on W , we can estimate an optimal decomposition (20) (u(S), r(S)) of an observed shape S by the minimization of u 2 m0 + r 2 W . When pure geometrical shapes are no longer involved but functional shapes instead, the previous setting breaks down with usual currents but is still valid if W is replaced by a RKHS space of fcurrents. The space M itself can be defined as M = { g · m 0 | g ∈ G} i.e. the orbit of a template m 0 under the action of a group of deformations G. The diffeomorphic transport discussed in subsection 3.2 offers several examples of such action. We will consider the simple situation of functional shapes with real valued signals (E = R d , M = R) where the action is given by (7) even if more complex actions as defined by (8) and (9) could be used. In this setting, the Riemannian structure on M is inherited from the optimisation of the kinetic energy 1 0 v t 2 V dt on a time dependant Eulerian velocity fields (t, x) → v(t, x) of the trajectory t → φ t · m 0 where φ t is the flow of the ODE y = v(t, y) starting from the identity. The overall framework has been popularized as the large deformation diffeomorphic mapping setting (LDDMM). The space V is a RKHS space of vector fields, here given by an isotropic Gaussian kernel, generating a right invariant distance on the group G of diffeomorphisms generated by flows of kinetic energy. This induces, by Riemannian submersion, a Riemannian structure on M (see [19,26] for a more extended presentation of this geometrical setting). In particular, if m 0 = C (X,f ) with X is a smooth manifold with finite volume or if m 0 = C is a more general element of W such that M (C) < ∞ (for instance a countable family of (X i , f i )'s with vol(X i ) < ∞) then the continuity result given by Proposition 4 or (17) gives the continuous embedding M → W . Obviously the RKHS norm plays the role of an attachment term and could be coupled with other matching approaches (even if we think that the previous setting is particularly attractive for further statistical studies). The reader not familiar with the above geodesic setting could replace the mapping u → m 1 (u) = Exp m0 (u) by any other mapping u → m 1 (u). Figure 8. Example of registration of two functional curves (top left) with binary signal (blue is zero and red is one). On top right, we show the classical matching with currents on the purely geometrical curves. On bottom left, the same curves are matched with our extension of LDDMM to functional currents. In both cases, the deformed curve fits closely to the target one but note the difference of the deformation field for the functional current's approach. Finally, on the right, we show the result of matching we obtain again with fcurrents' LDDMM but with a big value of λ f compared to the signal, in which case the matching is nearly similar to the current matching. With attachment distances provided by the RKHS norms on fcurrents, it is then possible to extend LDDMM algorithm to the registration of functional shapes. Leaving the technical details of implementation to a future paper, we just present some results of the method on simple examples. As we can expect, the resulting matching is driven both by the geometry of the shapes and by the functional values they carry accordingly to the scales of both kernels, which we first show on the example of figure 8. If we compare it now to the colored currents of section 2.2, we see that since functional currents clearly separate signal and geometry, we no longer have the same drawbacks : in the colored surfaces of figure 2, we have shown on the right the matching result with the functional currents' approach. In addition, the functional current representation is totally robust both to punctual outlying signal values and to missing connections between points, which is clear from the definition of the RKHS norm, because geometrically negligible subsets of the shape have zero norm. It was not the case for instance with the product current idea (cf 2.2) since variations of signal also carry non-zero norm. This has important consequences when trying to match curves with missing connections as we show on the example of figure 9. In our sense, it makes functional currents more fitted to the treatment of fiber bundles carrying signal, like the example given in figure 10. A second important thing to point out is that having a norm defined by the tensor product of two kernels K g and K f with two independent scales provides a total flexibility for the matching, geometrically and functionally. The choice of a bigger parameter λ f for instance allows the matching of signal values to be accurate only at a bigger scale, hence our method could still achieve matching under noisy or imprecise signals on shapes. The counterpart is of course the presence of an additional parameter that must be adapted to the data, based upon an a priori on the reliability of the signals we want to match. Multi-scale approaches can also be built by adding kernel at different scales in the spirit of [22] or [24]. But still, functional currents encompass usual currents' approach in the sense that for the limit case λ f → ∞, matching with fcurrents will reduce to a classical matching of purely geometrical parts of the data (cf bottom left figure 8). Figure 9. LDDMM matching of two planar curves with discontinuous signals and topological disconnections. Each curve has two points of functional discontinuity, one of them being also a disconnection of the geometrical support (point b on the source and b' on the target). On the right figure, the matching is performed by representing the colored curve as a current in the product space R 2 × R as explained in section 2.2. On the left, with the functional currents' representation. We see that the resulting deformation is much perturbed by the disconnections in the case of product currents : the algorithm intends to match connected part of the source shape on a connected part of the target shape although it leads to a very unnatural matching. Functional current matching Source and target Current matching Figure 10. A last example of matching on the case of a fiber bundle with signal. On the center figure, the source and target functional shapes. On the left, the resulting matching with the deformed shape and the deformation grid for the functional currents' setting. On the right, the result obtained by matching with currents. Note that even if the geometrical shapes are well matched in both cases, the two deformations are not the same. Functional currents elongate the dark blue part to fit with the target shape's colors whereas currents, by not taking signal into account, shrinks it. Conclusion and outlook We have presented in this paper a way to formally generalize the notion of currents in the purpose of integrating functional shapes into a coherent and robust representation. Functional currents provide a framework to model geometricallysupported signals of nearly any nature and regularity while preserving the interest of the current's approach for computational anatomy. The second main point of the study is the definition of an appropriate norm. The definition of a RKHS structure provides a distance between functional shapes that enjoys worthy control properties as stated in section 4.2. At the same time, the resulting Hilbert structure on fcurrents opens the way to a very wide class of applications. Although numerical issues that appear when computing with currents were not detailed in this paper, we have presented two examples of processing algorithms for functional shapes : a matching pursuit scheme to address fcurrents' compression and averaging as well as an adaptation of LDDMM algorithm for diffeomorphic registration of two functional shapes. Examples were provided essentially in the simplest cases of curves or surfaces with real-valued signal but same methods could easily apply to different kind of manifold, signal and deformation models. To sum up, the article has essentially the objective of setting a path to extend the scope of traditional computational anatomy to these kind of data structures we called functional shapes. This might constitute a serious possibility to improve registration and statistical estimation of deformable templates, which constitutes the future step of our work. In the case of brain anatomy for instance, by taking into account the additional information on the cortical surfaces provided by fMRI maps or estimations of cortical thickness. And last but not least, let us insist again on the point that the RKHS distances we have derived between functional shapes enable joint comparison of geometry and function without the usual curse of requiring a point to point correspondence between shapes or common coordinate systems, which opens interesting possibilities with respect to statistics on functional shapes. In a similar way, the same kind of control can be obtained for the differential of the flow as stated below : Theorem 4. For all v ∈ χ 1 and s, t ∈ [0, 1], φ v s,t is a C 1 function whose differential satisfies the integral equation : d x φ v s,t = Id + t s d x v(r, φ v s,r (x))dr Theorem 5. For all R > 0 there is a constant C(R) > 0 such that, for all v ∈ χ 1 with v χ 1 R : dφ v s,t − Id ∞ C(R). v χ 1 This last result, together with the multilinearity of exterior product and jacobian, leads to the following corollary : Corollary 1. For v χ 1 small enough, there exists constants α > 0 and β > 0 such that for all x ∈ E : |Jac x (φ v s,t ) − 1| ≤ α v χ 1 d x φ v s,t (ξ 1 ) ∧ ... ∧ d x φ v s,t (ξ d ) − ξ 1 ∧ ... ∧ ξ d ≤ β v χ 1 . ξ 1 ∧ ... ∧ ξ d Figure 1 . 1Representation of curve and surface in Dirac current Figure 3 . 3Product currents and topology. t, .)| 1,∞ dt, v χ 0 = 1 0 |v(t, .)| 0,∞ dt where |u(t, .)| 1,∞ = sup x |u(t, x)|+ i sup x | ∂u ∂xi (t, x)| and v(t, .) 0,∞ = sup m |v(t, m)|. Figure 4 . 4Comparison of the fcurrent's norm and the L p norm on a fixed geometrical support : example of crenellations on the unit circle. λ g = 0.04, λ f = 0.4, 47 Diracs λ g = 0.02, λ f = 0.4, 170 Diracs λ g = 0.01, λ f = 0.2, 565 Diracs λ g = 0.04, λ f = 0.2, 57 Diracs λ g = 0.02, λ f = 0.1, 161 Diracs λ g = 0.01, λ f = 0.1, 571 Diracs Figure 5 . 5Matching pursuit on a "painted" bunny with different parameters λ g and λ f . Geometrically, the surface has 0.16 × 0.22 × 0.12 extension in the 3D space and the signal goes from value zero (blue) to one (red). The original sampling of the fcurrent representation has 69451 Diracs and we choose a stopping criterion for the algorithm of = 5%. The resulting Dirac fcurrents δ ξ k (x k ,m k ) are here represented as colored vectors accordingly to the functional values m k . Vectors are all of same length covering an area proportional to the norm of ξ k . Notice that the sampling increases as λ g is smaller while the vector's colors are more accurate when λ f is smaller. Figure 6 . 6Close up on two of the previous results. Figure 7 . 7Matching pursuit on a 2D fiber bundle, each fiber carrying one value of signal represented by the color. On top, the initial object consisting of 300 fibers. Below, we show two results of matching-pursuit with the same λ g but two different values for λ f : λ f = 200 for the left figure, λ f = 20 for the right one. AcknowledgmentsThis work was made possible thanks to HM-TC (Hippocampus, Memory and Temporal Consciousness) grant from the ANR (Agence Nationale de la Recherche).Appendix A. Deformations' modelling in the LDDMM frameworkIn this appendix, we remind a few intermediate results which are necessary for the full proof of proposition 4. Most of them refer to deformations' modelling and can be found either in[11]or[27](chap. 12).Using notations of[27], for p ∈ N, let C p 0 (R n , R n ) be the Banach space of p-times continuously differentiable vector fields v on R n such that v, dv, .., d p v vanish at infinity, which is equipped with the normFor any v ∈ χ 1 , we consider the differential equation dy dt = v(t, y) with initial condition y(s) = x ∈ R n at time s ∈ [0, 1[. We have : In other words, the flow satisfies the following integral equation :We then have :Theorem 2. For all v ∈ χ 1 and all s, t ∈ [0, 1], φ v s,t is a C 1 -diffeomorphism of R n . In the special case where v = 0, φ v s,t is the identity application. From equation 21, using Gronwall inequality, it is easy to show that :Theorem 3. For all R > 0 there is a constant C(R) > 0 such that, for all v ∈ χ 1 with v χ 1 R :φ v s,t − Id ∞ C(R). v χ 0 C(R). v χ 1 Computational anatomy. J Ashburner, K J Friston, Statistical Parametric Mapping The Analysis of Functional Brain Images. K. J. Friston, J. Ashburner, S. J. Kiebel, T. E. Nichols, and W. D. PennyAcademic PressJ. Ashburner and K. J. Friston. Computational anatomy. In K. J. Friston, J. Ashburner, S. J. Kiebel, T. E. Nichols, and W. D. Penny, editors, Statistical Parametric Mapping The Analysis of Functional Brain Images, pages 49-100. Academic Press, 2007. Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. M F Beg, M I Miller, A Trouvé, L Younes, International Journal of Computer Vision. 612M. F. Beg, M. I. Miller, A. Trouvé, and L. Younes. Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. International Journal of Computer Vision, 61(2):139-157, 2005. Shape matching and object recognition using shape contexts. S Belongie, J Malik, J Puzicha, IEEE Transactions on Pattern Analysis and Machine Intelligence. 244S. Belongie, J. Malik, and J. Puzicha. Shape matching and object recognition using shape contexts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(4):509-522, 2002. Active appearance models. T F Cootes, G J Edwards, C J Taylor, T. F. Cootes, G. J. Edwards, and C. J. Taylor. Active appearance models, 2001. Statistical Shape Analysis, volume 4 of Wiley series in probability and statistics: Probability and statistics. I L Dryden, K V Mardia, John Wiley and SonsI. L. Dryden and K. V. Mardia. Statistical Shape Analysis, volume 4 of Wiley series in probability and statistics: Probability and statistics. John Wiley and Sons, 1998. Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. S Durrleman, PhD thesisS. Durrleman. Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. PhD thesis, 2009. Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. S Durrleman, P Fillard, X Pennec, A Trouvé, N Ayache, NeuroImage. S. Durrleman, P. Fillard, X. Pennec, A. Trouvé, and N. Ayache. Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage, 2010. Sparse approximations of currents for statistics on curves and surfaces. S Durrleman, A Trouvé, N Ayache, Proc. Medical Image Computing and Computer Assisted Intervention. Medical Image Computing and Computer Assisted InterventionS. Durrleman, A. Trouvé, and N. Ayache. Sparse approximations of currents for statistics on curves and surfaces. Proc. Medical Image Computing and Computer Assisted Intervention, sep 2008. Statistical models of sets of curves and surfaces based on currents. S Durrleman, A Trouvé, N Ayache, X Pennec, Medical Image Analysis. S. Durrleman, A. Trouvé, N. Ayache, and X. Pennec. Statistical models of sets of curves and surfaces based on currents. Medical Image Analysis, 2009. Geometric measure theory. H Federer, SpringerH. Federer. Geometric measure theory. Springer, 1969. Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l'anatomie numérique. J Glaunès, PhD thesisJ. Glaunès. Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l'anatomie numérique. PhD thesis, 2005. Surface matching via currents. J Glaunes, M Vaillant, Proceedings of Information Processing in Medical Imaging (IPMI). Information Processing in Medical Imaging (IPMI)3565J. Glaunes and M. Vaillant. Surface matching via currents. Proceedings of Information Pro- cessing in Medical Imaging (IPMI), Lecture Notes in Computer Science, 3565(381-392), 2006. Computational anatomy: An emerging discipline. U Grenander, M I Miller, Quarterly of Applied Mathematics. 564U. Grenander and M. I. Miller. Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, 56(4):617-694, 1998. Shape and Shape Theory. D G Kendall, D Barden, T K Carne, H Le, Wiley Series in Probability and Statistics. 11WileyD. G. Kendall, D. Barden, T. K. Carne, and H. Le. Shape and Shape Theory, volume 11 of Wiley Series in Probability and Statistics. Wiley, 1999. Differential and Riemannian Manifolds. S Lang, SpringerS. Lang. Differential and Riemannian Manifolds. Springer, 1995. Matching pursuits with time-frequency dictionaries. S Mallat, Z Zhang, IEEE Transactions on signal processing. S. Mallat and Z. Zhang. Matching pursuits with time-frequency dictionaries. IEEE Transac- tions on signal processing, 41(3397-3415), 1993. Riemannian geometries on spaces of plane curves. P W Michor, D Mumford, Journal of the European Mathematical Society. 845P. W. Michor and D. Mumford. Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society, 8(2006):45, 2003. The emerging discipline of Computational Functional Anatomy. M I Miller, A Qiu, NeuroImage. 451SupplM. I. Miller and A. Qiu. The emerging discipline of Computational Functional Anatomy. NeuroImage, 45(1 Suppl):S16-S39, 2009. On the metrics and euler-lagrange equations of computational anatomy. M I Miller, A Trouvé, L Younes, Annual Review of Biomedical Engineering. 41M. I. Miller, A. Trouvé, and L. Younes. On the metrics and euler-lagrange equations of computational anatomy. Annual Review of Biomedical Engineering, 4(1):375-405, 2002. From Riemannian Geometry to Computational Anatomy. X Pennec, Elements. X. Pennec. From Riemannian Geometry to Computational Anatomy. Elements, 2011. G D Rham, Variétés différentiables : formes, courants, formes harmoniques. Hermann. G. D. Rham. Variétés différentiables : formes, courants, formes harmoniques. Hermann, 1955. Simultaneous Multi-scale Registration Using Large Deformation Diffeomorphic Metric Mapping. L Risser, F.-X Vialard, R Wolz, M Murgasova, D D Holm, D Rueckert, IEEE Transactions on Medical Imaging. 3010L. Risser, F.-X. Vialard, R. Wolz, M. Murgasova, D. D. Holm, and D. Rueckert. Simultane- ous Multi-scale Registration Using Large Deformation Diffeomorphic Metric Mapping. IEEE Transactions on Medical Imaging, 30(10):1746-59, 2011. Function-based intersubject alignment of human cortical anatomy. M Sabuncu, B Singer, Cerebral Cortex. M. Sabuncu and B. Singer. Function-based intersubject alignment of human cortical anatomy. Cerebral Cortex, pages 130-140, jan 2010. A multi-scale kernel bundle for LDDMM: towards sparse deformation description across space and scales. S Sommer, M Nielsen, F Lauze, X Pennec, Information processing in medical imaging proceedings of the conference. 22S. Sommer, M. Nielsen, F. Lauze, and X. Pennec. A multi-scale kernel bundle for LDDMM: towards sparse deformation description across space and scales. Information processing in medical imaging proceedings of the conference, 22(17):624-635, 2011. A framework for computational anatomy. Computing and Visualization in Science. P M Thompson, A W Toga, 5P. M. Thompson and A. W. Toga. A framework for computational anatomy. Computing and Visualization in Science, 5(1):13-34, 2002. Local Geometry of Deformable Templates. A Trouve, L Younes, SIAM Journal on Mathematical Analysis. 37117A. Trouve and L. Younes. Local Geometry of Deformable Templates. SIAM Journal on Mathematical Analysis, 37(1):17, Nov. 2005. Shapes and diffeomorphisms. L Younes, SpringerL. Younes. Shapes and diffeomorphisms. Springer, 2010. Applied functional analysis. Applications to mathematical physics. E Zeidler, SpringerE. Zeidler. Applied functional analysis. Applications to mathematical physics. Springer, 1995.
[]
[ "Universal resources for approximate and stochastic measurement-based quantum computation", "Universal resources for approximate and stochastic measurement-based quantum computation" ]
[ "Caterina E Mora \nInstitut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria\n\nInstitute for Quantum Computing\nDepartment of Physics and Astronomy\nUniversity of Waterloo\nUniversity Ave. WN2L 3G1Canada\n", "Marco Piani \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria\n\nInstitute for Quantum Computing\nDepartment of Physics and Astronomy\nUniversity of Waterloo\nUniversity Ave. WN2L 3G1Canada\n", "Akimasa Miyake \nInstitut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria\n\nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria\n\nPerimeter Institute for Theoretical Physics\n31 Caroline St. NN2L 2Y5WaterlooONCanada\n", "Maarten Van Den Nest \nInstitut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria\n\nMax-Planck-Institut für Quantenoptik\nHans-Kopfermann-Str. 1D-85748GarchingGermany\n", "Wolfgang Dür \nInstitut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria\n\nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria\n", "Hans J Briegel \nInstitut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria\n\nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria\n" ]
[ "Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria", "Institute for Quantum Computing\nDepartment of Physics and Astronomy\nUniversity of Waterloo\nUniversity Ave. WN2L 3G1Canada", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria", "Institute for Quantum Computing\nDepartment of Physics and Astronomy\nUniversity of Waterloo\nUniversity Ave. WN2L 3G1Canada", "Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria", "Perimeter Institute for Theoretical Physics\n31 Caroline St. NN2L 2Y5WaterlooONCanada", "Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria", "Max-Planck-Institut für Quantenoptik\nHans-Kopfermann-Str. 1D-85748GarchingGermany", "Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria", "Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften\nInnsbruckAustria", "Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstraße 25A-6020InnsbruckAustria" ]
[]
We investigate which quantum states can serve as universal resources for approximate and stochastic measurement-based quantum computation, in the sense that any quantum state can be generated from a given resource by means of single-qubit (local) operations assisted by classical communication. More precisely, we consider the approximate and stochastic generation of states, resulting e.g. from a restriction to finite measurement settings or from possible imperfections in the resources or local operations. We show that entanglement-based criteria for universality obtained in [Van den Nest et al, New J. Phys. 9, 204 (2007)] for the exact, deterministic case can be lifted to the much more general approximate, stochastic case. This allows us to move from the idealized situation (exact, deterministic universality) considered in previous works, to the practically relevant context of non-perfect state preparation.We find that any entanglement measure fulfilling some basic requirements needs to reach its maximum value on some element of an approximate, stochastic universal family of resource states, as the resource size grows. This allows us to rule out various families of states as being approximate, stochastic universal. We prove that approximate, stochastic universality is in general a weaker requirement than deterministic, exact universality and provide resources that are efficient approximate universal, but not exact deterministic universal.We also study the robustness of universal resources for measurement-based quantum computation under realistic assumptions about the (imperfect) generation and manipulation of entangled states, giving an explicit expression for the impact that errors made in the preparation of the resource have on the possibility to use it for universal approximate and stochastic state preparation.Finally, we discuss the relation between our entanglement-based criteria and recent results regarding the uselessness of states with a high degree of geometric entanglement as universal resources [D. Gross et al., Phys. Rev. Lett. 102, 190501 (2009); M. J. Bremner et al., Phys. Rev. Lett 102, 190502 (2009)].
10.1103/physreva.81.042315
[ "https://arxiv.org/pdf/0904.3641v2.pdf" ]
119,198,315
0904.3641
01610920dbf78fb8fa0f56c3700536202f5bac64
Universal resources for approximate and stochastic measurement-based quantum computation 10 May 2010 (Dated: May 11, 2010) Caterina E Mora Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften InnsbruckAustria Institute for Quantum Computing Department of Physics and Astronomy University of Waterloo University Ave. WN2L 3G1Canada Marco Piani Institut für Theoretische Physik Universität Innsbruck Technikerstraße 25A-6020InnsbruckAustria Institute for Quantum Computing Department of Physics and Astronomy University of Waterloo University Ave. WN2L 3G1Canada Akimasa Miyake Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften InnsbruckAustria Institut für Theoretische Physik Universität Innsbruck Technikerstraße 25A-6020InnsbruckAustria Perimeter Institute for Theoretical Physics 31 Caroline St. NN2L 2Y5WaterlooONCanada Maarten Van Den Nest Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften InnsbruckAustria Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Str. 1D-85748GarchingGermany Wolfgang Dür Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften InnsbruckAustria Institut für Theoretische Physik Universität Innsbruck Technikerstraße 25A-6020InnsbruckAustria Hans J Briegel Institut für Quantenoptik und Quanteninformation derÖsterreichischen Akademie der Wissenschaften InnsbruckAustria Institut für Theoretische Physik Universität Innsbruck Technikerstraße 25A-6020InnsbruckAustria Universal resources for approximate and stochastic measurement-based quantum computation 10 May 2010 (Dated: May 11, 2010)arXiv:0904.3641v2 [quant-ph]numbers: 0367Lx0367Mn0365Ta We investigate which quantum states can serve as universal resources for approximate and stochastic measurement-based quantum computation, in the sense that any quantum state can be generated from a given resource by means of single-qubit (local) operations assisted by classical communication. More precisely, we consider the approximate and stochastic generation of states, resulting e.g. from a restriction to finite measurement settings or from possible imperfections in the resources or local operations. We show that entanglement-based criteria for universality obtained in [Van den Nest et al, New J. Phys. 9, 204 (2007)] for the exact, deterministic case can be lifted to the much more general approximate, stochastic case. This allows us to move from the idealized situation (exact, deterministic universality) considered in previous works, to the practically relevant context of non-perfect state preparation.We find that any entanglement measure fulfilling some basic requirements needs to reach its maximum value on some element of an approximate, stochastic universal family of resource states, as the resource size grows. This allows us to rule out various families of states as being approximate, stochastic universal. We prove that approximate, stochastic universality is in general a weaker requirement than deterministic, exact universality and provide resources that are efficient approximate universal, but not exact deterministic universal.We also study the robustness of universal resources for measurement-based quantum computation under realistic assumptions about the (imperfect) generation and manipulation of entangled states, giving an explicit expression for the impact that errors made in the preparation of the resource have on the possibility to use it for universal approximate and stochastic state preparation.Finally, we discuss the relation between our entanglement-based criteria and recent results regarding the uselessness of states with a high degree of geometric entanglement as universal resources [D. Gross et al., Phys. Rev. Lett. 102, 190501 (2009); M. J. Bremner et al., Phys. Rev. Lett 102, 190502 (2009)]. We investigate which quantum states can serve as universal resources for approximate and stochastic measurement-based quantum computation, in the sense that any quantum state can be generated from a given resource by means of single-qubit (local) operations assisted by classical communication. More precisely, we consider the approximate and stochastic generation of states, resulting e.g. from a restriction to finite measurement settings or from possible imperfections in the resources or local operations. We show that entanglement-based criteria for universality obtained in [Van den Nest et al, New J. Phys. 9, 204 (2007)] for the exact, deterministic case can be lifted to the much more general approximate, stochastic case. This allows us to move from the idealized situation (exact, deterministic universality) considered in previous works, to the practically relevant context of non-perfect state preparation. We find that any entanglement measure fulfilling some basic requirements needs to reach its maximum value on some element of an approximate, stochastic universal family of resource states, as the resource size grows. This allows us to rule out various families of states as being approximate, stochastic universal. We prove that approximate, stochastic universality is in general a weaker requirement than deterministic, exact universality and provide resources that are efficient approximate universal, but not exact deterministic universal. We also study the robustness of universal resources for measurement-based quantum computation under realistic assumptions about the (imperfect) generation and manipulation of entangled states, giving an explicit expression for the impact that errors made in the preparation of the resource have on the possibility to use it for universal approximate and stochastic state preparation. Finally, we discuss the relation between our entanglement-based criteria and recent results regarding the uselessness of states with a high degree of geometric entanglement as universal resources [D. Gross III. Universality in MQC 7 A. ε-approximate δ-stochastic universality 7 B. Efficient universality 8 IV. Criteria for universality and no-go results 9 A. ε-approximate deterministic universality 9 B. ε-approximate δ-stochastic universality 10 C. Efficiency in the approximate and stochastic case 10 V. Examples of ε-approximate and/or δ-stochastic universal resources 11 A. 2D cluster state with holes as an exact quasi-deterministic resource 11 B. Deformed 2D cluster state as an exact quasi-deterministic resource 11 C. A noisy cluster state as an ε-approximate deterministic resource 12 D. Stability of universal resources 12 VI. Conclusions and outlook 13 Acknowledgements 14 A. Proof of Proposition 2 14 B. Proof of Theorem 4 15 I. INTRODUCTION Quantum computation offers a new and exciting perspective to information processing, as it has been found that certain problems can be solved more efficiently on a quantum computer than on a classical device. Despite considerable effort it is however not fully understood which features of quantum mechanics are responsible for the apparent speedup. Basic questions regarding the nature and power of quantum computation remain largely unanswered to date. The existence of various models for quantum computation, among them the quantum Turing machine [1,2], the circuit model [3][4][5], adiabatic quantum computation [6,7] and measurement-based quantum computation [8][9][10][11][12][13][14][15][16], seems to indicate that a straightforward answer to these fundamental issues might be difficult to obtain. On the other hand, the different nature of the models allows one to study these fundamental issues from different perspectives, and it turns out that some models are better suited than others to study a certain aspect. For instance, the model of measurement-based quantum computation, with the one-way model [13] as most prominent representative, seems to be particularly well suited to investigate the role of entanglement in quantum computation. Such an investigation has been initiated in [17] and further developed in [18]. In one-way or measurementbased quantum computation (MQC) -which we use synonymously throughout this article-a highly entangled resource state, e.g. the 2D cluster state [19], is processed by sequences of single-qubit local measurements. As has been shown in [14], a proper choice of measurement directions allows one to generate -up to irrelevant local unitary correction operation-any quantum state deterministically and exactly on the unmeasured qubits. In this paper we aim at investigating the generalization of these previous results to the case in which stochastic and/or approximate quantum computation is allowed. The 2D cluster state is called a universal resource for MQC. In MQC, the role of entanglement is particularly highlighted, as all entanglement required in the computation already needs to be present in the initial resource state. This derives from the fact that no entanglement measure increases under local operations and classical communication (LOCC). This insight was recently used in [17,18] to investigate which other quantum states are universal resources for MQC. Entanglement-based criteria for universality have been established and many -otherwise highly entangled-resource states, including GHZ states [20], W states [21] and 1D cluster states [19], have been shown to be not universal for MQC. One should, however, emphasize that this does not mean that such non-universal resource states are useless for quantum information processing, as they might still serve to perform some specific quantum computation or as a re-source for some other task. On the positive side, several other states have been identified to be universal resources for MQC [9,18]. Notice that we use the term "universality" in its strongest form, i.e. we consider the generation of quantum states (universal state preparator). This has been termed CQ-universality (where CQ stands for classical input, quantum output) in Ref. [18] and we refer the interested reader to said work for an extended discussion on the different notions of universality. A. Approximate and stochastic universality In this article we will extend the results on universality obtained in [17,18] to a more general and realistic setting, which is motivated by experimental reality. More precisely, we will consider the approximate and probabilistic generation of quantum states from a given resource state, in contrast to the exact and deterministic generation discussed in [17,18]. In this work we therefore focus on the case in which the desired output states are required to be generated only with finite accuracy (that is the output of the computation is required to be within some distance ε of the desired state), and with probability 1 − δ. Such an extension needs to be considered naturally whenever the resource states are noisy, e.g. due to an imperfect generation process or due to decoherence, but also if the local operations used to process the state are imperfect. The latter may again be reflected in noisy single qubit operations, but may also result from a restriction to a finite number of measurement settings or local unitary operations. In all these cases, the resulting states can only be an approximation of the desired state. In addition, one might be interested in the generation of states with a probability of success (arbitrary) close to one -which we will call quasi-deterministic-, or even only with some (arbitrary) small success probability. In fact, similar issues are implicitly considered when one refers to universal gate sets in the circuit model for quantum computation: any finite universal gate set allows one to approximate any state with arbitrary accuracy. Notice that the issue of probabilistic computation has been deeply studied both in classical computation theory [22] and in the quantum setting [23]. On the one hand, if it is known when the computation succeeded, which happens, say, with probability p, then O(1/p) repetitions allow one to obtain a valid, confirmed outcome. On the other hand, even if it is not known whether the computation succeeded or not, but only that the correct outcome is obtained with some probability p > 1/2, this is still sufficient to extract the correct (classical) result of the computation with arbitrary high probability by considering many repetitions. The first scenario also applies without changes to the case where quantum states should be generated (CQ universality). The second scenario is restricted to the extraction of classical outputs (CC universality), while the resulting quantum states are in fact mixed. B. Summary of results We find that -analogously to the exact, deterministic case-entanglement based criteria for approximate and stochastic universality can be obtained. To formulate these criteria, we need to consider ε-measures of entanglement [24] and compute their extremal values over all states. Given any distance D on the set of states, and any entanglement measure E, the ε-measure of a state E ε (ρ) is defined as the minimal amount of entanglement of all states which are ε-close (with respect to D) to ρ, i.e. have a distance smaller than ε to ρ. We find the following necessary criteria for efficient, approximate stochastic universality: • For any entanglement measure E which is a strong extendable entanglement monotone (see below for exact definition), we have that an approximate, stochastic universal resource Σ which allows one to obtain an ε-approximation of any state with probability larger than 1 − δ, must have an amount of entanglement that is larger or equal than (1 − δ) times the maximum of the corresponding ε-measure E ε over all states of arbitrary size, E(Σ) ≥ (1 − δ)max ρ E ε (ρ). Roughly speaking, this means that any approximate, stochastic resource needs to be maximally entangled with respect to all such entanglement measures. • If one takes the efficiency of computation into account, we find that for any strong extendable entanglement monotone, the entanglement of the resource states does not only need to reach the maximum value of the corresponding ε-measure over all states, but needs to grow sufficiently fast with the system size. These two criteria allow one to rule out a large number of states as being not universal in an approximate an stochastic sense, e.g. GHZ states, W states and 1D cluster states. On the positive side, we present a number of approximate, quasi-deterministic resource. We find: • There exist efficient, approximate quasideterminist universal resources that are not believed to be exact, deterministic universal. For example, a 2D cluster state where particles are missing with a certain probability is an exact, quasi-deterministic universal resource, while an approximate 2D cluster state is an approximate deterministic universal resource. • Any state that is sufficiently close to an approximate stochastic universal resource is still an approximate stochastic universal resource, and the parameters quantifying approximation and stochasticity are quadratically related to the original ones. In particular, this last observation has implications in realistic (experimental) scenarios, where the preparation of the initial entangled states is imperfect. These errors in the preparation procedure still allow for the state to be used for MQC, in the approximate and stochastic scenario. While this might be considered intuititve and results of this type were already known for the 1-way model (where the initial state is a 2D cluster state) [25][26][27], in this paper we extend the observation to all approximate stochastic universal resources, computing an explicit expression for the interplay between the different parameters. C. Guideline through the paper The paper is organized as follows. In Section II we review some of the basic concepts, related to distance and entanglement measures respectively, which we use in the remaining of the paper. In Section III we recall the definition of universal resources for measurement-based quantum computation, and see how the definition can be generalized to the approximate and stochastic case. In Section IV we first review some of the results found in [18] and then show how they can be generalized in a very natural way obtaining necessary criteria for universal resources in the approximate and stochastic case. In this Section we also show how the issue of efficiency can be included in the analysis, obtaining in this way stronger versions of the above-mentioned criteria. Finally, in Section V, we give some experimentally relevant examples of resources that are approximate deterministic, exact stochastic and approximate stochastic universal, but not exact deterministic universal. In particular, we show that any family of states that is close to a universal family is still approximate stochastic universal. Section VI summarizes and concludes our results. II. ENTANGLEMENT MONOTONES In this section we review some essential features of entanglement monotones which are relevant in the study of universality in MQC. In Section II A we review the basic conditions which a function must satisfy in order to be considered an "entanglement monotone". Furthermore, we show how these conditions lead to the definitions of different "types" of entanglement measures. The distinction between different types of entanglement measures will be necessary to allow for a proper formulation of entanglement-based criteria for approximate and stochastic universality, as we will do in section IV. In section II B we consider a general class of monotones called "epsilon-measures". This class of measures was introduced in [24] in order to study the entanglement in states which are only known up to some approximation. For this reason they are suitable quantities to consider in the study of approximate universality. In Section II C, we focus on two examples of existing entanglement measures, namely the geometric measure and the Schmidt-rank width. We discuss in which sense these quantities are monotones, and we discuss their associated ε-measures. A. Properties of entanglement monotones The first examples of entanglement measures were built by first considering a particular application of entanglement (such as, e.g., distillation) and then deriving a quantifier based on such an operation. This approach led to measures that, while naturally having a clear physical interpretation, were often very hard to compute. To evaluate, for example, the entanglement of distillation [28] it is necessary to optimize over all purification protocols. A different approach to the problem, that one might define "axiomatic", has been proposed in [29]. The starting point of this work was the idea that an entanglement measure is some mathematical quantity that should somehow capture the essential features that we associate with entanglement. With this idea in mind, it is possible to identify a set of conditions that must be satisfied by any such measure E. The most fundamental of these conditions are: P1. Vanishing on separable states: separable states do not contain entanglement, therefore we require that E(σ sep ) = 0. Here Λ LOCC denotes an LOCC transformation. Note that property P2 also implies that E is invariant under local unitaries. Aside from these two postulates, other additional requirements for entanglement measures have been formulated. In particular, the following are among the most commonly found in literature. P3. Convexity: E(pρ 1 + (1 − p)ρ 2 ) ≤ pE(ρ 1 ) + (1 − p)E(ρ 2 ). P4. Monotonicity on average under LOCC: this condition is stronger than the monotonicity condition seen above, and is sometimes referred to as strong monotonicity. It requires that the following holds true E(ρ) ≥ i p i E(ρ i ),(1) where ρ i are the possible outputs of some LOCC protocol acting on ρ, and occur with probability p i . P5. Trivial extendability: in this case, one aims at comparing entanglement in states of different system size. The condition of trivial extendability states the following: let |ψ be an N -qubit state; then one requires that E(|ψ |0 ) = E(|ψ ). Here |ψ |0 is considered as an (N + 1)-party state (and not as an ancilla to one of the initial N parties), where the (N + 1)-th party is disentangled from the rest of the system. Conditions P3 and P4 are often found in literature as necessary requirements for entanglement measures. Condition P5 has been introduced more recently [18], in the context of the study of universality in MQC. Other different requirements have been formulated, and for a more detailed analysis of them we refer to [30]. Depending on the set of conditions that are satisfied by the quantity E, we can define different types of measures. In particular, we can distinguish the following types, which we will use in the following sections. Definition 1. Weak entanglement monotone. A real function E is called a weak entanglement monotone if it satisfies conditions P1 to P3. Strong entanglement monotone. A real function E is called a strong entanglement monotone if it satisfies conditions P1 to P4. Extendable weak/strong monotone. An extendable weak (strong) monotone is a weak (strong) entanglement monotone which additionally satisfies condition P5. Note that, in all these definitions, we are imposing the convexity of the function. This condition is not always deemed necessary, but the measures we consider in the following satisfy it. We also remark that every strong entanglement monotone is also a weak monotone. The notion of an extendable monotone was introduced in [18] under the name "type II monotone". We now define another property, related to monotonicity under LOCC operations, that will be relevant in the analysis of resources for approximate measurement-based quantum computation. P6. Weak non-increasing under LOCC: a function E is weakly non-increasing under LOCC if, for any state ρ and for any LOCC protocol Λ LOCC : ρ → {p i , ρ i }, we have E(ρ) ≥ min i E(ρ i ). In other words, monotones satisfying P6 are such that at least one of the outputs of an LOCC protocol acting on an initial state ρ has entanglement smaller than that of ρ. Such a condition is trivially satisfied by any strong entanglement monotone. We conjectured [31] that P6 is implied by weak monotonicity, but this still has not been proved. To end this section, we will introduce two quantities associated to any entanglement measure E, which play a fundamental role both in [18] and in the results contained in Section IV. The first notion is the asymptotic entanglement of a family of states. Let Σ = {σ i } i be an (infinitely large) family of many-qubit states, and E be entanglement monotone defined on N -qubit states, for all N . We define the asymptotic entanglement E(Σ) of the family as E(Σ) = sup σ∈Σ E(σ).(2) The case E(Σ) = ∞ is allowed. Second, the asymptotic entanglement E * of E is defined as E * = sup ρ∈S E(ρ),(3) where the supremum is taken over all N -qubit states, for all N ∈ N. The case E * = ∞ is allowed. Note that, if E is convex, one can restrict the set over which the supremum is taken to only the set of pure states (thus recovering the definition found in [18]). B. ε-measures of entanglement The ε-monotones [24] are a class of entanglement monotones which can be associated to any existing monotone, and which depend on a precision parameter ε. They have been introduced to address the issue of quantifying the entanglement contained in a state which is only partially known as in the case of, for example, a state prepared using an imperfect apparatus. Given any entanglement measure E, its ε-version is defined as E (D) ε (ρ) = min{E(σ) | D(σ, ρ) ≤ ε},(4) where D is a distance on the set S of states which is convex and contractive under completely positive trace preserving maps [32], and σ, ρ ∈ S. To lighten notation we will often omit the superscript in "E (D) ε " referring to the distance measure D when writing down an ε-measure, and we will simply write E ε . The quantity E ε quantifies the "guaranteed" entanglement contained in a state since, by definition, any state σ within an ε-distance of the desired state ρ has entanglement E(σ) ≥ E ε (ρ). In the following we will see that the ε-measure of a state is the crucial quantity to consider when studying approximate preparation of such a state. Indeed, if we aim at preparing a state which is ε-close to ρ, then E ε (ρ) is the minimum entanglement that we must be able to obtain from the initial resource state. In the remainder of this section, we highlight some relevant properties of ε-monotones. First, it has been shown [24] that E ε is always a weak entanglement monotone if E is. Moreover, also property P5 illustrated above is inherited by the ε-version of a monotone satisfying it. Therefore, the ε-version of an extendable weak monotone is again an extendable weak monotone. On the other side, the ε-version of an entanglement measure is never a strong monotone. We refer to [24] for details. Computing the asymptotic entanglement E * ε for arbitrary ε may be a difficult task. Nevertheless, it is often tractable to compute the asymptotic entanglement E * ε when we are interested in the limit ε → 0. This is particularly true in the case of continuous measures, where the following observation holds true. Proposition 1. If E is bounded (for any fixed dimension), convex, and continuous then lim ε→0 + E * ε = E * . Proof. Let E * ∈ (0, ∞]. To prove the statement we have to show that, for any µ > 0, there existsε(µ) > 0 such that ε ≤ε(µ) ⇒ E * ε ≥ E * − µ. Consider that, for any state ρ, we have that ε ′ ≤ ε ⇒ E ε ′ (ρ) ≥ E ε (ρ), which implies that ε ′ ≤ ε ⇒ E * ε ′ ≥ E * ε . Moreover, from the definition of E * ε it follows that, for any state ρ and for any choice of ε, E * ε ≥ E ε (ρ). This implies that, ∀ε ≤ε(µ) and ∀ρ, we have E * ε ≥ E * ε(µ) ≥ Eε (µ) (ρ). Therefore, it is sufficient to prove that ∀µ > 0 , ∃ε(µ), ρ(µ) such that Eε (µ) (ρ(µ)) ≥ E * − µ. In order to do so, we first recall that, since the family Ψ C = {|C Ni } i of two-dimensional cluster states (on N i = i×i qubits) is exact and deterministic universal, we have that E(Ψ C ) = E * , for any entanglement measure E [18]. This implies that, for any µ > 0, there exists N (µ) := N i(µ) such that E(|C N (µ) ) ≥ E * − µ/2. In [24], it has been shown that, if E satisfies the hypotheses above, then E ε is continuous in ε and ρ. Hence, it is always possible to find anε(µ, N (µ)) > 0 such that Eε (µ,N (µ)) (|C N (µ) ) ≥ E(|C N (µ) ) − µ/2. We have thus that, for any µ > 0, there exists a state |C N (µ) and anε(µ, N (µ)) > 0 such that E * ε(µ,N (µ)) ≥ Eε (µ,N (µ)) (|C N (µ) ) ≥ E(|C N (µ) ) − µ/2 ≥ E * − µ. In the case of discontinuous measures, such as the χwidth [33] or the Schmidt measure [34], one has to compute E * ε on a case by case basis. We will elaborate on the case of the χ-width in section II C. C. Two entanglement measures In this Section we consider two explicit examples of entanglement measures that we use in Section IV to construct criteria for approximate, non-deterministic universality. These are the geometric measure of entanglement and the Schmidt-rank width. We discuss in which sense these quantities are entanglement measures, what their asymptotic entanglement is, and how the ε-versions of these measures behave. Geometric measure of entanglement The geometric measure of entanglement was first introduced as a bipartite entanglement measure in [35] and then generalized in [36,37] to the multipartite setting. The intuition behind this measure is that the more entangled a state is, the more distinguishable it is from a separable state. The monotone can be defined as follows. Let |ψ be a state of N qubits, and let π(|ψ ) denote the maximum fidelity between |ψ and a factorized state on N qubits π(|ψ ) = max |ϕ =|ϕ1 ⊗···⊗|ϕN | ψ|ϕ | 2 .(5) The geometric measure E G is defined by E G (|ψ ) = 1 − π(|ψ ),(6) This measure, defined for pure states, can be generalized to the case of mixed states by the convex roof construction, that is: E G (ρ) = min {pi,|ψi }i i p i E G (|ψ i ),(7) where the minimum is taken over all {p i , |ψ i } i such that ρ = i p i |ψ i ψ i |. One can verify that such measure satisfies conditions P1 to P5 and is, thus, an extendable strong entanglement monotone (and therefore also an extendable weak monotone). Next we consider the ε-version of the geometric measure, and we focus on ε-measures based on distances that are "strictly related to the fidelity". Definition 2. A distance D on the set of states is said to be strictly related to the fidelity if, for any two states ρ and σ, D(ρ, σ) ≤ ε ⇒ F (ρ, σ) ≥ 1 − η(ε), with 0 ≤ η(ε) ≤ 1 a strictly monotonically increasing function of ε (for ε ≥ 0 and ε less than the maximum value that D can assume) such that η(0) = 0. An example of such a measure is the trace distance. The following is a technical result, which is a lower bound for (E G ) ε (|ψ ) in terms of E G (|ψ ). Proposition 2. Let D be a distance measure that is strictly related to the fidelity. Further, let (E G ) ε denote the corresponding ε-geometric measure. Then, for any pure state |ψ and for any choice of ε > 0 such that η = η(ε) 0.44, the quantity (E G ) ε (|ψ ) is not smaller than 1 − 3 √ η 2E G (|ψ ) 2/3 E G (|ψ ) − (18E G (|ψ )η) 1/3 .(8) The proof of Proposition 2 rather involved and will be given in Appendix A. The above result can be used to bound the asymptotic ε-geometric entanglement (E G ) * ε . We have: Proposition 3. Let D be a distance measure that is strictly related to the fidelity, and let ε > 0 be such that η(ε) ≤ 0.44, where η(ε) is such that D(ρ, σ) ≤ ε ⇒ F (ρ, σ) ≥ 1 − η(ε). If (E G ) ε denotes the ε-geometric measure with respect to distance D, then (E G ) * ε ≥ 1 − 4η 1/3 + 3.4η 2/3 (9) Proof. Since (E G ) * ε is defined as the supremum over all possible states, we have (E G ) * ε ≥ (E G ) ε (Ψ C ), where Ψ C = {|C Ni } i is the family of two-dimensional cluster states on N i = i × i qubits. The geometric measure for this class of states has been computed [38], and we have E G (|C Ni ) = 1 − 2 −Ni/2 . In order to prove the statement, we apply Proposition 2 to obtain (E G ) * ε ≥ (E G ) ε (Ψ C ) ≥ sup N 1 − 3 √ η 2(1 − 2 −N/2 ) 2/3 1 − 2 −N/2 − (18(1 − 2 −N/2 )η) 1/3 = 1 − 3 √ η 2 2/3 1 − (18η) 1/3 = 1 − ( 9η 4 ) 1/3 − (18η) 1/3 + ( 81 2 η 2 ) 1/3 ≥ 1 − 4η 1/3 + 3.4η 2/3 .(10) Note that this result implies that lim ε→0 (E G ) * ε = 1.(11) The latter also follows immediately from Proposition 1. Schmidt-rank width The Schmidt-rank width is an entanglement monotone which has been introduced and investigated in [17,18,33]. It has been proved that this measure is an extendable strong entanglement monotone, and it can be used to assess whether resources for MQC admit an efficient classical simulation [33]. The Schmidt-rank width χ wd of a pure state |ψ computes the minimum Schmidt rank χ of |ψ , where the minimum is taken over a specific class of bipartitions of Tree T \e obtained from T by removing edge e, and induced bipartition. the system. More precisely, χ wd (|ψ ) is defined as follows. Let |ψ be an N -partite state. We consider a subcubic tree T , i.e. a graph with no cycles, where each vertex has exactly 1 or 3 incident edges, with N leaves (N vertices with only 1 incident edge), which we identify with the N parties of the system (see Figure II χ wd (|ψ ) = min T max e∈T χ A e T ,B e T (|ψ ),(12) where the minimum is taken over all subcubic trees T with N leaves (identified with the N parties of the system), and χ A e T ,B e T (|ψ ) is the Schmidt rank of |ψ with respect to the bipartition (A e T , B e T ). The Schmidt rank width may be generalized to mixed states by a convex roof construction. Note that the Schmidt-rank width is not continuous, such that Proposition 1 cannot be used to compute the asymptotic behavior of its ε-version in the limit ε → 0. However, it is still relatively easy to gain insight in this matter, in the following way. First, note that χ wd (|ψ ) ≥ E wd (|ψ )(13) for every state |ψ . Here E wd (|ψ ) denotes the entropic entanglement width, as defined in [18]. The entropic entanglement width is defined via the same optimization procedure as the Schmidt-rank width, now with the entanglement entropy as the "basic measure". Note that (13) implies that (χ wd ) ε (|ψ ) ≥ (E wd ) ε (|ψ ),(14) and thus (χ wd ) * ε ≥ (E wd ) * ε . Furthermore, as the entropic entanglement width is a weak monotone which is moreover continuous, and since E * wd = ∞, one has (E wd ) * ε ε→0 − −− → E * wd = ∞(15) due to Proposition 1. We can therefore conclude that also lim ε→0 (χ wd ) * ε = ∞.(16) III. UNIVERSALITY IN MQC In the one-way model of computation, information is processed by means of single qubit measurements on an initial highly entangled state. In the original proposal [13], this state was chosen to be a cluster state, but there is no reason to assume that this is the only possible choice. Indeed, in recent works it has been shown that also other states can be used as a resource for measurement-based quantum computation [9,18]. Following [18], in this work we consider the case in which any LOCC operation can be performed on the initial state. This corresponds to allowing two way classical communication, whereas the original scheme only requires one-way communication. We report here the definition of universal CQ resources used in [18], and on which the following discussion will be based. Definition 3 (Exact universal resources.). A family Σ = {σ i } i of states is called a universal resource for measurement-based quantum computation if, for every N and for every N -qubit quantum state |ϕ out , there exists an M -qubit resource state σ ∈ Σ and an LOCC protocol Λ LOCC that acts in the following way σ ΛLOCC − −−− → P out ⊗ P ⊗(M−N ) 0 ,(17) where P out = |ϕ out ϕ out | and P 0 = |0 0|. A. ε-approximate δ-stochastic universality While previous works have considered the characterization of exact universal resources for MQC, we are here more interested in considering weaker forms of universality, where the output state can be generated stochastically (with some finite success probability) or with some finite accuracy. Note that the nature of the resource might not be the only reason for which exact universality cannot be achieved. Indeed, as well as the circuit model with a finite gate basis, one can consider the case of one-way quantum computation where there are, e.g., only finite possible measurement directions [39]. Also, one must consider the fact that any experimental implementation will introduce some source of error in the computation. In order to take these factors into account, in the following we define the concepts of δ-stochastic and ε-approximate universality. In a realistic scenario, one is expected to be interested mainly in approximate stochastic (or quasi-deterministic) universality. Definition 4 (ε-approximate δ-stochastic universal resources). A family of states Σ = {σ i } i is called εapproximate, relatively to a distance measure D, and δ-stochastic universal if for every N and for every Nqubit quantum state |ϕ out , there exists an M qubit state σ ∈ Σ and an LOCC protocol with output branches {p i , ρ i } such that the sum of the probabilities p i for the branches where D i = D(ρ i , P out ⊗ P ⊗M−N 0 ) ≤ ε (where P out = |ϕ out ϕ out | and P 0 = |0 0|) is larger than 1 − δ. First, as regards the case of δ-stochastic universality, we do not require that the output state is generated deterministically, but it is sufficient that this happens under stochastic LOCC (SLOCC) with sufficiently high probability, that is p success = i:ρi=Pout⊗P ⊗(M −N ) 0 p i ≥ 1 − δ. In particular, when δ can be made arbitrary small, we may call this quasi-deterministic universality, which is stronger than δ-stochastic universality for a fixed δ . Second, as regards ε-approximate universality, we require the output of the computation is generated approximately with accuracy ε, as is the case for the quantum circuits built from a finite universal set of elementary gates. Precisely, D can be any distance measure on the set of states, that is contractive under LOCC and convex. The choice of the appropriate measure might depend on the task for which the output state is required (see, for example, the related discussion in [40]). B. Efficient universality We now consider the issue of how to generalize the concept of efficient universality to the approximate and stochastic cases. In order to do so, let us first recall the definition of exact efficient universality [18]. Definition 5 (Exact efficient universal resources). A family of states Σ = {σ i } i is called an efficient exact universal resource for measurement-based quantum computation if, for every N and for every N -qubit quantum state |ϕ out which can be obtained by a poly-sized quantum circuit, there exists an M -qubit state σ ∈ Σ, with M ≤ O(poly(N )), such that the transformation σ → |ϕ out |0 ⊗(M−N ) is possible by means of LOCC in time that is at most poly(N ) and using classical processing that is polynomially bounded in space and time. This definition can be easily extended to the approximate and stochastic case, when the desired accuracy ε and success probability δ are fixed. In this case one has the following Definition 6 (Efficient ε-approximate δ-stochastic universal resources). Let |ϕ out be any N -qubit quantum state that can be generated efficiently, i.e. with a polysized quantum circuit, from a product state in the network model, and let P out = |ϕ out ϕ out |. For approximate and stochastic computation, in many cases it is meaningful and interesting to take into account also the scaling of the overhead with the desired accuracy ε and success probability δ. In the circuit model, the scaling with the accuracy is determined by the Solovay-Kitaev theorem [41]. Similarly, in the oneway model we require that the scaling of the overhead in spatial, temporal and computational resources with ε is O(poly(m, log(1/ε))) for states that can be produced with m gates in the network model. Notice that we allow for a polynomial increase of resources with respect to the number of elementary gates m, as it is also done in the definition of exact efficient universality. It follows that any state that can be generated efficiently in the network model, i.e. with poly(m) elementary gates, should be approximated with accuracy ε with overhead that scales O(poly(m, log(1/ε))) in the measurement-based model. As regards the scaling with the probability parameter δ, we claim that it should be treated in a way analogous to the accuracy, based on the following observation. Let us consider the following observation, in which we see that the two parameters δ and ε indeed play the same role when we try to determine the fidelity between the desired output of a computation on an ε-approximate δstochastic resource and the real output of the protocol. Observation 1. Let us consider a universal εapproximate δ-stochastic resource Σ = {σ i } i , and let Λ LOCC , such that σ → {p i , ρ i } i , be the LOCC protocol for some output |ϕ out . We can, almost equivalently, consider Λ LOCC to be performing the following transformation: σ → ρ = i p i ρ i . Computing the fidelity between the desired output state |ϕ out and ρ, one finds that this leads to the bound F (ρ, |ϕ out ) ≥ (1 − ε)(1 − δ). However, since the counterpart of the Solovay-Kitaev theorem for the success probability δ has not been found, it is not clear how we could attain an efficient scaling by poly(log(1/δ)) in practice. Thus, we provide here a natural definition for efficient approximate and stochastic universal resources [42]. Definition 7 (Efficient approximate stochastic universal resources). Let |ϕ out be any N -qubit quantum state that can be generated efficiently, i.e. with a poly-sized quantum circuit, from a product state in the network model, and let P out = |ϕ out ϕ out |. A family of states Σ = {σ i } i is efficient approximate (with respect to some distance D) stochastic universal if, for all ε, δ > 0, there exists an M -qubit state σ ∈ Σ, with M ≤ O(poly(N, 1 δ , 1 ε )), and an LOCC protocol with output branches {p i , ρ i } i such that δ , 1 ε )) time, using classical side processing that is bounded in space and time by poly(N, 1 δ , 1 ε ). IV. CRITERIA FOR UNIVERSALITY AND NO-GO RESULTS In this Section we prove some necessary conditions for ε-approximate δ-stochastic universality, based on some entanglement properties of the resource. These results can be interpreted as a generalization of the ones obtained in [18], even though in some cases they require stronger assumptions on the entanglement monotone used to quantify the entanglement of the resource. In [18] it was noticed that any deterministic exact universal resource Σ must be such that, for any extendable entanglement measure E, E(Σ) = E * . By evaluating E * in the case of different entanglement measures it was possible to show how some families of states (e.g. W states, 1-dimensional systems,...) could not be exact deterministic universal. In the case of ε-approximate and δ-stochastic universality, we show that a similar (but, naturally, weaker) result still holds true, where E * is substituted with E * ε . As we shall see in the following, though, in these more general cases it is necessary to consider entanglement measures E satisfying some properties in addition to those required from the measures considered in [18]. While we are interested in the most general case of ε-approximate and δ-stochastic resources, we shall first treat the issue of ε-approximate deterministic (i.e. εapproximate and δ-stochastic, with δ = 0) resources separately. A. ε-approximate deterministic universality Theorem 1 (Criterion for ε-approximate deterministic universality). Let E be an extendable monotone that is weakly non-increasing under LOCC (as defined in Section II A), and let Σ = {σ i } i be an ε-approximate universal resource, with respect to some distance D. Then E(Σ) ≥ E * ε . Furthermore, if Σ is an approximate universal resource, then E(Σ) ≥ lim ε→0 + E * ε .(18) Proof. Let us fix the distance measure D, let |ϕ out be any N -qubit state and P out = |ϕ out ϕ out |. Since Σ is ε-approximate deterministic universal, there exist an Mqubit state σ ∈ Σ and an LOCC protocol σ → {p i , ρ i } such that D(ρ i , P out ) ≤ ε ∀i. Thus, for all i, E(ρ i ) ≥ min ρ {E(ρ)|D(ρ, P out ⊗ P ⊗(M−N ) 0 ) ≤ ε} = E ε (|ϕ out ⊗ |0 ⊗M−N ) ≥ E ε (|ϕ out ),(19) where P 0 = |0 0| and in the last inequality we have used that, since E is an extendable monotone, also E ε is [24]. Since (19) holds for all ρ i , and we have assumed that E is weakly non-increasing under LOCC, we have: E(σ) ≥ min i E(ρ i ) ≥ E ε (|ϕ out ) .(20) The first part of the theorem is proved by considering the fact that |ϕ out is allowed to be any state. The second part of the theorem follows from the fact that E ε , and thus E * ε , is monotonically non-increasing with ε. If Σ is an approximate deterministic universal resource, then the previous result must hold true for any value of ε > 0. As we have mentioned above, computing E * ε can in general be a hard task. Nevertheless, we have seen how this is possible at least in some particular cases. Whenever this happens, we can use Theorem 1 to generalize the results obtained in the exact deterministic case also to the approximate (or even ε-approximate) deterministic one, and show that some classes of states are not universal even in the approximate cases. These include, for examples, all graph states whose underlying graph has bounded rank-width, such as, e.g., tree graphs or cycle graphs (which have bounded χ-width) [17,33]. Moreover, Proposition 3 also allows us to show that the family of W states [21] is not ε-approximate universal for values of ε smaller than some finiteε. Example 1. Let us consider the family Ψ W = {|W N } N , where |W N is the N -qubit W state |W N = 1 √ N N i=1 |e N,i , and where |e N,i is defined to be the N -qubit computational basis state with a |1 in the i-th position, and |0 elsewhere. If D is a distance measure that is strictly related to the fidelity (see Definition 2), then we have that Ψ W is not an ε-approximate universal resource for any ε <ε, whereε depends on the choice of distance and is such that η(ε) ≃ 0.1%, where η is defined as in Definition 2. Proof. If π(W N ) is defined as in Section II C, we can consider π(Ψ W ) = sup WN ∈ΨW π(W N ). Since it can be shown (c.f. Ref. [18]) that π(Ψ W ) = 1/e, it follows that E G (Ψ W ) = 1 − 1 e . The statement follows immediately from Proposition 3, since we have that E (ε) * G ≥ 1 − 4η 1/3 + 3.4η 2/3 > 1 − 1/e = E G (Ψ W ),(21) for any choice of ε such that η(ε) 0.1%, and where η = η(ε) is such that D(ρ, σ) ≤ ε ⇒ F (ρ, σ) ≥ 1 − η(ε). B. ε-approximate δ-stochastic universality Let us consider, now, the case of ε-approximate and δ-stochastic universality. Also in this case we can formulate a criterion which generalizes the results obtained in [18] for exact deterministic universal resources, even though it is necessary to impose further requirements on the entanglement measure from which the criterion is derived. Theorem 2 (Criterion for ε-approximate δ-stochastic universality). Let E be an extendable strong monotone, and let Σ = {σ i } i be an ε-approximate (with respect to a distance D) δ-stochastic universal resource. Then E(Σ) ≥ (1 − δ)E * ε ,(22) where E ε is the ε-generalization of E with respect to D. Proof. Let us fix the distance measure D, let |ϕ out be an N -qubit quantum state and P out = |ϕ out ϕ out |. Since Σ is ε-approximate deterministic universal, there exist an M -qubit state σ ∈ Σ and an LOCC protocol σ → {p i , ρ i } such that ε−close p i ≥ (1 − δ), where P 0 = |0 0| and where the sum is taken over all indices i such that D(ρ i , P out ⊗ P ⊗(M−N ) 0 ) ≤ ε. We have then E(σ) ≥ i p i E(ρ i ) ≥ ε−close p i E(ρ i ) ≥ ε−close p i min{E(ρ)|D(P out ⊗ P ⊗(M−N ) 0 , ρ) ≤ ε} = ε−close p i E ε (P out ⊗ P ⊗(M−N ) 0 ) ≥ (1 − δ)E ε (P out ⊗ P ⊗(M−N ) 0 ) ≥ (1 − δ)E ε (|ϕ out ), where in the first inequality we have used the fact that E is a strong monotone, and the last inequality follows from the fact that E and, consequently, E ε are extendable monotones. An immediate consequence of this Theorem is the following Corollary 1. Let us consider an strong monotone E, and let E ε be its ε-generalization (with respect to some distance D) such that E * ε = ∞. Then any ε-approximate δ-stochastic universal family of resources Σ is such that E(Σ) = ∞, for all fixed values of δ < 1. Note that this implies that those families that were shown not to be approximate deterministic universal in the previous family are also not approximate δ-stochastic universal, for all values of δ < 1. C. Efficiency in the approximate and stochastic case In the previous paragraphs, we have only considered criteria for universality, without taking efficiency into account. We will see now how these criteria can be strengthened to become necessary conditions for efficient ε-approximate and δ-stochastic universality. In order to do so, the strategy will be analogous to the one followed in [18] in the exact and deterministic case, and is based on the following observation: Observation. A set of states Σ = {σ i } i is an efficient ε-approximate δ-stochastic resource if and only if all 2-dimensional cluster states |C d×d (for all d) can be prepared efficiently from the set Σ by LOCC with success probability p ≥ (1 − δ) and with accuracy ε. Proof. The necessity of the condition is immediate. The sufficiency follows from the fact that a family composed of states each of which is close to a cluster state is εapproximate δ-stochastic universal for any choice of ε and δ (see Section V C). As in the exact deterministic case, we see that the scaling of entanglement plays a major role when one considers efficiency-related issues. Theorem 3 (Criterion for efficient ε-approximate δ-stochastic universality). Let Σ = {σ i } i be an εapproximate (with respect to some distance D) δstochastic universal family, where σ i is a state on N i qubits. Let us consider an extendable strong entanglement monotone E, and let f ε be a function such that, for every 2-dimensional cluster state |C d×d on N = d 2 qubits, one has E ε (|C d×d ) ≥ f ε (N ), where E ε is the ε-generalization of E with respect to D. If E(σ i ) scales as log f ε (N i ), then Σ cannot be an efficient ε-approximate and δ-stochastic universal resource. Proof. Since we have assumed that Σ is an ε-approximate δ-stochastic universal resource, for any N = d 2 there must exist a g(N )-qubit state σ g(N ) ∈ Σ and an LOCC protocol σ g(N ) → {p i , ρ i } i such that ε−close p i ≥ (1 − δ), where the sum is taken over the indices i such that D(|ϕ i , |C d×d |0 ⊗(g(N )−N ) ) ≤ ε. From what we have already seen (see the proof of Theorem 2), it follows that, necessarily E(σ g(N ) ) ≥ (1 − δ)E ε (|C d×d ) ≥ (1 − δ)f ε (N ). In order for Σ to be an efficient resource, though, it is necessary that g(N ) is at most polynomial in N , and thus, following an argument parallel to that in Theorem 9 of [18], we can conclude that E(Σ) cannot scale logarithmically with f ε (N ). We emphasize that the family of two-dimensional cluster states in principle does not play a distinguished role in Theorem 3, in the sense that it can be replaced -without weakening or strengthening the result-by any arbitrary efficient universal family or, in fact, any family of states which themselves also can efficiently be prepared. Example 2. Based on the criterion by Theorem 3, the states whose Schmidt-rank width have a polylogarithmic scaling in N are not efficient exact deterministic universal resources as shown in Ref. [18], nor efficient εapproximate and δ-stochastic universal resources. These include the cluster state on the 2D stripe d × log d, and the cluster state on the faulty 2D lattice with a site occupation probability p ≤ p c , as mentioned later in Sec. V. V. EXAMPLES OF ε-APPROXIMATE AND/OR δ-STOCHASTIC UNIVERSAL RESOURCES In this section we provide examples of families of states that are universal resource states when we relax our requirements for universal MQC to ε-approximate and/or δ-stochastic universality. A. 2D cluster state with holes as an exact quasi-deterministic resource Our model is a faulty 2D cluster state in which qubits get entangled after qubits are prepared with partial losses (called holes here) in the background 2D square lattice with total size M = N 2 , where N is the side length. The lattice-site occupation probability is denoted as p site , and thus the hole probability is given by 1 − p site . We assume here that every hole occurs independently according to the probability, and the locations of these holes are heralded. It is conceivable, for example in the implementations by optical lattice, that we may be able to check whether atoms are stored for each site before creating the 2D cluster state, and thus without destroying entanglement. That is why, our faulty 2D cluster state with holes is considered to be a pure graph state corresponding to a specific configuration of holes, in contrast with the statistical ensemble (classical mixture of several configurations) characterized by p site . All statistical statements, such as the percolation phenomenon, are meant to hold true almost with certainty (more precisely, with probability approaching unity in the thermodynamical limit), for all the possible realizations of the configuration of holes with a given p site . Proof. The detailed proof is available in Ref. [43], in which the phase transition of the computational power of the 2D cluster state with holes was proved at the above mentioned threshold p c . See also a preceding work [44] for the use of percolation theory to prepare cluster states by non-deterministic gates. In the supercritical phase (p site > p c ), it has been shown that if a preprocessing by polynomial-time classical computation is provided, we can construct an LOCC conversion which concentrates a perfect 2D cluster state from a faulty cluster state with a constant overhead (depending only on p site ). Such an LOCC conversion works almost with certainty (namely, with success probability of LOCC conversion approaching unity exponentially in L), and will produce the 2D cluster state with fidelity exactly one when it is available. That is why the resource is efficient exact quasi-deterministic universal. B. Deformed 2D cluster state as an exact quasi-deterministic resource We now give an example of universal resources which is not a graph state. Let us consider a local deformation of the 2D N × N cluster state |C N ×N , |dC N ×N = 2 1 + λ 2 N 2 /2 Λ ⊗N 2 |C N ×N ,(23) where Λ = diag(1, λ) is the local deformation parametrized by λ such that, without loss of generality, 0 ≤ λ ≤ 1. We call it a deformed 2D cluster state whereby the perfect 2D cluster state corresponds to λ = 1. The deformed 2D cluster state can be seen as a "noisy" 2D cluster state resulting probabilistically from the local filtering operation Λ. Note however that the fidelity with the perfect 2D cluster state is (1+λ) 2 2(1+λ 2 ) M , i.e., exponentially small in the number M = N 2 of the total qubits, so that the inverse transformation to the perfect 2D cluster state (with the same size) will succeed only with an exponentially small probability. Nevertheless, we show that one single copy of such a system can be an efficient resource, regardless of its size M , when λ lies above a certain threshold. Proof. We show that one can convert the deformed cluster state |dC N ×N by means of LOCC deterministically into a graph state corresponding to a 2D N × N square lattice with holes. We apply local 2-outcome measurements described by POVM {Λ −1 = diag(λ, 1), Λ −1 = diag( √ 1 − λ 2 , 0)} at each qubit. If the outcome Λ −1 occurs, we successfully "undo" the effect of deformation, while when the outcome Λ −1 happens, the qubit is projected into |0 so that it corresponds to a deletion of the vertex with attached edges (i.e., a hole) in the 2D cluster state. The probability of these successful events, which is independent of the position of qubits, determines the site occupation probability, p site = 2λ 2 1 + λ 2 .(24) It should be noted that this expression is independent of the system size M . According to the threshold p c of the 2D cluster state with holes, it is now clear that if λ > λ c ≈ 0.6490 . . . the resulting resource is efficient exact quasi-deterministic universal, so is true for the original deformed 2D cluster state. We remark that here λ > 0.6490 . . . is merely a sufficient condition for being efficiently universal. i = (1 − p)|C Ni C Ni | + p|C Ni C Ni |, where |C Ni is the 2-dimensional cluster state on N i = i × i qubits, and |C Ni is obtained from |C Ni by applying a phase flip σ z on a single qubit, so that |C Ni has a −1 eigenvalue only at the corresponding stabilizer operator. Note that p is independent of the total system size N i , because of the (unrealistic) assumption that only one phase flip can happen. Let D be a convex distance measure on the set of states such that D(ρ, σ) ≤ 1 for all ρ and σ [45]. Then Σ is an ε-approximate deterministic universal resource, relatively to D, for ε ≥ p. Proof. Let us consider any output state |ϕ out and let P out be the projector onto such a state. Since the family of cluster states is exact and deterministic universal, then there exist a state |C Ni and an LOCC protocol that, acting on |C Ni , generates the state |ϕ out . This means that there exists an LOCC protocol Λ LOCC such that Λ LOCC [|C Ni C Ni |] = k p k P (A) out ⊗P (R) k , where P k = |k k| are projectors onto orthogonal states of some register R. We have thus Λ LOCC [σ i ] = (1 − p)Λ LOCC [|C Ni C Ni |] + pΛ LOCC [|C Ni C Ni |] = (1 − p) k p k P (A) out ⊗ P (R) k + p kp k τ (A) k ⊗ P (R) k where Λ LOCC [|C Ni C Ni |] = kp k τ (A) k ⊗ P (R) k . Since both |C Ni and |C Ni are 2-dimensional cluster states on N i qubits, we have that the probability of each output branch is the same and is given by [14] p k =p k = 1 2 Ni−m , where N i − m is the number of qubits that are measured. We can thus write the final state of the system plus the register as Λ LOCC [σ i ] = k 1 2 Ni−m [(1 − p)P (A) out + pτ (A) k ] ⊗ P (A) k . The k-th output branch, thus, yields a state ρ k = (1 − p)P out + τ (k) such that D(ρ k , P out ) = D((1 − p)P out + pτ (k) , P out ) ≤ (1 − p)D(P out , P out ) + pD(τ (k) , P out ) ≤ p, where the first inequality derives from the convexity of the distance D, and the second one follows from the fact that D(ρ, σ) ≤ 1. Since this holds for all the output branches, we obtain that the state |ϕ out has been produced ε-approximately (for any ε ≥ p) and deterministically. The proof is completed by noticing that the argument holds for any desired output state |ϕ out . We remark that a similar result holds not only for mixtures of two cluster states, but also for states of the form σ i = (1 − p)|C Ni C Ni | + p k λ k |C k Ni C k Ni |,(25) where λ k = 1, k is a binary vector of length N i where k j ∈ {0, 1} corresponds to qubit j, and C k Ni is a 2D cluster state which is obtained from |C Ni by applying (σ j z ) kj to qubit j, i.e. |C k Ni = (σ j z ) kj |C Ni . Notice that the |C k Ni form a basis, and hence the noise term can also be the identity. Also the action of local Pauli noise channels acting on the individual qubits leads to states of this form [46]. The key insight is again that the success probability for each branch is the same for all noise terms, leading to a distance D(ρ k , P ) ≤ p for the output states, independent of the measurement outcomes. We also mention that a similar resource with the subsections V B and V C has been considered recently in Ref. [47] through the analysis of the thermal state for the cluster-state Hamiltonian with a local σ z field. D. Stability of universal resources Let us consider a scenario in which one wants to experimentally implement some measurement-based computation. In this case, it is natural to assume that the initial resource cannot be prepared exactly. In the following Theorem 4, we analyze this case, giving a proof of the stability of universal resources under initial perturbation, and determining an expression for the worsening of the probability and accuracy parameters as a function of the error in the initial preparation. Furthermore this also formally proves (taking into account the effect on both parameters ε and δ) the intuitive idea that the computation on the approximate states can take place by means of the same LOCC protocol, thus the exact knowledge of the state is not necessary. This also implies that if computation on the original states was efficient, then it remains so also on the new states. Notice however that we do not consider here the case in which the LOCC protocol itself is faulty. Theorem 4. Let D be a convex, bounded distance measure strictly related to the fidelity, such that the maximum distance between any two states be unity [48]. Let us consider an (efficient) ε-approximate (with respect to D) δstochastic universal resource Γ = {γ i } i , with δ + ε < 1. Moreover, let Σ = {σ j } j be a family of states such that, for any γ ∈ Γ, there exists a state σ ∈ Σ with D(σ, γ) ≤ µ (for some µ ≤ 1 − δ − ε). Then Σ is an (efficient) ε ′ -approximate δ ′ -stochastic universal resource for any choice of ε ′ and δ ′ such that δ ′ η(ε ′ ) ≥ η(ε + δ + µ), where η(ε) is such that D(ρ, σ) ≤ ε ⇒ F (ρ, σ) ≥ 1 − η(ε) [49]. Its proof is given in Appendix B. Note that, in general, δ ′ and ε ′ will have to be (polynomially) larger than δ and ε. If D is the trace distance, then we have η(ε) ≥ ε thus obtaining that one can always find ε ′ and δ ′ satisfying the condition: δ ′ ε ′ ≥ ε + δ + µ. Note that the condition we have found implies that δ ′ and ε ′ must be larger than, respectively, δ and ε. More importantly, though, Theorem 4 implies that, whenever Γ is an (efficient) deterministic exact universal resource, then one can choose any δ ′ and ε ′ such that δ ′ ε ′ ≥ µ. We have thus the following Corollary 2. Let Σ = {σ i } i be an (efficient) exact deterministic universal resource and D be any distance measure strictly related to fidelity. Then, for every δ, ε > 0 there exists a µ > 0 such that any familyΣ = {σ i } i with D(σ i ,σ i ) ≤ µ, ∀i is an (efficient) ε-approximate (with respect to D) δ-stochastic universal resource. Furthermore, if output |ϕ out is obtained by applying LOCC protocol Λ LOCC on a state σ i ∈ Σ, then the same protocol can be used on the corresponding stateσ i ∈Σ to produce an output that, with probability p ≥ (1 − δ), is within distance ε from |ϕ out . This implies, in particular, that any family composed of states that are close enough to, e.g., a cluster state is ε-approximate δ-stochastic universal for some non-trivial choice of ε and δ. VI. CONCLUSIONS AND OUTLOOK In this paper we have studied the issues of approximate and stochastic universality in measurement-based quantum computation. We have defined the concepts of approximate and stochastic universality, and shown how these concepts are not equivalent to each other by providing examples of resources that are approximate and deterministic universal, or exact stochastic universal. Generalizing the results obtained in [18], we have presented entanglement-based criteria that must be satisfied by any approximate (stochastic) universal resource. Moreover we have shown that such criteria are strong enough to allow us to discard some well-known families of states as non-universal, including e.g. GHZ states, W-states and 1D cluster states. The issue of efficiency has also been discussed, and we have shown how the previous results can be strengthened to include the request that a universal family of resources also allows for efficient computations. We found that entanglement needs to grow sufficiently fast for any approximate stochastic universal resource. On the other side, we have provided examples of resources that are approximate and/or stochastic universal. In particular, we have studied the case of a family of states that is only an approximation of some (εapproximate and/or δ-stochastic) universal family. We have given a formal proof of the fact that such a family is always ε ′ -approximate and δ ′ -stochastic universal, and found an explicit bound for the scaling of the parameters ε ′ and δ ′ as functions of the original parameters ε and δ, and of the degree of approximation of the family itself. The proof also formalizes the intuitive idea that the computation on the approximate family can be performed by means of the same protocol that was devised for the exact family. In particular this means that if the initial resource was efficient universal, then also the approximate one is. While we have found that basically any well behaved entanglement monotone can be used to obtain criteria for approximate and stochastic universality, one of the quantities considered in [18], the entropic entanglement width, does not fall under this category as it is not an entanglement monotone (in the terminology of [18], more precisely, not a type-I monotone). For this measure it is not clear whether the results obtained for the exact, deterministic case can be lifted to the approximate, stochastic case. This affects in particular results about nonuniversality of states with a bounded or logarithmically growing block-wise entanglement, such as ground states of strongly correlated 1D quantum systems. We have also not touched the issue of encoded universality [18], where the desired quantum states need only be generated in an encoded form. Also in this case it should be possible to obtain entanglement based criteria for approximate stochastic encoded universality, using the methods and techniques developed in this paper. Finally, we would like to comment in relation to the results presented in [50,51], where it is shown that a randomly chosen generic pure state (in other words the majority of all states) is no more useful as a resource for measurement-based quantum computation than a string of random classical bits, despite the fact that the former is colloquially often said to be almost maximally entangled. Particularly related to the results presented in this paper, is the fact (proved in Ref. [51]) that a family of states |ψ M on M qubits, whose geometric measure scales as E G (|ψ M ) ≥ 1 − 2 −M+O(log 2 M) cannot provide a super-polynomial speed-up over classical computation with the aid of randomness and thus it is conceivably not a universal resource (unless the class of decision problems solvable by a probabilistic Turing machine in polynomial time with bounded error (BPP) coincides with the class of decision problems solvable by a quantum computer in polynomial time with bounded error (BQP)). Note that it is required that the scaling of the geometric measure is even faster (by a constant factor in the front of M in the exponent) than that of the cluster state for any spatial dimension, E G (|C M ) = 1 − 2 −⌊M/2⌋ [38], and thus these states |ψ M can be considered highly entangled with regards to this measure (in the sense that such a family would not fail the criterion for universality based on the geometric measure). There are two kinds of examples in Ref. [51] which are shown to have such a scaling of the geometric measure. The first example is given by generic Haar-random pure states. It is not clear for us whether they also pass the necessary conditions illustrated in the previous sections if one considers other entanglement measures, although it is possible. However, it should be noted that these states already inherit "unphysical" complexity as resource states since it might not be possible to prepare them in a time polynomial in M . The second, efficiently preparable, example is given by a tree tensor network state. While in [51] it is shown that these states have indeed high geometric measure, it should be noted that its Schmidt-rank width is bounded without reaching the maximum (because of the constant tree width [52]). We could therefore interpret that its uselessness (as a universal resource for MQC) originates from being too little entangled in terms of the Schmidtrank width: the family would in fact fail the criteria illustrated in the previous sections when one bases them on this entanglement measure. It would be interesting to see whether it is possible to find necessary criteria such as the ones shown in this work that allows us to discard random pure states (and some pseudo random pure states which are efficiently preparable in case they are not universal either (cf. [53])) as nonuniversal. It is possible that such states already fail the criteria for some existing entanglement measure (other than the geometric measure), but it might prove necessary to identify a new one in order to obtain this result. We note that randomness in the description of the resource does not necessarily taint its usefulness immediately, as can be seen for instance by our Example 3. ) > ε respectively. In the last inequality we have used the fact that ε−close p i ≤ 1, and that Eq. (24) holds true. We have thus D(ρ A , P out ⊗ P ⊗(M−N ) 0 ) ≤ µ + ε + δ.(B2) Since D is strictly related to fidelity, this implies that F (ρ A , P out ⊗ P ⊗(M−N ) 0 ) ≥ 1 − η(µ + ε + δ). Defining two parameters δ ′ , ε ′ such that 1 − η(µ + ε + δ) := 1 − δ ′ η(ε ′ ), and applying Lemma 1, we obtain 1 − δ ′ ≤ i:| ψi|ϕout |0 ⊗(M −N ) | 2 ≥1−η(ε ′ ) q i = i:D(|ϕi ,|ϕout |0 ⊗(M −N ) )≤ε ′ q i , for any decomposition ofρ A into pure statesρ A = i q i |ψ i ψ i |. Noticing that the argument holds for any output state |ϕ out completes the proof. et al., Phys. Rev. Lett. 102, 190501 (2009); M. J. Bremner et al., Phys. Rev. Lett 102, 190502 (2009)]. PACS numbers: 03.67.Lx, 03.67.Mn, 03. P2 . P2Monotonicity under LOCC: entanglement cannot increase under LOCC, E(Λ LOCC [ρ]) ≤ E(ρ). Figure 1 : 1(a) Example of a subcubic tree T with six leaves. (b) C 2). If e = {i, j} is an arbitrary edge of T , we denote by T \e the graph obtained by deleting the edge e from T . The graph then consists of two connected components, which naturally induce a bipartition (A e T , B e T ) of the system. If χ A e T ,B e T (|ψ ) is the Schmidt rank of |ψ , with respect to the bipartition (A e T , B e T ), the Schmidt-rank width of |ψ is given by A family of states Σ = {σ i } i is efficient ε-approximate (with respect to some distance D) δ-stochastic universal if there exists an M -qubit state σ ∈ Σ, with M ≤ O(poly(N )), and an LOCC protocol with output branches {p i , ρ i } i such that i:D(ρi,Pout⊗P ⊗(M −N ) 0 )≤εp i ≥ 1 − δ (with P 0 = |0 0|) that can be implemented in O(poly(N )) time, using classical side processing that is bounded in space and time by poly(N ). i ≥ 1 − δ (with P 0 = |0 0|) that can be implemented in O(poly(N, 1 Example 3 ( 3[43]). A family of 2D cluster states with holes (characterized by increasing total size M ) is an efficient exact quasi-deterministic universal resource if and only if the site occupation probability p site is greater than the percolation threshold p c = 0.5927 . . . of the 2D square lattice. Example 4 . 4A family of the 2D deformed cluster states (with the total size M increasing) is an efficient exact quasi-deterministic universal resource if the deformation parameter λ is larger than 0.6490 . . . . D≤ (ρ A , ρ A ) ≤ D(ρ AR , ρ AR ) ≤ D(σ, γ) ≤ µ.This implies that D(ρ A ,P out ⊗ P D(ρ A , ρ A ) + D(ρ A , P out ⊗ P have used the triangle inequality in the second step. Since D is convex, D(ρ A , P out ⊗ P p i D(ρ i , P out ⊗ P Appendix A: Proof ofProposition 2In order to prove Proposition 2, we will first prove the following result.Proposition 4. Let D be a distance measure on the set of states that is strictly related to the fidelity, and let E (ε) G denote the corresponding ε-geometric measure. Then, for any pure state |ψ and for any ε > 0,This result will yield the proof of Proposition 2. Indeed, let us consider the right-hand-side of the inequality (A1). We have thatProposition 2 is then proved straightforwardly by showing that, for η ≤ 0.44 the above minimum is obtained byWe now prove Proposition 4. In order to do this we will need the following two lemmas. Lemma 1. Let ρ be a mixed state. Then, for any decomposition ρ = i p i |ψ i ψ i | of ρ into pure states, one has thatfor any choice of η ∈ [0, 1] and ∆ > 0.Proof. The statement is proved as follows.where by "close" and "far" we refer respectively to those for any ∆ > 0, we havewhere the second inequality derives from Lemma 2, the last one from Lemma 1, and η(ε) is such thatAppendix B: Proof of Theorem 4Proof. Let |ϕ out be any desired N -qubit output state, and let P out = |ϕ out ϕ out |. Since Γ is an ε-approximate δ-stochastic universal resource, there exist an M -qubit state γ ∈ Γ and an LOCC protocol acting on γ withwhere the sum is taken over all indices i such that) ≤ ε, and P 0 = |0 0|. This means that there exists an LOCC protocol Λ LOCC such that, where P i are projectors onto orthogonal states of some register R. We define ρ A = Tr R (ρ AR ) = i p i ρ (A) i . Let σ ∈ Σ be such that D(σ, γ) ≤ µ and let us definẽ ρ AR = Λ LOCC [σ] andρ A = Tr R (ρ AR ). Since D is contractive under CPT maps we have . D Deutsch, Proc. Roy. Soc. Lond. A. 400D. Deutsch, Proc. Roy. Soc. Lond. A 400 (1818), pp. 97- 117 (1985). . E Bernstein, U Vazirani, SIAM Journal on Computing. 265E. Bernstein and U. Vazirani. SIAM Journal on Com- puting 26(5), pp. 1411-1473 (1997). D Deutsch, Proc. R. Soc. Lond., A. R. Soc. Lond., AD. Deutsch. In Proc. R. Soc. Lond., A, page 425(1868) pp. 73-90 (1989). . A Barenco, C H Bennett, R Cleve, D P Divincenzo, N Margolus, P W Shor, T Sleator, J A Smolin, H Weinfurter, Phys. Rev. A. 523457A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P.W. Shor, T. Sleator, J.A. Smolin, H. We- infurter, Phys. Rev. A 52, 3457 (1995). A C , -C Yao, Proceedings of 34th Annual Symposium on the Foundation of Computer Science. 34th Annual Symposium on the Foundation of Computer ScienceLos AlamitosIEEE Computer SocietyA. C.-C. Yao, Proceedings of 34th Annual Symposium on the Foundation of Computer Science (IEEE Computer Society, Los Alamitos) pp. 352-361 (1993). . E Farhi, J Goldstone, S Gutmann, J Lapan, A Lundgren, D Preda, Science. 292472E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lund- gren, and D. Preda, Science, 292:472, 2001. . J I Latorre, R Orus, Phys. Rev. A. 6962302J. I. Latorre and R. Orus. Phys. Rev. A, 69:062302, 2004. . G K Brennen, A Miyake, Phys. Rev. Lett. 10110502G. K. Brennen and A. Miyake, Phys. Rev. Lett., 101: 010502, 2008. . D Gross, J Eisert, Phys. Rev. Lett. 98220503D. Gross and J. Eisert, Phys. Rev. Lett., 98:220503, 2007. . D Gottesman, I Chuang, Nature. 402390D. Gottesman and I. Chuang, Nature 402:390, 1999. . M A Nielsen, Phys. Lett. A. 30896M. A. Nielsen, Phys. Lett. A, 308:96, 2003. . S Perdrix, P Jorrand, quant-ph/0404146e-printS. Perdrix and P. Jorrand, e-print quant-ph/0404146. . R Raussendorf, H J Briegel, Phys. Rev. Lett. 865188R. Raussendorf and H. J. Briegel. Phys. Rev. Lett, 86:5188, 2001. . R Raussendorf, H J , Briegel. Quant. Inf. Comp. 26R. Raussendorf and H. J. Briegel. Quant. Inf. Comp., 2(6):443-486, 2002. . R Raussendorf, D E Browne, H J Briegel, Phys. Rev. A. 6822312R. Raussendorf, D. E. Browne and H. J. Briegel, Phys. Rev. A, 68:022312, 2003. . D Leung, Int. J. Quant. Info. 233D. Leung, Int. J. Quant. Info., 2:33, 2004. . M Van Den Nest, A Miyake, W Dür, H J Briegel, Phys. Rev. Lett. 97M. Van den Nest, A. Miyake, W. Dür, and H. J. Briegel. Phys. Rev. Lett., 97(150504), 2006. . M Van Den Nest, W Dür, A Miyake, H J Briegel, New J. Phys. 9204M. Van den Nest, W. Dür, A. Miyake, and H. J. Briegel. New J. Phys., 9(204), 2007. . H J Briegel, R Raussendorf, Phys. Rev. Lett. 86910H. J. Briegel and R. Raussendorf. Phys. Rev. Lett 86, 910 (2001). Bell's Theorem, Quantum Theory, and Conceptions of the Universe. D M Greenberger, M A Horne, A Zeilinger, M. KafatosKluwer, DordrechtD. M. Greenberger, M. A. Horne and A. Zeilinger, Bell's Theorem, Quantum Theory, and Conceptions of the Universe', M. Kafatos (Ed.), Kluwer, Dordrecht, 69- 72 (1989). . W Dür, G Vidal, J I Cirac, Phys. Rev. A. 6262314W. Dür, G. Vidal, and J. I. Cirac. Phys. Rev. A, 62:062314, 2000. . C M Papadimitriou, Computational Complexity. C. M. Papadimitriou. Computational Complexity. . Addison-Wesley, Reading, MassachusettsAddison-Wesley, Reading, Massachusetts, 1994. M A Nielsen, I L Chuang, Quantum Computation and Quantum Information. CambridgeCambridge University PressM. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. . C E Mora, M Piani, H J Briegel, New J. Phys. 1083027C. E. Mora, M. Piani, and H. J. Briegel. New J. Phys., 10:083027, 2008. . R Raussendorf, MunichLudwig-Maximilians UniversityPhD ThesisR. Raussendorf, PhD Thesis, Ludwig- Maximilians University, Munich (2003), online at http://edoc.ub.uni-muenchen.de/archive/00001367/. . M A Nielsen, C M Dawson, Phys. Rev. A. 7142323M. A. Nielsen and C. M. Dawson, Phys. Rev. A, 71:042323, 2005. . P Aliferis, D W Leung, Phys. Rev. A. 7332308P. Aliferis and D. W. Leung, Phys. Rev. A, 73:032308, 2006. . C H Bennett, D P Divincenzo, J A Smolin, W K Wootters, Phys. Rev. A. 543824C.H. Bennett, D.P. DiVincenzo, J. A. Smolin, and W.K. Wootters. Phys. Rev. A 54, 3824 (1996). . V Vedral, M B Plenio, M A Rippin, P L Knight, Phys. Rev. Lett. 782275V. Vedral, M.B. Plenio, M.A. Rippin, and P.L. Knight. Phys. Rev. Lett, 78:2275, 1997. . M B Plenio, S Virmani, Quant. Inf. Comp. 71M. B. Plenio and S. Virmani. Quant. Inf. Comp., 7(1), 2007. Entanglement and complexity in quantum information and computation. C.-E Mora, University of InnsbruckPhD thesisC.-E. Mora. Entanglement and complexity in quantum information and computation. PhD thesis, University of Innsbruck, 2007. A distance D on the set of states is said to be contractive under completely positive and trace preserving (CPTP) maps if, given any CPTP map Λ. then D(ρ, σ) ≥ D(Λ[ρA distance D on the set of states is said to be con- tractive under completely positive and trace preserv- ing (CPTP) maps if, given any CPTP map Λ, then D(ρ, σ) ≥ D(Λ[ρ], Λ[σ]), for any two states ρ and σ. . M Van Den Nest, W Dür, G Vidal, H J Briegel, Phys. Rev. A. 7512337M. Van den Nest, W. Dür, G. Vidal, and H. J. Briegel. Phys. Rev. A, 75:012337, 2007. . J Eisert, H J Briegel, Phys. Rev. A. 6422306J. Eisert and H. J. Briegel. Phys. Rev. A, 64:022306, 2001. . A Shimony, Ann. N. Y. Acad. Sci. 755675A. Shimony. Ann. N. Y. Acad. Sci, 755:675, 1995. . H Barnum, N Linden, J. Phys. A. 346787H. Barnum and N. Linden. J. Phys. A, 34:6787, 2001. . T C Wei, P M Goldbart, Phys. Rev. A. 6842307T. C. Wei and P. M. Goldbart. Phys. Rev. A, 68:042307, 2003. . D Markham, A Miyake, S Virmani, New J. Phys. 9194D. Markham, A. Miyake, and S. Virmani. New J. Phys., 9:194, 2007. One can show that the model is ε-approximate deterministic universal. One can show that the model is ε-approximate determin- istic universal. . A Gilchrist, N K Langford, M A Nielsen, Phys. Rev. A. 71A. Gilchrist, N. K. Langford, and M. A. Nielsen, Phys. Rev. A, 71(062310), 2005. . A Y Kitaev, Russ. Math. Surv. 5261191A. Y. Kitaev. Russ. Math. Surv., 52(6):1191, 1997. One could demand the polylogarithmic scaling for 1/δ and 1/ε in the definition. One could demand the polylogarithmic scaling for 1/δ and 1/ε in the definition. . D E Browne, M B Elliott, S T Flammia, S T Merkel, A Miyake, A J Short, New J. Phys. 1023010D. E. Browne, M. B. Elliott, S. T. Flammia, S. T. Merkel, A. Miyake, and A. J. Short, New J. Phys., 10:023010, 2008. . K Kieling, T Rudolph, J Eisert, Phys. Rev. Lett. 99130501K. Kieling, T. Rudolph, and J. Eisert Phys. Rev. Lett, 99:130501, 2007. Note that any bounded distance measure D can be rescaled so that it satisfies the condition D(ρ, σ) ≤ 1. Note that any bounded distance measure D can be re- scaled so that it satisfies the condition D(ρ, σ) ≤ 1. M Hein, W Dür, J Eisert, R Raussendorf, M Van Den Nest, H J Briegel, Proceedings of the International School of Physics "Enrico Fermi" on "Quantum algorithms and chaos. the International School of Physics "Enrico Fermi" on "Quantum algorithms and chaosM. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J. Briegel. In Proceedings of the Interna- tional School of Physics "Enrico Fermi" on "Quantum algorithms and chaos", 2005. . S D Barrett, S D Bartlett, A C Doherty, D Jennings, T Rudolph, arXiv:0807.4797S.D. Barrett, S.D. Bartlett, A.C. Doherty, D. Jennings, and T. Rudolph, arXiv:0807.4797. Note that, since the distance is chosen to be bounded. it is always possible to re-scale it so that the maximum distance between two states is 1Note that, since the distance is chosen to be bounded, it is always possible to re-scale it so that the maximum distance between two states is 1. The statement can be easily generalized to the case in which the distance is bounded by some constantD = 1. this case, one requires ε + δ + µ <D, and the bound becomes δ ′ η(ε ′ ) ≥ η(ε +Dδ + µ). The statement can be easily generalized to the case in which the distance is bounded by some constantD = 1. In this case, one requires ε + δ + µ <D, and the bound becomes δ ′ η(ε ′ ) ≥ η(ε +Dδ + µ). . M J Bremner, C Mora, A Winter, Phys. Rev. Lett. 102190502M.J. Bremner, C. Mora, A. Winter, Phys. Rev. Lett., 102:190502, 2009. . D Gross, S T Flammia, J Eisert, Phys. Rev. Lett. 102190501D. Gross, S.T. Flammia, and J. Eisert, Phys. Rev. Lett., 102:190501, 2009. The resource is a tree and thus its tree width is constant if we consider the local dimension is d, which actually scales as √ log M in the construction of Ref. Accordingly, the entanglement-based criterion for encoded universality should be applied. strictly speakingThe resource is a tree and thus its tree width is constant if we consider the local dimension is d, which actually scales as √ log M in the construction of Ref. [51]. Accordingly, the entanglement-based criterion for encoded universality should be applied, strictly speaking. . R A Low, arXiv:0903.5236pre-printR. A. Low, pre-print arXiv:0903.5236
[]
[ "Asteroseismic signatures of helium gradients in late F-type stars", "Asteroseismic signatures of helium gradients in late F-type stars" ]
[ "M Castro \nLaboratoire d'Astrophysique de Toulouse et Tarbes -UMR 5572\nUniversité Paul Sabatier Toulouse III -CNRS\n14, av. E. Belin31400ToulouseFrance\n", "S Vauclair \nLaboratoire d'Astrophysique de Toulouse et Tarbes -UMR 5572\nUniversité Paul Sabatier Toulouse III -CNRS\n14, av. E. Belin31400ToulouseFrance\n" ]
[ "Laboratoire d'Astrophysique de Toulouse et Tarbes -UMR 5572\nUniversité Paul Sabatier Toulouse III -CNRS\n14, av. E. Belin31400ToulouseFrance", "Laboratoire d'Astrophysique de Toulouse et Tarbes -UMR 5572\nUniversité Paul Sabatier Toulouse III -CNRS\n14, av. E. Belin31400ToulouseFrance" ]
[]
1We investigate signatures of helium diffusion in late F-type stars by asteroseismology. Stellar models were computed with different physical inputs (with or without element diffusion) and iterated in order to fit close-by evolutionary tracks for each mass. The theoretical oscillation frequencies were computed and compared for pairs of models along the tracks. Various asteroseismic tests (large separations, small separations, second differences) were used and studied for the comparisons. The results show that element diffusion leads to changes in the frequencies for masses larger than 1.2 M ⊙ . In particular the helium gradient below the convective zone should be detectable through the second differences.Abstract. Element diffusion is expected to occur in all kinds of stars : according to the relative effect of gravitation and radiative acceleration, they can fall or be pushed up in the atmospheres. Helium sinks in all cases, thereby creating a gradient at the bottom of the convective zones. This can have important consequences for the sound velocity, as has been proved in the sun with helioseismology.
10.1051/0004-6361:20064821
[ "https://arxiv.org/pdf/astro-ph/0605553v1.pdf" ]
17,481,981
astro-ph/0605553
0c0d4cd848947290e0b45f9dc6fb4bf6b68278a1
Asteroseismic signatures of helium gradients in late F-type stars May 2006 September 12, 2018 M Castro Laboratoire d'Astrophysique de Toulouse et Tarbes -UMR 5572 Université Paul Sabatier Toulouse III -CNRS 14, av. E. Belin31400ToulouseFrance S Vauclair Laboratoire d'Astrophysique de Toulouse et Tarbes -UMR 5572 Université Paul Sabatier Toulouse III -CNRS 14, av. E. Belin31400ToulouseFrance Asteroseismic signatures of helium gradients in late F-type stars May 2006 September 12, 2018Received ; acceptedarXiv:astro-ph/0605553v1 22 (DOI: will be inserted by hand later)asteroseismologydiffusionstars: oscillations (including pulsations)stars: interiors Astronomy & Astrophysics manuscript no CastroVauclairastroph 1We investigate signatures of helium diffusion in late F-type stars by asteroseismology. Stellar models were computed with different physical inputs (with or without element diffusion) and iterated in order to fit close-by evolutionary tracks for each mass. The theoretical oscillation frequencies were computed and compared for pairs of models along the tracks. Various asteroseismic tests (large separations, small separations, second differences) were used and studied for the comparisons. The results show that element diffusion leads to changes in the frequencies for masses larger than 1.2 M ⊙ . In particular the helium gradient below the convective zone should be detectable through the second differences.Abstract. Element diffusion is expected to occur in all kinds of stars : according to the relative effect of gravitation and radiative acceleration, they can fall or be pushed up in the atmospheres. Helium sinks in all cases, thereby creating a gradient at the bottom of the convective zones. This can have important consequences for the sound velocity, as has been proved in the sun with helioseismology. Introduction Among the different physical processes that occur in stellar interiors, element diffusion plays an important role. Under the oppositing effects of gravitation and thermal settling on the one hand, and radiative acceleration on the other hand, the various elements undergo relative separation. This process, which was described by Michaud (1970) to account for the abundance anomalies observed in chemically peculiar A stars, is now recognized to occur in all kinds of stars. Vauclair & Théado (2004) described in terms of the socalled "second differences" an asteroseismic signature of helium diffusion in main-sequence A stars (1.6 M ⊙ and 2.0 M ⊙ ). Théado et al. (2005) presented a first discussion of asteroseismic signatures of diffusion in stars between 1.1 and 1.3 M ⊙ . In the present paper we show evidence of differences in the internal structure of stars between 1.1 and 1.45 M ⊙ due to element diffusion. We compute evolutionary tracks of stellar models with similar external parameters (luminosities, effective temperatures and chemical compositions), with or without helium diffusion. We then study asteroseismic tests, particularly the signatures of helium gradients below the outer convective zones. In Section 2, we present the model computations and calibrations. The asteroseismic tests of internal structure between models with and without diffusion are discussed in Section 3. The helium gradients and their consequences for the second differences are studied in Section 4. Section 5 gives the discussion and conclusion. Calibration and characteristics of the models We computed models with the Toulouse-Geneva Evolutionary Code (TGEC) (Richard et al. 1996) Table 2). The physical inputs and parameters are : -Equation of state : MHD (Däppen 1992), -Opacities : OPAL (Iglesias & Rogers 1996) completed by the Alexander & Fergusson (1994) low temperature opacities, -Nuclear reaction rates : NACRE compilation of nuclear reaction rates (Angulo et al. 1999) with the Bahcall screening routine, -Convection treatment : mixing-length theory (Böhm-Vitense 1958), -Diffusion : diffusion coefficients computed as in Paquette et al. (1986). For a given stellar mass, we computed two series of models : with and without element diffusion. No mixing process or mass loss is taken into account. The series were calibrated to obtain evolutionary tracks very close in the HR diagram, such that, for the same luminosity, their temperature differences are never larger than 60 K. This calibration is achieved by adjusting two free parameters of the stellar evolution code : the initial helium abundance Y 0 and the mixing length parameter α. We used for all the homogeneous models the same values of the free parameters α and Y 0 . For each mass, models including diffusion are then calibrated to have evolutionary tracks very close to the homogeneous ones. The calibration parameters are presented in Table 1. Figure 1 displays the computed evolutionary tracks. For each mass the tracks obtained with and without diffusion lie very close to each other. Furthermore, up to 1.35 M ⊙ the age evolution is similar in the two cases, so that models with the same age lie close together in the HR diagram. Regarding the uncertainties, we can consider that these models are observationaly identical, i.e. with the same effective temperatures, luminosities and chemical compositions. This is no longer the case for larger masses : along track M1.45, of 1.45 M ⊙ , the ages for models with diffusion are quite different to those with- out diffusion. Indeed, for masses greater than 1.3 M ⊙ , more than half of the energy production is dominated by the CNO cycle, which is more sensitive to the central temperature than the pp chains. Since models with diffusion have a larger initial helium fraction than models without diffusion, the former are more evolved for the same age. Table 2 gives the characteristics of the models with and without diffusion for each mass and different ages. The calibrated models do not have strictly identical values of the surface parameters but they are very close, except for the case M1.45 for which models with similar parameters have different ages. Tests of internal structure The oscillation frequencies of our models have been computed with an updated version of the adiabatic code described in Brassard et al. (1992), for values of the azimuthal degree l between 0 and 3 and different values of the radial order n. We then studied in detail different combinations of these frequencies which could lead to observational tests of the internal structure and chemical composition of the stars. The "large separations" represent the frequency differences between two modes of the same degree l and successive radial orders n: ∆ν n,l = ν n,l − ν n−1,l(1) For p-modes in the asymptotic theory (n ≫ l), the large separations are nearly constant. The so-called "echelle diagrams" present the frequencies in ordinates, and the same frequencies modulo the average large separation in absissae. The asymptotic theory predicts a vertical line for each degree. The differences between the lines l = 0 and l = 2 on the one hand, l = 1 and l = 3 on the other hand, for each case, are the so-called "small separations", which we normalize as suggested by Roxburgh & Vorontsov (2001) : Dν l,l+2 = 1 2l + 3 (ν n,l − ν n−1,l+2 )(2) The deviations of the small separations to zero show departure from the asymptotic theory. The small separations are mainly sensitive to the stellar core. In Figure 2, we present as an example the echelle diagram for models M1.1 to M1.35 at different ages. For M1.45, we compare echelle diagrams for models at the same location on the HR diagram, which correspond to different ages for the models with and without diffusion. This is presented in Figure 3. The small separations of the same models are presented in Figure 4 for models M1.1 to M1.35 and in Figure 5 for models M1.45. We can see on these figures that the differences between the models with and without diffusion become more important during the evolution when the mass of the star increases. Up to 1.3 M ⊙ , the tracks in the large and small separations lie very close to each other whereas for higher masses significant deviations do appear. These effects are related to the internal differences Table 2. Physical characteristics of the models; L/L ⊙ : luminosity in solar unity; |∆L|/L : relative difference of luminosity between models with and without diffusion; T e f f : effective temperature; |∆T|/T : relative difference of effective temperature between models with and without diffusion; X S and Y S : surface hydrogen and helium mass fractions Models Ages (Gyr) According to the results of Bazot et al. (2005), which presents the seismic analysis of the star µ Arae, observed with the spectrograph HARPS, the uncertainties on the small separations are about 0.37 µHz. In the cases of 1.1 and 1.2 M ⊙ , the differences between models with and without helium diffusion are too small to be observable. For higher masses, this difference could be detectable. L/L ⊙ |∆L|/L T e f f (K) |∆T|/T X S Y S M1. Tests of helium gradients The helium gradients The largest differences between the models with and without diffusion lie just below the convective zone where helium drifts inward due to the effect of gravitation. Figures 6 to 8 a way similar to models M1.2 and models M1.35 are similar to M1.3 : they are available in the electronic version only. In the helium profiles, helium gradients due to diffusion are clearly seen. In models M1.1 and M1.2, this helium gradient remains smooth throughout the evolution in the main sequence, while in the case of models M1.3 to M1.45, it becomes sharper with time. For example, in models M1.3, the bottom of the convective zone deepens in the stellar interior around 2.5 Gyrs, which explains the discontinuity on the helium gradient and the increase of the helium mass fraction in Table 2 for the corresponding model, while in models M1.2, the convective zone sinks later and slower. Actually, the helium gradients in models M1.1 and M1.2 become steeper after the turnoff. In the later (Figure 6), the features around 700 s in the sound speed profile are due to the helium ionisation zones while the bottom of the convective zone leads to characteristic features at t=1900 s (1 Gyrs), t=2000 s (1.5 Gyrs), t=2200s (2.6 Gyrs) and t=2500 s (3.3 Gyrs). We can see in this example that the differences between the models with and without diffusion increase with time, and that this behaviour increases with increasing stellar mass. The second differences Asteroseismology provides tools to probe the base of the convective zone and the helium gradient. As we have seen, in these locations the sound velocity undergoes rapid variations, which involve partial reflections of the sound waves. These reflections modulate the computed frequencies of the stellar spectrum, which is better seen in the so-called "second differences" (Gough 1990;Monteiro & Thompson 1998;Roxburgh & Vorontsov 2001;Vauclair & Théado 2004;Théado et al. 2005). The second differences are defined as follows: δ 2 ν = ν n+1 + ν n−1 − 2ν n .(3) In order to identify the different components which modulate the oscillations, we have computed the Fourier transform of the second differences. The positions of the peaks correspond to the modulation periods of the second differences due to the partial reflections of the waves, which are equal to twice the acoustic depth of the corresponding features in the sound velocity (Vauclair & Théado 2004). In Figures 9 to 11 we can then identify for each model the peak due to the helium ionisation zone (the closest to the surface) and those due to the bottom of the convective zone, where the helium gradient lies. The differences between models with and without diffusion are clearly visible. In all the cases with diffusion (Figures 9 to 11), the peaks due to the helium ionisation zones decrease in amplitude during evolution, because of the helium concentration decreases in the convective zone due to diffusion. In the 1.1 M ⊙ and 1.2 M ⊙ stars, the diffusion process works on a time scale too long to be visible in the helium distribution during the main sequence. The amplitude of the peak due to the base of the convective zone slightly decreases because of the sinking of this zone (see Figure 9). For higher mass stars with diffusion ( Figures 10 and 11), the peak due to the base of the Fig. 6. Helium profiles (left column) and gradients of the sound velocity (right column) in the models with (solid lines) and without (dashed lines) element diffusion, for 1.2 M ⊙ as a function of the acoustic depth at 1, 1.5, 2.6 and 3.3 Gyrs convective zone undergoes a strong increase of amplitude when the convective zone becomes deeper and steepens the helium gradient. In the homogeneous models, there is no increase in the amplitude of this peak. This behaviour is clearly due to the diffusion process. Discussion The large and small separations represent asteroseismic tests of the different internal structure in models with and without diffusion. Our computations show that these differences increase during stellar evolution and that it becomes greater for The behaviour of the second differences of the models are clear : due to the effect of diffusion, helium falls below the outer convective zone and forms a gradient that increases during the evolution. But, depending on the mass of the star, the behaviour of the sound waves is different. For low masses (1.1 and 1.2 M ⊙ ), the diffusion process is slow, and the depth of the convective zone increases slowly. Hence, the helium gradient remains smooth during the main sequence and becomes steep only at the turnoff. When the mass of the star is large enough, the convective zone deepens more quickly and the helium gradient steepens more rapidly. There is a strong reflection of the pressure waves in the region of the helium gradient, which explains the large increase in the amplitude of the peak related to Fig. 6 for 1.45 M ⊙ at 1, 1.2, 1.6 and 1.8 Gyrs the base of the convective zone. Later, helium dilutes inside the convective zone, so that the helium concentration, which was very low after rapid diffusion, increases again (see Table 2). However, the helium gradient remains steep and the amplitude of the peak remains high (see Figures 10 and 11). The models presented here have been computed with pure gravitational settling. In real stars, the variation in the helium abundance and the steepening of the helium gradient resulting from diffusion will be smaller, due to the mixing processes below the convective zone and possible mass loss. We can however expect that such a signature of helium diffusion may be detected in stars slightly more massive than the Sun with CoRoT. The accuracy of the instrument is expected to reach 0.1 µHz and should allow detection of peaks in the Fourier transforms of the second differences of the oscillation frequencies as a signature of helium diffusion. display the helium profiles and the gradients of the sound velocity in models with (solid lines) and without (dashed lines) diffusion for M1.2, M1.3 and M1.45, as a function of the acoustic depth (time needed for the waves to travel from the surface down to the considered layer), at different ages. Models M1.1 behave in Fig. 2. Echelle diagram for models M1.1 to M1.35 at different ages. Crosses : l = 0, stars : l = 1, triangles : l = 2, diamonds : l = 3. The points connected by lines are for models with diffusion. Fig. 3 . 3Echelle diagram for models M1.45 at 1.2 Gyrs (with diffusion) and 1.6 Gyrs (without diffusion) on the left, and at 0.7 Gyrs (with diffusion) and 1.2 Gyrs (without diffusion) on the right. Crosses : l = 0, stars : l = 1, triangles : l = 2, diamonds : l = 3. The points connected by lines are for models with diffusion. Fig. 4 . 4Small separations for models M1.1 to M1.35 at different ages. The lines are for models with diffusion (solid lines : Dν 0,2 , dotted lines : Dν 1,3 ) and the symbols are for models without diffusion (triangles : Dν 0,2 , diamonds : Dν 1,3 ). Fig. 5 . 5Small separations for models M1.45 at 1.2 Gyrs (with diffusion) and 1.6 Gyrs (without diffusion) on the left, and at 0.7 Gyrs (with diffusion) and 1.2 Gyrs (without diffusion) on the right. The lines are for models with diffusion (solid lines : Dν 0,2 , dotted lines : Dν 1,3 ) and the symbols are for models without diffusion (triangles : Dν 0,2 , diamonds : Dν 1,3 ). Fig. 7 . 7Same as Fig. 6 for 1.3 M ⊙ at 0.5, 1.2, 2.2 and 2.6 Gyrs increasing mass. This behaviour should be detectable for stellar masses larger than 1.2 M ⊙ . Fig. 8 . 8Same as Fig. 9 . 9Second differences (left column), Fourier transform of the second differences (right column) of models M1.2 of 1.2 M ⊙ at 1, 1.5, 2.6 and 3.3 Gyrs, with (solid lines) and without (dashed lines) diffusion. , for masses from 1.1 to Send offprint requests to: M. Castro Correspondence to: [email protected] 1.45 M ⊙ , respectively noted M1.1 to M1.45 (see Table 1 . 1Calibration parameters of the models; α: mixing length parameter; Y 0 : initial helium abundanceModels Mass (M ⊙ ) α Y 0 M1.1-hom 1.1 1.75 0.268 M1.1-dif 1.1 1.84 0.267 M1.2-hom 1.2 1.75 0.268 M1.2-dif 1.2 1.88 0.268 M1.3-hom 1.3 1.75 0.268 M1.3-dif 1.3 1.99 0.268 M1.35-hom 1.35 1.75 0.268 M1.35-dif 1.35 1.99 0.268 M1.45-hom 1.45 1.75 0.268 M1.45-dif 1.45 1.88 0.278 M.Castro & S.Vauclair: Asteroseismic signatures of helium gradients M.Castro & S.Vauclair: Asteroseismic signatures of helium gradients 9 . D R Alexander, J W Ferguson, ApJ. 437879Alexander, D.R., Ferguson, J.W., 1994, ApJ, 437, 879 . C Angulo, NACRE collaborationM Arnould, NACRE collaborationM Rayet, NACRE collaborationNucl. Phys. 6561Angulo, C., Arnould, M., Rayet, M., (NACRE collaboration), 1999, Nucl. Phys. A656, 1 . M Bazot, S Vauclair, F Bouchy, N C Santos, A&A. 440615Bazot, M., Vauclair, S., Bouchy, F., Santos, N.C., 2005, A&A, 440, 615 . E Böhm-Vitense, 46108ZApBöhm-Vitense, E., 1958, ZAp, 46, 108 . P Brassard, C Pelletier, G Fontaine, F Wesemael, ApJS. 80725Brassard, P., Pelletier, C., Fontaine, G., Wesemael, F., 1992, ApJS, 80, 725 Same as Fig. 9 for models M1.3 of 1.3 M ⊙ at 0.5, 1.2, 2.2 and 2.6 Gyrs Däppen. Fig, RMxAA. 10141Fig. 10. Same as Fig. 9 for models M1.3 of 1.3 M ⊙ at 0.5, 1.2, 2.2 and 2.6 Gyrs Däppen, W., 1992, RMxAA, 23, 141 D O Gough, Progress of Seismology of the Sun and Stars Proc., Oji International Seminar (Hakone). Osaki, H.Shibahashi Iglesias, C.A., Rogers, F.J.JapanSpringer Verlag367943ApJGough, D.O., 1990, in Progress of Seismology of the Sun and Stars Proc., Oji International Seminar (Hakone)(Japan: Springer Verlag), Lect. Notes Phys., 367, 283, eds Osaki, H.Shibahashi Iglesias, C.A., Rogers, F.J., 1996, ApJ, 464, 943 . G Michaud, ApJ. 160641Michaud, G., 1970, ApJ, 160, 641 Mj P F G Monteiro, M J Thompson, New eyes to see inside the Sun and Stars. F.-LDordrechtKluwerMonteiro, MJ.P.F.G., Thompson, M.J., 1998, in New eyes to see inside the Sun and Stars (Dordrecht: Kluwer), eds F.-L. J Deubner, D W Christensen-Dalsgaard, Kurtz, Proc. IAU Symp. IAU Symp185317Deubner, J. Christensen-Dalsgaard, D.W. Kurtz, Proc. IAU Symp., 185, 317 . C Paquette, C Pelletier, G Fontaine, G Michaud, ApJS. 61177Paquette, C., Pelletier, C., Fontaine, G., Michaud, G., 1986, ApJS, 61, 177 Same as Fig. 9 for models M1. 4545 M ⊙ at 1, 1.2, 1.8 and 2.2 GyrsFig. 11. Same as Fig. 9 for models M1.45 of 1.45 M ⊙ at 1, 1.2, 1.8 and 2.2 Gyrs . O Richard, S Vauclair, C Charbonnel, W A Dziembowski, A&A. 3121000Richard, O., Vauclair, S., Charbonnel, C., Dziembowski, W.A., 1996, A&A, 312, 1000 . I W Roxburgh, S V Vorontsov, MNRAS. 32285Roxburgh, I.W., Vorontsov, S.V., 2001, MNRAS, 322, 85 . S Théado, S Vauclair, M Castro, S Charpinet, N Dolez, A&A. 437553Théado, S., Vauclair, S., Castro, M., Charpinet, S., Dolez, N., 2005, A&A, 437, 553 . S Vauclair, S Théado, A&A. 425179Vauclair, S., Théado, S., 2004, A&A, 425, 179
[]
[ "PAIR PLASMA DOMINANCE IN THE PARSEC-SCALE RELATIVISTIC JET OF 3C345", "PAIR PLASMA DOMINANCE IN THE PARSEC-SCALE RELATIVISTIC JET OF 3C345" ]
[ "Kouichi Hirotani ", "Satoru Iguchi ", "Moritaka Kimura ", "Kiyoaki Wajima ", "\nInstitute of Astronomy\nNational Astronomical Observatory\n181-8588OsawaMitakaJapan\n", "\nThe Institute of Space and Astronautical Science\nThe University of Tokyo\nYoshino-dai181-8588, 229-8510Osawa, SagamiharaMitaka, KanagawaJapan, Japan\n" ]
[ "Institute of Astronomy\nNational Astronomical Observatory\n181-8588OsawaMitakaJapan", "The Institute of Space and Astronautical Science\nThe University of Tokyo\nYoshino-dai181-8588, 229-8510Osawa, SagamiharaMitaka, KanagawaJapan, Japan" ]
[]
We investigate whether a pc-scale jet of 3C345 is dominated by a normal plasma or an electron-positron plasma. We present a general condition that a jet component becomes optically thick for synchrotron self-absorption, by extending the method originally developed by Reynolds et al. The general condition gives a lower limit of the electron number density, with the aid of the surface brightness condition, which enables us to compute the magnetic field density. Comparing the lower limit with another independent constraint for the electron density that is deduced from the kinetic luminosity, we can distinguish the matter content. We apply the procedure to the five components of 3C345 (C2, C3, C4, C5, and C7) of which angular diameters and radio fluxes at the peak frequencies were obtainable from literature. Evaluating the representative values of Doppler beaming factors by their equipartition values, we find that all the five components are likely dominated by an electron-positron plasma. The conclusion does not depend on the lower cutoff energy of the power-law distribution of radiating particles.
10.1086/317769
[ "https://arxiv.org/pdf/astro-ph/0005394v1.pdf" ]
17,274,015
astro-ph/0005394
7fd41c69f6d3e1655dbc86435bb299169267a2ba
PAIR PLASMA DOMINANCE IN THE PARSEC-SCALE RELATIVISTIC JET OF 3C345 May 2000 Kouichi Hirotani Satoru Iguchi Moritaka Kimura Kiyoaki Wajima Institute of Astronomy National Astronomical Observatory 181-8588OsawaMitakaJapan The Institute of Space and Astronautical Science The University of Tokyo Yoshino-dai181-8588, 229-8510Osawa, SagamiharaMitaka, KanagawaJapan, Japan PAIR PLASMA DOMINANCE IN THE PARSEC-SCALE RELATIVISTIC JET OF 3C345 May 2000arXiv:astro-ph/0005394v1 19Subject headings: galaxies: active -quasars: individual (3C345) -radio continuum: galaxies 1 present address: National Astronomical ObservatoryOsawaMitaka 181-8588Japan We investigate whether a pc-scale jet of 3C345 is dominated by a normal plasma or an electron-positron plasma. We present a general condition that a jet component becomes optically thick for synchrotron self-absorption, by extending the method originally developed by Reynolds et al. The general condition gives a lower limit of the electron number density, with the aid of the surface brightness condition, which enables us to compute the magnetic field density. Comparing the lower limit with another independent constraint for the electron density that is deduced from the kinetic luminosity, we can distinguish the matter content. We apply the procedure to the five components of 3C345 (C2, C3, C4, C5, and C7) of which angular diameters and radio fluxes at the peak frequencies were obtainable from literature. Evaluating the representative values of Doppler beaming factors by their equipartition values, we find that all the five components are likely dominated by an electron-positron plasma. The conclusion does not depend on the lower cutoff energy of the power-law distribution of radiating particles. Introduction The study of extragalactic jets on parsec scales is astrophysically interesting in the context of the activities of the central engines of AGN. In particlar, a determination of their matter content would be an important step in the study of jet formation, propagation and emission. The two main candidates are a 'normal plasma' consisting of protons and relativistic electrons (for numerical simulations of shock fronts in a VLBI jet, see Gómez et al. 1993Gómez et al. , 1994a, and a 'pair plasma' consisting only of relativistic electrons and positrons (for theoretical studies of two-fluid concept, see Sol, Pelletier & Asséo 1989;Despringre & Fraix-Burnet 1997). Distinguishing between these possibilities is crucial for understanding the physical processes occurring close to the central 'engine' (presumably a supermassive black hole) in the nucleus. VLBI is uniquely suited to the study of the matter content of pc-scale jets, because other observational techniques cannot image at milliarcsecond resolution and must resort to indirect means of studying the active nucleus. Recently, Reynolds et al. (1996) analyzed historical VLBI data of M87 jet at 5 GHz (Pauliny-Toth et al. 1981) and concluded that the core is probably dominated by an e ± plasma. In the analysis, they utilized the standard theory of synchrotron self-absorption to constrain the magnetic field, B [G], and the proper number density of electrons, N * e [1/cm 3 ] of the jet and derived the following condition for the core to be optically thick for self-absorption: N * e B 2 > 2δ −2 max , where δ max refers to the upper limit of the Doppler factor of the bulk motion. This condition is, however, applicable only for the VLBI observations of M87 core at epochs September 1972 andMarch 1973. Therefore, in order to apply the analogous method to other AGN jets or to M87 at other epochs, we must derive a more general condition. On these grounds, Hirotani et al. (1999) generalized the condition N * e B 2 > 2δ −2 max and applied it to the 3C 279 jet on parsec scales. In that paper, they revealed that core and components C3 and C4, of which spectra are obtained from the literature, are dominated by a pair plasma. It is interesting to note that the same concusion that 3C 279 jet is dominated by a pair plasma is derived from an independent method by Wardle et al. (1998), who studied the circularly polarized radio emission from 3C 279 jet. In the present paper, we apply the same method to the 3C 345 jet. The quasar 3C345 (redshift z=0.594) is one of a class of core-dominated flat-spectrum radio sources that are believed to emit X-rays via the synchrotron self-Compton (SSC) process. VLBI imaging observations of the "superluminal" quasar 3C345 have been made at 5 GHz every year since 1977 (Unwin & Wehrle 1992) while 10.5 and 22 GHz observations have occurred at more frequent intervals (e.g., Biretta et al. 1986). The apparent speeds of components C2, C3, C4, and C5 increase monotonically with time from ∼ 3c to ∼ 10c, consistent with a jet of constant Lorentz factor (Γ = 10) bending away from the line of sight (Zensus, Cohen, & Unwin 1995). Later, Unwin et al. (1997) studied the time evolution of spectral shapes and angular sizes of component C7 at a distance ∼ 0.5 mas (2 pc) from the nucleus. Using the physical parameters given in the literature above, and deducing the kinetic luminosity from its core-position offset, we conclude that all the five jet components are likely dominated by an e ± plasma. In the final section, we discuss the validity of assumptions. We use a Hubble constant H 0 = 65h km/s/Mpc and q 0 = 0.5 throughout this paper. These give a luminosity distance to 3C 345 of D L = 3.06h −1 Gpc. An angular size or separation of 1 mas corresponds to 5.83h −1 pc. A proper motion of 1mas yr −1 translates into a speed of β app = 30.3h −1 . Spectral index α is defined such that S ν ∝ ν α . Constraints on Magnetic Flux and Particle Number Densities We shall distinguish whether a radio-emitting component is dominated by a normal plasma or an e ± plasma, by imposing two independent constraints on N * e . First, in § 2.1, we give the synchrotron self-absorption constraint, which is obtained by extending the work by Reynolds et al. (1996). Secondly in § 2.2, the kinematic luminosity constraint is presented. Synchrotron Self-absorption Constraint In this paper, we model a jet component with redshift z as homogeneous spheres of angular diameter θ d , containing a tangled magnetic field B [G] and relativistic electrons which give a synchrotron spectrum with optically thin index α and maximum flux density S m [Jy] at frequency ν m . We can then compute the magnetic field density as follows (Cohen 1985;Ghisellini et al. 1992): B = 10 −5 b(α)S −2 m ν m GHz 5 θ d mas 4 δ 1 + z ,(1) where δ is the beaming factor defined by δ ≡ 1 Γ(1 − β cos ϕ) ,(2) Γ ≡ 1/ 1 − β 2 is the bulk Lorentz factor of the jet component moving with velocity βc, and ϕ is the orientation of the jet axis to the line of sight. The coefficient b(α) is given in Cohen (1985). Both Γ and ϕ can be uniquely computed from δ and β app as follows: Γ = β 2 app + δ 2 + 1 2δ ,(3)ϕ = tan −1 2β app β 2 app + δ 2 − 1 .(4) We assume that the electron number density between energies E and E + dE is expressed by a power law as dN * e dE = N 0 E 2α−1 .(5) Integrating dN * e /dE from γ min m e c 2 to γ max m e c 2 , and assuming γ max ≫ γ min and α < 0, we obtain the electron number density N * e = γ min 2α −2α (m e c 2 ) 2α N 0 .(6) Computing the optical depth along the line of sight, Marscher (1983) expressed N 0 in terms of θ d , S m , ν m , and α. Combining the result with equation (6), we finally obtain (see also Appendix B) N * e (SSA) = e(α) γ min 2α −2α h(1 + z) 2 q 0 2 sin ϕ zq 0 + (q 0 − 1)(−1 + √ 2q 0 z + 1) × θ d mas 4α−7 ν m GHz 4α−5 S m −2α+3 δ 1 + z 2α−3 ,(7) where e(α) ≡ 2.39 × 10 1−6.77α (0 < −α < 1.25). If the component is not resolved enough, this equation gives the lower bound of N * e . Kinetic luminosity constraint As described in Appendix B in detail, we can infer the kinetic luminosity, L kin , from the core-position offset, Ω rν , due to synchrotron self-absorption. For the core, we assume a conical geometry with a small half opening angle χ. Then L kin measured in the rest frame of the AGN becomes L kin ∼ C kin K r 1 2 r • 3 βΓ(Γ − 1)χ 2 Ω rν /ν • r 1 sin ϕ 2(5−2α)/(7−2α) × πC(α) χ sin ϕ K γ min r 1 r • −2α γ min 2α δ 1 + z 3/2−α −4/(7−2α) ,(8) where K is defined by equation (B13) and becomes 0.1 for α = −0.5 if an energy equipartition holds between the radiating particles and the magnetic field. For a pure pair plasma, we obtain C kin = π 2 γ − m e c 3 /γ min , where γ − is the averaged Lorentz factor of randomly moving electrons and positrons, which could be computed from equation (5) for a power-law distribution of radiating particles. For a normal plasma, on the other hand, we obtain C kin = π 2 m p c 3 /(2γ min ), where m p refers to the rest mass of a proton. It should be noted that γ min takes a different value from a pair plasma. Once L kin of a stationary jet is obtained, we can deduce N * e at an arbitrary position along the jet, even if the geometry deviates from a cone. When the jet has a perpendicular half width R ⊥ at a certain position, L kin and N * e are related by L kin = πR ⊥ 2 βc · ΓN * e · (Γ − 1) γ − m e c 2 + γ + m + c 2 ,(9) where γ − and γ + refer to the averaged Lorentz factors of electrons and positively charged particles, respectively; m + designates the mass of the positive charge. Replacing the angular diameter distance, 2R ⊥ /θ d , with the luminosity distance divided by (1 + z) 2 , we can solve equation (9) for N * e to obtain N * e (kin) = 3.42 × 10 2 h 2 q 0 4 (1 + z) 4 zq 0 + (q 0 − 1)(−1 + √ 2q 0 z + 1) 2 × θ d mas −2 1 βΓ(Γ − 1) L 46.5 γ − + γ + m + /m e cm −3 ,(10) where L 46.5 refers to the kinetic luminosity in the unit of 10 46.5 ergs s −1 . It should be noted that γ − + γ + m + /m e becomes roughly 2γ min ln(γ min /γ max ) for a pair plasma with α ∼ −0.5, while it becomes 1836 for a normal plasma. As a result, N * e (kin) for a pair plasma becomes about 100γ min −1 times greater than that for a normal plasma. Since N * e (SSA) is proportional to γ min 2α , the ratio N * e (kin) /N * e (SSA) for a pair plasma becomes about 100γ min −1−2α times greater than that for a normal plasma. For a jet component close to the VLBI core, we may put α ∼ −0.5; therefore, the dependence on γ min virtually vanishes. In short, we can exclude the possibility of a normal plasma dominance if 1 < N * e (pair) /N * e (SSA) ≪ 100 is satisfied, where N * e (pair) refers to the value of N * e (kin) computed for a pair plasma. On the other hands, N * e (pair) /N * e (SSA) < 1 implies that L kin is underestimated. The conclusion is invulnerable against the value of γ min of electrons and positrons. Application to the 3C345 Jet Let us apply the method described above to the 3C345 jet on parsec scales and investigate the matter content. It is, however, difficult to define α, ν m , and S m of each component well, because the spectral information for an individual component is limited by the frequency coverage and quality of VLBI measurements near a given epoch. Therefore, Zensus et al. (1996) chose self-consistent values that matched the data and gave a reasonable fit to the overall spectrum when the components C2, C3, and C4 (hereafter, C2-C4), and the core are considered together (Table 1). For C2 and C3 they used the highest value for ν m , while for C4 they used a representative possibility. Subsequently, Unwin et al. (1997) obtained these radio parameters for C5 and C7 by analogous method. We present these parameters together with their errors in Table 2. The jet half opening angle χ ∼ 2.4 • is calculated from measuring the jet size within 1 mas distance from the core ( § 4.3 in Lobanov 1998). We choose α = −0.65 as the spectral index of the core below the turnover frequency at 700 GHz ( §5.2 of Zensus et al. 1995). Kinetic Luminosity To estimate the kinetic luminosity from equation (8), we have to input Γ, ϕ, Ω rν , and δ for a given C kin , K, χ, and α. Let us first consider Γ, ϕ, and δ. As demonstrated in figure 4 in Unwin et al. (1995), a component (C7) accelerated as it moved away from the core: the Lorentz factor increased from Γ ∼ 5 to Γ > 10, and the viewing angle increased from ϕ ∼ 2 • to ϕ ∼ 10 • . It is inappropriate to consider the case ϕ ≪ χ; therefore, we assume ϕ ∼ 2 • for the core. In this case, δ ≫ 1 holds to give L kin ∝ Γ(Γ − 1)/δ ∝ δ. In the case of a newly born component (C7) at 1992.05, Unwin et al. (1995) derived a conservative limit δ > 11.7, by assuming that C7 was the origin of the observed X-rays. Therefore, it is likely that δ is much greater than 10 for the core, because δ decreased as the component moved away. The core-position offset of the 3C 345 jet was reported by Lobanov (1998), who derived the reference value Ω rν = 10.7pc · Hz. For a pair plasma with α ∼ −0.5, γ − ∼ γ min ln(γ max /γ min ) holds in the expression of C kin ; therefore, equation (8) gives L kin ∼ 10 46 ln(γ max /γ min ) 10 K 0.5 δ 20 ergs s −1 .(11) On the other hand, for a normal plasma, equation (8) gives L kin ∼ 10 46 γ min 100 −1 K 0.5 δ 20 ergs s −1 .(12) Unless the particles significantly dominates the magnetic field, K 0.5 does not exceed unity (see eqs. [B14] and [B15], which hold when an energy equipartition is realized between the radiating particles and the magnetic field). For a normal plasma jet, the energy distribution must cut off at γ min ∼ 100 ( § 4; see also Wardle et al. 1998). Since δ > 100 is unlikely for the 3C 345 jet, we adopt L kin = 10 46.5 ergs s −1 (or equivalently L 46.5 = 1) as the representative upper bound in this paper. If L kin becomes less than this value, the possibility of normal plasma dominance further decreases. Equipartition Doppler factor We estimate the value of δ by assuming an energy equipartition between the magnetic field and the radiating particles. In this case, K becomes of the order of unity and δ is given by the so-called "equipartition Doppler factor" (Readhead 1994), δ = δ eq ≡    10 3 F (α) (θ d /mas) 34 2(h/1.54) 1 − 1/ √ 1 + z 2 (1 + z) 15−2α S 16 m ν m MHz −35−2α    1/(13−2α) ,(13) where F (α) is given in Scott and Readhead (1977). There is much justice in adopting the equipartition Doppler factor as the representative value. First, as Güijosa & Daly (1996) pointed out, δ eq 's of various AGN jets have a high correlation with δ min , the minimum allowed Doppler factor derived by comparing the predicted and the observed self-Compton flux (Marscher 1983(Marscher , 1987Ghisellini et al. 1993). (If a homogeneous moving sphere emits all the observed X-ray flux via synchrotron self-Compton process, then δ equals δ min .) Secondly, the ratio δ eq /δ depends weakly on the ratio u p /u B , where u p and u B refer to the energy densities of radiating particles (i.e., electrons and positrons) and the magnetic field, respectively. For α = −0.75 for instance, we obtain δ eq /δ = (u p /u B ) 2/17 (Readhead 1994). It is noteworthy that N * e (SSA) depends relatively weakly on θ d , ν m , and α, if we adopt δ = δ eq . For example, we obtain N * e (SSA) ∝ θ d 2.9 ν m 5.5 S m −1.5 for α = −0.75. This forms a striking contrast with N * e (SSA) ∝ θ d −10 ν m −8 S m 4.5 δ −4.5 , which would be obtained from equation (7) without making any assumptions on δ. We present such representative values of δ eq , Γ eq ≡ (β 2 app + δ 2 eq + 1)/(2δ eq ), B, and N * e (SSA) for C2-C4 in Table 1, and those for C5 and C7 in Table 2. We first compare the values of δ eq with δ min . It follows from Tables 1 and 2 that δ eq > δ min is satisfied for all the eight cases, as expected. Moreover, the values of δ eq for C2-C4 at 1982.0 and those for C7 at the four epochs, decrease with increasing projected distance, ρ [mas], from the core. As a result, the viewing angle computed from β app and δ eq (see eq. [4]), ϕ eq , increases with increasing ρ. (We exclude C5, for which the trajectory appears in a different position angle from those for C2-C4.) The results are qualitatively consistent with Zensus et al. (1995) and Unwin et al. (1997). Let us next consider N * e (SSA) . This variable is roughly constant at ∼ 0.2 cm −3 for C2-C4, whereas it increases from 0.5cm −3 at 1992.05 to 10cm −3 at 1993.55 for C7. We consider that this tendency comes from insufficient angular resolution in particular when a component is close to the core. We can alternatively compute N * e from N * e = (K/γ min m e c 2 )(B 2 /8π), the energy equipartition. Reminding K ∼ 0.1 for α ∼ −0.5, we find that N * e computed in this way is consistent with N * e (SSA) . We can compute N * Table 1 that C2 and C3 are likely dominated by a pair plasma. It is also suggested that C4 is dominated by pair plasma unless L kin exceeds 10 46.5 ergs/s. Unfortunately, the errors in B, N * e (SSA) and N * e (pair) cannot be calculated, because those in ν m and S m are not presented in Zensus et al. (1995). Furthermore, Table 2 indicates that C5 and C7 at all the four epochs are likely dominated by a pair plasma. Unfortunately, the meaningful errors in B, N * e (SSA) , and N * e (pair) for C5 cannot be calculated, because its error in θ d (or ξ in their notation) is not presented in Unwin et al. (1997). Nevertheless, the results of N * e (pair) /N * e (SSA) strongly suggest that the jet components of 3C 345 on parsec scales are dominated by a pair plasma. Discussion In summary, we derive the proper electron number density, N * e (SSA) , of a homogeneous radio-emitting component of which spectral turnover is due to synchrotron self-absorption. Comparing N * e (SSA) with the density derived from the kinetic luminosity of the jet, we can investigate whether we can exclude the possibility of normal plasma (e − -p) dominance. Applying this method to the "superluminal" quasar 3C345, using the published spectrum data of C2, C3, C4, C5, and C7, we find that all the five components are likely dominated by a pair plasma. As demonstrated in the last part of §2, the conclusion is invulnerable against the undetermined value of γ min of electrons and positrons. However, if γ min for a normal plasma were to be significantly less than 100, then the possibility of a normal plasma dominance could not be ruled out in general. In the case of the 3C 345 jet, equation (12) would give L kin ∼ 10 48 ergs s −1 for a normal plasma with γ min ∼ 1. In this case, the large kinetic luminosity (∼ 10 48 ergs s −1 ) is carried by protons, because γ − m e c 2 ∼ γ min K m e c 2 ≪ m p c 2(14) holds. Nevertheless, we consider that such a jet is unlikely, because the protons carry about two orders of magnitude more energy than is seen to be dissipated as synchrotron radiation (∼ 10 46 ergs s −1 ). Electrons on parsec scales will not be cooled down so rapidly shortly after being heated-up at the shock fronts. It is interesting to consider the case when δ is estimated by other methods than the energy equipartition. As an example, let us consider a jet motion with a roughly constant Lorentz factor; Zensus et al. (1995) derived that Γ ∼ 10 is close to the smallest value that is consistent with all their available kinematic constraints. Such values of δ and ϕ are denoted by the solid dots in Fig. 12 of their paper and tabulated again in table 3 in the present paper. Using those data, we can compute B and N * e (SSA) of each component (table 3). For C2, we adopt Γ = 13 rather than 10, because β app = 12.9 for h = 1 (or equialently H 0 = 65) gives Γ > 1 + β 2 app = 12.9. The results of N * e (pair) /N * e (SSA) show again that C2-C4 at 1982.0 are likely dominated by a pair plasma. A. Derivation of the Synchrotron Self-absorption Constraints We assume that the parsec-scale jet close to the core propagates conically with a half opening angle χ in the observer's frame. Then the optical depth τ for synchrotron self absorption is given by τ ν (R) = 2R sin χ sin(ϕ + χ) α ν ,(A1) where R is the distance of the position from the injection point of the jet and α ν [1/cm] refers to the absorption coefficient. For a small half opening angle (χ ≪ 1), this equation can be approximated as τ ν (R) = 2R χ sin ϕ α ν (A2) Since τ and Rχ are Lorentz invariants, we obtain α ν sin ϕ = α * ν sin ϕ * ,(A3) where a quantity with an asterisk is measured in the co-moving frame, while that without an asterisk in the observer's frame. Since να ν is also Lorentz invariant, equation (A3) gives sin ϕ * sin ϕ = ν ν * = δ 1 + z .(A4) Combining equations (A2) and (A4), we obtain τ ν = 1 + z δ 2Rχ sin ϕ α * ν = 1 + z δ 1 sin ϕ θ d D L (1 + z) 2 α * ν ,(A5) where the angular diameter distance of the jet, 2Rχ/θ d , is rewritten with the luminosity distance, D L , divided by (1 + z) 2 ; here, θ d is the angular diameter of the component in the perpendicular direction of the jet propagation. If we observe τ ν at the turnover frequency, ν m , it becomes a function of the optical thin spectral index α, which is tabulated in Scott and Readhead (1977). Averaging over pitch angles of the isotropic electron power-law distribution (eq. [5]), we can write down the absorption coefficient in the co-moving frame as (Le Roux 1961, Ginzburg & Syrovatskii 1965 α * ν = C(α)r • 2 k * e ν • ν * ν B ν * (−2α+3)/2 ,(A6) where ν • ≡ c/r • ≡ c/[e 2 /(m e c 2 )] and ν B ≡ eB/(2πm e c). The coefficient C(α) is given in Table 1 of Gould (1979). Substituting equation (A6) into (A5), and assuming α < 0 and γ min ≪ γ max , we obtain with the aid of (5) N * e B −α+1.5 = m e c e 2 e 2πm e c −1.5+α τ ν (α) C(α) γ min 2α −2α × (1 + z) 2 D L sin ϕ θ d 1 + z δ −α+1.5 ν −α+2.5 .(A7) Evaluating ν at the turnover frequency, ν = ν m , and combining with equation (1), we obtain N * e presented in equation (7), which equals (γ min m e c 2 ) 2α /(−2α) times N 0 given in equation (3) in Marscher (1983). It is noteworthy that electron number density in the observer's frame can be obtained if we multiply (1 + z)/δ on N * e . B. Kinetic luminosity inferred from core-position offset In this appendix, we deduce the kinetic luminosity of a jet from its core-position offset due to synchrotron self-absorption. This method was originally developed by Lobanov (1988). However, our results somewhat differs from his results; therefore, we explicitly describe the derivation so that the readers can check it. B.1. Scaling Law First, we assume that N * e and B scale on r in the following manner: N * e = N 1 r −n , B = B 1 r −m ,(B1) where N 1 and B 1 refer to the values of N * e and B at r 1 = 1 pc, respectively; r ≡ R/r 1 . Introducing dimensionless variables x N ≡ r 1 r • 2 N 1 x B ≡ ν B 1 /ν • = eB 1 2πm e c ,(B2) and utilizing equation (A6), we obtain from the left equality in equation (A5) τ ν = C(α) 2χ sin ϕ −2α γ min 2α 1 + z δ −ǫ ν ν • −1−ǫ r 1−n−mǫ x N x B ǫ ,(B3) where ǫ ≡ 3/2 − α. At a given frequency ν, the flux density will peak at the position where τ ν becomes unity. Thus setting τ = 1 and solving equation (B3) for r, we obtain the distance from the VLBI core observed at frequency ν from the central engine as r(ν) = x B k b F ν • ν 1/kr (B4) where F (α) ≡ C(α) 2χ sin ϕ −2α γ min 2α δ 1 + z ǫ x N 1/(ǫ+1) (B5) k b ≡ 3 − 2α 5 − 2α , (B6) k r ≡ (3 − 2α)m + 2n − 2 5 − 2α . (B7) B.2. Core-Position Offset If we mesure r(ν) at two different frequencies (say ν a and ν b ), equation (B4) gives the dimensionless, projected distance of r(ν a ) − r(ν b ) as ∆r proj = [r(ν a ) − r(ν b )] sin ϕ = (x B k b F ν • ) 1/kr ν 1/kr b − ν 1/kr a ν 1/kr a ν 1/kr b sin ϕ.(B8) Defining the core-position offset as Ω rν ≡ r 1 ∆r proj ν 1/kr a ν 1/kr b ν 1/kr b − ν 1/kr a ,(B9) we obtain Ω rν r 1 = (x k b B F ν • ) 1/kr sin ϕ (B10) To express x B in terms of x N and Ω rν , we can invert equation (B10) as x B = Ω rν r 1 sin ϕ kr/k b (F ν • ) −1/k b .(B11) Note that x N is included in F = F (α). Setting ν b → ∞ in equation (B8), we obtain the absolute distance of the VLBI core measured at ν from the central engine as r core (ν) = Ω rν r 1 sin ϕ ν −1/kr . That is, once Ω rν is obtained from multi-frequency VLBI observations, we can deduce the distance of the synchrotron-self-absorbing VLBI core from the central engine, assuming the scaling laws of N * e and B as equation (B1). We next represent x N and x B (or equivalently, N 1 and B 1 ) as a function of Ω rν . To this end, we relate N * e and B as follows: N * e γ min m e c 2 = K B 2 8π .(B13) When an energy equipartition between the radiating particles and the magnetic field holds, equation (5) gives for α = −0.5 K = 1 ln(γ max /γ min ) ∼ 0.1,(B14) whereas for α < −0.5 K = 2α + 1 2α γ max 2α − γ min 2α γ max 2α+1 − γ min 2α+1 .(B15) Substituting N * e = N 1 r −2 and B = B 1 r −1 into (B13), and replacing N 1 and B 1 with x N and x B , we obtain x N = π 2 K γ min r 1 r • x B 2 (B16) It is noteworthy that the assumptions of n = 2 and m = 1, which results in k r = 1, guarantees the energy equipartition at an arbitrary distance, r. Combining equations (B11) and (B16), we obtain x B = Ω rν /ν • r 1 sin ϕ (5−2α)/(7−2α) × πC(α) χ sin ϕ K γ min r 1 r • −2α γ min 2α δ 1 + z ǫ −2/(7−2α) . (B17) The particle number density, x N , can be readily computed from equation (B16). B.3. Kinetic luminosity We can now relate the kinetic luminosity with the core-position offset. The factor N e * R 2 in equation (9) can be expressed in terms of x N and hence x B as N * e R 2 = N 1 r 1 2 = r 1 r • 2 x N = π 2 K γ min r 1 2 r • 3 x B 2 .(B18) For a pure pair plasma, we obtain γ + = γ − and m + = m e . Therefore, for a conical geometry, we can put R ⊥ = Rχ in equation (9) to obtain equation (8), where C kin = π 2 γ − m e c 3 /γ min . In the same manner, for a normal plasma, we have γ + = 1 and m + = m p . In this case, we obtain C kin = π 2 m p c 3 /(2γ min ). Zensus et al. (1995). b The values for h = 1.0 are presented. Kinetic luminosity is normalized as L46.5 ≡ L kin /10 46.5 ergs · s −1 (see text). 2.7 ± 0.5 12.8 ± 0.5 12.5 ± 1.0 11.6 ± 0.5 11.0 ± 1.5 Sm [Jy] b 3.2 ± 0.5 4.6 ± 0.5 7.0 ± 0.5 5.1 ± 0.5 3. Unwin et al. (1997). Errors are nominally 1 σ but are dominated by systematic errors, which are included in the estimate. c The values for h = 1 are presented. L46.5 ≡ L kin /10 46.5 ergs · s −1 . Errors are 90% confidence regions for a single parameter of interest. likely yes likely yes likely yes a h = 1 is assumed. L46.5 ≡ L kin /10 46.5 ergs · s −1 . ) for a pair plasma, from equation (8). The results of N * e (pair) are presented in Tables 1 and 2, together with the ratio N * e (pair) /N * e (SSA) . It follows from Table 1 . 1Magnetic field and electron density of each componentcomponent C2 C3 C4 epoch 1982.0 1982.0 1982.0 ρh [mas] a 4.9/0.65 2.2/0.65 0.40/0.65 βapph a 8.4/0.65 6.0/0.65 4.0/0.65 νm [GHz] a 1.5 2.6 14.6 Sm [Jy] a 2.0 2.1 7.6 α a −0.6 −0.7 −0.3 θ d [mas] a 2.15 0.97 0.29 δmin a 2.1 3.6 14.3 δeq b 6.7 13 17 Γeq b 16 9.8 9.6 ϕeq [rad] b 0.12 0.072 0.039 B [mG] b 5.7 6.9 18 N * e (SSA) [cm −3 ] b 0.11 0.33 0.19 N * e (pair) [cm −3 ] b 0.095L46.5 1.3L46.5 16L46.5 N * e (pair) /N * e (SSA) b 0.86L46.5 4.0L46.5 80L46.5 e ± dominated? likely yes likely yes maybe yes a From Table 2 . 2Mangnetic field and electron density of each componentcomponent C5 C7 C7 C7 C7 epoch 1990.55 1992.05 1992.67 1993.19 1993.55 ρh [mas] b 1.75/0.65 0.14/0.65 0.22/0.65 0.38/0.65 0.52/0.65 βapph 5.7/0.65 a 1.8/0.65 b 3.9/0.65 b 6.8/0.65 b 9.4/0.65 b νm [GHz] b From Unwin & Wehrle (1992).1 ± 0.5 α b −0.75 −0.75 −0.75 −0.75 −0.75 θ d [mas] b 0.80 0.20 ± .04 0.35 ± .02 0.41 ± .02 0.38 ± .02 δmin b 8.0 ± 3.5 11.7 ± 4.1 6.5 ± 0.9 5.5 ± 0.6 4.0 ± 1.1 δeq c 33 39 +43 −24 15 +6 −5 8.4 +2.5 −2.3 7.1 +6.7 −4.7 Γeq c 18 20 +21 −12 8.8 +2.6 −2.1 11.0 +0.8 −0.8 25 +19 −10 ϕeq [rad] c .022 .008 +.014 −.007 .05 +.04 −.03 .12 +.02 −.03 .11 +.02 −.04 B [mG] c 4.4 19 +11 −11 31 +12 −12 43 +12 −12 62 +61 −39 N * e (SSA) [cm −3 ] c 0.09 0.5 +1.9 −0.4 4.5 +6.6 −2.9 11 +5 −4 12 +10 −8 N * e (pair) [cm −3 ] c 0.63L46.5 11 +17 −8 L46.5 14 +7 −6 L46.5 5.9 +1.4 −1.8 L46.5 1.9 +1.9 −1.6 L46.5 N * e (pair) /N * e (SSA) c 7.2L46.5 23 +40 −12 L46.5 3.0 +1.9 −1.2 L46.5 0.53 +0.44 −0.27 L46.5 0.17 +0.81 −0.15 L46.5 e ± dominated? likely yes likely yes likely yes likely yes likely yes a b From Table 3 . 3Electron densities when Γ is givencomponent C2 C3 C4 epoch 1982.0 1982.0 1982.0 ρ [mas] 4.9/0.65 2.2/0.65 0.40/0.65 βapph 8.4/0.65 6.0/0.65 4.0/0.65 Γ 13 10 10 δ 14 13 18 B [mG] a 11 7.2 19 N * e (SSA) [cm −3 ] a 0.0055 0.77 1.2 N * e (pair) [cm −3 ] a 0.15L46.5 1.3L46.5 14L46.5 N * e (pair) /N * e (SSA) a 27L46.5 1.6L46.5 11L46.5 e ± dominated? . J A Biretta, R L Moore, M H Cohen, 30893Biretta, J. A., Moore, R. L., & Cohen, M. H. 1986, ApJ308, 93 . R D Blandford, A Levinson, ApJ. 44179Blandford, R. D., & Levinson, A. 1995, ApJ 441, 79 . V Despringre, D Fraix-Burnet, AA. 32026Despringre, V. & Fraix-Burnet, D. 1997, AA 320, 26. . C D Dermer, ApJ. 44663Dermer, C. D. 1995, ApJ 446, L63 . C D Dermer, R Schlickeiser, A Mastichiadis, AA. 25627Dermer, C. D., Schlickeiser, R., & Mastichiadis, A. 1992, AA 256, L27 . C D Dermer, R Schlickeiser, ApJ. 416484Dermer, C. D., Schlickeiser, R. 1993, ApJ 416, 484 . J L Gómez, A Alberdi, J M Marcaide, AA. 27455Gómez, J. L., Alberdi, A., & Marcaide, J. M. 1993, AA 274, 55 . J L Gómez, A Alberdi, J M Marcaide, AA. 28451Gómez, J. L., Alberdi, A., & Marcaide, J. M. 1994a, AA 284, 51 . J L Gómez, A Alberdi, J M Marcaide, AA. 29233Gómez, J. L., Alberdi, A., & Marcaide, J. M. 1994b, AA 292, 33 . G Ghisellini, P Padovani, A Celotti, L Maraschi, 776Ghisellini, G., Padovani, P., Celotti, A., & Maraschi, L. 1992 MNRAS258, 776 . A Güijosa, R A Daly, ApJ. 461600Güijosa, A., & Daly, R. A. 1996, ApJ 461, 600 . Gould, AA. 76306Gould 1979, AA 76, 306 . K Hirotani, S Iguchi, M Kimura, K Wajima, PASJ. 51263Hirotani, K., Iguchi, S., Kimura, M., & Wajima, K. 1979, PASJ 51, 263 . W H Ku, D J Helfand, L B Lucy, Nature. 288323Ku, W. H., Helfand, D. J., & Lucy, L. B. 1980 Nature 288, 323 . A Levinson, ApJ. 467546Levinson, A. 1996, ApJ 467, 546 F Makino, AGN and the X-ray Background, Proc. 23d ESLAB Symp. J. Hunt & BattrichNoordwijkESA803Makino, F. 1989, in AGN and the X-ray Background, Proc. 23d ESLAB Symp. ed. J. Hunt & Battrich (Noordwijk:ESA), 803 . A P Marscher, ApJ. 264296Marscher, A. P. 1983, ApJ 264, 296 A P Marscher, Superluminal Radio Sources. A. Zensus & T. J. PearsonCambridge Univ. Press280Marscher, A. P. 1987, in Superluminal Radio Sources, ed. A. Zensus & T. J. Pearson (Cambridge Univ. Press), 280 . I I K Pauliny-Toth, E Preuss, A Witzel, D Graham, I I Kellerman, B Ronnang, ApJ. 86371Pauliny-Toth I. I. K., Preuss E., Witzel A., Graham D., Kellerman I. I., Ronnang B. 1981, ApJ 86, 371 . G Pelletier, H Sol, 635Pelletier, G., & Sol, H. 1992 MNRAS254, 635 . C S Reynolds, A C Fabian, A Celotti, M J Rees, 283873Reynolds, C. S., Fabian, A. C., Celotti, A., & Rees, M. J. 1996 MNRAS283, 873 . Scott &amp; Readhead, 180539Scott & Readhead 1977, MNRAS180, 539 . M Sikora, M C Begelman, M J Rees, ApJ. 421153Sikora, M., Begelman, M. C., & Rees, M. J. 1994, ApJ 421, 153 . H Soll, G Pelletier, E Asséo, MNRAS. 237411Soll, H. Pelletier, G., & Asséo, E. 1989, MNRAS 237, 411 . S C Unwin, A E Wehrle, 39874Unwin, S. C., & Wehrle, A. E. 1992, ApJ398, 74 . S C Unwin, A E Wehrle, A P Lobanov, J A Zensus, G M Madejski, 480596Unwin, S. C., Wehrle, A. E., Lobanov, A. P., Zensus, J. A., & Madejski, G. M. 1997, ApJ480, 596 . J F C Wardle, D C Homan, R Ojha, D H Roberts, Nature. 395457Wardle, J. F. C., Homan, D. C., Ojha, R, Roberts, D. H. Nature 395, 457 . D M Worrall, B J Wilkes, ApJ360. 396Worrall, D. M., & Wilkes, B. J. 1990 ApJ360, 396 . J A Zensus, M H Cohen, S C Unwin, 44335Zensus, J. A., Cohen, M. H., & Unwin, S. C. 1995, ApJ443, 35
[]
[ "Unitarity Bounds in AdS 3 Higher Spin Gravity", "Unitarity Bounds in AdS 3 Higher Spin Gravity" ]
[ "Alejandra Castro [email protected]:[email protected]:[email protected] \nMcGill Physics Department\n3600 rue UniversityH3A 2T8MontréalQCCanada\n", "Eliot Hijano \nMcGill Physics Department\n3600 rue UniversityH3A 2T8MontréalQCCanada\n", "Arnaud Lepage-Jutier \nMcGill Physics Department\n3600 rue UniversityH3A 2T8MontréalQCCanada\n" ]
[ "McGill Physics Department\n3600 rue UniversityH3A 2T8MontréalQCCanada", "McGill Physics Department\n3600 rue UniversityH3A 2T8MontréalQCCanada", "McGill Physics Department\n3600 rue UniversityH3A 2T8MontréalQCCanada" ]
[]
We study SL(N, R) Chern-Simons gauge theories in three dimensions. The choice of the embedding of SL(2, R) in SL(N, R), together with asymptotic boundary conditions, defines a theory of higher spin gravity. Each inequivalent embedding leads to a different asymptotic symmetry group, which we map to an OPE structure at the boundary. A simple inspection of these algebras indicates that only the W N algebra constructed using the principal embedding could admit a unitary representation for large values of the central charge.
10.1007/jhep06(2012)001
[ "https://arxiv.org/pdf/1202.4467v3.pdf" ]
118,619,873
1202.4467
9550623648e188050e40cc8034b90f8cb8a87a72
Unitarity Bounds in AdS 3 Higher Spin Gravity 20 Feb 2012 Alejandra Castro [email protected]:[email protected]:[email protected] McGill Physics Department 3600 rue UniversityH3A 2T8MontréalQCCanada Eliot Hijano McGill Physics Department 3600 rue UniversityH3A 2T8MontréalQCCanada Arnaud Lepage-Jutier McGill Physics Department 3600 rue UniversityH3A 2T8MontréalQCCanada Unitarity Bounds in AdS 3 Higher Spin Gravity 20 Feb 2012 We study SL(N, R) Chern-Simons gauge theories in three dimensions. The choice of the embedding of SL(2, R) in SL(N, R), together with asymptotic boundary conditions, defines a theory of higher spin gravity. Each inequivalent embedding leads to a different asymptotic symmetry group, which we map to an OPE structure at the boundary. A simple inspection of these algebras indicates that only the W N algebra constructed using the principal embedding could admit a unitary representation for large values of the central charge. Introduction Higher spin theories provide a new venue to examine our expectations about quantum gravity. The pioneer work of Vasiliev gives a background independent formulation of a classical theory of AdS gravity coupled to an infinite tower of higher spin fields (see e.g. [1,2] and references within). An immediate consequence is that the gauge symmetries of the theory encompass both diffeomorphisms and the higher spin transformations, providing a non-linear and nonlocal theory. This is one of many features that have the potential to address puzzles such as singularity resolution and the significance of black hole horizons. In relation to the holographic principle, higher spin theories allow us to investigate in more depth the dictionary and consequences of this correspondence. For AdS 4 /CFT 3 , the first version of the duality was conjectured by Klebanov and Polyakov in [3], and further refined and tested in e.g. [4,5,6]. Without going into details, these complicated bulk Vasiliev theories are conjectured to be dual to simple, and in principle, solvable theories. This opens the possibility of tracking the emergence of space-time from the boundary theory, among other effects. Our focus here will be in the three-dimensional version of AdS higher spin gravity, and hence its two-dimensional dual CFT. This is arguably the simplest setup of the correspondence from the bulk point of view, which has allowed a better understanding of physical phenomena in Vasiliev theory. Starting with the construction of a classical phase space [7,8], the advances include understanding the quantization of the theory [9,10,11,12,13], a non-geometric definition of black holes [14,15,16], construction of novel solutions [17,18,19], generalizations to de Sitter space [20], and much more [21,22,23,24]. Further, the duality proposed in [25] between a specific Vasiliev theory and a large N 't Hooft limit of W N minimal models is providing new insights in the field [26,27,28,29,30,31,32,33,34,35,36,20,37,38,39,40]. The advantage of AdS 3 gravity, and its higher spin generalizations, is due to the absence of local degrees of freedom. The construction of these theories is straightforward by using the Chern-Simons (CS) formulation of 3D gravity, and as we will review below, coupling higher spin fields to gravity is as simple as studying a SL(N, R) × SL(N, R) CS theory. While it seems almost trivial, the theory still contains both perturbative and non-perturbative configurations which characterize the global dynamics of the theory. Our aim is to understand the perturbative spectrum of the higher spin theory, and from here identify which of these classical theories are well-defined after canonical quantization. There is a systematic and complete construction of the perturbative spectrum, and the essence of this construction is based on the original work of Brown and Henneaux [41]. The idea is to construct the non-trivial gauge transformations, and the group generated by this set labels all physical states smoothly connected to the identity. In the bulk language, this is known as the asymptotic symmetry group. Based on [42,43], the analysis in [8] provides a systematic implementation of the Brown-Henneaux construction adapted to higher spin theories. The remarkable observation is that the resulting algebra is a conformal extension of the Virasoro algebra, known as W-algebras [44], with central charge c. As in the Drinfeld-Sokolov reduction [45] -which is an algebraic construction of W-the resulting algebra constructed in the bulk depends on the gauge group of the CS theory, the embedding of sl 2 in the gauge group and the coupling constant k, where c ∼ 6k. The majority of the literature listed above focus on the principal embedding of sl 2 in sl N ; here the gravitational theory has a simple interpretation as an interacting theory for a non-degenerate tower of massless spin s fields with s = 2, . . . , N. Our focus is on the physical interpretation of secondary (non-principal) embeddings, which we will infer by exploiting some basic features of the conformal algebras. Specific examples of non-principal embeddings have been discussed in the context of higher spin theories in [18,15,32,16]. A point that has been overlooked is that the Walgebra, obtained after imposing asymptotically AdS boundary conditions, is universally ill-defined in the following sense. All non-principal embeddings contain either Abelian or non-Abelian subalgebras generated by spin 1 fields. These subalgebras are enhanced to chiral Kac-Moody algebras at level κ inside the W-algebra, and κ is mostly fixed by the central charge c. 1 Our construction shows that κ is strictly negative for large positive values of c. As we show explicitly in the text, a negative level implies that the spectrum contains negative norm states. The snapshot of the argument is that the Kac-Moody subalgebra is schematically of the form [U n , U m ] = −|κ|nδ n+m + · · · , (1.1) and therefore the state |ψ = U −1 |0 has negative norm. The dots in (1.1) are additional terms appearing for non-Abelian currents, but those terms don't interfere with the logic. The details are discussed sections 3.2 and 3.3. Hence, the semiclassical W-algebra for any secondary embedding does not admit a unitary representation. 2 However, a class of SL(N) theories that escapes our fatal conclusion are those built using the principal embedding. Our analysis provides a simple selection principle that places the principle embedding as perhaps the only consistent framework of higher spin gravity in three dimensions. The organization of the paper is as follows. We first work out the matter content of our theory in section 2, focusing on two classes of non-principal embeddings: the sum and product embedding. We then construct the asymptotic symmetry algebra in section 3 which we use to map our fields to operators at the boundary. The relevant operator product expansions are then related to norms of descendents of the spin 1 current in the different subsections. In appendix A we introduce notation and conventions, and in appendix B we discuss in more depth the unitary representation of W for finite values of the central charge. Matter Content of Higher Spin Gravity Three dimensional Einstein gravity with a negative cosmological constant can be recast as a SL(2, R) ×SL(2, R) Chern-Simons theory [46,47]. To fix some of the notation, we will briefly review this statement here. We will then proceed to discuss the construction of higher spin theories using the language of the Chern-Simons theory. See also [48,49,50,51,8] for more details and generalizations of this construction. The key observation is that by rewriting the dreibein and spin connection as e = ℓ 2 (A −Ā) , ω = 1 2 (A +Ā) ,(2.1) with ℓ the AdS radius, the Einstein-Hilbert action can be written as a SL(2, R) × SL(2, R) Chern-Simons theory S = S CS [A] − S CS [Ā] , (2.2) where S CS [A] = k 4π tr M A ∧ dA + 2 3 A ∧ A ∧ A . (2.3) The trace is with respect to the invariant quadratic form of SL(2, R), and the integration is over the 3-manifold M. The level of the Chern-Simons action k is related to the AdS radius (ℓ) and Newton's constant (G 3 ) by matching the normalization to agree with the Einstein-Hilbert action k tr L 2 0 = ℓ 8G 3 , (2.4) where L 0 is the generator for the center of SL(2, R). The Einstein's equations are then flatness conditions for the connections, dA + A ∧ A = 0 , dĀ +Ā ∧Ā = 0 . (2.5) The absence of local degrees of freedom for three-dimensional gravity is evident in the Chern-Simons formulation. Coupling matter to this theory can be easily done in this framework. In particular we can include spin fields by looking at extensions of the Chern-Simons gauge group. We require that SL(2, R) sits as a subgroup in this extension to guarantee gravitational dynamics. Much effort has been put recently into the study of SL(N, R) × SL(N, R) Chern-Simons theory, which has been coined higher spin gravity for reasons that will become clear in our treatment. The interpretation of this theory in terms of metric-like fields depends on the choice of embedding of SL(2, R) in SL(N, R), and we will review here what is known about such embeddings. For example, consider the case N = 3. Here we have only two possible embeddings. The principal embedding of SL(2, R) in SL(3, R) contains a spin 2 field and spin 3 field, hence it is a description of Einstein gravity coupled to a spin 3 field. The other inequivalent embedding, denoted non-principal embedding, contains a spin 2 field -analogue to the one found in the principal embedding -a spin 1 current, and two bosonic spin 3/2 fields [18]. Even though both theories are loosely speaking "SL(3) gravitational theory", the two inequivalent embeddings have different matter content, and therefore a different interpretation when written locally in terms of metric-like fields. The number of inequivalent embeddings and the field content complexity of the theory increases with N, and hence the gravitational interpretation of an SL(N) Chern-Simons theory is not unique. The number of possible embeddings of the algebra sl 2 in sl N is given by the partition of N [52]. We can denote such an embedding by the branching of the fundamental representation under the choice of sl 2 N N → ⊕ {j} n j · 2j + 1 . (2.6) Here we have partitioned N = j n j (2j + 1); also we denoted the d-dimensional representation of sl m by d m and dropped the index for sl 2 . The centralizer C of the embedded sl 2 subalgebra will also play a role, and in this case is given by C = ⊕ {j} sl n j , (2.7) up to U(1) factors. To illustrate our discussion, let's define the principal embedding. This is the embedding relevant for the duality proposed by [9], and studied initially in [7,8]. We have N N → N . (2.8) The centralizer C is trivial, and the adjoint representation (ad N + 1 N ≡ N N ×N N ) branches as ad N → 3 ⊕ · · · ⊕ 2N − 1 . (2.9) From this branching, one can see the degrees of freedom organize into a tower of massless fields of spin from 2 to N [8]. Therefore one can argue that this embedding gives a description of gravity coupled to a finite tower of massless higher spin fields. The non-principal embeddings are more involved since the centralizer of sl 2 into sl N will be non-trivial. Under C ⊗ sl 2 the adjoint representation will branch as ad N + 1 N →           ⊕ {i =0} ad n i + 1 n i ⊗ 1 + n 0 ⊗n 0 ⊗ 1 ⊕ {i =0} 1 C ⊗ (3 ⊕ ... ⊕ 4i + 1) ⊕ {i =0} ad n i ⊗ (3 ⊕ ... ⊕ 4i + 1) ⊕ {i =j} n i n i ⊗ n j n j ⊗ 2 |i − j| + 1 ⊕ ... ⊕ 2 |i + j| + 1           . (2.10) The matter content is given by the branching of ad N , and thus one of the singlet on the right-hand side of (2.10) will be canceled by the singlet constraint on the left-hand side. The first line represents the possible spin 1 fields. The very first term will lead to non-Abelian currents associated to C, and U(1) singlets will also be present if n i =0 > 1. The last term in this line will contain a singlet unless n 0 = 0. The second line in this expression represents singlets under C of spin 2 up to (2i + 1), which we identify as the metric and the higher spin fields in the bulk. The third line represents multiplets of spin from 2 to (2i + 1) that transform in the adjoint representation of a sl n i algebra. The fourth term contains fields of spin from (|i − j| + 1) to (|i + j| + 1) that transform non-trivially under n i n i ⊗ n j n j . We point out that the only embedding that lacks spin 1 currents is the one with n 0 = 0 and a unique n i =0 = 0, which we single out as the principal embedding. Any other embedding will contain either singlets of spin 1 with U(1) gauge symmetry or multiplets of spin 1 transforming in the adjoint representation of a sl n i algebra. More about non-principal embeddings The presence of spin 1 currents will be crucial to understand aspects of these higher spin theories. For sake of simplicity, we will carry out explicit computations for only two class of embeddings: the sum and the product embedding [53]. The sum embedding will serve as an example for the embeddings with U(1) currents, while the product embedding is a nice setting to study non-Abelian currents. Taking N = P + M, the structure of the sum embedding is P + M P +M → 1 · M + P · 1. Its adjoint decomposition is ad P +M → ad P ⊗ 1 + P P ⊗ M + P P ⊗ M + 1 P ⊗ (1 ⊕ 3 ⊕ · · · ⊕ 2M − 1) . (2.11) The theory contains a sl P algebra, P fields of spin ( M +1 2 ) that transform in the fundamental representation of the sl P algebra, and another P that transform in the conjugate representation. Finally we have spin from 1 to M fields that transform as singlets under sl P . Note that the latter with spin greater than 1 can also be found in the principal embedding of sl 2 in sl M ad M → 3 ⊕ · · · ⊕ 2M − 1 . (2.12) The product embedding will be used to study non-Abelian current algebras. Here we take N = P M, and the structure is P · M P ·M → P · M . Its adjoint decomposition is ad P ·M → ad P ⊗ 1 + 1 P ⊗ (3 ⊕ · · · ⊕ 2M − 1) + ad P ⊗ (3 ⊕ · · · ⊕ 2M − 1) . (2.13) The theory contains a sl P current algebra, singlets of spin from 2 up to M, and a multiplet of P 2 − 1 fields of spins from 2 up to M that transform in the adjoint representation of sl P . Operator Product Expansions and Unitarity In this section we will describe some general features of the asymptotic symmetry algebra for any embedding of sl 2 in sl N through the Drinfeld-Sokolov procedure. Our goal is to show that the presence of spin 1 currents in any embedding implies that the algebra does not admit a unitary representation in the classical limit. 3 Asymptotic symmetry group, W-algebras and OPEs The asymptotic symmetry group is the set of non-trivial gauge transformations that preserve specific boundary conditions. For each generator of the group there is a finite conserved charge associated to it, and the perturbative spectrum of the theory is obtained by acting with these charges. With asymptotically AdS boundary conditionsà la Brown-Henneaux [41], the familiar example is pure AdS 3 gravity. For this theory the rigid sl 2 × sl 2 algebra is enhanced to two copies of the Virasoro algebra. A similar statement holds for the higher spin theories: starting from an sl N × sl N theory with asymptotic AdS boundary conditions, the resulting asymptotic symmetry group is two copies of the W-algebra [7,8]. 4 By construction these W-algebras contain the Virasoro generators, and the additional spin currents depend on the embedding of sl 2 in sl N . In this subsection, we will review this construction following the conventions in [53,8]. In the Chern-Simons formulation of the theory, we first need to specify the connections and topologies under consideration. We start by introducing light-cone coordinates x ± = t/ℓ ± θ, where t stands for the time direction and θ parametrizes the circle at the boundary. Constant time slices have the topology of a disc parametrized by θ and the radial coordinate ρ. We will choose a radial gauge for the connections where A ρ = b −1 ∂ ρ b ,Ā ρ = b ∂ ρ b −1 . (3.1) This is always possible as there exists a local gauge transformation that brings any possible connection to this form. Here b is an arbitrary function of ρ valued in the SL(N, R) group. The connection one forms are now given by A = b −1 a(x + , x − )b + b −1 d b , A = bā(x + , x − )b −1 + b d b −1 ,(3.2) where a(x + , x − ) andā(x + , x − ) are flat sl N valued one forms. In order to ensure the flatness of those one forms, we impose that a(x + , x − ) = a(x + )dx + ,ā(x + , x − ) =ā(x − )dx − . (3.3) And from now on, we focus on the connection A, since the treatment forĀ is parallel. Following the discussion of section 2, the sl N algebra splits into irreducible representations of sl 2 , i.e. ad N → ⊕ j n j 2j + 1. This means that a generic connection can be specified as a(x + ) = j j m=−j Φ j,m (n j ) (x + )T (n j ) j,m ,(3.4) where the T labels the different representations of weight j. 5 For example, we denote the sl 2 generators as (T 1,−1 , T 1,0 , T 1,1 ). We can now define asymptotically AdS boundary conditions. The connection which corresponds to the AdS background is A AdS = e ρ T(1)1,1 dx + + T (1) 1,0 dρ ,(3.5) from which we read the radial parameter b = e ρT (1) 1,0 . Choosing the same radial parameter for all connections, an asymptotically AdS configuration satisfies (A − A AdS )| ρ→∞ = O(1) . (3.6) In terms of the expansion (3.4), this effectively cuts down the fields that would have negative conformal weights, i.e. Φ j,m (n j ) = 0 for m > 0 , j > 1 . (3.7) One can then use gauge transformations that respect the boundary conditions to bring the connection to the highest-weight gauge, i.e. a = T(1)1,1 + a f ix , a f ix = j 1 c j Φ j (n j ) T (n j ) j,−j , (3.8) where Φ j (n j ) ≡ Φ j,−j (n j ) , and we introduced c j = tr(T (n j ) j,−j T (n j ) j,j ) to assure conventional normalization of the fields. Despite appearances, there is some residual gauge symmetry. More concretely, consider a gauge transformation j,m , this implements only a transformation on Φ j (n j ) . Hence it will preserve conditions (3.6) and (3.8). And for those transformations that do not vanish near the boundary -physical symmetries -we will have the corresponding conserved charges where µ (n j ) j ≡ µ (n j ) j,j is the source conjugate to Φ j (n j ) . The level of the Chern-Simons action is related again to the AdS radius via (2.4), where now the trace is the bilinear invariant of sl(N, R). δ Λ a = dΛ + [a, Λ] ,(3.Q(µ) = k 2π dθ tr (a f ix Λ) = k 2π dθ j µ (n j ) j Φ j (n j ) ,(3. Further the Poisson bracket of these charges generate gauge transformation on the fields Φ, i.e. δ µ Φ = {Φ, Q(µ)} ,(3.12) and most importantly δ µ ′ Q(µ) = {Q(µ), Q(µ ′ )} = Q([µ, µ ′ ]) + K(µ, µ ′ ) . (3.13) This is the asymptotic symmetry group. For pure AdS 3 gravity it would lead to the Virasoro algebra, where K is the central term. For the sl N higher spin theories K contains in addition nonlinear terms on Q. These are the W-algebras, and examples of the explicit construction for N = 3, 4 can be found in e.g. [8,32,16]. For any of these algebras, one can then read off the central charge in (3.13) and recover the celebrated result of Brown and Henneaux c = 12ktr(L 2 0 ) = 3ℓ 2G 3 ,(3.14) with L 0 ≡ T 1,0 . For our purposes it will be convenient to translate (3.12) to a statement concerning OPEs of the fields as in [14]. At the boundary, we will use Euclidean coordinates (x + , x − ) → (z,z) and use the holomorphic nature of the fields in our theory. We then use Noether's theorem to write δΦ(z) = Res z→0 [J(z)Φ(0)] ,(3.15) where J(z) = j µ (n j ) j Φ j (n j ) . By adjusting the sources, one can extract from (3.9) and (3.15) the different OPEs. From simple contour integrations the latter are mapped to commutation relations. In the following sections we use this procedure to characterize the asymptotic symmetry algebra corresponding to non-principal embeddings. This procedure is equivalent to the Drinfeld-Sokolov reduction that leads to extended conformal algebras. One important ingredient of this procedure, which we will later use in this section, is the definition of the stress tensor L. In the presence of currents, the stress tensor contains a Sugawara density in addition to the spin 2 field, and in our notation it reduces to 6 L = − k 2 tr(a 2 ) ,(3.16) where the proportionality factor is fixed by (3.12). With this definition, the fields Φ j (n j ) are primaries of weight (j + 1). Sum embedding We now proceed to apply this rather abstract discussion of the asymptotic symmetry group to a concrete setup. Consider the sum embedding as defined in (2.11). Taking N = M + P , we can write the embedding as the branching of the fundamental representation of sl N , i.e. P + M P +M → 1 · M + P · 1 . Constructing the full asymptotic symmetry group is tedious, and not very illuminating. Instead, we consider a truncation that involves only the gravitational sector and a U(1) currents of this embedding. For the sum embedding, the gauge fixed one-form (3.8) turns to be a f ix = − 1 c 0 M P U1 P ×P 0 M ×P 0 P ×M 1 c 1 T W (2) −1 + 1 c 0 U1 M ×M . (3.17) with c j are the traces defined below (3.8). Here T is the spin 2 field and U is the U(1) current, and stand for the fields Φ 1 (1) and Φ 0 (1) respectively. The stress tensor (3.16) is given by L = c 12 1 c 1 2T + 1 c 0 U 2 . (3.18) The explicit expressions for the generators, as introduced in (3.4), are T (1) 1,{0,±1} = 0 P ×P ⊕ W (2) {0,±1} , T (0) 0 = − M P 1 P ×P ⊕ 1 M ×M , (3.19) where W (2) 0,±1 are traceless M × M matrices (see appendix A). To construct the asymptotic symmetry group, it will be sufficient to start with a gauge parameter of the form Λ = − M P µ (0) 0 1 P ×P 0 M ×P 0 P ×M µ (1) 1 W (2) 1 + µ (1) 1,0 W (2) 0 + µ (1) 1,−1 W (2) −1 + µ (0) 0 1 M ×M . (3.20) Here µ 1,−1 = 1 2 ∂ 2 µ (1) 1 + T c 1 µ (1) 1 , (3.21) where we used the notation ∂ ≡ ∂ x + and∂ ≡ ∂ x − . As a result, the non-trivial gauge transformations (3.20)-(3.21) act on the fields as δ Λ L = c 12 ∂ 3 ǫ + ǫ∂L + 2L∂ǫ + U∂η , δ Λ U = − M 2 c M P + 1 M 2 − 1 ∂η + U∂ǫ + ǫ∂U . (3.22) In (3.22) we introduced ǫ = µ (1) 1 , η = − c 6c 0 c 1 µ (1) 1 U − c 0 µ (0) 0 . (3.23) in order to get the canonically normalized transformations. We can reconstruct the OPEs from (3.22). Using J(z) = ǫL + ηU in (3.15) we obtain where n, m ∈ Z. Even though this is not the complete asymptotic algebra, we can make some precise assertions about the entire Fock space. It is clear from (3.25) that for c > 0, which is our case due to (3.14), the level of the U(1) current is negative. This immediately indicates that the spectrum is sick: by simple inspection negative norm states will be present in the theory. More explicitly, in a highest weight representation, the vacuum is defined as L n |0 = U n |0 = 0 , n ≥ 0 , (3.26) and analogous expressions for the other spin generators. The above also implies that the vacuum is annihilated by the rigid sl N generators, and in particular L −1 |0 = 0. Descendants are constructed by acting on the vacuum with creation operators associated to each spin generator, e.g. L −n−1 and U −n with n > 0. L (z) L (0) ∼ c 2 1 z 4 + 2L (0) z 2 + ∂L (0) z , L (z) U (0) ∼ U (0) z 2 + ∂U (0) z , U (z) U (0) ∼ − M 2 c M P + 1 M 2 − 1 1 z 2 . The first excited state is given solely by |ψ = U −1 |0 ; note that this is true only for the vacuum state and it is unaffected by the other spin generators with j = 0. 7 According to (3.25), the state has norm ψ|ψ = − M 2 c M P + 1 M 2 − 1 0|0 < 0 . (3.27) Hence, the physical spectrum associated to this embedding does not admit unitary representations. As a gravitational higher spin theory, the presence of U(1) fields makes the theory pathological. This conclusion is generic to any embedding containing U(1) currents, not only the sum embedding. Including other fields with j = 0 in the branching of the adjoint representation of SL(N, R) (2.10) does not affect the UU OPE; this OPE depends only on the variation δU with respect to its associated source η. In the next sub-section, we will generalize the analysis for multiple currents, but first we will work out an explicit example. Example: U(1) singlet in W (2) 3 algebra One example of the sum embedding that can be carried out explicitly -without leaving any field behind-is the diagonal embedding of SL(2, R) in SL(3, R). The partition is 3 → 1 · 2 + 1 · 1 and the adjoint representation of the algebra is ad 3 → 1 ⊗ 2 + 1 ⊗ 2 + 1 ⊗ 3 + 1 ⊗ 1 . (3.28) The theory contains two bosonic spin 3/2 fields, a spin 2 field that carries the gravitational dynamics, and a U(1) current. One can do the same analysis as we have done from (3.17) to (3.25), by simply setting P = 1, M = 2. In the highest weight gauge we can represent the one form (3.8) 8 a f ix =    −2j 0 Ḡ G j T 0 0 j    ,(3.30) where j is the U(1) current, G andḠ are the spin 3/2 fields and T is the graviton. They play the role of Φ in equation (3.8). The OPE was computed in [18] 7 The presence of additional states at level 1 would require computing the Kac determinant, and from there assure that the matrix is positive definite. In our setup, this will be the case if and only if we have additional spin 1 fields, and it is the subject of section 3.3. 0 , Φ(1)1 , Φ(1) 8 In [18], a different representation was used. In their notation we would have 29) using the techniques outlined above. We will not repeat the derivation here and just state the final answer found in [18] L (z) L (0) ∼ c 2 a f ix,there =   jḠ T 0 −2j G 0 0 j   .(3.1 z 4 + 2L(0) z 2 + ∂L(0) z , L (z) U (0) ∼ U(0) z 2 + ∂U(0) z , U (z) U (0) ∼ − c 9 1 z 2 , L (z) G ± (0) ∼ ∂G + (0) z + 3 2 G + (0) z 2 , G + (z) G − (0) ∼ − c 3 1 z 3 + 3U(0) z 2 − 1 z L(0) + 18 c U(0) 2 − 3 2 ∂U(0) , G ± (z) U (0) ∼ ± G + (0) z ,(3.31) where T = − 6 c L + 9 2c U 2 , j = 3 c U , G = − 6 c G − ,Ḡ = − 6 c G + . (3.32) This is known as the W or Polyakov-Bershadsky algebra [55,56]. The relevant structure of L and U is preserved after the inclusion of the spin 3/2 fields. In particular the negative sign in the UU OPE is not affected. Hence, negative norm states will appear, just as in the general sum embedding. Note that using the Chern-Simons theory we will always obtain the classical limit (large central charge limit) of the W-algebra. Some of these algebras have a known quantum version, and the claims about unitary representations we make here could change. We refer the interested reader to appendix B, where we compute the Kac determinant for the first levels of the quantum W (2) 3 algebra. Product embedding We consider now an embedding of sl 2 in sl N that contains non-Abelian currents. A good representative is the product embedding, where the partition of N = P · M is P · M P ·M → P · M and the adjoint representation of the algebra is explained below equation (2.13). For sake of simplicity, we turn off the fields that transforms in ad P ⊗ (3 ⊕ · · · ⊕ 2M − 1). The connection will only contain the graviton and non-Abelian currents, i.e. we choose (3.8) as a f ix = 1 c 0 U a σ a ⊗ 1 M ×M + 1 c 1 T 1 P ×P ⊗ W (2) −1 . (3.33) The fields U a are currents that transform under sl P and T is again the gravitational spin 2 field. We pick a representation of the sl P algebra {σ a } such that tr (σ a σ b ) ( c tr (σ 2 c )) = γ ab , [σ a , σ b ] = f ab c σ c , (3.34) where γ ab is the Killing form and f ab c are the structure constants. The explicit expression for the generators are T (a) 0 = σ a ⊗ 1 M ×M , T(1)1,{0,±1} = 1 P ×P ⊗ W (2) {0,±1} . (3.35) To ease the notation, we modified slightly the normalization of the sl P currents to c 0 = a tr(T (a) 0 T (a) 0 ) = M a tr σ 2 a . (3.36) The stress tensor is then (3.16) L = c 12 1 c 1 2T + 1 c 0 U a U b γ ab . (3.37) The gauge parameter that preserves our boundary conditions takes the form Λ = µ (a) 0 σ a ⊗ 1 M ×M + µ (1) 1 1 P ×P ⊗ W (2) 1 +µ (1) 1,0 1 P ×P ⊗ W (2) 0,M ×M + µ (1) 1,−1 1 P ×P ⊗ W (2) −1 ,(3.38) where µ U a n , U b m = − M 2 P c M 2 − 1 nδ n+m γ ab + MP c M 2 − 1 f ab c U c m+n . (3.42) where it is clear that the sl P currents form now a Kac-Moody algebra. One can check that the Kac-Moody currents are problematic, providing with a generalization of the single U(1) case. To construct the spectrum of the theory, we follow the arguments at the end of section 3.2. At level 1, we now have a collection of the states |ψ a = U a −1 |0 , and the norms of these states are given by ψ a |ψ b = − M 2 c M P + 1 M 2 − 1 γ ab 0|0 . (3.43) The Killing matrix γ ab generically will contain both positive and negative eigenvalues, hence the matrix is not positive definite. This captures the basic pathology of having spin 1 fields in SL(N) higher spin theories. We conclude that any non-principal embedding will contain negative norm states for positive central charge c. We next present an explicit example of the simplest product embedding, keeping track of the inclusion of all the fields that are present in the adjoint representation. In this section we check that the truncation for the product embeddings does not interfere with the results for the non-Abelian currents. We consider the decomposition 4 4 → 2·2. The field content consists of a sl 2 current, a multiplet of 3 spin 2 fields that transform under the adjoint representation of sl 2 , and a spin 2 field in the trivial representation of sl 2 that we will call the graviton. We denote the asymptotic algebra W (2,2) 4 which is the simplest instance of a non-principal, non-diagonal embedding. Note that this algebra is the case P = 2, M = 2 of a product embedding. The gauge fixed current can be written as a f ix = J T 0 J , J = U a 2 c (tr (σ 2 c )) σ a , T = − T 4 1 + t a 2 c (tr (σ 2 c )) σ a . (3.44) The fields T, U a and t a are the graviton, the sl 2 currents and the multiplet of spin 2 fields respectively. The gauge parameter is now written as follows Λ = γ + 1 2 λ 0 λ −1 λ 1 γ − 1 2 λ 0 , γ = µ (a) 0 σ a , λ i = µ(1)1,i 1 + µ a 1,i σ a .(3.45) The conjugate sources are encoded in λ 1 and µ (a) 0 . And λ 0,−1 are solved for in a manner similar to (3.21). The stress tensor (3.37) is not affected by the multiplet of spin 2 fields t a , nor is the redefinition of the sources (3.40). We include here the variations of the stress tensor and the currents under the residual gauge transformations δL = c 12 ∂ 3 ǫ + 2L∂ǫ + ǫ∂L + U a ∂η b γ ab + 2t a ∂µ b 1 γ ab , δU a = − 4 c c 0 ∂η a + ∂ (ǫU a ) − 4 c f a bc U b η c + f a bc t b µ c 1 . (3.46) In particular, we have that the t a fields are indeed spin 2 multiplets transforming in the adjoint of the sl 2 currents. Moreover, one can see that the U a U b OPE will not be affected by the multiplets, hence the same type of states as in the general case will preclude unitary representations of the W (2,2) 4 algebra. A Conventions An explicit M-dimensional representation of the sl M generators is [19] W (2) 1 = −            0 0 0 0 0 0 √ M − 1 0 0 0 0 0 0 . . . 0 0 0 0 0 0 i (M − i) 0 0 0 0 0 0 . . . 0 0 0 0 0 0 √ M − 1 0            (A.1) W (2) −1 =            0 √ M − 1 0 0 0 0 0 0 . . . 0 0 0 0 0 0 i (M − i) 0 0 0 0 0 0 . . . 0 0 0 0 0 0 √ M − 1 0 0 0 0 0 0            (A.2) W (2) 0 = 1 2            M − 1 0 0 0 0 0 0 M − 3 0 0 0 0 0 0 . . . 0 0 0 0 0 0 M + 1 − 2i 0 0 0 0 0 0 . . . 0 0 0 0 0 0 −(M − 1)            (A.3) and where W (s) m = (−1) s−m−1 (s + m − 1)! (2s − 2)! [W (2) −1 , [Wt (s) m = (−1) m (s − 1)! 2 (s + m − 1)!(s − m − 1)! (2s − 1)!(2s − 2)! M s−1 i=1 M 2 − i 2 . (A.6) A.1 Sum embedding construction The generalization of (3.17) to include the remaining matter fields is a f ix =    J P ×P − M P c 0 U1 P ×P 0 (M −1)×P G 1×P G P ×1 0 P ×(M −1) W M ×M + 1 c 0 U1 M ×M    . (A.7) Here J P ×P contains the sl P currents, G (G) is the multiplet of fields that transform in the fundamental (conjugate) representation of sl P and U is the spin 1 singlet. W contains the information for the rest of higher spin fields W M ×M = W Here 1 is the unit matrix, J is a traceless matrix that contains the sl P currents, and Ψ s contains both the spin s field and the multiplets of P 2 − 1 spin s fields that transform in the adjoint representation of the sl P currents. Any of these matrices can be decomposed as Ψ s = Ψs P 1 P ×P + ψ a s σ a where σ a are generators of the sl P algebra. B Quantum W 3 algebra The algebras we have used in this work are semiclassical limits of W-algebras. The full quantum algebra receives corrections that affect the relation between the level of the current algebras and the central charge; it also affects the coefficients of non-linear terms. An illustrative example is the Polyakov-Bershadsky algebra W (2) 3 [55,56]. Here we will explore if our conclusions in section 3.2.1 are modified in the quantum regime. In the quantum regime, the matter content does not change, but the commutation relations get modified slightly. We have and κ is a real parameter. The classical limit is given by −κ ≫ 1, reproducing the OPEs (3.31). In the highest-weight representation, the vacuum satisfies From the algebra it is also true that L −1 |0 = 0. We define the hermitian conjugate as L † −n ≡ L n ; U † −n ≡ U n ; (G + −n ) † ≡ G − n . (B.5) At level 1 the only state is U −1 |0 and its norm reads Imposing positive norms at the two lowest level, the allowed values for κ are − 3 2 < κ < − 11 12 . (B.8) n j )j,m are generators combined into representations of weight j, and the index n j 4 See[24] for a modification of these boundary conditions. non-trivial relations among the functions µ (n j ) 11) 5 5According to (2.10) the different T (nj) j,m can have non-trivial relations through the centralizer of sl 2 . Hence the field is specified by its weight j and its transformation with respect to the centralizer. the sources for T and U, in accordance with(3.11). The components µ and U in a Laurent expansion, the singular parts of the operator expansions yield commutation relations by contour integration [L n , L m ] = (n − m) L n+m + c 12 n 3 − n δ n+m , [L n , U m ] = −mU m+n , [U n , U m ] are again simply given by(3.21). The variation of the fields under residual gauge transformations (3.38) can be s−1 ] . . .]] . (A.4) with m = −(s − 1) . . . (s + 1) and s = 3 . . . M. The Killing metric of this algebra is tr W (s) m W (r) n = t (s) m δ r,s δ m,−n , (A.5) of the gauge fixed one-form (3.33) is a f ix = J P ×P ⊗ 1 M ×M + 1 P ×P ⊗ L 1,M ×M + n, m) ∈ Z and (r, s) ∈ (Z + 1/2). The central charge of the algebra is corrected by L n |0 = 0 , U n |0 = 0 , n ≥ 0 , 0|U 1 U − 1 11|0 as simple as the first, since the only state is G + −3/2 |0 . The norm is then0|(G + −3/2 ) † G + −3/2 |0 = (2κ + 3) [1 − 12(κ + 1)] . (B.7) κ also depends on the rank of the gauge group and other minor details of the embedding.2 The algebra could still admit non-unitary irreducible representations, but in our opinion it is not an interesting scenario. By classical limit we mean one for which the central charge of the boundary theory is large, i.e. the AdS radius is large in Planck units. The nature of W-algebras in a quantum regime has been addressed previously in[54]. We use the example of the W(2)3 algebra in appendix B to discuss the possibility of unitary representations for small central charge. In a general gauge, the stress tensor could contain a linear improvement term proportional to the Cartan element of the singled-out sl 2 subalgebra. In the highest-weight gauge the latter drops[53]. AcknowledgementsWe are very grateful to Alex Maloney for useful discussions and comments. We also thank Jan de Boer, Michael Gutperle and Per Kraus for discussions. This work was supported by the National Science and Engineering Research Council of Canada. E.H. acknowledges support from Fundación Caja Madrid.(3.39)Again, we redefine the sources asin order to get the transformations in a canonical form. Using J = ǫL + η a U a , the above transformations completely determines the singular part of the OPEs Note that in this range c is valued between 0 and 1, and therefore this bound is not consistent with a semiclassical limit. 0We now compute the inner product matrix at level 2, in the basis given by L −2 |0Note that in this range c is valued between 0 and 1, and therefore this bound is not consistent with a semiclassical limit. We now compute the inner product matrix at level 2, in the basis given by L −2 |0 , U −2 |0 Progress in higher spin gauge theories. M Vasiliev, hep-th/0104246M. Vasiliev, "Progress in higher spin gauge theories," hep-th/0104246. Nonlinear higher spin theories in various dimensions. X Bekaert, S Cnockaert, C Iazeolla, M Vasiliev, hep-th/0503128X. Bekaert, S. Cnockaert, C. Iazeolla, and M. Vasiliev, "Nonlinear higher spin theories in various dimensions," hep-th/0503128. AdS dual of the critical O(N) vector model. I R Klebanov, A M Polyakov, hep-th/0210114Phys. Lett. 550I. R. Klebanov and A. M. Polyakov, "AdS dual of the critical O(N) vector model," Phys. Lett. B550 (2002) 213-219, hep-th/0210114. Massless higher spins and holography. E Sezgin, P Sundell, hep-th/0205131Nucl. Phys. 644E. Sezgin and P. Sundell, "Massless higher spins and holography," Nucl. Phys. B644 (2002) 303-370, hep-th/0205131. Higher Spin Gauge Theory and Holography: The Three-Point Functions. S Giombi, X Yin, 0912.3462JHEP. 11509S. Giombi and X. Yin, "Higher Spin Gauge Theory and Holography: The Three-Point Functions," JHEP 09 (2010) 115, 0912.3462. Higher Spins in AdS and Twistorial Holography. S Giombi, X Yin, JHEP. 1104S. Giombi and X. Yin, "Higher Spins in AdS and Twistorial Holography," JHEP 1104 (2011) 086, 1004.3736. Nonlinear W inf inity as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity. M Henneaux, S.-J Rey, 1008.4579JHEP. 10127M. Henneaux and S.-J. Rey, "Nonlinear W inf inity as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity," JHEP 1012 (2010) 007, 1008.4579. Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields. A Campoleoni, S Fredenhagen, S Pfenninger, S Theisen, 1008.4744JHEP. 117A. Campoleoni, S. Fredenhagen, S. Pfenninger, and S. Theisen, "Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields," JHEP 11 (2010) 007, 1008.4744. Quantum W-symmetry in AdS 3. M R Gaberdiel, R Gopakumar, A Saha, 1009.6087M. R. Gaberdiel, R. Gopakumar, and A. Saha, "Quantum W-symmetry in AdS 3 ," 1009.6087. Higher Spin Theories in AdS 3 and a Gravitational Exclusion Principle. A Castro, A Lepage-Jutier, A Maloney, 1012.0598JHEP. 1101A. Castro, A. Lepage-Jutier, and A. Maloney, "Higher Spin Theories in AdS 3 and a Gravitational Exclusion Principle," JHEP 1101 (2011) 142, 1012.0598. One loop partition function for Topologically Massive Higher Spin Gravity. A Bagchi, S Lal, A Saha, B Sahoo, JHEP. 1112A. Bagchi, S. Lal, A. Saha, and B. Sahoo, "One loop partition function for Topologically Massive Higher Spin Gravity," JHEP 1112 (2011) 068, 1107.2063. The Gravitational Exclusion Principle and Null States in Anti-de Sitter Space. A Castro, T Hartman, A Maloney, Class.Quant.Grav. 28A. Castro, T. Hartman, and A. Maloney, "The Gravitational Exclusion Principle and Null States in Anti-de Sitter Space," Class.Quant.Grav. 28 (2011) 195012, 1107.5098. Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole. S Datta, J R David, 1112.4619. 47S. Datta and J. R. David, "Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole," 1112.4619. 47 pages. Higher Spin Black Holes. M Gutperle, P Kraus, JHEP. 05M. Gutperle and P. Kraus, "Higher Spin Black Holes," JHEP 05 (2011) 022, 1103.4304. Black Holes and Singularity Resolution in Higher Spin Gravity. A Castro, E Hijano, A Lepage-Jutier, A Maloney, 1110.4117JHEP. 120131A. Castro, E. Hijano, A. Lepage-Jutier, and A. Maloney, "Black Holes and Singularity Resolution in Higher Spin Gravity," JHEP 1201 (2012) 031, 1110.4117. Aspects of Three-dimensional Spin-4 Gravity. H Tan, 1111.2834H. Tan, "Aspects of Three-dimensional Spin-4 Gravity," 1111.2834. BTZ Black Hole as Solution of 3-D Higher Spin Gauge Theory. V Didenko, A Matveev, M Vasiliev, hep-th/0612161Theor.Math.Phys. 153V. Didenko, A. Matveev, and M. Vasiliev, "BTZ Black Hole as Solution of 3-D Higher Spin Gauge Theory," Theor.Math.Phys. 153 (2007) 1487-1510, hep-th/0612161. Spacetime Geometry in Higher Spin Gravity. M Ammon, M Gutperle, P Kraus, E Perlmutter, 1106.4788M. Ammon, M. Gutperle, P. Kraus, and E. Perlmutter, "Spacetime Geometry in Higher Spin Gravity," 1106.4788. Conical Defects in Higher Spin Theories. A Castro, R Gopakumar, M Gutperle, J Raeymaekers, 1111.3381A. Castro, R. Gopakumar, M. Gutperle, and J. Raeymaekers, "Conical Defects in Higher Spin Theories," 1111.3381. Toward Higher Spin dS3/CFT2. P Ouyang, 1111.0276P. Ouyang, "Toward Higher Spin dS3/CFT2," 1111.0276. Spin-3 Topological Massive Gravity. B Chen, J Long, J.-B Wu, 1106.5141Phys.Lett. 705B. Chen, J. Long, and J.-B. Wu, "Spin-3 Topological Massive Gravity," Phys.Lett. B705 (2011) 513-520, 1106.5141. High Spin Topologically Massive Gravity. B Chen, J Long, 1110.5113JHEP. 1112B. Chen and J. Long, "High Spin Topologically Massive Gravity," JHEP 1112 (2011) 114, 1110.5113. Higher Spins in D = 2+1. A Campoleoni, 1110.5841A. Campoleoni, "Higher Spins in D = 2+1," 1110.5841. Towards non-AdS holography in 3-dimensional higher spin gravity. M Gary, D Grumiller, R Rashkov, 1201.0013M. Gary, D. Grumiller, and R. Rashkov, "Towards non-AdS holography in 3-dimensional higher spin gravity," 1201.0013. An AdS 3 Dual for Minimal Model CFTs. M R Gaberdiel, R Gopakumar, Phys.Rev. 83M. R. Gaberdiel and R. Gopakumar, "An AdS 3 Dual for Minimal Model CFTs," Phys.Rev. D83 (2011) 066007, 1011.2986. Large-N limits of 2d CFTs, Quivers and AdS 3 duals. E Kiritsis, V Niarchos, 1011.5900JHEP. 1104E. Kiritsis and V. Niarchos, "Large-N limits of 2d CFTs, Quivers and AdS 3 duals," JHEP 1104 (2011) 113, 1011.5900. Symmetries of Holographic Minimal Models. M R Gaberdiel, T Hartman, 1101.2910JHEP. 0531M. R. Gaberdiel and T. Hartman, "Symmetries of Holographic Minimal Models," JHEP 05 (2011) 031, 1101.2910. The Large N 't Hooft Limit of Coset Minimal Models. C Ahn, 1106.0351JHEP. 1110C. Ahn, "The Large N 't Hooft Limit of Coset Minimal Models," JHEP 1110 (2011) 125, 1106.0351. Partition Functions of Holographic Minimal Models. M R Gaberdiel, R Gopakumar, T Hartman, S Raju, 1106.1897M. R. Gaberdiel, R. Gopakumar, T. Hartman, and S. Raju, "Partition Functions of Holographic Minimal Models," 1106.1897. Higher Spin Gravity with Matter in AdS 3 and Its CFT Dual. C.-M Chang, X Yin, 1106.2580C.-M. Chang and X. Yin, "Higher Spin Gravity with Matter in AdS 3 and Its CFT Dual," 1106.2580. Minimal Model Holography for SO(2N). M R Gaberdiel, C Vollenweider, 1106.2634JHEP. 1108M. R. Gaberdiel and C. Vollenweider, "Minimal Model Holography for SO(2N)," JHEP 1108 (2011) 104, 1106.2634. Asymptotic W-symmetries in three-dimensional higher-spin gauge theories. A Campoleoni, S Fredenhagen, S Pfenninger, 1107.0290JHEP. 1109A. Campoleoni, S. Fredenhagen, and S. Pfenninger, "Asymptotic W-symmetries in three-dimensional higher-spin gauge theories," JHEP 1109 (2011) 113, 1107.0290. . A Bagchi, S Lal, A Saha, B Sahoo, 1107.0915Topologically Massive Higher Spin Gravity. 1110JHEPA. Bagchi, S. Lal, A. Saha, and B. Sahoo, "Topologically Massive Higher Spin Gravity," JHEP 1110 (2011) 150, 1107.0915. Partition functions of higher spin black holes and their CFT duals. P Kraus, E Perlmutter, 1108.2567P. Kraus and E. Perlmutter, "Partition functions of higher spin black holes and their CFT duals," 1108.2567. Correlation Functions in Holographic Minimal Models. K Papadodimas, S Raju, 1108.3077Nucl.Phys. 856K. Papadodimas and S. Raju, "Correlation Functions in Holographic Minimal Models," Nucl.Phys. B856 (2012) 607-646, 1108.3077. The Coset Spin-4 Casimir Operator and Its Three-Point Functions with Scalars. C Ahn, 1111.0091JHEP. 120227C. Ahn, "The Coset Spin-4 Casimir Operator and Its Three-Point Functions with Scalars," JHEP 1202 (2012) 027, 1111.0091. Higher spin AdS 3 supergravity and its dual CFT. T Creutzig, Y Hikida, P B Ronne, 1111.2139T. Creutzig, Y. Hikida, and P. B. Ronne, "Higher spin AdS 3 supergravity and its dual CFT," 1111.2139. Scalar fields and three-point functions in D=3 higher spin gravity. M Ammon, P Kraus, E Perlmutter, 1111.3926M. Ammon, P. Kraus, and E. Perlmutter, "Scalar fields and three-point functions in D=3 higher spin gravity," 1111.3926. Limits of minimal models and continuous orbifolds. M R Gaberdiel, P Suchanek, 1112.1708M. R. Gaberdiel and P. Suchanek, "Limits of minimal models and continuous orbifolds," 1112.1708. Correlators in W N Minimal Model Revisited. C.-M Chang, X Yin, 1112.5459C.-M. Chang and X. Yin, "Correlators in W N Minimal Model Revisited," 1112.5459. Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity. J D Brown, M Henneaux, Commun. Math. Phys. 104J. D. Brown and M. Henneaux, "Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity," Commun. Math. Phys. 104 (1986) 207-226. Global charges in Chern-Simons field theory and the (2+1) black hole. M Banados, hep-th/9405171Phys. Rev. 52M. Banados, "Global charges in Chern-Simons field theory and the (2+1) black hole," Phys. Rev. D52 (1996) 5816, hep-th/9405171. Three-dimensional quantum geometry and black holes. M Banados, hep-th/9901148M. Banados, "Three-dimensional quantum geometry and black holes," hep-th/9901148. W symmetry in conformal field theory. P Bouwknegt, K Schoutens, hep-th/9210010Phys.Rept. 223P. Bouwknegt and K. Schoutens, "W symmetry in conformal field theory," Phys.Rept. 223 (1993) 183-276, hep-th/9210010. Lie algebras and equations of Korteweg-de Vries type. V Drinfeld, V Sokolov, J.Sov.Math. 30V. Drinfeld and V. Sokolov, "Lie algebras and equations of Korteweg-de Vries type," J.Sov.Math. 30 (1984) 1975-2036. A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories. A Achucarro, P K Townsend, Phys. Lett. 18089A. Achucarro and P. K. Townsend, "A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories," Phys. Lett. B180 (1986) 89. (2+1)-Dimensional Gravity as an Exactly Soluble System. E Witten, Nucl. Phys. 31146E. Witten, "(2+1)-Dimensional Gravity as an Exactly Soluble System," Nucl. Phys. B311 (1988) 46. M P Blencowe, A CONSISTENT INTERACTING MASSLESS HIGHER SPIN FIELD THEORY IN D = (2+1). 6443M. P. Blencowe, "A CONSISTENT INTERACTING MASSLESS HIGHER SPIN FIELD THEORY IN D = (2+1)," Class. Quant. Grav. 6 (1989) 443. The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant. O Coussaert, M Henneaux, P Van Driel, gr-qc/9506019Class.Quant.Grav. 12O. Coussaert, M. Henneaux, and P. van Driel, "The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant," Class.Quant.Grav. 12 (1995) 2961-2966, gr-qc/9506019. Anti-de Sitter / CFT correspondence in three-dimensional supergravity. M Banados, K Bautier, O Coussaert, M Henneaux, M Ortiz, hep-th/9805165Phys.Rev. 5885020M. Banados, K. Bautier, O. Coussaert, M. Henneaux, and M. Ortiz, "Anti-de Sitter / CFT correspondence in three-dimensional supergravity," Phys.Rev. D58 (1998) 085020, hep-th/9805165. Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity. M Henneaux, L Maoz, A Schwimmer, hep-th/9910013Annals Phys. 282M. Henneaux, L. Maoz, and A. Schwimmer, "Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity," Annals Phys. 282 (2000) 31-66, hep-th/9910013. Semisimple subalgebras of semisimple Lie algebras. E Dynkin, Trans.Am.Math.Soc. 6111E. Dynkin, "Semisimple subalgebras of semisimple Lie algebras," Trans.Am.Math.Soc. 6 (1957) 111. Covariantly coupled chiral algebras. F Bais, T Tjin, P Van Driel, Nucl.Phys. 357F. Bais, T. Tjin, and P. van Driel, "Covariantly coupled chiral algebras," Nucl.Phys. B357 (1991) 632-654. The Relation between quantum W algebras and Lie algebras. J Boer, T Tjin, hep-th/9302006Commun.Math.Phys. 160J. de Boer and T. Tjin, "The Relation between quantum W algebras and Lie algebras," Commun.Math.Phys. 160 (1994) 317-332, hep-th/9302006. Gauge Transformations and Diffeomorphisms. A M Polyakov, Physics and Mathematics of Strings. 5833Int.J.Mod.Phys.A. M. Polyakov, "Gauge Transformations and Diffeomorphisms," Int.J.Mod.Phys. A5 (1990) 833. Submitted to V. Knizhnik Memorial volume, Physics and Mathematics of Strings. Conformal field theories via Hamiltonian reduction. M Bershadsky, Commun.Math.Phys. 139M. Bershadsky, "Conformal field theories via Hamiltonian reduction," Commun.Math.Phys. 139 (1991) 71-82.
[]
[ "Pseudo-real-time retinal layer segmentation for high-resolution adaptive optics optical coherence tomography", "Pseudo-real-time retinal layer segmentation for high-resolution adaptive optics optical coherence tomography" ]
[ "Worawee Janpongsri \nBiomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada\n", "Joey Huang \nBiomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada\n", "| Ringo Ng ", "| Daniel \nBiomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada\n", "J Wahl \nBiomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada\n", "| Marinko ", "V Sarunic \nBiomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada\n", "Yifan Jian [email protected] \nCasey Eye Institute\nOregon Health & Science University\nPortlandORUSA\n\nDepartment of Biomedical Engineering\nOregon Health & Science University\nPortlandORUSA\n\nCorrespondence\nCenter for Ophthalmic Optics & Lasers\nOregon Health & Science University\nCasey Eye Institute, 515 S.W. Campus Dr97229PortlandOR\n", "Yifan Jian " ]
[ "Biomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada", "Biomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada", "Biomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada", "Biomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada", "Biomedical Optics Research Group\nSchool of Engineering Science\nSimon Fraser University\nBurnabyBCCanada", "Casey Eye Institute\nOregon Health & Science University\nPortlandORUSA", "Department of Biomedical Engineering\nOregon Health & Science University\nPortlandORUSA", "Correspondence\nCenter for Ophthalmic Optics & Lasers\nOregon Health & Science University\nCasey Eye Institute, 515 S.W. Campus Dr97229PortlandOR" ]
[]
We present a pseudo-real-time retinal layer segmentation for high-resolution Sensorless Adaptive Optics-Optical Coherence Tomography (SAO-OCT). Our pseudo-real-time segmentation method is based on Dijkstra's algorithm that uses the intensity of pixels and the vertical gradient of the image to find the minimum cost in a geometric graph formulation within a limited search region. It segments six retinal layer boundaries in an iterative process according to their order of prominence. The segmentation time is strongly correlated to the number of retinal layers to be segmented. Our program permits en face images to be extracted during data acquisition to guide the depth specific focus control and depth dependent aberration correction for high-resolution SAO-OCT systems. The average processing times for our entire pipeline for segmenting six layers in a retinal B-scan of 496x400 pixels and 240x400 pixels are around 25.60 ms and 13.76 ms, respectively. When reducing the number of layers segmented to only two layers, the time required for a 240x400 pixel image is 8.26 ms.
10.1002/jbio.202000042
[ "https://arxiv.org/pdf/2004.05264v1.pdf" ]
215,745,652
2004.05264
8b695664ab92605dbfd7b2d49666b9696e8d8d1f
Pseudo-real-time retinal layer segmentation for high-resolution adaptive optics optical coherence tomography Worawee Janpongsri Biomedical Optics Research Group School of Engineering Science Simon Fraser University BurnabyBCCanada Joey Huang Biomedical Optics Research Group School of Engineering Science Simon Fraser University BurnabyBCCanada | Ringo Ng | Daniel Biomedical Optics Research Group School of Engineering Science Simon Fraser University BurnabyBCCanada J Wahl Biomedical Optics Research Group School of Engineering Science Simon Fraser University BurnabyBCCanada | Marinko V Sarunic Biomedical Optics Research Group School of Engineering Science Simon Fraser University BurnabyBCCanada Yifan Jian [email protected] Casey Eye Institute Oregon Health & Science University PortlandORUSA Department of Biomedical Engineering Oregon Health & Science University PortlandORUSA Correspondence Center for Ophthalmic Optics & Lasers Oregon Health & Science University Casey Eye Institute, 515 S.W. Campus Dr97229PortlandOR Yifan Jian Pseudo-real-time retinal layer segmentation for high-resolution adaptive optics optical coherence tomography FULL ARTICLEK E YWO RDS Retinal layer segmentationgraph searchimage processingpseudo-real-time We present a pseudo-real-time retinal layer segmentation for high-resolution Sensorless Adaptive Optics-Optical Coherence Tomography (SAO-OCT). Our pseudo-real-time segmentation method is based on Dijkstra's algorithm that uses the intensity of pixels and the vertical gradient of the image to find the minimum cost in a geometric graph formulation within a limited search region. It segments six retinal layer boundaries in an iterative process according to their order of prominence. The segmentation time is strongly correlated to the number of retinal layers to be segmented. Our program permits en face images to be extracted during data acquisition to guide the depth specific focus control and depth dependent aberration correction for high-resolution SAO-OCT systems. The average processing times for our entire pipeline for segmenting six layers in a retinal B-scan of 496x400 pixels and 240x400 pixels are around 25.60 ms and 13.76 ms, respectively. When reducing the number of layers segmented to only two layers, the time required for a 240x400 pixel image is 8.26 ms. | INTRODUCTION Optical Coherence Tomography (OCT) is a noninvasive medical imaging technique based on low coherence interferometry, typically using near-infrared light, to capture high-resolution cross-sectional views of biological tissues (e.g., the retina). Ocular imaging with OCT allows ophthalmic clinicians to view and measure the distinctive retinal layer structure to diagnose retinal diseases such as glaucoma and age-related macular degeneration. Also, the diseases related to retinal vessels, such as diabetic retinopathy, can be diagnosed through the retinal vasculature and its hierarchical structure from en face OCT Angiography (OCTA) images. OCT technology has continuously improved its acquisition speed; however, due to the complexity of OCT data processing from interferometric fringe data into images, the signal processing is computationally burdensome. Thus, powerful computational resources such as Graphics Processing Units (GPUs) were used to perform the parallelizable aspects of processing interferometric fringes into A-scans, and rendering the resulting volumes [1][2][3][4][5][6][7][8]. Our custom GPU pipeline could perform OCT processing at 2.24 MHz axial scan rate [1] and was also demonstrated for displaying flow contrast en face images extracted from the selected depth region on speckle variance OCT (svOCT) angiography [2], [9][10] in real-time. Rather than perform post-processing analysis, there are ample opportunities and motivations to process the B-scan images to extract additional information as they are acquired. For example, segmentation of the retinal layer boundaries provides an opportunity to perform thickness measurements, which has clinical implications in monitoring glaucoma. In other clinical applications such as computer-assisted surgery, [11][12][13][14][15][16][17][18] high-resolution ocular images can be produced in real time to evaluate dynamic anatomical changes and thus assist clinicians during surgery to potentially increase the rate of success. Moreover, retinal layer segmentation can extract en face images (taken in the C-scan directions) from the OCT volumes, and real-time retinal layer segmentation can provide effective focus control and direct feedback of aberration correction performance with image-guided AO techniques [19][20][21][22][23]. This last application is the main focus of this report. Optical aberrations caused by imperfections in the cornea and the intraocular lens reduce the resolution of retinal imaging. Adaptive optics have been integrated with optical retinal imaging to correct ocular aberrations and allow diffraction limited imaging [24]. In particular, our interest is in the combination of AO with OCT [24][25][26][27][28][29][30][31]. The conventional approach to AO uses a Hartmann-Shack Wavefront Sensor (HS-WFS) to detect wavefront distortions and compensate them using a deformable mirror. However, the HS-WFS is sensitive to back-reflections, causing most of the conventional AO to use curved mirrors instead of lenses. We are developing a lens-based sensorless adaptive optics (SAO) approach to correct optical aberrations up to 21 st order Zernike polynomials, starting from a defocus [32]. The SAO-OCT can directly evaluate an image quality metric using the extracted en face projections to drive the AO correction. Hence, SAO-OCT systems can be used for applications where there are multiple reflecting surfaces from the sample (which could confound a wavefront sensor measurement) or requiring depth resolved aberration correction. While confocal AO scanning laser ophthalmoscope also have depth section capability depending on the confocal pinhole size, AO-SLO is not a volumetric imaging technique, images from only one depth plane can be visualized at a time. With AO-OCT or SAO-OCT, the entire retina can be visualized with a volumetric scan despite limited depth of focus. In addition, combing with real-time retinal layer segmentation, en face images generated from any depth plane even if they are out of focus, can be used to drive the SAO correction to shift the focus and sharpen the image. To extract the thickness measurements and en face images from OCT volumes, many segmentation methods have been introduced. Active contours segmentation [33][34][35] uses an energy formulation; however, it requires a good initialization, and the constraints on the boundaries can cause errors when the retinal layers are in irregular shapes. Besides, the active contours approach also has a high computational cost which is not suitable for real-time applications. Machine learning approaches [36], have been recently introduced, performing retinal segmentation based on learning data representations. They give accurate results; however, they require substantial amount of labelled learning data sets supplied to the networks and it is computationally expensive to train the networks. Automated segmentation methods based on graph theory use pixel intensity and the gradient of the image to find the minimum cost in a geometric graph information for each retinal layer boundary [37][38][39][40][41]. The accuracy and the speed of segmentation based on graph theory depend on the algorithmic implementation used. To integrate segmentation in real-time OCT imaging applications, a robust segmentation algorithm with low computational cost and low complexity is required. There are several fast retinal layer segmentation attempts for OCT images. Fabritius et al. reduced the processing time of segmenting Inner Limiting Membrane (ILM) and Retinal Pigment Epithelium (RPE) by heavily down-sampling each Bscan [42]. The total processing time using a computer with 2.4 GHz CPU to segment a healthy macula volume (1024x320x140) was 6.7 seconds: the average time needed for the ILM was 4.0 seconds and for the RPE was 1.9 seconds. Tian et al. developed a faster (~10x) automatic segmentation program called OCTRIMA 3D [40] based on the previous work by Chiu et al. [37]. In a follow up paper [43], Tian et al. evaluated different segmentation software on a computer with an Intel® Core TM i7-2600 @3.4 GHz CPU. The result showed that OCTRIMA 3D could perform the segmentation of an OCT volume (768 x 496 x 610) in 28 seconds. In this paper, the main goal is to minimize the processing time of the automated retinal layer program while maintaining the reliability of the segmentation results. We present a pseudo-real-time retinal layer boundaries segmentation program in mice modified from the Caserel software [38]. The pseudo-real-time retinal layer segmentation was integrated in our custom GPU pipeline [1] with SAO algorithm [32] , which provides focus control and aberration correction for high-resolution real-time visualization of vascular network. The organization of the rest of this report is as follows. The details of the algorithm are presented in Section 2 followed by how we set up the environment for running and testing the algorithm in Section 3. Section 4 shows the results of our segmentation algorithm along with the speed performance and compares the results of AO optimization with and without the segmentation. Discussion about the segmentation performance is covered in Section 5 and the conclusion and future work are presented in Section 6. | METHODS This section describes the implementation of the pseudo-real-time retinal layer segmentation in C/C++ for integration with an OCT acquisition system. The Method description is divided into six subsections: a) image cropping to contain only the area of interest, b) logarithmic scaling and noise reduction, c) layer endpoint initialization and weights calculation, d) ILM and RPE layers segmentation, and e) limiting the search region for the other layers. Figure 1 shows the flowchart outline of the steps for our retinal layer segmentation. Our segmentation software uses graph-cut algorithm to delineate the retinal layers which was first introduced by Chiu et al [37]. We perform image cropping on the GPU following the generation of the B-scan image, and then the rest of the program is performed on the CPU in order to gain performance in heterogeneous computing according to the capability of each type of the processor. In our previous work where we chose to perform a graph-cut on the GPU to segment ILM and RPE of human retina using Push-Relabel Graph-Cut (PR GC) algorithm [44], we found that sometimes the PR GC generates an unwanted region along with the two retinal layers. As a result, we performed connected component labeling (CCL) after segmentation, which is a computational expensive method to identify the two largest connected groups and remove smaller artifacts. According to M. Miao [44], the segmentation pipeline with multiple GPU could only segment ILM and RPE layers on a single OCT B-scan (1024x300x900 pixels) in 57.26 ms which was largely affected by the CCL (12.45 ms). Nevertheless, due to the discontinuity of the pre-built NVIDIA Performance Primitives (NPP) graph-cut APIs after Compute Unified Device Architecture (CUDA) 7.5 version [45], other segmentation methods are needed to replace NPP graph-cut APIs with a better speed performance and without generating artifacts with retinal layers. Although there are several alternatives with both CPU and GPU implementations available for the NPP graph-cut APIs [46][47][48][49][50], these maxflow/min-cut algorithm requires the CCL to remove artifacts which is quite slow. Conversely, Dijkstra's algorithm performs a Graphcut by finding the shortest path or the minimum cost path between two nodes. It produces only a single path without any artifacts. Besides, Dijkstra's algorithm from the Boost Graph Library [51], implemented on the CPU, has a generic interface and can be utilized easily using a header file in C/C++. Therefore, we implemented our retinal layer segmentation program using Dijkstra's algorithm from the Boost Graph Library. | Pre-processing In OCT images, particularly when imaging the retina, the data in the axial direction (e.g, the retinal depth) is contained in only a relatively small number of pixels. The pixels that do not contain image information affect the speed performance of the segmentation. In order to decrease the computational cost and make the delineation more reliable, cropping the image to contain only the region of interest (ROI) is necessary. Image cropping was performed on the GPU, immediately after the B-scan image has been generated from the interference signal, and is performed as follows: Firstly, the average of pixels in each row of the image is computed along the lateral direction: 1 -1 ( ) ( , ) 0 N avg y f x y j i j i N = ∑ =(1) where f(x i ,y j ) is the grayscale intensity of pixel (x i ,y j ), i and j are the horizontal index and vertical index respectively, and N is the width of the image. Secondly, we generate a histogram on the row average where the number of bins is chosen to be ten and the size of the bins is calculated as: max( _ ) min( _ ) 1 _ _ _ row avg row avg bin size number of bins − + = (2) After the histogram is computed, the value of the bin that contains the most common elements is selected as the threshold. Empirically, the first and the last indices of the rows for which the average values are greater than the threshold correspond to the position where the retinal structure begins (ILM layer) and ends (RPE layer), respectively. In order to include all the retinal characteristics in the ROI, the indices where the ROI begins, and ends are calculated as: _ begin index image ROI i offset height = − × (3) and _ begin index image ROI i offset height = − ×(4) where i and j are the first and the last indices where the value of the row average are greater than the threshold. In this study, we set the offset = 0.1. We only applied the cropping step on images with height greater than 200 pixels; otherwise, we skip this step because limiting the ROI may crop out some important retinal characteristics for segmentation. Figure 2 shows the original B-scan, its cropped image and the Gaussian blurred image. In addition, we implemented this step on the GPU for faster parallel operations performance on a large set of data. Before segmenting the OCT retinal images, we employed a logarithm operation on each pixel on the image to compress the dynamic range of the image and enhance low intensity pixel values. Then we down-sampled the cropped image by a factor of two and referred to this as a resized image. Removing the noise before further processing the images is essential as the noise could affect the quality of the automated segmentation. The most common noise in OCT images is the speckle noise which is produced by constructive/destructive interference. Speckle noise appears as white and black intensity fluctuations and can be reduced in appearance by applying a Gaussian blur filter. After down-sampling the | Layer endpoint initialization and weights calculation Each image is considered as a graph of nodes, in which a node equates to a pixel on the image having edges connecting to other nodes [49]. A graph may consist of multiple layered structures, and segmenting a particular layer requires the selection of the start and the end nodes. The start and the end nodes are automatically initialized by assuming that the retinal layers to be segmented extend across the entire width of the image. One vertical column is added to each side of the Gaussian blurred image, and they are assigned with zero values. The start node is the left top corner pixel and the end node is the right bottom pixel. These additional columns are removed after the segmentation is completed. In the retinal images, the foreground is defined as the retinal layers and the background as the vitreous and posterior chamber. The transition in pixel intensity from the background to the foreground is large so a graph can be simply constructed based on calculating the vertical gradient of the image. Figure 3 shows the two gradient images (the positive and negative gradient) of size 105x202 pixels. Both of them are generated because some retinal layer boundaries, such as the vitreous/ILM appears to have a darker layer above a brighter layer, whereas other boundaries, such as the Nerve Fiber Layer/Ganglion Cell Layer (NFL/GCL), have a lighter layer above a darker layer. The weight of the edges usually represents the geometric distance and/or the intensity difference between the neighboring pixels. However, in OCT retinal images, the features of interest have a smooth transition between neighboring pixels and each pixel is only connected with its eight nearest neighboring pixels and disconnected with other nodes. Hence, the weight of the edges is a function of the intensity difference between the neighboring pixels. Because the retinal layers in OCT images are horizontal structures distinguishable by a vertical change in pixel intensity, the weights are calculated based on the vertical gradient. The formula used in this method for calculating the weights is: min 2 ( ) ab a b w g g w =− − + (5) where w ab is the weight assigned to the edge connecting nodes a and b, g a is the vertical gradient of the image at node a, g b is the vertical gradient of the image at node b and w min is the minimum weight in the graph (1E-5). The weights are also calculated based on the directionality of the gradient. As the result, we have two sparse adjacency matrices of intensity difference graph weights of size [MN x MN] with MNC filled entries where [M x N] is the image size and C is the number of the nearest neighbors (in this case is eight). The light-to-dark sparse adjacency matrix of graph weights was created using the gradient whereas the darkto-light sparse adjacency matrix of graph weights used the negative gradient.-As mentioned above, we added one additional column on each side of the image, we assigned the weight values in those additional columns to be wmin so the shortest path calculation would not be affected by the additional columns. The edge weight of zero indicates that the two nodes are unconnected. | ILM and RPE layer segmentation The retinal layers are segmented in an iterative process according to their order of prominence. The ILM and RPE layers are segmented first due to their high contrast in pixel intensity relative to the background. To begin the retinal segmentation, the Gaussian blurred image is again resized by a factor of two to roughly segment the ILM and RPE layers. This twice down-sampled image is referred to as the rough image. Then we produce the gradient image and the dark-tolight sparse adjacency matrix of graph weights for the rough image. A ROI matrix of the same size as the rough image with two additional columns is generated and each pixel value of the ROI is set to 1 if the corresponding pixel on the rough image is greater than the mean value of the rough image, otherwise zero. We set the value of dark-to-light sparse adjacency matrix of graph weights to zero where the ROI is zero, otherwise the value is not changed. The procedure described above helps with indicating the region to find the shortest path because zero edge weight means unconnected nodes. Next, we used Dijkstra's algorithm and the dark-to-light sparse adjacency matrix of graph weights to find the shortest paths. After the first layer was segmented, we set the pixels of the first found layer on the ROI matrix to zero in order to segment the second rough layer. We iterated the process to find the second rough layer, setting the value of the dark-to-light sparse adjacency matrix of graph weights to zero where the ROI is zero; otherwise the same value is kept. After the two rough layers are found, both of the rough layers were interpolated to have the same dimensions as the resized image. Next, we set the layer where the mean value of the y-coordinate is smaller to be the ILM and the other layer to be the RPE. Then we use the ROI matrix of the same dimensions as the resized image with the additional columns to find the precise ILM and RPE layers. First, we segment the ILM by setting the region of the ROI matrix near the rough ILM layer to one where the rest are zero. Also, we change the value of the dark-to-light sparse adjacency matrix of graph weights to zero where the ROI matrix is zero. Then again, we use Dijkstra's algorithm and the dark-to-light sparse adjacency matrix of graph weights to find the precise ILM layer. The precise RPE is found in the same manner as the ILM. Figure 4 shows the ROI images for finding the rough layers and the precise ILM and RPE layers on the resized image (with two additional columns). The ROI images are 53x102 pixels whereas the resized image is 105x202 pixels. | Limiting the search region for the other layers As mentioned earlier, due to the hyper-reflectivity of the ILM and RPE layers, they are easily segmented. In contrast, the remaining layers are not as prominent because their characteristics (relative intensity) are similar. In order to correctly segment the targeted layer, the search region should be limited such that the irrelevant features are excluded. This exclusion is accomplished by setting the weight of the nontargeted features to zero before segmenting the graph using Dijkstra's algorithm. The search space of each layer is selected based on the previously segmented layers. The order of layer boundaries to be segmented is Inner Nuclear Layer/Outer Plexiform Layer (INL/OPL), NFL/GCL, Inner Plexiform Layer/Inner Nuclear Layer (IPL/INL), and Outer Plexiform Layer/Outer Nuclear Layer (OPL/ONL). Table 1 shows the upper and lower boundaries and the sparse adjacency matrix of graph weights for segmentation. Each boundary requires two previously segmented boundaries to be the upper and the lower bounds to limit the search region. Once all the retinal layer boundaries are segmented, the additional column on each side of the image is removed, leaving the accurate six retinal layer boundaries. Nonetheless, these retinal layers are not the same size as the original input image, and they need to be interpolated, smoothed and offset adjusted based on the cropped background to correctly delineate these features on the original uncropped image. Figure 5 shows representative result of the retinal layer boundaries delineated on mouse cross-sectional retinal image. | EXPERIMENTAL SETUP This section provides an overview of our SAO-OCT imaging system for small animals [32] and our processing and display program for the OCT retinal images [1]. The environment used for implementing and testing our retinal layer segmentation program is described, and the imaging results along with speed performance are presented. | Overview of our OCT processing and display program Real-time application of the OCT requires high throughput and low overhead (latency). In this research, we used the parallelization strategies introduced by Jian et al. [1] to accelerate OCT processing. To fully utilize the PCIe bandwidth, we transferred the interferometric data from the host to the device as a batch rather than a single frame. In order to hide memory transfer latency, the memory transfer from the host to the device and the data processing on the device was implemented using two CUDA streams concurrently; one to transfer the data processing on the device and another to process the interferometric fringe data on the device. While the small batch of the interferometric fringe data was being transferred from the host to the device by the transfer stream, the previous batch that is already in the device is simultaneously being processed by the kernel stream. These two CUDA streams, which are executed simultaneously, are synchronized after processing each batch. The original implementation by Jian et al. [1] demonstrated a high throughput; however, it suffered from a large latency as its processing pipeline was completely asynchronous with the acquisition. In this project, we improved the program latency by synchronizing the data acquisition and processing at the batch level. | SAO-OCT imaging system for small animals Our imaging system is a multi-modal small animal retina imaging system that includes OCT, OCT-A, confocal scanning laser ophthalmoscopy (cSLO), and fluorescence detection [32]. It is a compact lens-based system incorporating the SAO technique to correct the optical aberrations instead of using a wavefront sensor to measure the aberrations. For this project, we only used the modified OCT subsystem for mouse imaging. The OCT subsystem in this project used a light source with a central wavelength of 810 nm and a bandwidth of 100 nm. We also integrated our retinal layer segmentation program in our OCT processing and display program along with the SAO-OCT imaging system. The retinal layers were segmented on the cross-sectional images and these results were used to project en face images from the selected retinal layers. Then the SAO-OCT used the extracted en face projections as the input of its merit function defined as the sum of the intensity squared of each pixel of the en face image to drive the optimization algorithm. The OCT acquisition modality provided a 100 kHz A-scan rate for retina imaging and the OCT volumes are acquired with user selected dimensions. Two B-scans were acquired at the same lateral location to generate an OCT-A B-scan image [2]. Our retinal segmentation algorithm was implemented in C/C++ using Microsoft Visual Studio 2013, running on a personal computer with the CPU of Intel® Core TM i9-9900K CPU @3.6 GHz with a Graphics Processing Unit of NVIDIA GeForce RTX 2060. In this study, we used the mouse SD-OCT volume datasets of different dimensions (as indicated below), but each dataset contained 800 frames. The mouse imaging experiments were performed under protocols compliant to the Canadian Council on Animal Care, and with the approval of the University Animal Care Committee at Simon Fraser University. | RESULTS For the speed performance, we compared our segmentation in C++ with the modified Caserel software using MATLAB R2019a. Since our OCTViewer processes the interferometric fringe data into the processed floating-point data, we modified the Caserel software to read these types of input. We also modified the resize scale of rough layers to 0.5 instead of 0.2 because if the image size is too small, the path of the rough second layer cannot be found due to the broken path on the ROI. Table 2 shows the accumulated average speed performance of our segmentation up to the specified layer using our C++ implementation and the modified version of Caserel software. The order of retinal layers to be segmented is rough ILM and RPE, precise ILM, precise RPE, INL/OPL, NFL/GCL, IPL/INL and OPL/ONL. On average, our implementation can achieve a ~5x acceleration compared to the Caserel software. We also compared the segmentation accuracy of the two implementations on 800 B-scans, the mean pixel difference in retinal layer segmentation between the two methods is negligible (ILM, 0.89 ± 0.19; RPE, 2.05 ± 0.32; INL/OPL, 0.12 ± 0.97; NFL/GCL, 0.24 ± 0.23; IPL/INL, 0.07 ± 0.76; OPL/ONL, 0.11 ± 0.93). Figure 6 compares the results of the intensity and speckle variance en face images when using static (fixed) depth locations to generate the en face images versus using the pseudo-real-time retinal layer segmentation. With the static user-selected depth option, the operator selects two horizontal lines on the B-scan image. Figure 6 (a) shows a representative B-scan with the static user-selected depth at NFL layer and its corresponding en face images. Figure 6 (b) shows a representative B-scan with the pseudo-real-time retinal layer segmentation results of ILM and NFL/GCL layers with its OCT and OCT-A en face images. Although the images are Mouse retinal axial motion during imaging is typically on the order of 4.34 pixels during imaging due to breathing and related motion. Hence, without tracking of the retina position, the region of interest may shift in and out of the bounding box. Figure 7 shows a representative case of when the static user-selected depth option retinal layers option failed to detect the retinal layer of interest during data acquisition due to motion in vivo mouse. The results for NFL imaging with OCT and OCTA (e.g., not aberration corrected), SAO-OCT, and SAO-OCT-A visualization with and without our pseudo-real-time segmentation are shown in Figure 8. Without our segmentation, an operator selected two straight lines within the region of interest in order to obtain en face images. Figure 8 shows en face OCT images in (a) and OCT-A images in (b) with and without the segmentation and with and without SAO optimization. Figure 8 (c) shows a representative plot of the SAO merit function value during optimization with and without the segmentation. Figure 9 shows the segmentation results of different retinal layers from different mice. The left column on each row shows the cross-sectional mouse retina images with two segmented retinal layers. The middle and the right columns displays the intensity and speckle variance en face images generated from the segmented layers shown on the corresponding B-scan. Figure 9 (a) shows the results from segmenting the ILM (green line) and RPE layers (red line). In Figure 9 (b), the en face images are generated from the OPL layer (the green line is the boundary between INL and OPL and the red line is the boundary between the OPL and ONL). | DISCUSSION The main objective of this project is to implement a pseudo-real-time retinal layer segmentation program that provides en face images generation based on anatomical retinal layer structures. Our retinal layer segmentation method was based graph-cut that use pixel intensity and the gradient of the image, this algorithm has been proven to be robust and reliable for both diseased and non-diseased retina, and it has been used in multiple clinical studies [37][38][39][40][41], [52][53][54][55]. Although in this report the in vivo experiments performed with wild type mice without retinal pathologies, we expect that our retinal segmentation method can achieve similar performance in segmenting retinas with diseases and abnormalities. We employed heterogeneous computing to gain the performance based on the nature of each task. CPUs are good at performing complex tasks, and in the context of Graph-cut search, are more reliable in terms of global convergence. In contrast, GPUs are optimized for performing tasks of lesser complexity that can be broken down into smaller independent parts that need less or no communication among tasks. We parallelized the code for cropping the image to contain only the retinal structure and to remove the redundant data by implementing it into CUDA for NVIDIA GPU. Removing redundant data helps Dijkstra's algorithm, which is performed on the CPU due to its capability to perform complex tasks, and to search for the shortest path faster because of the smaller (resized) image size. Moreover, our retinal layer segmentation program utilizes arrays and Intel® Math Kernel Library (MKL) rather than vectors and their operations. Arrays take less time to access their elements because of their contiguous property, and permit access to the elements efficiently with a constant time irrespective of the element location. Since the array is a fixed size data structure, and all elements must be of the same type, hence, it is type safe and the most efficient in terms of speed and performance. The specifications of the segmentation time requirements (including the OCT signal processing time) were set by the acquisition system parameters. The image acquisition system used in this report provided a 100 kHz Aline scan rate, and completed a B-scan consisting of 400 A- lines in 4 ms and a B-scan consisting of 300 A-lines in 3 ms, corresponding to B-scan frame rates of ~250-333Hz. In order to maximize the data transfer speed across the PCIe bus, the Bscans are acquired and transferred in batches. Therefore, the combination of our CUDA and segmentation pipelines (the processing pipeline) must guarantee the segmentation outcome of the current batch before the next batch is completely acquired. Our segmentation using C/C++ showed a significant improvement in the processing time of a B-scan by more than 74% compared to the processing speed of Caserel, and is able to perform the rough segmentation on each (cropped) B-scan at the rate that it is acquired. We found that the lateral movement of the retinal layers within the acquisition time of a batch is negligible; this is particular true for OCT-A where multiple B-scans are acquired at the same position to generate flow contrast. Thus, we reduced the number of frames to be segmented by applying the full resolution segmentation using graph-cuts to the first frame of each batch and we applied segmentation results of the first frame in each batch to all frames in that batch. If we chose to segment only the ILM and RPE layers and chose a batch to contain 4 frames, the acquisition time is about 16 ms for the image with width of 400 pixels and 12 ms for the image with a width of 300 pixels. As shown in Table 2, the execution time to segment the rough and the precise ILM and RPE on an image of size 496x400 takes about 13.57 ms, and on an image of size 240x300 takes about 8.26 ms. As a result, our segmentation pipeline can run at least at 62.5 Hz which is faster than a video rate of 60 Hz. However, we must consider the OCT acquisition and signal processing time along with the image segmentation time to respond within the batch acquisition time. Thus, we chose a batch size to contain 6 frames which extends the acquisition time to 24 ms for the images with 400 pixels width and 18 ms for images with 300 pixels width to include the OCT signal acquisition and processing time. This feature would provide axial tracking and extraction of the shape of the retina for visualization during acquisition. If we wanted to segment a specific retinal layer, we could offset the width between ILM and RPE layers to generate an en face image that is close to the anatomical shape of that layer. The segmentation of additional layers would require larger batch sizes. For example, we could also segment all six retinal layers and choose the batch size to be 10 frames to generate the results before the acquisition time of 40 ms and 30 ms for the images where the width is 400 pixels and 300 pixels, respectively. In addition, in most cases, we do not need all six retinal layers delineated on the image while performing a real-time application. Instead, we often need to segment a particular layer by using two segmented retinal layer boundaries. For instance, the boundaries between INL/OPL and OPL/ONL are needed for segmenting the OPL layer. We could speed up our segmentation pipeline by segmenting only the rough ILM and RPE layers and use these rough layers to limit the search region to segment INL/OPL and again use INL/OPL and rough RPE to get OPL/ONL. Similar to the NFL layer, we could segment the rough layers and use the rough layers to segment INL/OPL and again use rough ILM and INL/OPL to segment NFL/GCL. Alternative computational hardware could also be used for OCT signal processing and retinal layer segmentation. Our approach focused on the use of GPU and CPU because of the ease of implementation, ability for rapid changes to accommodate emerging applications, and for integration with relatively complex control of adaptive optics systems. Future work may involve the use of FPGAs. The OCT processing pipeline has been developed on FPGA previously [56][57][58][59][60]. With its advantage of the execution time and power consumption, an FPGA platform could also be a suitable option for speeding up the retinal layer segmentation. To the best of our knowledge, there is no work of retinal layer segmentation that uses FPGA; however, there are some works on image segmentation using graph-cut on FPGA [61][62]. In this report, we implemented a pseudo-real-time 2D Graph-cut for retinal layer segmentation which was integrated into a complete OCT acquisition and processing software. We demonstrated the system performance for generating layer segmentations to guide an image-based SAO optimization [63][64][65]. With the ability to segment the retinal layers, our processing program created anatomically correct en face images by using a maximum intensity projection (MIP) between the two selected retinal layers for guiding the SAO optimization in real-time. The use of a static user-selected depth region of interest is not ideal to detect specific retinal layers because, for in vivo mouse imaging, the retinal position moves during the acquisition as shown in Figure 8. Figure 6 shows that the top of the NFL en face images are dark because during the acquisition, the retina moved out of the depth that is on focus. In contrast, the pseudo-real-time retinal segmentation generated en face images contained the correct retinal features because the segmented lines followed the motion of the retina in vivo. The pseudo-real-time retinal layer segmentation tracks the layer of interest even though there is motion during acquisition which leads to a better en face image as the input for the SAO optimization as shown in Figure 8. Figure 8 (c) shows the merit plots with and without our segmentation for SAO optimization of 18 modes in total. The merit plot with segmentation shows a better improvement in the image quality after each mode is optimized than the merit plot without the segmentation. | CONCLUSION We have utilized consumer grade PC to control the acquisition of real-time OCT signals and perform image processing. A GPU was used for OCT processing and for generating the B-scan images. The CPU was used for the processes such as shortest-path graph search to segment the retinal layers on the acquisition computer and extracting a depth resolved layer from the volumetric data as it gets acquired. We employed down-sampling and parallel processing to improve the speed of the application and as a result our retinal segmentation program can be used as a pseudo-real-time application. Segmentation of the retinal layers permits OCT and OCTA en face images to be extracted during data acquisition and used to guide the depth specific focus control for high-solution OCT systems. We demonstrated our retinal segmentation program in real-time in vivo mouse retinal imaging, and it can be applied in human retinal imaging in the future. FIGURE 1 1Flowchart of the pseudo-real-time retinal layer segmentation. FIGURE 2 2(a) Original B-scans (496x400 pixels), (b) its cropped image (210x400 pixels) and (c) Gaussian blurred image (105x200). FIGURE 3 3(a) The gradient and (b) the negative gradient images. FIGURE 4 4(a) ROI for the first rough layer (b) ROI for the second rough layer and (c) precise ILM and RPE layers on the resized image with two additional columns. FIGURE 5 5The SD-OCT mouse B-scan with the retinal layer boundaries segmented by the pseudo-real-time segmentation. Figure 9 ( 9c) shows the segmentation results from the boundary between NFL/GCL (green line) and IPL/INL (red line) layers, whereas Figure 9 (d) displays the segmented layers of IPL/INL (green line) and OPL/ONL (red line) FIGURE 6 6Structural and OCTA NFL en face images using static user-selected depth (segmentation OFF) of retinal layers and using the pseudo-real-time retinal layer segmentation (segmentation ON). FIGURE 7 7Axial motion of the retina during acquisition causes failure in detecting NFL layer using two fixed horizontal lines and the plot showing the motion of the ILM position (in pixel) during acquisition. FIGURE 8 8(a) and (b) En face images of NFL layer with and without segmentation and SAO. (c) Image quality of each step in the SAO optimization with segmentation (red plot) and without segmentation (blue plot). FIGURE 9 9Mouse B-scan images with selected depths of interest with OCT and OCT-A enface images. TABLE 1 1Upper and lower bounds, and the sparse adjacency (S.A.) matrix used for performing graph search for a particular layer. TABLE 2 2Speed performance of C/C++ segmentation and Caserel in milliseconds on different image sizes. ACK NOW LE DGM E NT S . Y Jian, K Wong, M V Sarunic, J. Biomed. Opt. 1826002Y. Jian, K. Wong, and M. V. Sarunic, J. Biomed. Opt., 2013, 18, 026002. . J Xu, K Wong, Y Jian, M V Sarunic, J. Biomed. Opt. J. Xu, K. Wong, Y. Jian, and M. V. Sarunic, J. Biomed. Opt., 2014, 19, 026001. . D Xu, Y Huang, J U Kang, Opt. Express. D. Xu, Y. Huang, and J. U. Kang, Opt. Express, 2014, 22, 14871. . N H Cho, U Jung, S Kim, W Jung, J Oh, H Kang, J Kim, J. Opt. Soc. Korea. 1768N. H. Cho, U. Jung, S. Kim, W. Jung, J. Oh, H. Kang, and J. Kim, J. Opt. Soc. Korea, 2013, 17, 68. . Y Wang, C M Oh, M C Oliveira, M S Islam, A Ortega, B , Y. Wang, C. M. Oh, M. C. Oliveira, M. S. Islam, A. Ortega, and B. . H Park, Opt. Express. H. Park, Opt. Express, 2012, 20, 14797. . M Sylwestrzak, D Szlag, M Szkulmowshi, I Gorczynska, D , M. Sylwestrzak, D. Szlag, M. Szkulmowshi, I. Gorczynska, D. . M Bukowka, P Wojtkowski, Targowski, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVII. 851785710Bukowka, M. Wojtkowski, and P. Targowski, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVII, 2013, 8517, 85710Y. . X Wei, A Camino, P Shaohua, T T Hormel, W Cepurna, D , X. Wei, A. Camino, P. Shaohua, T. T. Hormel, W. Cepurna, D. . J C Huang, Y Morrison, Jia, Opt. Lett. 1431Huang, J. C. Morrison, and Y. Jia, Opt. Lett., 2019, 44, 1431. . J Li, P Bloch, J Xu, M V Sarunic, L Shannon, Appl. Opt. J. Li, P. Bloch, J. Xu, M. V. Sarunic, and L. Shannon, Appl. Opt., 2011, 50, 1832. . M Heisler, S Lee, Z Mammo, Y Jian, M J Ju, A Merkur, E Navajas, C Balaratnasingam, M F Beg, M V Sarunic, J. Biomed. Opt. 36007M. Heisler, S. Lee, Z. Mammo, Y. Jian, M. J. Ju, A. Merkur, E. Navajas, C. Balaratnasingam, M. F. Beg, and M. V. Sarunic, J. Biomed. Opt., 2017, 22, 036007. . J Xu, K Wong, V Wong, M Heisler, S Lee, M Cua, Y Jian, M V Sarunic, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX. 93122J. Xu, K. Wong, V. Wong, M. Heisler, S. Lee, M. Cua, Y.Jian, and M. V. Sarunic, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XIX, 2015, 9312, 93122H. . J P Ehlers, S K Srivastava, D Feiler, A I Noonan, A , J. P. Ehlers, S. K. Srivastava, D. Feiler, A. I. Noonan, A. M. . Y K Rollins, Tao, PLoS One. 9105224Rollins, and Y. K. Tao, PLoS One, 2014, 9, 0105224. . J P Ehlers, Y K Tao, S K Srivastava, Current Opinion in Ophthalmology. J. P. Ehlers, Y. K. Tao, and S. K. Srivastava, Current Opinion in Ophthalmology. 2014, 25, 221. . Y K Tao, J P Ehlers, C A Toth, J A Izatt, Opt. Lett. 3315Y. K. Tao, J. P. Ehlers, C. A. Toth, and J. A. Izatt, Opt. Lett., 2010, 35, 3315. . J P Ehlers, Y K Tao, S Farsiu, R Maldonado, J A Izatt, C , J. P. Ehlers, Y. K. Tao, S. Farsiu, R. Maldonado, J. A. Izatt, and C. Investigative Ophthalmology and Visual Science. A Toth, 523153A. Toth, Investigative Ophthalmology and Visual Science. 2011, 52, 3153. . C Toth, O Carrasco-Zevallos, B Keller, L Shen, C Viehland, D , C. Toth, O. Carrasco-Zevallos, B. Keller, L. Shen, C. Viehland, D. . P Nam, A Hahn, J Kuo, Izatt, Investig. Ophthalmol. Vis. Sci. 563512Nam, P. Hahn, A. Kuo,and J. Izatt, Investig. Ophthalmol. Vis. Sci., 2015, 56, 3512 . O M Carrasco-Zevallos, B Keller, C Viehland, L Shen, M , O. M. Carrasco-Zevallos, B. Keller, C. Viehland, L. Shen, M. . J Seider, C Izatt, Toth, Investig. Ophthalmol. Vis. Sci. 37Seider, J. Izatt, and C. Toth, Investig. Ophthalmol. Vis. Sci., 2016, 57, OCT37. . Y Huang, Z Ibrahim, W P A Lee, G Brandacher, J , Y. Huang, Z. Ibrahim, W. P. A. Lee, G. Brandacher, and J. U. Kang, Image-Guided Procedures, Robotic Interventions, and Modeling. 86711Kang, Medical Imaging 2013: Image-Guided Procedures, Robotic Interventions, and Modeling, 2013, 8671, 86711H. . M T El-Haddad, Y K Tao, Current Opinion in Biomedical Engineering. 337M. T. El-Haddad and Y. K. Tao, Current Opinion in Biomedical Engineering, 2017, 3, 37. . Y Jian, S Lee, M Ju, M Heisler, W Ding, R Zawadzki, S Bonora, M V Sanuric, Sci. Rep. Y. Jian, S. Lee, M. Ju, M. Heisler, W. Ding, R. Zawadzki, S. Bonora, and M. V. Sanuric, Sci. Rep., 2016, 6, 27620. Y Jian, A Issaei, R J Zawadzki, M V Sarunic, Proc. of SPIE. of SPIE821382130Y. Jian, A. Issaei, R. J. Zawadzki, and M. V. Sarunic, Proc. of SPIE., 2012, 8213, 82130L. . S Bonora, Y Jian, E N Pugh, M V Sarunic, R J Zawadzki, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVIII. 893489340S. Bonora, Y. Jian, E. N. Pugh, M. V. Sarunic, and R. J. Zawadzki, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVIII, 2014, 8934, 89340Q. . A Zam, P Zhang, Y Jian, M V Sarunic, S Bonora, E Pugh, R Zawadzki, 93122A. Zam, P. Zhang, Y. Jian, M. V. Sarunic, S. Bonora, E. Pugh, and R. Zawadzki, 2015, 9312, 93122I. . P Zhang, D Wahl, J Mocci, S Manna, R Meleppat, S Bonora, M V Sarunic, E Pugh, R Zawadzki, Ophthalmic Technologies XXVIII. P. Zhang, D.Wahl, J. Mocci, S. Manna, R. Meleppat, S. Bonora, M. V. Sarunic, E. Pugh, R. Zawadzki, Ophthalmic Technologies XXVIII, 2018, 10474, 1047427. D R Williams, Vision Research. 1379D. R. Williams, Vision Research. 2011, 1379 . . M Pircher, R J Zawadzki, Biomed. Opt. Express. M. Pircher and R. J. Zawadzki, Biomed. Opt. Express, 2017, 8, 2536. . P Godara, A M Dubis, A Roorda, J L Duncan, J Carroll, Optometry and Vision Science. 87930P. Godara, A. M. Dubis, A. Roorda, J. L. Duncan, and J. Carroll, Optometry and Vision Science, 2010, 87, 930. . Y Zhang, J Rha, R S Jonnal, D T Miller, Opt. Express. 4792Y. Zhang, J. Rha, R. S. Jonnal, and D. T. Miller, Opt. Express, 2005, 13, 4792. . R J Zawadzki, S Jones, S Olivier, M Zhao, B Bower, J Izatt, S Choi, S Laut, J Werner, Opt. Express. 8532R. J. Zawadzki, S. Jones, S. Olivier, M. Zhao, B. Bower, J. Izatt, S. Choi, S. Laut, and J. Werner, Opt. Express, 2005, 13, 8532. . K Kurokawa, Z Liu, D T Miller, Biomed. Opt. Express. K. Kurokawa, Z. Liu, and D. T. Miller, Biomed. Opt. Express, 2017, 8, 1803. . Z Liu, J Tam, O Saeedi, D X Hammer, Biomed. Opt. Express. 94246Z. Liu, J. Tam, O. Saeedi, and D. X. Hammer, Biomed. Opt. Express, 2018, 9, 4246. . R S Jonnal, O P Kocaoglu, R J Zawadzki, Z Liu, D T Miller, J S Werner, Investig. Ophthalmol. Vis. Sci. 51R. S. Jonnal, O. P. Kocaoglu, R. J. Zawadzki, Z. Liu, D. T. Miller, and J. S. Werner, Investig. Ophthalmol. Vis. Sci., 2016, 57, OCT51. . D J Wahl, R Ng, M J Ju, Y Jian, M V Sarunic, Biomed. Opt. Express. 10252D. J. Wahl, R. Ng, M. J. Ju, Y. Jian, and M. V. Sarunic, Biomed. Opt. Express, 2019, 10, 252. . M Mujat, R Chan, B Cense, B Park, C Joo, T Akkin, T Chen, J Boer, Opt. Express. 9480M. Mujat, R. Chan, B. Cense, B. Park, C. Joo, T. Akkin, T. Chen, J. de Boer, Opt. Express, 2005, 13, 9480. . A Yazdanpanah, G Hamarneh, B R Smith, M V Sarunic, IEEE Trans. Med. Imaging. A. Yazdanpanah, G. Hamarneh, B. R. Smith, and M. V. Sarunic, IEEE Trans. Med. Imaging, 2011, 30, 484. . M A Mayer, J Hornegger, C Y Mardin, R P Tornow, Biomed. Opt. Express. M. A. Mayer, J. Hornegger, C. Y. Mardin, and R. P. Tornow, Biomed. Opt. Express, 2010, 1, 1358. . G M Somfai, E Tátrai, L Laurik, B Varga, V Ölvedy, H , https:/bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-15-106#auth-6G. M. Somfai, E. Tátrai, L.Laurik, B. Varga, V. Ölvedy, H. . J Jiang, W Wang, A Smiddy, D C Somogyi, Debuc, https:/bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-15-106#auth-10BMC Bioinformatics. 15106Jiang, J. Wang, W. E Smiddy, A. Somogyi, and D. C. DeBuc, BMC Bioinformatics, 2014, 15, 106 . S J Chiu, X T Li, P Nicholas, C A Toth, J A Izatt, S , S. J. Chiu, X. T. Li, P. Nicholas, C. A. Toth, J. A. Izatt, and S. . Farsiu, Opt. Express. 18Farsiu, Opt. Express, 2010, 18, 19413. Caserel -An Open Source Software for Computer-aided Segmentation of Retinal Layers in Optical Coherence Tomography Images. P Teng, NovP. Teng, Caserel -An Open Source Software for Computer-aided Segmentation of Retinal Layers in Optical Coherence Tomography Images, https://www.researchgate.net/publication/308776805_Caserel_- _An_Open_Source_Software_for_Computer- aided_Segmentation_of_Retinal_Layers_in_Optical_Coherence_T omography_Images (accessed: Nov, 2019). . Q Yang, C Reisman, A Wang, Y Fukuma, M Hangai, N , Q. Yang, C. Reisman, A. Wang, Y. Fukuma, M. Hangai, N. . A Yoshimura, M Tomidokoro, A Araie, D Raza, K Hood, Yoshimura, A. Tomidokoro, M. Araie, A. Raza, D. Hood, and K. . Chan , Opt. Express. 21293Chan, Opt. Express, 2010, 18, 21293. . J Tian, B Varga, G M Somfai, W H Lee, W E Smiddy, D , J. Tian, B. Varga, G. M. Somfai, W. H. Lee, W. E. Smiddy, and D. . C Debuc, PLoS One. 10C. DeBuc, PLoS One, 2015, 10. . B Keller, M Draelos, G Tang, S Farsiu, A N Kuo, K Hauser, J A Izatt, Biomed. Opt. Express. 92716B. Keller, M. Draelos, G. Tang, S. Farsiu, A. N. Kuo, K. Hauser, J. A. Izatt, Biomed. Opt. Express, 2018, 9, 2716. . T Fabritius, S Makita, M Miura, Y Yasuno, R Myllylä, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XV. 788933T. Fabritius, S. Makita, M. Miura, Y. Yasuno, and R. Myllylä, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XV, 2011, 7889, 788933. . J Tian, B Varga, E Tatrai, P Fanni, G M Somfai, W E Smiddy, D C Debuc, Journal of Biophotonics. 9478J. Tian, B. Varga, E. Tatrai, P. Fanni, G. M. Somfai, W. E. Smiddy, and D. C. Debuc, Journal of Biophotonics. 2016, 9, 478. . M Miao, Masc , Simon Fraser UniversityM. Miao, MASc, Simon Fraser University, 2016. Cuda Samples, CUDA Toolkit Documentation. CUDA Samples :: CUDA Toolkit Documentation, https://docs.nvidia.com/cuda/cuda-samples/index.html#new- features-in-cuda-toolkit-6-5 (accessed: Nov, 2019). . L Data, Y Peng, L Chen, W Chen, J Yong, IEEE Trans. Image Process. 655L. Data, Y. Peng, L. Chen, W. Chen, and J. Yong, IEEE Trans. Image Process., 2013, 24, 655. . V Vineet, P J Narayanan, 10.1109/CVPRW.2008.4563095Compute. V. Vineet and P. J. Narayanan, Compute, 2008, DOI: 10.1109/CVPRW.2008.4563095. GPU Computing Gems Emerald Edition. W M W Hwu, W. M. W. Hwu, GPU Computing Gems Emerald Edition. 2011. . Y Boykov, V Kolmogorov, Pattern Anal. Mach. Y. Boykov and V. Kolmogorov, Pattern Anal. Mach., 2004. O Jamriska, D Sykora, A Hornung, Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. the IEEE Computer Society Conference on Computer Vision and Pattern Recognition3673O. Jamriska, D. Sykora, and A. Hornung, Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2012, 3673. Boost Graph Library: Dijkstra's Shortest Paths -1.65.0. NovBoost Graph Library: Dijkstra's Shortest Paths -1.65.0., https://www.boost.org/doc/libs/1_65_0/libs/graph/doc/dijkstra_sho rtest_paths.html (accessed: Nov, 2019). B I Dodo, Y Li, K Eltayef, X Liu, Proceedings of the 11 th International Joint Conference on Biomedical Engineering Systems and Technologies. the 11 th International Joint Conference on Biomedical Engineering Systems and Technologies235B. I. Dodo, Y. Li, K. Eltayef and X. Liu, Proceedings of the 11 th International Joint Conference on Biomedical Engineering Systems and Technologies -Volume 2 BIOIMAGING: BIOIMAGING, 2018, 35. . Y Guo, A Camino, M Zhang, J Wang, D Huang, T Hwang, Y Jia, Biomed. Opt. Express. 94429Y. Guo, A. Camino, M. Zhang, J. Wang, D. Huang, T. Hwang and Y. Jia, Biomed. Opt. Express, 2018, 9, 4429. . A Lang, A Carass, M Hauser, E S Sotirchos, P A Calabresi, H S Ying, J L Prince, Biomed. Opt. Express. A. Lang, A. Carass, M. Hauser, E. S. Sotirchos, P. A. Calabresi, H. S. Ying and J. L. Prince, Biomed. Opt. Express, 2013, 4, 1133. . M Zhang, J Wang, A D Pechauer, T S Hwang, S S Gao, L , M. Zhang, J. Wang, A. D. Pechauer, T. S. Hwang, S. S. Gao, L. . L Liu, S T Liu, D J Bailey, D Wilson, Y Huang, Jia, Biomed. Opt. Express. 64661Liu, L. Liu, S. T. Bailey, D. J. Wilson, D. Huang and Y. Jia, Biomed. Opt. Express, 2015, 6, 4661. L Shannon, J Li, M R Mohammadnia, M V Sarunic, Conference Record -Asilomar Conference on Signals. 483L. Shannon, J. Li, M. R. Mohammadnia, and M. V. Sarunic, Conference Record -Asilomar Conference on Signals, Systems and Computers, 2011, 483. J Li, M V Sarunic, L Shannon, Proceedings -IEEE International Symposium on Field-Programmable Custom Computing Machines, FCCM. -IEEE International Symposium on Field-Programmable Custom Computing Machines, FCCMJ. Li, M. V. Sarunic, and L. Shannon, Proceedings -IEEE International Symposium on Field-Programmable Custom Computing Machines, FCCM 2011, 2011, 49. . T E Ustun, N V Iftimia, R D Ferguson, D X Hammer, Rev. Sci. Instrum. 114301T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, Rev. Sci. Instrum., 2008, 79, 114301. K C Jin, K S Lee, G H Kim, 3rd IEEE International Conference on Computer and Communications. K. C. Jin, K. S. Lee, and G. H. Kim, 2017 3rd IEEE International Conference on Computer and Communications, ICCC 2017, 2018. . V Bandi, J Goette, M Jacomet, T Von Niederhäusern, A H Bachmann, M Duelk, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVII. 85712V. Bandi, J. Goette, M. Jacomet, T. von Niederhäusern, A. H. Bachmann, and M. Duelk, Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVII, 2013, 8571, 85712Z. D Kobori, T Maruyama, Proceedings -22nd International Conference on Field Programmable Logic and Applications, FPL 2012. -22nd International Conference on Field Programmable Logic and Applications, FPL 2012407D. Kobori and T. Maruyama, Proceedings -22nd International Conference on Field Programmable Logic and Applications, FPL 2012, 2012, 407. S Saha, K H Uddin, M S Islam, M Jahiruzzaman, A B M A Hossain, SKIMA 2014 -8th International Conference on. S. Saha, K. H. Uddin, M. S. Islam, M. Jahiruzzaman, and A. B. M. A. Hossain, SKIMA 2014 -8th International Conference on . Knowledge Software, Information Management and Applications. Software, Knowledge, Information Management and Applications, 2014. D J Wahl, M J Ju, Y Jian, M V Sarunic, Optical Coherence Imaging Techniques and Imaging in Scattering Media III. 56D. J. Wahl, M. J. Ju, Y. Jian, and M. V. Sarunic, Optical Coherence Imaging Techniques and Imaging in Scattering Media III, 2019, 56. . D J Wahl, R Ng, J Huang, W Janpongsri, M V Sarunic, Y , D. J. Wahl, R. Ng, J. Huang, W. Janpongsri, M. V. Sarunic, and Y. Jian, Optical Coherence Imaging Techniques and Imaging in Scattering Media III. Jian, Optical Coherence Imaging Techniques and Imaging in Scattering Media III, 2019, 11078, 20. . M J Ju, M Heisler, D Wahl, Y Jian, M V Sarunic, J. Biomed. Opt. 121703M. J. Ju, M. Heisler, D. Wahl, Y. Jian, and M. V. Sarunic, J. Biomed. Opt., 2017, 22, 121703.
[]
[ "Realising Mutated Hilltop Inflation in Supergravity", "Realising Mutated Hilltop Inflation in Supergravity" ]
[ "Tony Pinhero \nPhysics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B. T. Road700108KolkataIndia\n", "Supratik Pal \nPhysics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B. T. Road700108KolkataIndia\n" ]
[ "Physics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B. T. Road700108KolkataIndia", "Physics and Applied Mathematics Unit\nIndian Statistical Institute\n203 B. T. Road700108KolkataIndia" ]
[]
We present N = 1 supergravity models of mutated hilltop inflation (MHI) for both large and small field sectors. Models with canonical kinetic terms are developed based on a shift symmetric Kähler potential in inflaton superfield, and with a superpotential linear in Goldstino superfield. We also construct models with non-canonical kinetic terms for MHI by generalizing the shift symmetry. We found that a good fraction of the models can address the entire branch of MHI in a single framework. *
10.1016/j.physletb.2019.07.042
[ "https://arxiv.org/pdf/1905.04737v2.pdf" ]
152,282,766
1905.04737
8231aca8bc239b76f2d2758fa2caefde35d2a09c
Realising Mutated Hilltop Inflation in Supergravity Tony Pinhero Physics and Applied Mathematics Unit Indian Statistical Institute 203 B. T. Road700108KolkataIndia Supratik Pal Physics and Applied Mathematics Unit Indian Statistical Institute 203 B. T. Road700108KolkataIndia Realising Mutated Hilltop Inflation in Supergravity We present N = 1 supergravity models of mutated hilltop inflation (MHI) for both large and small field sectors. Models with canonical kinetic terms are developed based on a shift symmetric Kähler potential in inflaton superfield, and with a superpotential linear in Goldstino superfield. We also construct models with non-canonical kinetic terms for MHI by generalizing the shift symmetry. We found that a good fraction of the models can address the entire branch of MHI in a single framework. * I. INTRODUCTION Among a plethora of models for cosmological inflation [1], one of the very interesting models that is wellappreciated after the latest Planck 2015 and 2018 data [2,3], is the Mutated Hilltop Inflation (MHI) [4] model. First proposed around a decade back, the salient features of these models have been explored at length at different stages (see, e.g., [1,5,6]). MHI models generically have a potential of the form V = V 0 (1 − sech µφ)(1) for the inflaton field φ with Minkowski minimum at φ = 0. This model belongs to the class of chaotic inflation with super-Planckian inflaton field value φ ≥ 1. Depending on the value of model parameter µ, MHI can occur in two branches: one belongs to large field excursion sector ∆φ ≥ 1 for µ 2.8 and the other one belongs to small field sector ∆φ ≤ 1 for µ 2.8 [5]. Moreover, the model has a subset in the large field sector which is belongs to the class of α-attractors [7][8][9][10][11][12][13][14][15] for the limiting case µφ 1 [5]. Spectral index n s is almost independent of the parameter µ, but with a slight negative running, and the tensor to scalar ratio r can address the value from 10 −4 to 10 −1 depending upon the value of µ. Subsequently, some more interesting features of these models have been studied in [1,6]. These predictions are in good agreement with the latest Planck 2015 and 2018 data [2,3]. However, despite all its successes, a complete description of MHI in the context of supergravity is still unavailable. The aim of the present article is to construct a supergravity model that would lead to the mutated hilltop inflation. The form of the potential mentioned in Eq.(1) can be considered as the functional form of tanh φ and hence it belongs to the class of α-attractors and these α-attractors are well formulated and studied in the context of supergravity and string theory [7][8][9][10][11][12][13][14][15][16][17][18]. In fact, the T-model variant of α-attractor gives rise to a potential of the form Eq.(1) and, for suitable choice of the parameter µ (and hence α), one can realize the MHI model for small-field and large-field within the framework of α-attractor. The goal of this letter is to construct a supergravity framework that can account for MHI, with all noninflaton moduli fields stabilized, and to demonstrate that it can address each and every branch of the model therefrom. One can accomplish this in the context of general inflaton potentials in supergravity for the chaotic inflation [19], since MHI falls under the category of chaotic inflation. In such a scenario one has to choose a Kähler potential which is invariant under the shift of inflaton superfield T and a superpotential which is linear in Goldstino superfield S [20]. More specifically, Kähler potential should be a function of T ± T * and the component (i.e., real or imaginary part of inflaton superfield), which is not appearing in the Kähler potential should be treated as the real inflaton field: T ∓ T * . This is to avoid the usual η-problem in supergravity. Explicit functional form for these super-and Kähler potentials in such a construction reads W = S f (T) K ± = K ± (T ± T * ) 2 , SS * .(2) This Kähler potential is invariant under the following shift transformation: T → T + ic for K + , T → T + c for K − .(3) Next step is to stabilize the Goldstino superfield S at S = 0 during inflation. This will assure the F-term SUGRA potential is positive definite via the vanishing of superpotential. i.e., V = e K D T i WK ij * D T j * W * − 3 |W| 2 S=0 > 0.(4) Further, inflaton partner field T ∓ T * will be stabilized at zero along the inflationary trajectory by attaining the mass greater than Hubble scale. As a result, the final potential will take the form V = | f (T)| 2 .(5) In the following sections we are going to elaborate on this. Specifically, we will show that our construction arXiv:1905.04737v2 [hep-th] 29 Jul 2019 of canonical MHI models in supergravity are based on the above approach. Hence one can conclude that these formulations are the special examples of the general class of supergravity models of chaotic inflation [19]. Moreover, if one can generalize the above shift symmetry Eq.(3), into the form: N ∑ n=1 K (n) T n → N ∑ n=1 K (n) T n + C N(6) and construct a Kähler potential invariant under this shift, one will end up at a non-canonical kinetic term for the model. In the subsection (II A 3) we construct MHI model in supergravity in such a direction also. II. MUTATED HILLTOP INFLATION IN SUPERGRAVITY In this section, we discuss different possibilities of supergravity embedding of MHI. As mentioned above, these models are based on two superfields: one is inflaton superfield T, and another one is Goldstino superfield S. For the standard (canonical) MHI construction we use the Kähler potentials K ± respecting either of the shift symmetry Eq.(3) and representing them explicitly as follows: K ± = ± 1 2 (T ± T * ) 2 + SS * − ζ(SS * ) 2(7) with T ∓ T * as the inflaton. Here the term ζ(SS * ) 2 has been added to the Kähler potential in order to render the mass of the stabilizer field S greater than Hubble scale. So that S will be stabilized during inflation. In the absence of this term, mass of the S field will be light and comparable to the mass of inflaton and it will be added to the inflationary fluctuations. Hence the dynamics of the inflation cannot be regulated with single field. We explicitly discuss two branches of MHI i.e., large and small field inflation separately in the following subsections. A. Branch-I: MHI in Large Field Sector Model-I Let us begin with the following superpotential and Kähler potential W = Λ 2 S e aT/2 − e −aT/2 (e aT + e −aT ) 1/2 , K = K −(8) where a is the real model parameter and it should satisfy the condition a 3.959, in order to materialise large field inflation. Next we represent the complex superfields in terms of the real variables T = 1 √ 2 (φ + iχ) , S = 1 √ 2 (s + iβ)(9) where real part of T is considered as inflaton. Masses of the fields along the inflationary trajectory χ = S = 0 is computed as follows: m 2 χ = 6H 2 1 + a 2 8 sinh −2 aφ 2 √ 2 × 1 + sech aφ √ 2 − sech 2 aφ √ 2 (10) m 2 S = H 2 12ζ + 3a 2 8 sinh −2 aφ 2 √ 2 × sech aφ √ 2 1 + sech aφ √ 2 .(11) Thus during inflation, sinflaton attains T − T * = χ = 0 and Goldstino S stabilizes at S = 0 for 12ζ > 1 and the final potential, along the inflationary trajectory, reads V| S=0,T−T * =0 = Λ 4 1 − sech aφ √ 2(12) This potential is shown in the fig.(1) for the field variables φ and χ. Model-II The second model in this branch can be realised by performing a change of variables T → iT in model-I, so that the imaginary part of the superfield T is assumed as inflaton. As a result superpotential and Kähler potential turn out to be W = iΛ 2 S e iaT/2 − e −iaT/2 e iaT + e −iaT 1/2 , K = K + .(13) Also, here we consider the field variables have an explicit form: T = 1 √ 2 (χ + iφ) , S = 1 √ 2 (s + iβ) .(14) It can be readily found that, in this setup, the inflationary dynamics and the mass matrix for the non-inflaton fields are same as that of the previous model (II A 1). With no apparent change in the above parameters but with the crucial difference of the interpretation of inflation field as above, the final potential during inflation in this model is given by V| S=0,T+T * =0 = Λ 4 1 − sech aφ √ 2(15) As apparent, this potential will have nearly the same behavior as the one in model-I as shown in the Fig.(1). Model-III Once again we start from the model-I and perform the transformation aT → sech −1 T. As a result this will end up with a similar form of Kähler potential and superpotential which appear in T-model supergravity α-attractor setups [12,15]. Under the above mentioned transformation super and Kähler potential takes the form: W = Λ 2 S √ 1 − T,(16)K = − 3α 2 log 2   T * 1 + √ 1 − T 2 T 1 + √ 1 − T * 2   + SS * − ζ(SS * ) 2 ,(17) where we have used 1/a = √ 3α with a 3.96. Under this change of variables the new Kähler potential also preserves the shift symmetry. This is quite obvious from the behavior of vanishing of first term of Kähler potential Eq.(17) during inflation due to the inflaton partner T − T * that attains the zero vev. The Kähler potential Eq. (17) is invariant under the following shift transformation: Corresponding kinetic term and potential for this model are, respectively, as follows: log 1 + √ 1 − T 2 T → log 1 + √ 1 − T 2 T + C(18)1 √ −g L kin = − 3α TT * √ 1 − T 2 √ 1 − T * 2 ∂ µ T∂ µ T * − (1 − 4ζS * S) ∂ µ S∂ µ S * ,(19)V| S=0 = Λ 4 e − 3α 2 log 2 T * ( 1+ √ 1−T 2 ) T ( 1+ √ 1−T * 2 ) √ 1 − T √ 1 − T * (20) Decomposing the superfields into real and imaginary parts T = 1 √ 6α (φ + iχ) , S = 1 √ 2 (s + iβ)(21) we get the following Lagrangian at the inflationary trajectory S = χ = 0: L = −g   R 2 − 3α φ 2 1 − φ 2 6α ∂ µ φ∂ µ φ − Λ 4 1 − φ √ 6α  (22) Further, under the field redefinition into the canonical real variable ψ: φ = √ 6α sech(ψ/ √ 6α), the final Lagrangian reads: L = −g R 2 − 1 2 ∂ µ ψ∂ µ ψ − Λ 4 1 − sech ψ √ 6α .(23) The scalar potential for the model Eq. (17) and Eq.(16) is shown in Fig.(2). One may wonder if the flat direction is visible in these non-canonical variables φ and χ. In order to clarify that, we represent them in a more convenient variableT = (ψ + iθ) √ 6α, which is related to T as T = sechT. Consequently, the kinetic terms become canonical for both fields ψ and θ and the potential will look approximately same as that of model-I as shown in the Fig.(1), thereby reassuring flat directions. Nevertheless, the moduli space associated with this model, i.e., Kähler manifold associated with the Kähler potential Eq. (17) is geometrically flat. This can be demonstrated as follows. The metric of the moduli space based on the Kähler potential Eq.(17) is defined as ds 2 = g TT * dTdT * (24) where g TT * = K TT * = − 3α TT * √ 1 − T 2 √ 1 − T * 2 .(25) From this Kähler metric non-vanishing Levi-Civita connection coefficients, Riemannian tensors, and the curvature of the moduli space are computed as follows: Γ T TT = − 1 − 2T 2 T − T 3 , Γ T * T * T * = − 1 − 2T * 2 T * − T * 3 (26) R T TT * T = ∂ T * Γ T TT = 0. (27) R Kähler = 0 (28) Alternatively, from the definition of curvature of Kähler manifold via the metric one can straightaway show that R Kähler = −g −1 ΦΦ * ∂ Φ ∂ Φ * log g ΦΦ * = 0.(29) From the above, one can conclude that geometry associated with our Kähler manifold is indeed flat. Model-IV Till now, we were discussing about MHI models in supergravity based on a single parameter. A two parameter realisation of the scenario in supergravity is also possible for MHI model. In such a scenario one can consider a combination of superpotential and Kähler potential as follows: W = Λ 2 S e −aT/2 − e −bT/2 e −aT + e −bT 1/2 , K = K −(30) This will end up with a potential of the form for the real part of the inflaton superfield T: V| S=0,T−T * =0 = Λ 4 1 − sech (a − b)φ 2 √ 2(31) for the variables defined in Eq. (9). Thus, although we started with two parameters in supergravity, the form of the potential boils down to such a form so as to be represented by a single parameter, namely, (a − b). As a result, in this model the parameters a and b can take any value so far as the constraint −7.191 (a − b) 7.191 is satisfied in order to guarantee large field excursion. The mass matrix for the non-inflaton fields along the inflationary trajectory are computed as follows: m 2 χ = 6H 2 1 + (a − b) 2 16 sinh −2 (a − b)φ 4 √ 2 +2 sech 2 (a − b)φ 2 √ 2 (32) m 2 S = H 2 12ζ + 3(a − b) 2 32 sinh −2 (a − b)φ 4 √ 2 × sech (a − b)φ 2 √ 2 1 + sech (a − b)φ 2 √ 2 . (33) The same will also work for the field replacement of all scalars of the type Z → iZ in Eq.(30) (as in model-(II A 2)). In that case the super-and Kähler potentials take the form: W = iΛ 2 S e −iaT/2 − e −ibT/2 e −iaT + e −ibT 1/2 , K = K +(34) where the inflaton field will be the imaginary part of the superfield T and it will end up with the potential Eq.(31) for the variables Eq. (14). B. Branch-II: MHI in Small Field Sector Model-I The first kind of models in Small Field Sector is rather trivial. It is a straightforward exercise to show that all the models presented in section-(II A) in the context of large field inflation can also govern small field excursion if the value of the model parameter(s) a 3.959 or (a − b) 7.191. In this scenario all noninflaton moduli fields are stabilized in the inflationary trajectory with a slightly different mass. This is quite evident from the expressions of mass matrix of the fields Eq.(10) and Eq. (11). There is a small change in the shape of the potential along χ direction at small φ region. We have demonstrated the results in Fig.(3) that reflects distinctive features of small field models. Model-II Until now, we have supersymmetric models which can simultaneously explain both large field and small field MHI. In what follows we will construct an exclusive model for small field MHI. For this let us consider superand Kähler potentials as: W = Λ 2 Se 3 2 α log 2 T 1+ √ 1−T 2 √ 1 + T,(35)K = −3α log T 1 + √ 1 − T 2 2 + SS * − ζ(SS * ) 2 . (36) Where α 0.0212 to satisfy the condition of small field inflation. The associated kinetic and potential terms for this model are given by 1 √ −g L kin = 3α TT * √ 1 − T 2 √ 1 − T * 2 ∂ µ T∂ µ T * − (1 − 4ζS * S) ∂ µ S∂ µ S * , (37) V = Λ 4 e 3 2 α log 2 T 1+ √ 1−T 2 +log 2 T * 1+ √ 1−T * 2 e −3α log T 1+ √ 1−T 2 2 × √ 1 − T √ 1 − T * (38) where the potential is calculated at S = 0. Decomposing these superfields into real and imaginary parts T = 1 √ 6α (φ + iχ) , S = 1 √ 2 (s + iβ)(39) the total Lagrangian at the inflationary trajectory S = χ = 0 takes the form Consequently, under the following field redefinition in the canonical real variable ψ: φ = √ 6α sech(ψ/ √ 6α), the final Lagrangian reads: L = −g   R 2 + 3α φ 2 1 − φ 2 6α ∂ µ φ∂ µ φ − Λ 4 1 − φ √ 6α  (40)L = −g R 2 − 1 2 ∂ µ ψ∂ µ ψ − Λ 4 1 − sech ψ √ 6α . (41) Which represents the mutated hilltop inflation (MHI). Scalar potential for this theory is shown in the fig (4) in terms of the more adequate variablesT = (ψ + iθ) √ 6α, which is related to T as T = sechT. The canonical masses for all those non-inflaton stabilized fields are computed in terms of the variableT along the inflationary trajectory as follows: m 2 θ = 6H 2 1 − 1 24α sinh ψ 2 √ 6α −2 × 1 + sech ψ √ 6α − sech 2 ψ √ 6α (42) m 2 S = H 2 12ζ − 1 4α sinh −2 ψ √ 6α × sech ψ √ 6α 1 + sech ψ √ 6α .(43) III. CONCLUDING REMARKS In this letter, we have presented various supergravity embedding of mutated hilltop inflation (MHI) model for both large and small field branches with stable Minkowski vacuum. Models constructed with canonical kinetic terms are the special examples of general class of models describing chaotic inflation in supergravity [19]. These canonical models are developed based on a shift symmetric Kähler potential in inflaton superfield, and with a superpotential linear in Goldstino superfield. By generalizing this shift symmetry, we also constructed models for MHI with non-canonical kinetic terms. We also found that many of the models described in the letter can address the entire branch (large field and small field sectors) of MHI in a single framework. ACKNOWLEDGMENTS TP would like to thank Barun Kumar Pal and Abhishek Naskar for the enlightening discussions. T.P gratefully acknowledge the support from Senior Research fellowship (Order No. DS/18-19/0616) of the Indian Statistical Institute (ISI), Kolkata. Figure 1 . 1Scalar potential for the theory Eq.(8), for the variables Eq.(9) with a = 1. Potential has a de Sitter valley of constant depth and width for large values of φ and has nearly Minkowski minimum at small values of φ. Figure 2 . 2Scalar potential for the theory Eq.(16) and Eq.(17), for the variables Eq.(21). Although the existence of flat direction in the potential is not obvious in this set of variables, there indeed is a flat direction as demonstrated in Sec.II A 3. Figure 3 . 3Scalar potential for the theory Eq.(8), for the variables Eq.(9) with a = 4.242. This potential also has a de Sitter valley of constant depth and width for large values of φ and has nearly Minkowski minimum at small values of φ. Figure 4 . 4Scalar potential for the theory Eq.(35) and Eq.(36) for α = 0.008. . J Martin, C Ringeval, V Vennin, Encyclopaedia Inflationaris, arXiv:1303.3787Phys.Dark Univ. astroph.COJ. Martin, C. Ringeval, V. Vennin, Encyclopaedia Inflationaris, Phys.Dark Univ. 5-6 (2014) 75-235, [arXiv:1303.3787 [astro- ph.CO]] P A R Ade, Planck CollaborationarXiv:1502.02114Planck 2015 results. XX. Constraints on inflation. 594astro-ph.COPlanck Collaboration, P. A. R. Ade et al., Planck 2015 results. XX. Constraints on inflation, Astron.Astrophys. 594 (2016) A20, [arXiv:1502.02114 [astro-ph.CO]] Y Akrami, Planck CollaborationarXiv:1807.06211Planck 2018 results. X. Constraints on inflation. astroph.COPlanck Collaboration, Y. Akrami et al., Planck 2018 res- ults. X. Constraints on inflation, [ arXiv:1807.06211 [astro- ph.CO]] Mutated Hilltop Inflation : A Natural Choice for Early Universe. B K Pal, S Pal, B Basu, https:/iopscience.iop.org/article/10.1088/1475-7516/2010/01/029/metaarXiv:0908.2302JCAP. 100129hep-thB. K. Pal, S. Pal, B. Basu, Mutated Hilltop Inflation : A Natural Choice for Early Universe, JCAP 1001 (2010) 029, [arXiv:0908.2302 [hep-th]] Mutated Hilltop Inflation Revisited. B K , arXiv:1711.00833Eur.Phys.J. C78. 3585gr-qcB. K. Pal, Mutated Hilltop Inflation Revisited, Eur.Phys.J. C78 (2018) no.5, 358, [arXiv:1711.00833 [gr-qc]] A semi-analytical approach to perturbations in mutated hilltop inflation. B K Pal, S Pal, B Basu, https:/www.worldscientific.com/doi/abs/10.1142/S0218271812500174arXiv:1010.5924Int.J.Mod.Phys. 21astro-ph.COB. K. Pal, S. Pal, B. Basu,A semi-analytical approach to per- turbations in mutated hilltop inflation, Int.J.Mod.Phys. D21 (2012) 1250017, [arXiv:1010.5924 [astro-ph.CO]] Universality Class in Conformal Inflation. R Kallosh, A Linde, http:/iopscience.iop.org/article/10.1088/1475-7516/2013/07/002/meta;jsessionid=0C76449ABC7D9A14139309271C43F61E.c2.iopscience.cld.iop.orgarXiv:1306.5220JCAP. 13072hep-thR. Kallosh and A. Linde, Universality Class in Conformal Inflation, JCAP 1307 (2013) 002, [arXiv:1306.5220 [hep-th]] Superconformal Inflationary α-Attractors. R Kallosh, A Linde, D Roest, arXiv:1311.0472JHEP. 1311hepthR. Kallosh, A. Linde, D. Roest, Superconformal Inflationary α-Attractors, JHEP 1311 (2013) 198, [arXiv:1311.0472 [hep- th]] The Unity of Cosmological Attractors. M Galante, R Kallosh, A Linde, D Roest, https:/journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.141302arXiv:1412.3797Phys.Rev.Lett. 11414hep-thM. Galante, R. Kallosh, A. Linde, D. Roest, The Unity of Cosmological Attractors, Phys.Rev.Lett. 114 (2015) no.14, 141302, [arXiv:1412.3797 [hep-th]] Escher in the Sky. R Kallosh, A Linde, arXiv:1503.06785Comptes Rendus Physique. 16hepthR. Kallosh and A. Linde, Escher in the Sky, Comptes Ren- dus Physique 16 (2015) 914-927, [arXiv:1503.06785 [hep- th]] Cosmological Attractors from α-Scale Supergravity. D Roest, M Scalisi, https:/journals.aps.org/prd/abstract/10.1103/PhysRevD.92.043525arXiv:1503.07909Phys.Rev. 9243525hep-thD.Roest, M. Scalisi, Cosmological Attractors from α-Scale Su- pergravity, Phys.Rev. D92 (2015) 043525, [arXiv:1503.07909 [hep-th]] The Hyperbolic Geometry of Cosmological Attractors. J J M Carrasco, R Kallosh, A Linde, D Roest, https:/journals.aps.org/prd/abstract/10.1103/PhysRevD.92.041301arXiv:1504.05557Phys.Rev. 924hep-thJ. J. M. Carrasco, R. Kallosh, A. Linde, D. Roest, The Hyperbolic Geometry of Cosmological Attractors, Phys.Rev. D92 (2015) no.4, 041301, [arXiv:1504.05557 [hep-th]] Cosmological Attractors and Initial Conditions for Inflation. J J M Carrasco, R Kallosh, A Linde, https:/journals.aps.org/prd/abstract/10.1103/PhysRevD.92.063519arXiv:1506.00936Phys.Rev. 926hep-thJ. J. M. Carrasco, R. Kallosh, A. Linde, Cosmological At- tractors and Initial Conditions for Inflation, Phys.Rev. D92 (2015) no.6, 063519, [arXiv:1506.00936 [hep-th]] Single-field α-attractors. A Linde, http:/iopscience.iop.org/article/10.1088/1475-7516/2015/05/003/metaarXiv:1504.00663JCAP. 15053hep-thA. Linde Single-field α-attractors, JCAP 1505 (2015) 003 [arXiv:1504.00663 [hep-th]] T Pinhero, arXiv:1812.05406Natural α-Attractors from N = 1 Supergravity via flat Kähler Manifolds. hep-thT. Pinhero, Natural α-Attractors from N = 1 Supergravity via flat Kähler Manifolds, [arXiv:1812.05406 [hep-th]] Seven-Disk Manifold, alpha-attractors and B-modes. Sergio Ferrara, Renata Kallosh, https:/journals.aps.org/prd/abstract/10.1103/PhysRevD.94.126015arXiv:1610.04163Phys.Rev. 9412hep-thSergio Ferrara and Renata Kallosh, Seven-Disk Manifold, alpha-attractors and B-modes, Phys.Rev. D94 (2016) no.12, 126015, [arXiv:1610.04163 [hep-th]] Maximal Supersymmetry and B-Mode Targets. R Kallosh, A Linde, T Wrase, Y Yamada, arXiv:1704.04829JHEP. 1704hep-thR. Kallosh, A. Linde, T. Wrase, Y. Yamada, Maximal Su- persymmetry and B-Mode Targets, JHEP 1704 (2017) 144, [arXiv:1704.04829 [hep-th]] M Dias, J Frazer, A Retolaza, M Scalisi, A Westphal, arXiv:1805.02659Pole N-flation. hep-thM. Dias, J. Frazer, A. Retolaza, M. Scalisi, A. Westphal, Pole N-flation, [arXiv:1805.02659 [hep-th]] General inflaton potentials in supergravity. R Kallosh, A Linde, T Rube, http:/journals.aps.org/prd/abstract/10.1103/PhysRevD.83.043507arXiv:1011.5945Phys.Rev. 8343507hep-thR. Kallosh, A. Linde, T. Rube, General inflaton potentials in supergravity, Phys.Rev.D83:043507,2011, [arXiv:1011.5945 [hep-th]] Natural Chaotic Inflation in Supergravity. M Kawasaki, M Yamaguchi, T Yanagida, http:/journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.3572arXiv:hep-ph/0004243Phys.Rev.Lett. 85M. Kawasaki, M. Yamaguchi, T. Yanagida, Natural Chaotic Inflation in Supergravity, Phys.Rev.Lett. 85 (2000) 3572-3575, [arXiv:hep-ph/0004243]
[]
[ "The QUESO Library User's Manual Quantification of Uncertainty for Estimation, Simulation, and Optimization (QUESO)", "The QUESO Library User's Manual Quantification of Uncertainty for Estimation, Simulation, and Optimization (QUESO)" ]
[ "Kemelli C Estacio-Hiroms \nCenter for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA\n", "Ernesto E Prudencio \nCenter for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA\n", "Nicholas P Malaya \nCenter for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA\n", "Manav Vohra \nCenter for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA\n", "Damon Mcdougall \nCenter for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA\n" ]
[ "Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA", "Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA", "Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA", "Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA", "Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES)\nThe University of Texas at Austin Austin\n78712TXUSA" ]
[]
QUESO stands for Quantification of Uncertainty for Estimation, Simulation and Optimization and consists of a collection of algorithms and C++ classes intended for research in uncertainty quantification, including the solution of statistical inverse and statistical forward problems, the validation of mathematical models under uncertainty, and the prediction of quantities of interest from such models along with the quantification of their uncertainties.QUESO is designed for flexibility, portability, easy of use and easy of extension. Its software design follows an object-oriented approach and its code is written on C++ and over MPI. It can run over uniprocessor or multiprocessor environments.QUESO contains two forms of documentation: a user's manual available in PDF format and a lower-level code documentation available in web based/HTML format. This is the user's manual: it gives an overview of the QUESO capabilities, provides procedures for software execution, and includes example studies.iv v Disclaimer
null
[ "https://arxiv.org/pdf/1611.07521v1.pdf" ]
88,520,688
1611.07521
84b844bfedbfa21868121dc53376c37e830c7802
The QUESO Library User's Manual Quantification of Uncertainty for Estimation, Simulation, and Optimization (QUESO) 22 Nov 2016 Kemelli C Estacio-Hiroms Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES) The University of Texas at Austin Austin 78712TXUSA Ernesto E Prudencio Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES) The University of Texas at Austin Austin 78712TXUSA Nicholas P Malaya Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES) The University of Texas at Austin Austin 78712TXUSA Manav Vohra Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES) The University of Texas at Austin Austin 78712TXUSA Damon Mcdougall Center for Predictive Engineering and Computational Sciences (PECOS) Institute for Computational and Engineering Sciences (ICES) The University of Texas at Austin Austin 78712TXUSA The QUESO Library User's Manual Quantification of Uncertainty for Estimation, Simulation, and Optimization (QUESO) 22 Nov 2016Version 0.51.0 http://pecos.ices.utexas.edu Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with the Invariant Sections being "GNU General Public License" and "Free Software Needs Free Documentation", the Front-Cover text being "A GNU Manual", and with the Back-Cover text being "You have the freedom to copy and modify this GNU Manual". A copy of the license is included in the section entitled "GNU Free Documentation License". iii This document was prepared by The University of Texas at Austin. Neither the University of Texas at Austin, nor any of its institutes, departments and employees, make any warranty, ex-press or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represent 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 Univer-sity of Texas at Austin or any of its institutes, departments and employees thereof. The views and opinions expressed herein do not necessarily state or reflect those of The University of Texas at Austin or any institute or department thereof. QUESO library as well as this material are provided as is, with absolutely no warranty expressed or implied. Any use is at your own risk. vi QUESO stands for Quantification of Uncertainty for Estimation, Simulation and Optimization and consists of a collection of algorithms and C++ classes intended for research in uncertainty quantification, including the solution of statistical inverse and statistical forward problems, the validation of mathematical models under uncertainty, and the prediction of quantities of interest from such models along with the quantification of their uncertainties.QUESO is designed for flexibility, portability, easy of use and easy of extension. Its software design follows an object-oriented approach and its code is written on C++ and over MPI. It can run over uniprocessor or multiprocessor environments.QUESO contains two forms of documentation: a user's manual available in PDF format and a lower-level code documentation available in web based/HTML format. This is the user's manual: it gives an overview of the QUESO capabilities, provides procedures for software execution, and includes example studies.iv v Disclaimer Referencing the QUESO Library When referencing the QUESO library in a publication, please cite the following: Introduction QUESO is a parallel object-oriented statistical library dedicated to the research of statistically robust, scalable, load balanced, and fault-tolerant mathematical algorithms for the quantification of uncertainty (UQ) of mathematical models and their predictions. The purpose of this chapter is to introduce relevant terminology, mathematical and statistical concepts, statistical algorithms, together with an overall description of how the user's application may be linked with the QUESO library. Preliminaries Statistical inverse theory reformulates inverse problems as problems of statistical inference by means of Bayesian statistics: all quantities are modeled as random variables, and probability distribution of the quantities encapsulates the uncertainty observed in their values. The solution to the inverse problem is then the probability distribution of the quantity of interest when all information available has been incorporated in the model. This (posterior) distribution describes the degree of confidence about the quantity after the measurement has been performed [33]. Thus, the solution to the statistical inverse problem may be given by Bayes' formula, which express the posterior distribution as a function of the prior distribution and the data represented through the likelihood function. The likelihood function has an open form and its evaluation is highly computationally expensive. Moreover, simulation-based posterior inference requires a large number of forward calculations to be performed, therefore fast and efficient sampling techniques are required for posterior inference. It is often not straightforward to obtain explicit posterior point estimates of the solution, since it usually involves the evaluation of a high-dimensional integral with respect to a possibly non-smooth posterior distribution. In such cases, an alternative integration technique is the Markov chain Monte Carlo method: posterior means may be estimated using the sample mean from a series of random draws from the posterior distribution. QUESO is designed in an abstract way so that it can be used by any computational model, as long as a likelihood function (in the case of statistical inverse problems) and a quantity of interest (QoI) function (in the case of statistical forward problems) is provided by the user application. QUESO provides tools for both sampling algorithms for statistical inverse problems, following Bayes' formula, and statistical forward problems. It contains Monte Carlo solvers (for autocorrelation, kernel density estimation and accuracy assessment), MCMC (e.g. Metropolis Hastings [39,23]) as well as the DRAM [21] (for sampling from probability distributions); it also has the capacity to handle many chains or sequences in parallel, each chain or sequence itself demanding many computing nodes because of the computational model being statistically explored [41]. Key Statistical Concepts A computational model is a combination of a mathematical model and a discretization that enables the approximate solution of the mathematical model using computer algorithms and might be used in two different types of problems: forward or inverse. Any computational model is composed of a vector θ of n parameters, state variables u, and state equations r(θ, u) = 0. Once the solution u is available, the computational model also includes extra functions for e.g. the calculation of model output data y = y(θ, u), and the prediction of a vector q = q(θ, u) of m quantities of interest (QoI), Parameters designate all model variables that are neither state variables nor further quantities computed by the model, such as: material properties, coefficients, constitutive parameters, boundary conditions, initial conditions, external forces, parameters for modeling the model error, characteristics of an experimental apparatus (collection of devices and procedures), discretization choices and numerical algorithm options. In the case of a forward problem, the parameters θ are given and one then needs to compute u, y and/or q. In the case of an inverse problem, however, experimental data d is given and one then needs to estimate the values of the parameters θ that cause y to best fit d. The process of parameter estimation is also referred to as model calibration or model update, and it usually precedes the computation of a QoI, a process called model prediction. Computational models can be classified as either deterministic or stochastic -which are the ones of interest here. In deterministic models, all parameters are assigned numbers, and no parameter is related to the parametrization of a random variable (RV) or field. As a consequence, a deterministic model assigns a number to each of the components of quantities u, y and q. In stochastic models, however, at least one parameter is assigned a probability density function (PDF) or is related to the parametrization of a RV or field, causing u, y and q to become random variables. Note that not all components of θ need to be treated as random. As long as at least one component is random, θ is a random vector, and the problem is stochastic. In the case of forward problems, statistical forward problems can be represented very similarly to deterministic forward problems, as seen in Figure 1.2.2. In the case of inverse problems, as depicted in Figure 1.2.3, however, the conceptual connection between deterministic and statistical problems is not as straightforward. QUESO adopts a Bayesian analysis [33,42] for statistical inverse problems, interpreting the posterior PDF π posterior (θ|d) = π prior (θ)π likelihood (d|θ) π(d) (1.2.1) as the solution. Such solutions combine the prior information π prior (θ) of the parameters, the information π(d) on the data, and the likelihood π likelihood (d|θ) that the model computes certain data values with a given set of input parameters. This semantic interpretation of achieving a posterior knowledge on the parameters (on the model) after combining some prior model knowledge with experimental information provides an important mechanism for dealing with uncertainty. Although mathematically simple, is not computationally trivial. The Software Stack of an Application Using QUESO An application using QUESO falls into three categories: a statistical inverse problem (IP), a statistical forward problem (FP), or combinations of both. In each problem the user might deal with up to five vectors of potentially very different sizes: parameters θ, state u, output y, data d and QoIs q. Algorithms in the QUESO library require the supply of a likelihood routine π like : R n → R + for statistical inverse problems and of a QoI routine q : R n → R m for statistical forward problems. These routines exist at the application level and provide the necessary bridge between the statistical algorithms in QUESO, model knowledge in the model library and scenario and experimental data in the disk space. Figure 1.3.1 shows the software stack of a typical application that uses QUESO. In the figure, the symbol θ represents a vector of n 1 parameters. Even though QUESO deals directly with θ and q only, it is usually the case the one of the other three vectors (u, y and d) will have the biggest number of components and will therefore dictate the size of the minimum parallel environment to be used in a problem. So, for example, even though one processor might be sufficient for handling θ, y, d and q, eight processors at least might be necessary to solve for u. QUESO currently only requires that the amounts n and m can be handled by the memory available to one processor, which allows the analysis of problems with thousands of parameters and QoIs, a large amount even for state of the art UQ algorithms. QUESO currently supports three modes of parallel execution: an application user may simultaneously run: An application software stack. QUESO requires the input of a likelihood routine π like : R n → R + for IPs and of a QoI routine q : R n → R m for FPs. These application level routines provide the bridge between the statistical algorithms in QUESO, physics knowledge in the model library, and relevant experimental (calibration and validation) data. For example, suppose an user wants to use the Metropolis-Hastings (MH) algorithm to solve a statistical IP, and that 1,024 processors are available. If the physical model is simple enough to be handled efficiently by a single processor, then the user can run 1,024 chains simultaneously, as in case (a). If the model is more complex and requires, say, 16 processors, then the user can run 64 chains simultaneously, as in case (b), with 16 processors per chain. QUESO treats this situation by using only 1 of the 16 processors to handle the chain. When a likelihood evaluation is required, all 16 processors call the likelihood routine simultaneously. Once the likelihood returns its value, QUESO puts 15 processors into idle state until the routine is called again or the chain completes. Case (c) is useful, for instance, in the case of a computational procedure involving two models, where a group of processors can be split into two groups, each handling one model. Once the two-model analysis end, the combined model can use the full set of processors. 1 Algorithms for solving Statistical Inverse Problems The goal of inference is to characterize the posterior PDF, or to evaluate point or interval estimates based on the posterior [27]. Samples from posterior can be obtained using Markov chain Monte Carlo (MCMC) which require only pointwise evaluations of the unnormalized posterior. The resulting samples can then be used to either visually present the posterior or its marginals, or to construct sample estimates of posterior expectations. Examples of MCMC are: the Metropolis-Hastings (MH) algorithm [39,23], the Delayed Rejection (DR) algorithm [17,40], and Adaptive Metropolis (AM) [22] which are combined together in the Delayed Rejection Adaptive Metropolis, DRAM, algorithm [21]. The DRAM is implemented in QUESO and available for the solution of SIP. MCMC methods are well-established and documented [4,17,21,22,23,33,36,39,40]; thus only brief description of the DRAM algorithm is presented in Section 1.4.1. During model construction, errors arising from imperfect modeling and uncertainties due to incomplete information about the system and its environment always exist; thus, there has been a crescent interest in Bayesian model class updating and selection [9,7,8]. Model updating refers to the methodology that determines the most plausible model for a system, given a prior PDF. One stochastic method that handles model updating successfully is the multilevel method. Throughout the years, sereveral versions of the same method have been implemented as improvements of its predecessors [3,9,8]. QUESO hosts the novel Adaptive Multilevel Stochastic Simulation Algorithm (AMSSA) [8], which is described in Section 1. 4 .2. For details about the method, please refer to [8]. DRAM Algorithm DRAM is a combination of two ideas for improving the efficiency of Metropolis-Hastings type Markov chain Monte Carlo (MCMC) algorithms, Delayed Rejection and Adaptive Metropolis [34]. Random walk Metropolis-Hasting algorithm with Gaussian proposal distribution is useful in simulating from the posterior distribution in many Bayesian data analysis situations. In order for the chain to be efficient, the proposal covariance must somehow be tuned to the shape and size of the target distribution. This is important in highly nonlinear situations, when there are correlation between the components of the posterior, or when the dimension of the parameter is high. The problem of adapting the proposal distribution using the chain simulated so far is that when the accepted values depend on the history of the chain, it is no longer Markovian and standard convergence results do not apply. One solution is to use adaptation only for the burn-in period and discard the part of the chain where adaptation has been used. In that respect, the adaptation can be thought as automatic burn-in. The idea of diminishing adaptation is that when adaptation works well, its effect gets smaller and we might be able to prove the ergodicity properties of the chain even when adaptation is used throughout the whole simulation. This is the ideology behind AM adaptation. On the other hand, the DR method allows the use of the the current rejected values without losing the Markovian property and thus allows to adapt locally to the current location of the target distribution. In Adaptive Metropolis [22] the covariance matrix of the Gaussian proposal distribution is adapted on the fly using the past chain. This adaptation destroys the Markovian property of the chain, however, it can be shown that the ergodicity properties of the generated sample remain. How well this works on finite samples and on high dimension is not obvious and must be verified by simulations. Starting from initial covariance C (0) , the target covariance is updated at given intervals from the chain generated so far. C (i) = s d cov(chain 1 : chain i ) + s d εI d , the small number ε prevents the sample covariance matrix from becoming singular. For the scaling factor, the value s d = 2.4 2 /d is standard optimal choice for Gaussian targets, d being the dimension of the target [15]. A standard updating formula for the sample covariance matrix can be used, so that the whole chain does not need to reside in the computer memory. With the Delayed rejection method [40], it becomes possible to make use of several tries after rejecting a value by using different proposals while keep the reversibility of the chain. Delayed rejection method (DR) works in the following way. Upon rejection a proposed candidate point, instead of advancing time and retaining the same position, a second stage move is proposed. The acceptance probability of the second stage candidate is computed so that reversibility of the Markov chain relative to the distribution of interest is preserved. The process of delaying rejection can be iterated for a fixed or random number of stages, let's say n stages . The higher stage proposals are allowed to depend on the candidates so far proposed and rejected. Thus DR allows partial local adaptation of the proposal within each time step of the Markov chain still retaining the Markovian property and reversibility. The first stage acceptance probability in DR is the standard MH acceptance and it can be written as α 1 (a, x (1) ) = min 1, π(x (1) ) π(a) · q 1 (x (1) , a) q 1 (a, x (1) ) , Here a is the current point, x (1) is the proposed new value drawn from the distribution q 1 (a, ·), and π is the target distribution. If x (1) is rejected, a second candidate x (2) is drawn from q 2 (a, x (1) , ·) using the acceptance probability α 2 (a, x (1) , x (2) ) = min 1, π(x (2) )q 1 (x (2) , x (1) )q 2 (x (2) , x (1) , a)[1 − α 1 (x (2) , x (1) )] π(a)q 1 (a, x (1) )q 2 (a, x (1) , x (2) )[1 − α 1 (a, x (1) )] i.e., it depends not only on the current position of the chain but also on what we have just proposed and rejected. As the reversibility property is preserved, this method also leads to the same stationary distribution π as the standard MH algorithm. The procedure can be iterated further for higher-stage proposals. The Gaussian proposal at each stage i is defined as: q i (a, x (1) , . . . , x (i−1) i terms , z) = e − 1 2 [z − a] T · [C] −1 · [z − a] (1.4.1) where the covariance matrix C and the scalings for the higher-stage proposal covariances 1 = γ 1 γ 2 . . . γ nstages are given. If q i denotes the proposal at the i-th stage, the acceptance probability at that stage is: (1) , . . . , x (i) ) = min 1, π(x (i) ) π(a) · q fraction · α fraction . (1.4.2) where the expressions q fraction and α fraction are given by (1) , x (2) ) . . . q i (x (i) , x (i−1) , . . . , x (1) , a) q i (a, x (1) , . . . , x (i−1) , x (i) ) and (1) , . . . , x (i−1) )] . α i (a, xq fraction = q 1 (x (i) , x (i−1) ) q 1 (a, x (1) ) q 2 (x (i) , x (i−1) , x (i−2) ) q 2 (a, xα fraction = [1 − α 1 (x (i) , x (i−1) )] [1 − α 1 (a, x (1) )] [1 − α 2 (x (i) , x (i−1) , x (i−2) )] [1 − α 2 (a, x (1) , x (2) )] . . . [1 − α i−1 (x (i) , x (i−1) , . . . , x (1) )] [1 − α i−1 (a, x Since all acceptance probabilities are computed so that reversibility with respect to π is preserved separately at each stage, the process of delaying rejection can be interrupted at any stage that is, we can, in advance, decide to try at most, say, 3 times to move away from the current position, otherwise we let the chain stay where it is. Alternatively, upon each rejection, we can toss a p-coin (i.e., a coin with head probability equal to p), and if the outcome is head we move to a higher stage proposal, otherwise we stay put [21]. The smaller overall rejection rate of DR guarantees smaller asymptotic variance of the estimates based on the chain. The DR chain can be shown to be asymptotically more efficient that MH chain in the sense of Peskun ordering (Mira, 2001a). Haario, et al. 2006 [21] combine AM and DR into a method called DRAM, in what they claim to be a straightforward possibility amongst the possible different implementations of the idea, and which is described in this section. In order to be able to adapt the proposal, all you need some accepted points to start with. One "master" proposal is tried first -i.e., the proposal at the first stage of DR is adapted just as in AM: the covariance C (1) is computed from the points of the sampled chain, no matter at which stage these points have been accepted in the sample path. After rejection, a try with modified version of the first proposal is done according to DR. A second proposal can be one with a smaller covariance, or with different orientation of the principal axes. The most common choice is to always compute the covariance C (i) of the proposal for the i-th stage (i = 2, . . . , n stages ) simply as a scaled version of the proposal of the first stage, (1) where the scale factors γ i can be somewhat freely chosen. Then, the master proposal is adapted using the chain generated so far, and the second stage proposal follows the adaptation in obvious manner. C (i) = γ i C The requirements for the DRAM algorithm are: • Number n pos 2 of positions in the chain; • Initial guess m (0) ; • Number of stages for the DR method: n stages 1; • For 1 i n stages , functions q i : R N × . . . × R N (i+1) times → R + , such that q i (a, x (1) , . . . , x (i−1) , ·) is a PDF for any (a, x (1) , . . . , x (i−1) ) ∈ R N × . . . Recalling that a sample is defined as: a sample = a + C 1/2 N (0, I). a simple, but useful, implementation of DRAM is described in Algorithm 1. There are six variables in the QUESO input file used to set available options for the DRAM algorithm, which are described in 3.3.4. Here, they are presented presented bellow together with their respective definition in Algorithm 1. ip mh dr maxNumExtraStages: defines how many extra stages should be considered in the DR loop (n stages ); ip mh dr listOfScalesForExtraStages: defines the list s of scaling factors that will multiply the covariance matrix (values of γ i ); ip mh am adaptInterval: defines whether or not there will be a period of adaptation; ip mh am initialNonAdaptInterval: defines the initial interval where the proposal covariance matrix will not be changed (n 0 ); ip mh am eta: is a factor used to scale the proposal covariance matrix, usually set to be 2.4 2 /d, where d is the dimension of the problem [36,21] (s d ); ip mh am epsilon: is the covariance regularization factor (ε). Adaptive Multilevel Stochastic Simulation Algorithm In this section we rewrite the Bayesian formula (1.2.1) by making explicit all the implicit model assumptions. Such explication demands the use of probability logic and the concept of a stochastic system model class (model class for short); as these concepts enable the comparison of competing model classes. Let M j be one model class; the choice of θ specifies a particular predictive model in M j , and, for brevity, we do not explicitly write θ j to indicate that the parameters of different model classes may be different, but this should be understood. Based on M j , one can use Algorithm 1 DRAM algorithm [36]. Input: Number of positions in the chain n pos 2; initial guess m (0) ; initial first stage proposal covariance C (0) ; n stages 1; and functions q i : R N × . . . × R N (i+1) times → R + 1: Select s d scaling factor 2: Select ε covariance regularization factor 3: Select n 0 initial non-adaptation period 4: for i ← 1 to n stages do n stages is the number of tries allowed 5: Select γ i scalings for the higher-stage proposal covariances 6: end for 7: do 8: Set ACCEP T ← f alse Set i ← 1 After an initial period of simulation, adapt the master proposal (target) covariance using the chain generated so far: 9: if k n 0 then 10: C (1) = s d Cov(m (0) , . . . , m (k−1) ) + s d εI d 11: end if n stages -DR loop: 12: do 13: Generate candidate c (i) ∈ R N by sampling q i (m (k) , c (1) , . . . , c (i−1) , ·) q i is the proposal probability density 14: if c (i) / ∈ supp(π) then 15: i ← i + 1 16: end if 17: if c (i) ∈ supp(π) then 18: Compute α i (m (k) , c (1) , . . . , c (i−1) , c (i) ) acceptance probability 19: Generate a sample τ ∼ U ((0, 1]) 20: if (α i < τ ) then i ← i + 1 21: if (α i τ ) then ACCEPT←true 22: end if 23: C (i) = γ i C (1) Calculate the higher-stage proposal as scaled versions of C (1) , according to the chosen rule 24: while (ACCEPT=false) and (i n stages ) 25: if (ACCEPT=true) then 26: Set m (k+1) ← c (i) 27: end if 28: if (ACCEPT=false) then 29: Set m (k+1) ← m (k) 30: end if 31: Set k ← k + 1 32: while (k + 1 < n pos ) data D to compute the updated relative plausibility of each predictive model in the set defined by M j . This relative plausibility is quantified by the posterior PDF π(θ|D, M j ). Bayes theorem allows the update of the probability of each predictive model M j by combining measured data D with the prior PDF into the posterior PDF: π posterior (θ|D, M j ) = f (D|θ, M j ) · π prior (θ|M j ) π(D, M j ) = f (D|θ, M j ) · π prior (θ|M j ) f (D|θ, M j ) · π prior (θ|M j ) dθ (1.4.3) where the denominator expresses the probability of getting the data D based on M j and is called the evidence for M j provided by D; π prior (θ|M j ) is the prior PDF of the predictive model θ within M j ; and the likelihood function f (D|θ, M j ) expresses the probability of getting D given the predictive model θ within M j -and this allows stochastic models inside a model class M j to be compared. When generating samples of posterior PDF π posterior (θ|D, M j ) in order to forward propagate uncertainty and compute QoI RV's, it is important to take into account potential multiple modes in the posterior. One simple idea is to sample increasingly difficult intermediate distributions, accumulating "knowledge" from one intermediate distribution to the next, until the target posterior distribution is sampled. In [8], an advanced stochastic simulation method, referred to as Adaptive Multi Level Algorithms, is proposed which can generate posterior samples from π posterior (θ|D, M j ) and compute the log of the evidence p(D|θ, M j ) at the same time by adaptively moving samples from the prior to the posterior through an adaptively selected sequence of intermediate distributions [7]. Specifically, the intermediate distributions are given by: for a given L > 0 and a given sequence 0 = τ 0 < τ 1 < . . . < τ L = 1 of exponents. In order to compute the model evidence π(D|M j ) where: π(D|M j ) = f (θ|D, M j ) · π prior (θ|M j ) dθ, (1.4.5) the use of intermediate distribution is also beneficial. For that, recall that π(D|M j ) = f (θ)π(θ) dθ = f π dθ = f 1−τ L−1 f τ L−1 −τ L−2 . . . f τ 2 −τ 1 f τ 1 π dθ = c 1 f 1−τ L−1 f τ L−1 −τ L−2 . . . f τ 2 −τ 1 f τ 1 π c 1 dθ = c 2 c 1 f 1−τ L−1 f τ L−1 −τ L−2 . . . f τ 2 −τ 1 f τ 1 π c 2 c 1 dθ = c L c L−1 · · · c 2 c 1 . (1. 4.6) Assuming that the prior PDF is normalized (it integrates to one) and if τ is small enough, then Monte Carlo method can be efficiently applied to calculate c in Equation (1.4.6). Due to numerical (in)stability, it is more appropriate to calculate the estimators: c i = ln c i , i = 1, . . . , L. (1.4.7) Combining Equations (1.4.6) and (1.4.7), we have: ln[π(D|M j )] =c L +c L−1 + . . . +c 2 +c 1 . Computing the log of the evidence instead of calculating the evidence directly is attractive because the evidence is often too large or too small relative to the computer precision. The posterior probability can be calculated readily in terms of the log evidence, allowing overflow and underflow errors to be avoided automatically [7]. Now let's define some auxiliary variables for k = 1, . . . , n ( ) total : • k-th sample at the -th level: θ ( )[k] , = 0, 1, . . . , L (1.4.8) • Plausibility weight: w ( )[k] = f (θ ( )[k] |D, M j ) τ · π prior (θ ( )[k] , M j ) f (θ ( )[k] |D, M j ) τ −1 · π prior (θ ( )[k] , M j ) = f (τ ) (D|θ ( )[k] , M j ) f (τ −1 ) (D|θ ( )[k] , M j ) , = f (τ −τ −1 ) (D|θ ( )[k] , M j ), = 0, 1, . . . , L (1.4.9) • Normalized plausibility weight: w ( )[k] = w ( )[k] n ( ) total s=1 w ( )[s] , = 0, 1, . . . , L (1.4.10) • Effective sample size: n ( ) eff = 1 n ( ) total s=1 (w ( )[s] ) 2 (1.4.11) • Estimate for the sample covariance matrix for π ( ) int : Σ = n ( −1) total m=1w m (θ ( −1)[m] − θ)(θ ( −1)[m] − θ) t , where θ = n ( −1) total m=1w m θ ( −1)[m] (1.4.12) so we can define the discrete distribution: P ( ) (k) =w ( )[k] , k = 1, 2, . . . , n ( ) total . (1.4.13) The ML algorithm consists of a series of resampling stages, with each stage doing the following: given n ( ) total samples from π ( ) int (θ|D), denoted by θ ( )[k] , k = 1...n ( ) total obtain samples from π ( +1) int (θ|D), denoted by θ ( +1)[k] , k = 1...n ( +1) total . This is accomplished by: given the samples θ ( )[k] , k = 1...n ( ) total , in Equation (1.4.8), from π ( ) int (θ|D), we compute the plausibility weights w ( ) [k] given in Equation (1.4.9) with respect to π ( +1) int (θ|D). Then we re-sample the uncertain parameters according to the normalized weightsw ( )[k] , given in Equation (1.4.10), through the distribution in Equation (1.4.13). This is possible due to the fact that for large n ( ) total and n ( +1) total , then θ ( +1)[k] , k = 1...n ( +1) total will be distributed as π ( +1) int (θ|D) [9]. The choice of τ , = 1, . . . , L − 1 is essential. It is desirable to increase the τ values slowly so that the transition between adjacent PDFs is smooth, but if the increase of the τ values is too slow, the required number of intermediate stages (L value) will be too large [9]. More intermediate stages mean more computational cost. In the ML method proposed by [8] and implemented in QUESO, τ is computed through a bissection method so that: β ( ) min < n ( ) eff n ( ) total < β ( ) max (1.4.14) AMSSA Algorithm Based on the above results, and recalling that the series of intermediate PDFs, π ( ) int (θ|D), start from the prior PDF and ends with the posterior PDF, Algorithm 2 can be applied both to draw samples from the posterior PDF, π posterior (θ|D, M j ), and to estimate the evidence π(D, M j ). Steps 44 and 45 in Algorithm 2 are accomplished by sampling the distribution in Equation (1.4.13) a total of n ( ) total times. The selected indices k determine the samples θ ( )[k] to be used as initial positions, and the number of times an index k is selected determines the size of the chain beginning at θ ( ) [k] . At each level , many computing nodes can be used to sample the parameter space collectively. Beginning with = 0, the computing nodes: (a) sample π ( ) int (θ|D, M j ); (b) select some of the generated samples ("knowledge") to serve as initial positions of Markov chains for the next distribution π ( +1) int (θ|D, M j ); and (c) generate the Markov chains for π ( +1) int (θ|D, M j ). The process (a)-(b)-(c) continues until the final posterior distribution is sampled. As increases, the selection process tends to value samples that are located in the regions of high probability content, which gradually "appear"as τ increases. So, as increases, if the "good" samples selected from the -th level to the ( +1)-th level are not redistributed among computing nodes before the Markov chains for the ( +1)-th level are generated, the "lucky" computing nodes (that is, the ones that had, already at the initial levels, samples in the Algorithm 2 AMSSA Algorithm proposed by [8]. Input: for each = 0, . . . , L: the total amount of samples to be generated at -th level (n ( ) total > 0) and the thresholds (0 < β ( ) min < β ( ) max < 1) on the effective sample size of the -th level Output: θ (m)[k] , k = 1, . . . , n(m) total ; which are asymptotically distributed as π posterior (θ|D, M j ) Output: c ; which is asymptotically unbiased for π(D, M j ) 33: Set = 0 34: Set τ = 0 35: Sample prior distribution, π prior (θ|M j ), n (0) total times i.e, obtain θ (0)[k] , k = 1, . . . , n (0) total 36: while τ < 1 do At the beginning of the -th level, we have the samples θ ( −1)[k] , k = 1...n ( −1) total from π ( −1) int (θ|D),Compute c = 1 n ( −1) total n ( −1) total s=1 w s recall that π(D|M j ) = c , Equation (1.4.5) 49: end while final posterior regions of high probability content) will tend to accumulate increasingly more samples in the next levels. This possible issue is avoided maintaining a balanced computational load among all computing nodes, which is handled in the ML by the step in Line 46. Running the step in Line 46 of Algorithm 2 is then equivalent of solving the following problem: given the number of processors N p , the total number of runs n total and the number of runs n j (to be) handled by the j-th processor; distribute N t tasks among the N p processors so that each processor gets its total number n j of program runs, j = 1, . . . , N p , the closest possible to the meann = n total /N p . This parallel implementation of the algorithm is proposed in [8], and it has been implemented in QUESO by the same authors/researchers. Algorithms for solving the Statistical Forward Problem The Monte Carlo method is commonly used for analyzing uncertainty propagation, where the goal is to determine how random variation, lack of knowledge, or error affects the sensitivity, performance, or reliability of the system that is being modeled [43]. Monte Carlo works by using random numbers to sample, according to a PDF, the 'solution space' of the problem to be solved. Then, it iteratively evaluates a deterministic model using such sets of random numbers as inputs. Suppose we wish to generate random numbers distributed according to a positive definite function in one dimension P (x). The function need not be normalized for the algorithm to work, and the same algorithm works just as easily in a many dimensional space. The random number sequence x i , i = 0, 1, 2, . . . is generated by a random walk as follows: 1. Choose a starting point x 0 2. Choose a fixed maximum step size δ. 3. Given a x i , generate the next random number as follows: (a) Choose x trial uniformly and randomly in the interval [x i − δ, x i + δ]. (b) Compute the ratio w = P (x trial ) P (x i ) . Note that P need not be normalized to compute this ratio. (c) If w > 1 the trial step is in the right direction, i.e., towards a region of higher probability. Accept the step x i+1 = x trial . (d) If w < 1 the trial step is in the wrong direction, i.e., towards a region of lower probability. We should not unconditionally reject this step! So accept the step conditionally if the decrease in probability is smaller than a random amount: i. Generate a random number r in the interval [0, 1]. ii. If r < w accept the trial step x i+1 = x trial . iii. If w ≤ r reject the step x i+1 = x i . Note that we don't discard this step! The two steps have the same value. There are essentially two important choices to be made. First, the initial point x 0 must be chosen carefully. A good choice is close to the maximum of the desired probability distribution. If this maximum is not known (as is usually the case in multi-dimensional problems), then the random walker must be allowed to thermalize i.e., to find a good starting configuration: the algorithm is run for some large number of steps which are then discarded. Second, the step size must be carefully chosen. If it is too small, then most of the trial steps will be accepted, which will tend to give a uniform distribution that converges very slowly to P (x). If it is too large the random walker will step right over and may not ' 'see" important peaks in the probability distribution. If the walker is at a peak, too many steps will be rejected. A rough criterion for choosing the step size is for the Acceptance ratio = Number of steps accepted Total number of trial steps to be around 0.5. An implementation of Monte Carlo algorithm is described in Algorithm 3. Algorithm 3 Monte Carlo Algorithm proposed by [39]. for j = 0 . . . N do 51: Set x trial ← x i + (2 RAND([0,1]) − 1)δ 52: Set x trial ← x i + (2 RAND([0,1]) − 1)δ Set w = P (x trial )/P (x) Set accepts ← 0 53: if w ≥ 1 then uphill 54: x i+1 ← x trial accept the step 55: accepts ← accepts+1 56: else downhill 57: Set r ← RAND([0,1]) but not too far 58: if r < w then 59: x i+1 ← x trial accept the step CHAPTER 2 Installation This chapter covers the basic steps that a user will need follow when beginning to use QUESO: how to obtain, configure, compile, build, install, and test the library. It also presents both QUESO source and installed directory structure, some simple examples and finally, introduces the user on how to use QUESO together with the user's application. This manual is current at the time of printing; however, QUESO library is under active development. For the most up-to-date, accurate and complete information, please visit the online QUESO Home Page 1 . Getting started In operating systems which have the concept of a superuser, it is generally recommended that most application work be done using an ordinary account which does not have the ability to make system-wide changes (and eventually break the system via (ab)use of superuser privileges). Thus, suppose you want to install QUESO and its dependencies on the following directory: $HOME / LIBRARIES / so that you will not need superuser access rights. The directory above is referred to as the QUESO installation directory (tree). There are two main steps to prepare your LINUX computing system for QUESO library: obtain and install QUESO dependencies, and define a number of environmental variables. These steps are discussed bellow. Obtain and Install QUESO Dependencies QUESO interfaces to a number of high-quality software packages to provide certain functionalities. While some of them are required for the successful installation of QUESO, other may be used for enhancing its performance. QUESO dependencies are: 1. C and C++ compilers. Either gcc or icc are recommended [50,53]. 2. Autotools: The GNU build system, also known as the Autotools, is a suite of programming tools (Automake, Autoconf, Libtool) designed to assist in making source-code packages portable to many Unix-like systems [51]. 3. STL: The Standard Template Library is a C++ library of container classes, algorithms, and iterators; it provides many of the basic algorithms and data structures of computer science [28]. The STL usually comes packaged with your compiler. GSL: The GNU Scientific Library is a numerical library for C and C++ programmers. It provides a wide range of mathematical routines such as random number generators, special functions and least-squares fitting [14]. The lowest version of GSL required by QUESO is GSL 1.10. 5. Boost: Boost provides free peer-reviewed portable C++ source libraries, which can be used with the C++ Standard Library [49]. QUESO requires Boost 1.35.0 or newer. 6. MPI: The Message Passing Interface is a standard for parallel programming using the message passing model. E.g. Open MPI [56] or MPICH [54]. QUESO requires MPI during the compilation step; however, you may run it in serial mode (e.g. in one single processor) if you wish. QUESO also works with the following optional libraries: 1. GRVY: The Groovy Toolkit (GRVY) is a library used to house various support functions often required for application development of high-performance, scientific applications. The library is written in C++, but provides an API for development in C and Fortran [52]. QUESO requires GRVY 0.29 or newer. 2. HDF5: The Hierarchical Data Format 5 is a technology suite that makes possible the management of extremely large and complex data collections [18]. The lowest version required by QUESO is HDF5 1.8.0. 3. GLPK: The GNU Linear Programming Kit package is is a set of routines written in ANSI C and organized in the form of a callable library for solving large-scale linear programming, mixed integer programming, and other related problems [38]. QUESO works with GLPK versions newer than or equal to GLPK 4.35. Trilinos: The Trilinos Project is an effort to develop and implement robust algorithms and enabling technologies using modern object-oriented software design, while still leveraging the value of established libraries. It emphasizes abstract interfaces for maximum flexibility of component interchanging, and provides a full-featured set of concrete classes that implement all abstract interfaces [25,24]. QUESO requires Trilinos release to be newer than or equal to 11.0.0. Remark: An additional requirement for QUESO work with Trilinos is that the latter must have enabled both Epetra and Teuchos libraries. The majority of QUESO output files is MATLAB /GNU Octave compatible [19,55]. Thus, for results visualization purposes, it is recommended that the user have available either one of these tools. Prepare your LINUX Environment This section presents the steps to prepared the environment considering the user LINUX environment runs a BASH-shell. For other types of shell, such as C-shell, some adaptations may be required. Before using QUESO, the user must first set a number of environmental variables, and indicate the full path of the QUESO's dependencies (GSL and Boost) and optional libraries. For example, supposing the user wants to install QUESO with two additional libraries: HDF5 and Trilinos. Add the following lines to append the location of QUESO's dependencies and optional libraries to the LD_LIBRARY_PATH environment variable: Naturally, for versions of QUESO other than 0.51.0, the file names in the above commands must be adjusted. $ Recommended Build Directory Structure Via Autoconf and Automake, QUESO configuration facilities provide a great deal of flexibility for configuring and building the existing QUESO packages. However, unless a user has prior experience with Autotools, we strongly recommend the following process to build and maintain local builds of QUESO (as an example, see note on Section 2.6). To start, we defined three useful terms: Source tree -The directory structure where the QUESO source files are located. A source tree is is typically the result of expanding an QUESO distribution source code bundle, such as a tarball. Build tree -The tree where QUESO is built. It is always related to a specific source tree, and it is the directory structure where object and library files are located. Specifically, this is the tree where you invoke configure, make, etc. to build and install QUESO. Installation tree -The tree where QUESO is installed. It is typically the prefix argument given to QUESO's configure script; it is the directory from which you run installed QUESO executables. Although it is possible to run ./configure from the source tree (in the directory where the configure file is located), we recommend separate build trees. The greatest advantage to having a separate build tree is that multiple builds of the library can be maintained from the same source tree [25]. Configure QUESO Building Environment QUESO uses the GNU Autoconf system for configuration, which detects various features of the host system and creates Makefiles. The configuration process can be controlled through environment variables, command-line switches, and host configuration files. For a complete list of switches type: $ ./ configure --help from the top level of the source tree (exemplified as $HOME/queso_download/queso-0.47.1 in this report). This command will also display the help page for QUESO options. Many of the QUESO configure options are used to describe the details of the build. For instance, to include HDF5, a package that is not currently built by default, append --with-hdf5=DIR, where DIR is the root directory of HDF5 installation, to the configure invocation line. QUESO default installation location is '/usr/local', which requires superuser privileges. To use a path other than '/usr/local', specify the path with the '--prefix=PATH' switch. For instance, to follow the suggestion given in Section 2.1, the user should append '--prefix=$HOME/LIBRARIES'. Therefore, the basic steps to configure QUESO using Boost, GSL (required), HDF5 and Trilinos (optional) for installation at '$HOME/LIBRARIES/QUESO-0.51.0' are: $ ./ configure --prefix = $HOME / LIBRARIES / QUESO -0.51.0 \ --with -boost = $HOME / LIBRARIES / boost -1.53.0 \ --with -gsl = $HOME / LIBRARIES / gsl -1.15 \ --with -hdf5 = $HOME / LIBRARIES / hdf5 -1.8.10 \ --with -trilinos = $HOME / LIBRARIES / trilinos -11.2.4 Note: the directory '$HOME/LIBRARIES/QUESO-0.51.0' does not need to exist in advance, since it will be created by the command make install described in Section 2.4. Compile, Check and Install QUESO In order to build, check and install the library, the user must enter the following three commands sequentially: $ make $ make check # o p t i o n a l $ make install Here, make builds the library, confidence tests, and programs; make check conducts various test suites in order to check the compiled source; and make install installs QUESO library, include files, and support programs. The files are installed under the installation tree (refer to Section 2.2.1), e.g. the directory specified with '--prefix=DIR' in Section 2.3. The directory, if not existing, will be created automatically. By running make check, several printouts appear in the screen and you should see messages such as: The tests printed in the screen are tests under your QUESO build tree, i.e., they are located at the directory $HOME/queso_download/queso-0.47.1/test (see Section 2.7 for the complete list of the directories under QUESO build tree). These tests are used as part of the periodic QUESO regression tests, conducted to ensure that more recent program/code changes have not adversely affected existing features of the library. --------------------------------------------------------------------( QUESO Developer's Documentation QUESO code documentation is written using Doxygen [57], and can be generated by typing in the build tree: $ make docs A directory named docs will be created in $HOME/queso_download/queso-0.47.1 (the build tree; your current path) and you may access the code documentation in two different ways: 1. HyperText Markup Language (HTML) format: docs/html/index.html, and the browser of your choice can be used to walk through the HTML documentation. 2. Portable Document Format (PDF) format: docs/queso.pdf, which can be accessed thought any PDF viewer. Summary of Installation Steps Supposing you have downloaded the file 'queso-0.47.1.tar.gz' into $HOME/queso download/. In a BASH shell, the basic steps to configure QUESO using GRVY, Boost and GSL for installation at '$HOME/LIBRARIES/QUESO-0.51.0' are: $ export LD_LIBRARY_PATH = $LD_LIBRARY_PATH :\ $HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib /:\ $HOME / LIBRARIES / trilinos -11.2.4/ lib : $ export CC = gcc $ export CXX = g ++ $ export MPICC = mpicc $ export MPICXX = mpic ++ $ export F77 = fort77 $ export FC = gfortran $ cd $HOME / queso_download / #e n t e r s o u r c e d i r $ gunzip < queso -0.47.1. tar . gz | tar xf -$ cd $HOME / queso_download / queso -0.47.1 #e n t e r t h e b u i l d d i r $ ./ configure --prefix = $HOME / LIBRARIES / QUESO -0.51.0 \ --with -boost = $HOME / LIBRARIES / boost -1.53.0 \ --with -gsl = $HOME / LIBRARIES / gsl -1.15 \ --with -hdf5 = $HOME / LIBRARIES / hdf5 -1.8.10 \ --with -trilinos = $HOME / LIBRARIES / trilinos -11.2.4 $ make $ make check $ make install $ make docs $ ls $HOME / LIBRARIES / QUESO -0.51.0 # l i s t i n g QUESO i n s t a l l a t i o n d i (a) gravity/: inference of the acceleration of gravity via experiments and a solution of a SIP; and forward propagation of uncertainty in the calculation of the distance traveled by a projectile. It is presented in detail in Section 6.3. (b) simpleStatisticalForwardProblem/: simplest example of how to use QUESO to solve a SFP, described in detail in Section 6.2. (c) simpleStatisticalInverseProblem/: simplest example of how to use QUESO to solve a SIP, thoroughly described in Section 6.1. (d) validationCycle/: presents a combination of SIP and SFP to solve a thermogravimetric analysis problem; this problem has the majority of its code in *.h files, with templated routines. This example is described in Section 6.4. (e) validationCycle2/: also presents a combination of SIP and SFP to solve a thermogravimetric analysis problem; but the majority of its code is in *.C files. All the examples presented in Chapter 6 come with the mathematical formulation, their translation into code, the options input file required by QUESO and auxiliary Matlab (GNU Octave compatible) files for data visualization. The Installed Directory Structure After having successfully executed steps described in Sections 2.1-2.4, the QUESO installed directory will contain four subdirectories: 1. bin: contains the executable queso_version, which provides information about the installed library. The code bellow presents a sample output: = GLPK DIR = HDF5 DIR = --------------------------------------------------------------- keme Create your Application with the Installed QUESO Prepare your environment by either running or saving the following command in your .bashrc (supposing you have a BASH-shell): Suppose your application code consists of the files: example_main.C, example_qoi.C, example_likelihood.C, example_compute.C and respective .h files. Your application code may be linked with QUESO library through a Makefile such as the one displayed as follows: QUESO_DIR = $HOME / LIBRARIES / QUESO -0.51.0/ BOOST_DIR = $HOME / LIBRARIES / boost -1.53.0/ GSL_DIR = $HOME / LIBRARIES / gsl -1.15/ GRVY_DIR = $HOME / LIBRARIES / grvy -0.32.0 TRILINOS_DIR = $HOME / LIBRARIES / trilinos -11. C++ Classes in the Library QUESO is is a parallel object-oriented statistical library dedicated to the research of statistically robust, scalable, load balanced, and fault-tolerant mathematical algorithms for the quantification of uncertainty in realistic computational models and predictions related to natural and engineering systems. Classes in QUESO can be divided in four main groups: core, templated basic, templated statistical and miscellaneous. The classed that handle environment (and options), vector and matrix classes are considered core classes. Classes implementing vector sets and subsets, vector spaces, scalar functions, vector functions, scalar sequences and vector sequences are templated basic classes; they are necessary for the definition and description of other entities, such as RVs, Bayesian solutions of IPs, sampling algorithms and chains. Vector realizer, vector RV, statistical IP (and options), MH solver (and options), statistical FP (and options), MC solver (and options) and sequence statistical options are part of templated statistical classes. And finally, the miscellaneous classes consist of C and FORTRAN interfaces. Core Classes QUESO core classes are the classes responsible for handling the environment (and options), vector and matrix operations. They are described in the following sections. Environment Class (and Options) The Environment class sets up the environment underlying the use of the QUESO library by an executable. This class is virtual. It is inherited by EmptyEnvironment and FullEnvironment. The QUESO environment class is instantiated at the application level, right after MPI_Init(&argc,&argv). The QUESO environment is required by reference by many constructors in the QUESO library, and is available by reference from many classes as well. The constructor of the environment class requires a communicator, the name of an options input file, and the eventual prefix of the environment in order for the proper options to be read (multiple environments can coexist, as explained further below). The environment class has four primary tasks: 1. Assigns rank numbers, other than the world rank, to nodes participating in a parallel job, 2. Provides MPI communicators for generating a sequence of vectors in a distributed way, 3. Provides functionality to read options from the options input file (whose name is passed in the constructor of this environment class), and 4. Opens output files for messages that would otherwise be written to the screen (one output file per allowed rank is opened and allowed ranks can be specified through the options input file). Let S 1 be the number of problems a QUESO environment will be handling at the same time, in parallel. S has default value of 1 and is an option read by QUESO from the input file provided by the user. The QUESO environment class manages five types of communicators, referred to as: A subenvironment in QUESO is the smallest collection of processors necessary for the proper run of the model code. An environment in QUESO is the collection of all subenvironments, if there is more than one subenvironment. Each subenvironment is able to generate a statistical inverse problem and/or a statistical forward problem; that is, each subenvironment is able to handle a "sub" Markov chain (a sequence) of vectors and/or a "sub" Monte Carlo sequence of output vectors. The "sub" sequences can be seen as forming a "unified" sequence in a distributed way. Indeed, the virtual class VectorSequence provides "sub" and "unified" statistical operations. Thus, if the model code requires 16 processors to run and the user decides to run 64 Markov chains in parallel, then the environment will consist of a total of F = 1024 processors and S = 64 subenvironments, each subenvironment with F/S = 16 processors. Any given computing node in a QUESO run has potentially five different ranks. Each subenvironment is assigned a subid varying from 0 (zero) to S − 1, and is able to handle a statistical IP and/or a statistical FP. That is, each subenvironment is able to handle a sub Markov chain (a sequence) of vectors and/or a sub MC sequence of output vectors. The sub sequences form an unified sequence in a distributed way. QUESO takes care of the unification of results for the application programming and for output files. Of course, if the user is solving just one statistical problem with just one MPI node, then all ranks are equal to zero. A QUESO subenvironment eventually prints messages to its own output file. In order for that to happen, the requirements are: 1. option m_subDisplayFileName, a string, must be different than the default value "."; 2. option m_subDisplayAllowedSet, a set of sub ids, must contain the id of the sub environment wanting to write a message to the output file; 3. the previous requirement is automatically satisfied if the option m_subDisplayAllowAll, a boolean, is set to 1 (the default value is 0); 4. the processor wanting to write a message to the output file must have sub rank 0 (zero). If all requirements are satisfied, then QUESO will generate a file with name <m_subDisplayFileName>_sub<sub id>.txt. For instance, if m_subDisplayFileName is 'pROblem_775_' then a node of sub rank 0 in sub environment 17 will write a message to the file 'pROblem_775_sub17.txt'. The class responsible for reading options one can pass to a QUESO environment through an input file is the EnvironmentOptions class. Finally, the input file options for a QUESO environment, i.e., the options the user may set in his/her input file when using QUESO together with the application of interest, is presented in Table 3.1.1. Vector The Vector class handles all the vector operations carried out in QUESO, and currently has two derived classes: GslVector and TeuchosVector. GslVector is based on the GSL vector structure; whereas TeuchosVector is based on Trilinos Teuchos vector structure [25], and therefore, it is only available if QUESO was compiled with Trilinos. A class diagram for Vector class is presented in Figure 3.1.4. Matrix The Matrix class handles all the matrix operations carried out in QUESO. Analogously to the vector class case described in the previous section, matrix class currently has two derived classes: GslMatrix and TeuchosMatrix. GslMatrix is based on the GSL matrix structure; whereas TeuchosMatrix is based on Trilinos Epetra matrix structure. A class diagram for Matrix is presented in Figure 3.1.5; it displays its protected attributes together with its member functions. Again, the diagram displays in some detail the inherited classes GslMatrix and TeuchosMatrix. Templated Basic Classes The classes in this group are: vector sets, subsets and spaces (Section 3.2.1), scalar and vector function classes (Section 3.2.2), and scalar and vector sequences (Section 3.2.3). These classes constitute the core entities necessary for the formal mathematical definition and description of other entities, such as random variables, Bayesian solutions of inverse problems, sampling algorithms and chains. Vector Set, Subset and Vector Space Classes The vector set class is fundamental for the proper handling of many mathematical entities. Indeed, the definition of a scalar function such as π : B ⊂ R n → R requires the specification of the domain B, which is a subset of the vector space R n , which is itself a set. Additionally, SIPs need a likelihood routine π like : R n → R + , and SFPs need a QoI routine q : R n → R m ; the sets R n , R m , etc., are vector spaces. The relationship amongst QUESO classes for handling sets, namely VectorSet; subsets, namely VectorSubset; and vector spaces, namely VectorSpace is sketched in Figure 3.2.1. An attribute of the subset class is the vector space which it belongs to, and in fact a reference to a vector space is required by the constructor of the subset class. An example of this case is the definition of a scalar function such as π : B ⊂ R n → R above. The power of an object-oriented design is clearly featured here. The intersection subset derived class IntersectionSubset is useful for handling a posterior PDF on Equation (1.2.1), since its domain is the intersection of the domain of the prior PDF with the domain of the likelihood function. Scalar Function and Vector Function Classes Scalar Sequence and Vector Sequence Classes The scalar sequence class contemplates scalar samples generated by an algorithm, as well as operations that can be done over them, e.g., calculation of means, variances, and convergence indices. Similarly, the vector sequence class contemplates vector samples and operations such as means, correlation matrices and covariance matrices. Templated Statistical Classes The classes in this group are: vector realizer, vector random variable, statistical inverse problem (and options), Metropolis-Hastings solver (and options), statistical forward problem (and options), Monte Carlo solver (and options), and Sequence statistical options. For QUESO, a SIP has two input entities, a prior RV and a likelihood routine, and one output entity, the posterior RV, as shown in Chapter 1, Figure 1.2.3. Similarly, a SFP has two input entities, a input RV and a QoI routine, and one output entity, the output RV, as shown in Figure 1 Vector Realizer Class A realizer is an object that, simply put, contains a realization() operation that returns a sample of a vector RV. QUESO currently supports several realizers: • uniform, implemented in UniformVectorRealizer, • Gaussian, implemented in GaussianVectorRealizer, • Log Normal, implemented in LogNormalVectorRealizer, • Gamma, implemented in GammaVectorRealizer, • Inverse Gamma, implemented in InverseGammaVectorRealizer, and • Beta, , implemented in BetaVectorRealizer, which are all derived from the base class BaseVectorRealizer. QUESO conveniently provides the class ConcatenatedVectorRealizer, which allows two distinct realizers to be concatenated. It also contains a sequence realizer class for storing samples of a MH algorithm. Vector Random Variable Class Vector RVs are expected to have two basic functionalities: compute the value of its PDF at a point, and generate realizations following such PDF. The joint PDF (BaseJointPdf and derived classes, see Section 3.2.2) and vector realizer (BaseVectorRealizer and derived classes, see Section 3.3.1) classes allow a straightforward definition and manipulation of vector RVs. Similarly to the vector realizer class above, QUESO also allows users to form new RVs through the concatenation of existing RVs (class ConcatenatedVectorRV). QUESO currently supports a few vector RVs such as uniform, Gaussian, Gamma and Beta, as depicted in Diagram 3.3.1. A derived class called generic vector RV allows QUESO to store the solution of an statistical IP: a Bayesian joint PDF becomes the PDF of the posterior RV, while a sequence vector realizer becomes the realizer of the same posterior RV. Statistical Inverse Problem (and Options) Similarly to its mathematical concepts, a SIP in QUESO also expects two input entities, a prior RV and a likelihood routine, and one output entity, the posterior RV. The SIP is represented in QUESO through the templated class StatisticalInverseProblem<P_V,P_M>, which is illustrated in Figure 3 Metropolis-Hastings Solver (and Options) The templated class that represents a Metropolis-Hastings generator of samples in QUESO is MetropolisHastingsSG<P_V,P_M>, where SG stands for 'Sequence Generator'. This class implements the DRAM algorithm of Haario, Laine, Mira and Saksman [21] together with an operation named generateSequence() based on the core routine at the MCMC toolbox for MATLAB [35]. The Metropolis-Hastings sequence generator class is depicted in Figure 3.3.3a; the Metropolis-Hastings sequence generator options class is depicted in Figure 3 Multilevel Solver (and Options) The templated class that represents a Multilevel generator of samples in QUESO is Option Name Default Value 〈PREFIX〉mh dataOutputFileName "." 〈PREFIX〉mh dataOutputAllowAll 0 〈PREFIX〉mh initialPositionDataInputFileName "." 〈PREFIX〉mh initialPositionDataInputFileType "m" 〈PREFIX〉mh initialProposalCovMatrixDataInputFileName "." 〈PREFIX〉mh initialProposalCovMatrixDataInputFileType "m" 〈PREFIX〉mh rawChainDataInputFileName "." 〈PREFIX〉mh rawChainDataInputFileType "m" 〈PREFIX〉mh rawChainSize 100 〈PREFIX〉mh rawChainGenerateExtra 0 〈PREFIX〉mh rawChainDisplayPeriod 500 〈PREFIX〉mh rawChainMeasureRunTimes 1 〈PREFIX〉mh rawChainDataOutputPeriod 0 〈PREFIX〉mh rawChainDataOutputFileName "." 〈PREFIX〉mh rawChainDataOutputFileType "m" 〈PREFIX〉mh rawChainDataOutputAllowAll 0 〈PREFIX〉mh filteredChainGenerate 0 〈PREFIX〉mh filteredChainDiscardedPortion 0. 〈PREFIX〉mh filteredChainLag 1 〈PREFIX〉mh filteredChainDataOutputFileName "." 〈PREFIX〉mh filteredChainDataOutputFileType "m" 〈PREFIX〉mh filteredChainDataOutputAllowAll 0 〈PREFIX〉mh displayCandidates 0 〈PREFIX〉mh putOutOfBoundsInChain 1 〈PREFIX〉mh tkUseLocalHessian 0 〈PREFIX〉mh tkUseNewtonComponent 1 〈PREFIX〉mh drMaxNumExtraStages 0 〈PREFIX〉mh drDuringAmNonAdaptiveInt 1 〈PREFIX〉mh amKeepInitialMatrix 0 〈PREFIX〉mh amInitialNonAdaptInterval 0 〈PREFIX〉mh amAdaptInterval 0 〈PREFIX〉mh amAdaptedMatricesDataOutputPeriod 0 〈PREFIX〉mh amAdaptedMatricesDataOutputFileName "." 〈PREFIX〉mh amAdaptedMatricesDataOutputFileType "m" 〈PREFIX〉mh amAdaptedMatricesDataOutputAllowAll 0 〈PREFIX〉mh amEta 1. 〈PREFIX〉mh amEpsilon 1 × 10 −5 〈PREFIX〉mh enableBrooksGelmanConvMonitor 0 〈PREFIX〉mh BrooksGelmanLag 100 Option Name Default Value 〈PREFIX〉ml restartOutput levelPeriod 0 〈PREFIX〉ml restartOutput baseNameForFiles "." 〈PREFIX〉ml restartOutput fileType "m" 〈PREFIX〉ml restartInput baseNameForFiles "." 〈PREFIX〉ml restartInput fileType "m" 〈PREFIX〉ml stopAtEnd 0 〈PREFIX〉ml dataOutputFileName "." 〈PREFIX〉ml dataOutputAllowAll 0 〈PREFIX〉ml loadBalanceAlgorithmId 2 〈PREFIX〉ml loadBalanceTreshold 1.0 〈PREFIX〉ml minEffectiveSizeRatio 0.85 〈PREFIX〉ml maxEffectiveSizeRatio 0.91 〈PREFIX〉ml scaleCovMatrix 1 〈PREFIX〉ml minRejectionRate 0.50 〈PREFIX〉ml maxRejectionRate 0.75 〈PREFIX〉ml covRejectionRate 0.25 〈PREFIX〉ml minAcceptableEta 0. 〈PREFIX〉ml totallyMute 1 〈PREFIX〉ml initialPositionDataInputFileName "." 〈PREFIX〉ml initialPositionDataInputFileType "m" 〈PREFIX〉ml initialProposalCovMatrixDataInputFileName "." 〈PREFIX〉ml initialProposalCovMatrixDataInputFileType "m" 〈PREFIX〉ml rawChainDataInputFileName "." 〈PREFIX〉ml rawChainDataInputFileType "m" 〈PREFIX〉ml rawChainSize 100 〈PREFIX〉ml rawChainGenerateExtra 0 〈PREFIX〉ml rawChainDisplayPeriod 500 〈PREFIX〉ml rawChainMeasureRunTimes 1 〈PREFIX〉ml rawChainDataOutputPeriod 0 〈PREFIX〉ml rawChainDataOutputFileName "." 〈PREFIX〉ml rawChainDataOutputFileType "m" 〈PREFIX〉ml rawChainDataOutputAllowAll 0 〈PREFIX〉ml filteredChainGenerate 0 〈PREFIX〉ml filteredChainDiscardedPortion 0. 〈PREFIX〉ml filteredChainLag 1 〈PREFIX〉ml filteredChainDataOutputFileName "." 〈PREFIX〉ml filteredChainDataOutputFileType "m" 〈PREFIX〉ml filteredChainDataOutputAllowAll 0 〈PREFIX〉ml displayCandidates 0 〈PREFIX〉ml putOutOfBoundsInChain 1 〈PREFIX〉ml tkUseLocalHessian 0 〈PREFIX〉ml tkUseNewtonComponent 1 〈PREFIX〉ml drMaxNumExtraStages 0 〈PREFIX〉ml drScalesForExtraStages 0 〈PREFIX〉ml drDuringAmNonAdaptiveInt 1 〈PREFIX〉ml amKeepInitialMatrix 0 〈PREFIX〉ml amInitialNonAdaptInterval 0 〈PREFIX〉ml amAdaptInterval 0 〈PREFIX〉ml amAdaptedMatricesDataOutputPeriod 0 〈PREFIX〉ml amAdaptedMatricesDataOutputFileName "." 〈PREFIX〉ml amAdaptedMatricesDataOutputFileType "m" 〈PREFIX〉ml amAdaptedMatricesDataOutputAllowAll 0 〈PREFIX〉ml amEta 1. 〈PREFIX〉ml amEpsilon 1.e-5 Statistical Forward Problem (and Options) A SFP in QUESO also has two input entities, the input (parameter) RV and a QoI function, and one output entity, the QoI RV. The SIP is represented through the templated class StatisticalForwardProblem<P_V,P_M,Q_V,Q_M >, which diagram is presented in Figure 3.3.7a. Again, the types P_V and Q_V of vectors and types P_M and Q_M of matrices, where P_ stands for 'parameter' and Q_ stands for 'quantities of interest'. The input RV and the output QoI RV are instances of the BaseVectorRv<P_V,P_M> class, while the QoI function is an instance of BaseVectorFunction<P_V,P_M,Q_V,Q_M>. In the template parameters, the prefix P_ refers to the parameters, whereas the prefix Q_ refers to the QoIs. In order to find the solution of a SFP, one must call the solveWithMonteCarlo() member function of the StatisticalForwardProblem<P_V,P_M> class. Upon return from a solution operation, the QoI RV is available through the qoiRv() member function. Such QoI RV is able to provide: a vector realizer through the operation 'qoiRv().realizer()', which returns an instance of the class 'uqBaseVectorRealizer<Q_V,Q_M>'. Figure 3.3.7b displays the statistical forward problem options class, i.e. that class that handles a variety of options for solving the SFP. Such options may be provided to QUESO at the user's input file; and they are listed in Table 3.3.4. In the table, p-q stands for parameterquantity of interest. Compute p-q correlations 〈PREFIX〉fp dataOutputFileName "." Name of data output file 〈PREFIX〉fp dataOutputAllowedSet "" Subenvironments that will write to data output file Monte Carlo Solver (and Options) The templated class that implements a Monte Carlo generator of samples within QUESO is MonteCarloSG<P_V,P_M,Q_V,Q_M>, as illustrated in Figure 3.3.8a. This class has the requirement that the image set of the vector random variable and the domain set of the QoI function belong to vector spaces of equal dimensions. If the requirements are satisfied, the class constructor reads input options that begin with the string '<PREFIX>_mc_' (See Table 3.3.5). Options reading is handled by class MonteCarloOptions, which is illustrated in Figure 3.3.8b. Option Name Default Value 〈PREFIX〉mc dataOutputFileName "." 〈PREFIX〉mc dataOutputAllowedSet 〈PREFIX〉mc pseq dataOutputFileName "." 〈PREFIX〉mc pseq dataOutputAllowedSet 〈PREFIX〉mc qseq dataInputFileName "." 〈PREFIX〉mc qseq size 100 〈PREFIX〉mc qseq displayPeriod 500 〈PREFIX〉mc qseq measureRunTimes 0 〈PREFIX〉mc qseq dataOutputFileName "." 〈PREFIX〉mc qseq dataOutputAllowedSet QUESO::StatisticalForwardProblem< P_V, P_M, Q_V, Q_M > -m_env -m_paramRv -m_qoiFunction -m_qoiRv -m_paramChain -m_qoiChain -m_mcSeqGenerator -m_solutionRealizer -m_solutionPdf -m_alternativeOptionsValues -m_optionsObj + StatisticalForwardProblem() + ~StatisticalForwardProblem() + computeSolutionFlag() + solveWithMonteCarlo() + qoiRv() + getParamChain() + print() -commonConstructor() (a) StatisticalForwardProblem QUESO::StatisticalForwardProblemOptions + m_ov + m_prefix -m_env -m_optionsDesc -m_option_help -m_option_computeSolution -m_option_computeCovariances -m_option_computeCorrelations -m_option_dataOutputFileName -m_option_dataOutputAllowedSet + StatisticalForwardProblemOptions() + StatisticalForwardProblemOptions() + ~StatisticalForwardProblemOptions() + scanOptionsValues() + print() -defineMyOptions() -getMyOptionValues() (b) StatisticalForwardProblemOptions Miscellaneous Classes and Routines As the name suggests, QUESO miscellaneous classes and routines have a variety of routines. For instance, the function MiscReadDoublesFromString is used for reading the options input files and assigning the values to the respective variables, in uqMonteCarloSGOptions:: getMyOptionValues and in MetropolisHastingsSGOptions::getMyOptionValues. QUESO class BaseOneDGrid generates grids necessary for calculating the CDF of a RV; it is required by class ArrayOfOneDGrids, which, in turn, is used in both classes: StatisticalForwardProblem and StatisticalInverseProblem. Important Remarks At this point, the user may feel comfortable and ready to start his/her validation and calibration exercises using QUESO. There are, however, a few quite important remarks that will make the linkage of the QUESO Library with the user application code possible. They are addressed in the following sections. Revisiting Input Options Input options are read from the QUESO input file, whose name is required by the constructor of the QUESO environment class. Herein, suppose that no prefix is defined, i.e., nothing will precede the input variables names (PREFIX = "" in Tables 3.1.1 -3.3.5). An example of the use of prefixes may be found in the input file tgaCycle.inp under the subdirectory /examples/validationCycle/ of QUESO installation tree. The first part of a input file commonly handles the environment options. he variable assignment env_numSubEnvironments = 1 indicates to QUESO that only one subenvironment should be used. The variable assignment env subDisplayFileName = outputData/ display create both the subdirectory outputData/ and a file named display_sub0.txt that contains all the options listed in the input file together with more specific information, such as the chain run time and the number of delayed rejections. The existence of file display_sub0.txt allows, for instance, the user in verifying the actual parameters read by QUESO. For an SIP, the user may set up variables related to the DRAM algorithm. Six important variables are: ip mh dr maxNumExtraStages defines how many extra candidates will be generated; ip mh dr listOfScalesForExtraStages defines the list s of scaling factors that will multiply the covariance matrix. The variable ip mh am initialNonAdaptInterval de-fines the initial interval in which the proposal covariance matrix will not be changed; whereas ip mh am adaptInterval defines the size of the interval in which each adapted proposal covariance matrix will be used. ip mh am eta is a factor used to scale the proposal covariance matrix, usually set to be 2.4 2 /d, where d is the dimension of the problem [36,21]. Finally, ip mh am epsilon is the covariance regularization factor used in the DRAM algorithm. For a SFP, the variable assignment fp_computeSolution = 1 tells QUESO to compute the solution process; the assignment fp_computeCovariances = 1, instructs QUESO to compute parameter-QoI covariances, and analogously, fp_computeCorrelations = 1 inform QUESO to compute parameter-QoI correlations. The name of the data output file can be set with variable fp_dataOutputFileName arg; and fp_dataOutputAllowedSet defines which subenvironments will write to data output file. An example a complete input file used by QUESO to solve a SIP-SFP is presented in Section 6.3.5; however every application example included in QUESO build and installation directories examples has an options input file and the user is invited to familiarize him/herself with them. Revisiting Priors QUESO offers a variety of prior distributions: uniform, Gaussian, Beta, Gamma, Inverse Gamma, and Log Normal. Also, QUESO presents the option of concatenating any of those priors, through the Concatenated prior. Concatenated priors are employed in problems with multiple random parameters. They allow one random parameter to have a different prior distribution then other; i.e., one variable may have a uniform prior distribution whereas other may have a Gaussian prior distribution. It is important to notice that, in order to set a Gaussian prior, besides providing the mean, the user must also supply the variance, not the standard deviation. Running with Multiple Chains or Monte Carlo Sequences As presented in the previous section, the variable env_numSubEnvironments determines how many subenvironments QUESO will work with. Thus, if env_numSubEnvironments=1, then only one subenvironment will be used, and QUESO will use only one set on Monte Carlo chains of size defined by ones of the variables ip_mh_rawChain_size or fp_mc_qseq_size, depending either the user is solving a SIP or a SFP. If the user wants to run QUESO with multiple chains or Monte Carlo sequences, then two variables have to be set in QUESO input file: env_numSubEnvironments = N s , with N s > 1 is the number of chains and/or Monte Carlo sequences of samples; and env_seed = −z, with z 1, so that each processor sets the seed to value MPI RANK+z. It is crucial that env_seed takes a negative value, otherwise all chain samples are going to be the same. Also, the total number N p of processors in the full communicator, usually named MPI COMM WORLD, needs to be a multiple of N s . Running with Models that Require Parallel Computing It is possible to run QUESO with models that require parallel computing as long as total number of processors N p is multiple of the number of subenvironments N s . QUESO will internally create N s subcommunicators, each of size N p /N s , and make sure that the likelihood and QoI routines are called for all processors in these subcommunicators -the likelihood/QoI routine will have available a communicator of size N p /N s . For instance, if N p = 2048 and N s = 16, then each likelihood/QoI will have available a communicator of size 128. s Each subcommunicator is accessible through env.subComm(). At the end of the simulation, there will be a total of N s chains. The user, however, must keep in mind the possible occurrence of race condition, especially in the case where the application is a black box and files are being accessed constantly (e.g. data is being written and read). A Requirement for the DRAM Algorithm Besides setting up the variables related to the DRAM algorithm in the input file, as described in Section 4.1 above, the user must also provide an initialized covariance matrix before calling the DRAM solver, solveWithBayesMetropolisHastings(...), in his/her application code. It is worth to note that this is rather a DRAM requirement [36], not a QUESO limitation. An example of the instantiation and initialization of a proposal covariance matrix and its subsequent use is presented in lines 145-147 of Listings 6.27, Section 6.3.4. Global sensitivity analysis (GSA) involves a quantitative assessment of variability in the model output or quantity of interest (QoI) due to uncertain model parameters. Variance based approaches relying on pseudo-random sampling of prior distributions of the parameters have been used effectively [11,47,45]. However, it can be understood that estimating the sensitivity indices ('first order effect' and 'total effect') can be computationally intensive especially in situations where a complex multiphysics model is simulated for a considered set of parameter samples. In order to mitigate such computational costs, alternative strategies involving construction and application of cheap surrogates for the models have been developed. Examples include polynomial chaos expansions [61,16] which have been used extensively for physics-based models and admit simple analytical expressions for computing the sensitivity indices [59,58]. Other examples include response surfaces based on Kriging, and radial basis functions [20]. Surrogate models, however, are not the central theme of this chapter. Instead, we focus our attention on exploiting QUESO to perform a prior based, parametric GSA. As mentioned earlier, the analysis helps determine relative contribution to the variance of the QoI and thus the relative importance of the stochastic model parameters. Potentially, such an analysis could help reduce the dimensionality of an inverse problem. In the following section, we provide a mathematical framework for the first order effect and total effect sensitivity indices as well as a brief survey of existing estimators for these indices. In section 5.2, we provide an algorithm based on [44] for using QUESO to perform a prior-based parametric GSA and further demonstrate its implementation using a simple exercise involving sensitivity analysis of the slope and intercept of a straight line. Sensitivity Indices Consider a model, G(θ), where θ denotes a vector of model parameters. Variance based measures for the first order effect and total effect sensitivity indices can be computed as discussed below. First Order Effect The first order effect sensitivity index S(θ i ) for a specific model parameter (θ i ) quantifies relative contribution to the variance of the QoI strictly due to (θ i ) and does not consider its interactions with other parameters. Mathematically, this is expressed as follows: S(θ i ) = V θ i (E θ ∼i [G|θ i ]) V(G) (5.1.1) where θ i is the i th parameter for which the first order effect index is to be computed and θ ∼i denotes a vector of all parameters except θ i . The quantity, E θ ∼i [G|θ i ] denotes the mean estimate of the model output taken over all possible values of θ ∼i while using a fixed value of θ i . The outer variance of this expectation is hence computed over all possible values of θ i . The quantity, V θ i (E θ ∼i [G|θ i ]) can also be understood as the expected reduction in variance due to fixed θ i . It is normalized by, V(G), i.e. the total variance of of the model output. Total Effect The total effect sensitivity index T (θ i ) for a specific model parameter (θ i ) quantifies relative contribution to the variance of the QoI due to (θ i ) and accounts for its interactions with other parameters. Mathematically, this is expressed as follows: T (θ i ) = E θ ∼i [V θ i (G|θ ∼i )] V(G) (5.1.2) = 1 − V θ ∼i (E θ i [G|θ ∼i ]) V(G) (5.1.3) where E θ ∼i [V θ i (G|θ ∼i )) is the expected variance when all parameters except θ i could be fixed. We can also interpret T (θ i ) using the quantity, V θ ∼i (E θ i [G|θ ∼i ]) which denotes the expected reduction in variance when all parameters except θ i could be fixed. Estimation of S(θ i ) and T (θ i ) In order to estimate the first order effect and the total effect sensitivity indices, we need to numerically estimate the quantities, V θ i (E θ ∼i [G|θ i ]) and E θ ∼i [V θ i (G|θ ∼i )] respectively. Tabulated below are commonly used estimators. As discussed in [44], we consider two independent set of samples denoted by the matrices, A and B. Additionally, we consider derived sets of samples denoted by the matrices, A (i) B where all columns are from A except the i th column which is from B. Similarly, we can construct the matrix, B (i) A as well. Estimator Reference V θ i (E θ ∼i [G|θ i ]) 1 N N k=1 f (A) k f (B (i) A ) k − f 2 0 Sobol 1990 [47] 1 N N k=1 f (B) k (f (A (i) B ) k − f (A) k ) Saltelli 2010 [44] V(G) − 1 2N N k=1 (f (B) k − f (A (i) B ) k ) 2 Jansen 1999 [30] E θ ∼i [V θ i (G|θ ∼i )] V(G) − 1 N N k=1 f (A) k f (A (i) B ) k + f 2 0 Homma 1996 [26] 1 N N k=1 f (A) k (f (A) k − f (A (i) B ) k ) Sobol 2007 [48] 1 2N N k=1 (f (A) k − f (A (i) B ) k ) 2 Jansen 1999 [30]V θ i (E θ ∼i [G|θ i ]) and E θ ∼i [V θ i (G|θ ∼i )]. Statistical forward problem (SFP) can be solved with QUESO by computing the QoI for pseudo-random samples drawn from the posterior distribution as discussed in 3.3.6. However, for GSA, we need to generate two independent data sets comprising pseudo-random samples for the model parameters, drawn from their individual prior distributions. QoIs are estimated for both sets of samples as well derived matrices for the model parameters, as discussed earlier. In the following section, we present a simple application involving sensitivity analysis of the slope and y-intercept of a straight line. SFP on samples from prior distributions of the model parameters is solved to generate the data which can further be used to compute the first order effect and total effect sensitivity indices. Application We consider the following equation for a straight line: y = mx + c (5.2.1) The slope, m and the y-intercept, c are considered to be uniformly distributed in the intervals, [2,5] and [3,7] respectively. For reference purposes, we provide an algorithm followed by the C++ code which interfaces with QUESO to generate the required data for GSA as follows. In order to estimate the first order effect and the total effect sensitivity indices, we solve the forward problem in QUESO to generate the required set of data. Specifically, we need to generate (2n+2) data files for n model parameters. Hence, in the present case, we need 6 data files as listed and described below. Note that the pseudorandom samples pertaining to the individual model parameters are given by their respective columns. In this case, column 1 corresponds to the slope, m and column 2 corresponds to the y-intercept, c. Moreover, y in the above equation is regarded as the QoI. 1. qoi samplesA.txt: Pseudo-random samples and corresponding QoI estimates, regarded as set A. 2. qoi samplesB.txt: Pseudo-random samples and corresponding QoI estimates, regarded as set B. 3. m qoi samplesAi.txt: All columns from set A except the i th (i=1) column which is from set B and corresponding QoI estimates. 4. m qoi samplesBi.txt: All columns from set B except the i th (i=1) column which is from set A and corresponding QoI estimates. 5. c qoi samplesAi.txt: All columns from set A except the i th (i=2) column which is from set B and corresponding QoI estimates. 6. c qoi samplesBi.txt: All columns from set B except the i th (i=2) column which is from set A and corresponding QoI estimates. In the above list, let us denote files in 3-6 as the set of derived files. The following algorithm provides a sequence of steps as well as snippets of code which could be used to generate the set of data files to compute the sensitivity indices. Relevant source files have also been included later in this section. Algorithm: Generating data for GSA 1: procedure Solving SFP with QUESO 2: Instantiate a QoI object (qoi mc): Qoi mc<> qoi mc("qoi ", paramDomain, qoiSpace); 3: Instantiate the forward problem (fp mc): QUESO::StatisticalForwardProblem<> fp mc("", NULL, priorRv, qoi mc, qoiRv); 4: Solve the forward problem to generate the data file, qoi samplesA.txt i.e. set A. fp mc.solveWithMonteCarlo(NULL); 5: Repeat steps 2-4 to generate the data file, qoi samplesB.txt i.e. set B. 8: Compute S(θ i ) and T (θ i ) using the set of 6 data files generated in previous steps. 9: end procedure The source code for generating the required set of data files is provided by the header file, sensitivity mc.h and the corresponding source file, sensitivity mc.C as follows. As shown in lines, 42-54 in the above listing for sensitivity mc.C, in order to generate the set of derived files, we compute the QoI by reading samples from corresponding intermediate files (such as c samples Ai.txt in this case). Whereas, for generating the pair of files, qoi samplesA.txt and qoi samplesB.txt, we compute the QoI for pseudo-random samples drawn from the prior distributions for m and c as shown in lines, 57-59 which are commented in the present case. Results In this section, we provide results for the first order effect sensitivity index as computed using approximations from Sobol [47] and Saltelli et al. [44] for the quantity, Table 5.1.1. In Figure 5.3.1(a), we perform a convergence study for the first order effect sensitivity index, S(θ i ). It is observed that for a small number of samples (< 5000), estimates from both, Sobol and Saltelli estimators exhibit large oscillations with increase in sample size indicating that the estimates have not yet converged to a stable value. Moreover, we observe large discrepancies are observed between estimates obtained from the two estimators in this regime. However, as we increase the sample size above 10000, it the two estimators seem to converge to stable values that are in close agreement. This phenomenon underscores the need for a large enough sample size for computing the sensitivity indices using pseudo-random sampling techniques. Optimizing the required number of samples in a way that the sensitivity indices are estimated within reasonable accuracy with the least possible sample size is a challenging task and depends on the map from the uncertain model parameters to the quantity of interest. Figure 5.3.1(b), illustrates estimates for S(θ i ) for the slope, m and the y-intercept, c based on 25000 samples. For both parameters, estimates from Sobol [47] and Saltelli et al. [44] are in close agreement. Moreover, the QoI (y) is observed to be much more sensitive to the uncertainty in the slope as compared to the y-intercept. V θ i (E θ ∼i [G|θ i ]) as provided in Concluding Remarks Global Sensitivity Analysis can be a computationally challenging task especially if it involves model estimates for a complex multiphysics problem. However, in case the forward solve of the model is inexpensive, one can exploit the SFP machinery in QUESO to compute the sensitivity indices as demonstrated with the help of a simple example in this chapter. Moreover, when stochastic formulations are proposed to capture the inadequacy in a model, parametric sensitivity analysis based on prior distributions of the stochastic parameters can potentially reduce the dimensionality of an inverse problem. Depending upon the nature of the problem, one can benefit from valuable insight into relative importance of the parameters with much fewer samples than required for convergence of the estimates as observed in the case of the straight line problem discussed in this chapter. • the mathematical models for the SIP and/or the SFP; • the application codes that translate the mathematical language into C++ using the QUESO classes and algorithms; • the input file that contains a list of options for either the Markov chain Monte Carlo algorithm or the Multilevel algorithm (in case of SIPs) and the Monte Carlo algorithm (in case of SFPs) which will be used by QUESO classes and algorithms; • examples of Makefiles which may be used to link the code with QUESO library; • how to plot figures using Matlab/GNU Octave and the output data generated by each application. All the examples presented in this chapter may be found under the directory examples in both QUESO installation and build directories and are compatible with QUESO 0.51.0. Note: Even though the Multilevel method is a methodology very useful for stochastic system model class comparison (model updating, model selection, model validation) [6], such tasks are not discussed in this manual. Thus the explicit dependency of the statistical variables on the predictive model in the set M j as presented in Section 1.4.2 are omitted herein. simpleStatisticalInverseProblem According to the Bayesian paradigm, the unobservable parameters in a statistical model are treated as random. When no data is available, a prior distribution is used to quantify our knowledge about the parameter. When data are available, we can update our prior knowledge using the conditional distribution of parameters, given the data. The transition from the prior to the posterior is possible via the Bayes theorem: π posterior (θ|d) = π prior (θ)π likelihood (d|θ) π(d) In this example, suppose a random variable of interest with two parameters θ ∈ R 2 has a uniform prior distribution, and suppose that a suitable likelihood has normal distribution with mean µ and covariance matrix C, given by: µ = −1 2 and C = 4 0 0 1 . (6.1.1) Therefore, we have: π prior (θ) ∝ 1 and π like (θ) ∝ exp − 1 2 (θ − µ) T [C −1 ](θ − µ) , where θ = θ 1 θ 2 ∈ R 2 . Therefore, posterior PDF is given by: π post (θ) ∝ e − 1 2 {(θ−µ) T [C −1 ](θ−µ)} . (6.1.2) In this example, we can replace the values for the mean and covariance matrix given in Equation (6.1.1) into Equation (6.1.2), in order to analytically compute both the posterior PDF: π post (θ) = 1 4π exp − 1 2 (θ − µ) T [C −1 ](θ − µ) = 1 4π exp − 1 8 (θ 1 + 1) 2 − 1 2 (θ 2 − 2) 2 , and the marginal results for θ 1 and θ 2 : π post (θ 1 ) = 1 2 √ 2π exp − 1 8 (θ 1 + 1) 2 , π post (θ 1 ) = 1 √ 2π exp − 1 2 (θ 2 − 2) 2 . (6.1.3) Recall that the posterior PDF given in Equation (6.1.2) can be sampled through the expression: µ + C 1/2 N (0, I), (6.1.4) where N (0, I) designates a Gaussian joint PDF of zero mean and unit covariance matrix, and C 1/2 is given by: C 1/2 = 2 0 0 1 . Thus, in this simple statistical inverse problem, we use QUESO implementation of the Markov chain algorithm to sample the posterior (6.1.2) via Expression (6.1.4) and compare the calculated marginal results for θ 1 and θ 2 against the analytical formulas given in Equation (6.1.3). Note: Due to the possibility to compare QUESO sampling algorithms to analytical expressions, this example is also used in the verification procedures and regression tests within QUESO, and it is reproduced in the directory tests/t02_sip_sfp. Running the Example To run the executable provided (available after QUESO installation), enter the following commands: It is worth noting presence of an argument passed to the executable in the example, namely 'example.inp'. The argument is a input file to be provided to QUESO with options for the solution of the SIP and/or SFP; and it is always required. Each option in the input file is related to one (or more) of the QUESO classes, and is presented throughout Chapter 3. Example Code The source code for the example is composed of 5 files: example main.C (Listing 6.1), example likelihood.h and example likelihood.C (Listings 6.2 and 6.3), example compute.h and example compute.C (Listings 6.4 and 6.5). Input File QUESO reads an input file for solving statistical problems. In the case of a SIP, it expects a list of options for MCMC (or Multilevel), together with options for QUESO environment; such as the amount of processors to be used and the seed for its random algorithms. Note that the names of the variables have been designed to be informative: env: refers to QUESO environment; ip: refers to inverse problem; mh: refers to Metropolis-Hastings; dr: refers to delayed rejection; am: refers to adaptive Metropolis; rawChain: refers to the raw, entire chain; filteredChain: refers to a filtered chain (related to a specified lag); The options used for solving this simple SIP are displayed in Listing 6.6. Create your own Makefile Makefiles are special format files that together with the make utility will help one to compile and automatically build and manage projects (programs). Listing 6.7 presents a Makefile, named 'Makefile sip example margarida', that may be used to compile the code and create the executable simple_sip_example. Naturally, it must be adapted to the user's settings, i.e., it has to have the correct paths for the user's libraries that have actually been used to compile and install QUESO (see Sections 2.1-2.4). The 'export' instruction above is only necessary if the user has not saved it in his/her .bashrc file. Data Post-Processing and Visualization There are a few Matlab-ready commands that are very helpful tools for post-processing the data generated by QUESO when solving statistical inverse problems. This section discusses the results computed by QUESO with the code of Section 6.1.2, and shows how to use Matlab for the post-processing of such results. Only the essential Matlab commands are presented; for the complete/detailed codes, please refer to file 'simple_ip_plots.m'. According to the specifications of the input file in Listing 6.6, a folder named 'outputData' containing the following files should be created: display_sub0.txt, ip_filt_chain_sub0.m, ip_raw_chain_sub0.m, sipOutput_sub0.m, ip_filt_chain.m, ip_raw_chain.m The code bellow shows how to load the data provided by QUESO during the solution process of the SIP described, in the form of chains of positions. % inside Matlab >> clear all >> si mp le _ ip _p lo t s Listing 6.8: Matlab code for loading the data in both raw and filtered chains of the SIP, by calling the file simple ip plots.m. Autocorrelation Plots The code presented in Listing 6.9 uses Matlab function autocorr to generate Figure 6.1.1 which presents the autocorrelation of the parameters θ 1 and θ 2 in both cases: raw and filtered chain. KDE Plots Matlab function [f,xi] = ksdensity(x) (kernel smoothing density estimate) computes a probability density estimate of the sample in the vector x. f is the vector of density values evaluated at the points in xi. The estimate is based on a normal kernel function, using a window parameter ('width') that is a function of the number of points in x. The density is evaluated at 100 equally spaced points that cover the range of the data in x. In order to estimate the KDE of the parameters, it is used together with the option 'pdf'. % Inside Matlab % Raw chain >> [f , x ] = ksdensity ( i p _ m h _ r a w C h a i n _ u n i f i e d (: ,1) ,' function ' , ' pdf ') ; >> [ f2 , x2 ] = ksdensity ( i p _ m h _ r a w C h a i n _ u n i f i e d (: ,2) ,' function ' , ' pdf ') ; >> x_p1 = sort ( i p _ m h _ r a w C h a i n _ u n i f i e d (: ,1) ) ; % analytical >> f_p1 =( exp ( -( x_p1 +1) .*( x_p1 +1) /8) ) /2/ sqrt (2* pi ) ; >> x_p2 = sort ( i p _ m h _ r a w C h a i n _ u n i f i e d (: ,1) ) ; >> f_p2 =( exp ( -( x_p2 -2) .*( x_p2 -2) /2) ) / sqrt (2* pi ) ; >> plot (x ,f , 'b ' , x2 , f2 , 'g ' , ' linewidth ' ,4) ; >> hold ; >> plot ( x_p1 , f_p1 , ' --k ' , x_p2 , f_p2 , ' -k ' , ' linewidth ' ,2) ; >> h = legend ( '\ theta_1 ' , '\ theta_2 ' , ' analytical (\ theta_1 ) ', ' analytical (\ theta_2 ) ',1 ) = 1 2 √ 2π exp − 1 8 (θ 1 + 1) 2 and π post (θ 2 ) = 1 √ 2π exp − 1 2 (θ 2 − 2) 2 , respectively. Covariance and Correlation Matrices Matlab function cov calculates the covariance matrix for a data matrix (where each column represents a separate quantity), and corr calculates the correlation matrix. Listing 6.39 presents the Matlab steps for calculating the covariance and correlation matrices for the parameters θ 1 and θ 2 . Listing 6.11: Matlab code for finding covariance and correlation matrices. simpleStatisticalForwardProblem In this simple statistical forward problem (SFP), suppose that the quantity of interest q is a function of a random variable θ of two parameters, namely q : R 2 → R such as: q(θ) = θ 1 + θ 2 , ∀θ = (θ 1 , θ 2 ) ∈ R 2 . (6.2.1) Suppose also that the parameters in θ have Gaussian distribution with mean µ and covariance matrix C given by: µ = −1 2 and C = 4 0 0 1 . (6.2.2) Notice that since the solution Q of this SFP is the sum of two random variables Θ 1 and Θ 2 , and since these two random variables independent Gaussian by assumption, should have: E[Q] = E[Θ 1 ] + E[Θ 2 ] = −1 + 2 = 1 and V [Q] = V [Θ 1 ] + V [Θ 2 ] = 4 + 1 = 5 (6.2.3) where E and V indicate expectation and variance, respectively. Thus the analytical expression for the solution Q is this SFP is the one-dimensional Gaussian distribution of mean 1 and variance 5: Q(x) = 1 √ 10π exp − 1 10 (x − 1) 2 (6.2.4) In this example, we use QUESO Monte Carlo algorithm to sample from the QoI given in Equation (6.2.1) and analyze it. Since the parameters have known independent Gaussian distributions, the results obtained by QUESO via sampling the QoI, in Equation (6.2.1), should match the Gaussian distribution given in Equation (6.2.4). Note: Due to the possibility to compare QUESO sampling algorithms to an analytical expression, this example is also used in the verification procedures and regression tests within QUESO. In fact it is the second part of the test tests/t02_sip_sfp. Running the Example To run the executable provided (available after QUESO installation), enter the following commands: Example Code The source code for the SFP example is composed of 5 files: simple sfp example main.C (Listing 6.12), simple sfp example qoi.h and simple sfp example qoi.C (Listings 6.13 and 6.14), simple sfp example compute.h and simple sfp example compute.C (Listings 6.15 and 6.16). Input File In the case of a SFP, QUESO expects a list of options for Monte Carlo algorithm, together with options for QUESO environment; such as the name of the output files and which subenvironments will write to to them. Note that the names of the variables have been designed to be informative: env: refers to QUESO environment; fp: refers to forward problem; mc: refers to Monte Carlo; pseq: refers to the parameter sequence; and qseq: refers to the quantity of interest sequence. The options used for solving this simple SFP are displayed in Listing 6.17. Create your own Makefile Listing 6.33 presents a Makefile, named 'Makefile sfp example margarida', that may be used to compile the code and create the executable simple_sfp_example. Naturally, it must be adapted to the user's settings, i.e., it has to have the correct paths for the user's libraries that have actually been used to compile and install QUESO. Thus, to compile, build and execute the code, the user just needs to run the following commands in the same directory where the files are: The 'export' instruction above is only necessary if the user has not saved it in his/her .bashrc file. Data Post-Processing and Visualization This section discusses the results computed by QUESO with the code of Section 6.2.2, and shows how to use Matlab for the post-processing of the data generated by QUESO when solving SFPs. Only the essential Matlab commands are presented; for the complete/detailed codes, please refer to file 'simple_fp_plots.m'. According to the specifications of the input file in Listing 6.17, a folder named 'outputData' containing the following files should be created: display_sub0.txt, fp_p_seq.m, fp_p_seq_sub0.m, fp_q_seq.m, fp_q_seq_sub0.m, and sfpOutput_sub0.m. The code below shows how to load the data provided by QUESO during the solution process of the SFP described, in the form of chains of positions. Histogram Plots In order to plot a histogram of the QoI, you may use the pre-defined Matlab function hist. The Matlab code presented in Listing 6.21 below shows how to create the Figure 6 KDE Plot Matlab function ksdensity (Kernel smoothing density estimate) together with the option 'pdf' may be used to estimate the KDE of the QoI. CDF Plot Matlab function ksdensity with 'cdf' option may also be used for plotting the Cumulative Distribution Function of the QoI. gravity This section presents an example of how to use QUESO in order to develop an application that solves a statistical inverse problem (SIP) and a statistical forward problem (SFP), where the solution of the former serves as input to the later. During the SIP, the acceleration due to gravity for an object in free fall near the surface of the Earth is inferred. During the SFP, the distance traveled by a projectile launched at a given angle and altitude is calculated using the calibrated magnitude of the acceleration of gravity. In this section we describe a statistical forward problem of predicting the described in Section 6.3.1. Statistical Inverse Problem A possible deterministic mathematical model for the vertical motion of an object in free fall near the surface of the Earth is given by h(t) = − 1 2 gt 2 + v 0 t + h 0 . (6.3.1) where v 0 [m/s] is the initial velocity, h 0 [m] is the initial altitude, h(t) [m] is the altitude with respect to time, t [s] is the elapsed time, and g [m/s 2 ] is the magnitude of the acceleration due to gravity (the parameter which cannot be directly measured and will be statistically inferred). Experimental Data We assume that the experiment of allowing an object to fall from different altitudes with zero initial velocity has been repeatedly conducted (See Figure 6.3.1). The data collected, e.g. d, is displayed in Table 6.3.1; the standard deviations, σ's, refer to the uncertainties in the measured times during the experiment execution [1]. In a straightforward classical interpretation of Bayesian inference, the prior signifies the modeler's honest opinion about the unknown. For the gravity inference problem, let's assume that gravity varies uniformly in the interval [8,11], or, in other words, we chose uniform prior distribution in that interval: π prior = U(8, 11). (6.3.2) We choose the usual likelihood function: π like (d|θ) ∝ exp − 1 2 [y(θ) − d] T [C(θ)] −1 [y(θ) − d] ,(6. 3.3) where C(θ) is a given covariance matrix, d denotes experimental data, y(θ) is the model output data. Recalling the deterministic model for the acceleration of gravity (6.3.1) with zero initial velocity, the information provided in Table 6.3.1, and Equation (6.3.3); and, additionally, invoking the nomenclature used in Section 1.2, we have: θ def. = g, y(θ) =            2h 1 g 2h 2 g . . . 2h n d g            , d =      t 1 t 2 . . . t n d      , C(θ) =      σ 2 1 0 · · · 0 0 σ 2 2 · · · 0 . . . . . . . . . 0 0 0 · · · σ 2 n d      , (6.3.4) where n d = 14 is the number of data points in Table 6.3.1. Now we are ready to evoke Bayes' formula in order to obtain the posterior PDF π post (θ): π post (θ|d) ∝ π like (d|θ) π prior (θ). (6.3.5) Statistical Forward Problem Projectile motion refers to the motion of an object projected into the air at an angle, e.g. a soccer ball being kicked, a baseball being thrown, or an athlete long jumping. Supposing the object does not have a propulsion system and neglecting air resistance, then the only force acting on the object is a constant gravitational acceleration g. A possible deterministic two-dimensional mathematical model for the vertical motion of an object projected from near the surface of the Earth is given by Figure 6.3.2 displays the projectile motion of an object in these conditions. v x = v 0x (6.3.6) v y = v 0y − gt (6.3.7) x = v 0x t (6.3.8) h = h 0 + v 0y t − 1 2 gt 2 (6.3.9) where h 0 is the initial height, x = x(t) is the distance traveled by the object, v 0 = (v 0x , v 0y ) is the initial velocity, v 0x = v 0 cos(α), v 0y = v 0 sin(α), and v 0 = v 0 2 . For this example, we assume that h 0 = 0 m, α = π/4 radians, v 0 = 5 m/s, all deterministic variables; and g is the solution of the SIP described in Section 6.3.1. Since a PDF is assigned to parameter g; thus, the output of the mathematical model (6.3.6) becomes a random variable, thus we have a statistical forward problem. The Input RV, QoI Function and Output RV The input random variable for the statistical forward problem is the acceleration of gravity g, which is also the solution (posterior PDF) of the inverse problem described in Section 6.3.1. The output random variable for this example is the distance x traveled by an object in projectile motion. Note that, since there is uncertainty in the parameter g (g is given as a PDF), one can expect that this uncertainty will be propagated to x, which will also be given as a PDF. Combining the expressions in Equation 6.3.6 and rearranging them, we have that QoI function for x is: x = v 0 cos α g v 0 sin α + (v 0 sin α) 2 + 2g y 0 . (6.3.10) where y is the distance traveled and our quantity of interest (QoI). Running the Example To run the executable provided (available after QUESO installation), enter the following commands: Example Code The source code for the SIP and the SFP is composed of 7 files. Three of them are common for both problems: gravity main.C, gravity compute.h and gravity compute.C; they combine both problems and use the solution of the SIP (the posterior PDF for the gravity) as an input for the SFP and are presented, respectively, in Listings 6.25, 6.26 and 6.27. Two of files specifically handle the SIP: gravity likelihood.h, and gravity likelihood.C, and are displayed in Listings 6.28 and 6.29. Finally, the files specific for the SFP are gravity qoi.h and gravity qoi.C, and they are presented in Listings 6.30 and 6.31. Moreover, for the gravity inverse problem, one may notice that QUESO will use the Metropolis-Hastings algorithm to sample the posterior PDF (indicated by the prefix mh in the variable names) without adaptive steps (indicated by the zero value assigned to the variable ip mh am initialNonAdaptInterval, which can also be achieved by setting zero to ip_mh_am_adaptInterval) and with delayed rejection (indicated by the one-value assigned to the variable ip mh dr maxNumExtraStages). p _ m h _ f i l t e r e d C h a i n _ d i s c a r d e d P o r t i o n = 0. i p _ m h _ f i l t e r e d C h a i n _ l a g = 20 i p _ m h _ f i l t e r e d C h a i n _ d a t a O u t p u t F i l e N a m e = outputData / s i p _ g r a v i t y _ f i l t e r e d _ c h a i n i p _ m h _ f i l t e r e d C h a i n _ d a t a O u t p u t Create your own Makefile Running the Gravity Example with Several Processors Even though the application described in Section 6.3.4 is a serial code, it is possible to run it using more than one processor, i.e., in parallel mode. Supposing the user's workstation has N p = 8 processors, then, the user my choose to have N s = 8, 4 or 2 subenvironments. This complies with the requirement that the total number of processors in the environment must be a multiple of the specified number of subenvironments. Thus, to build and run the application code with N p = 8, and N s = 8 subenvironments, the must set the variable env numSubEnvironments = 8 in the input file (Listing 6.32) and enter the following command lines: cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / gravity / mpirun -np 8 ./ gravity_gsl g r av it y _i nv _f w d . inp The steps above will create a total number of 8 raw chains, of size defined by the variable ip mh rawChain size. QUESO internally combines these 8 chains into a single chain of size 8 × ip mh rawChain size and saves it in a file named according to the variable ip mh rawChain dataOutputFileName. QUESO also provides the user with the option of writing each chain -handled by its corresponding processor -in a separate file, which is accomplished by setting the variable ip mh rawChain dataOutputAllowedSet = 0 1 ... Ns-1. Note: Although the discussion in the previous paragraph refers to the raw chain of a SIP, the analogous is true for the filtered chains (SIP), and for the samples employed in the SFP (ip mh filteredChain size, fp mc qseq size and fp mc qseq size, respectively). Data Post-Processing and Visualization According to the specifications of the input file in Listing 6.32, both a folder named outputData and a the following files should be generated: In this section, a convenient capability of QUESO of internally handling possible conflicts in chain size is presented. Recalling the input file gravity_inv_fwd.inp presented in Listing 6.32, one may notice that the raw chain size for the SIP is chosen to have 20000 positions (ip_mh_rawChain_size = 20000); the lag of the filtered chain is chosen to be 20 (ip_mh_filteredChain_lag = 20) and the chain size for the SFP has 16384 positions (fp_mc_qseq_size = 16384). Because the solution of the SIP, ie, the posterior PDF, is used as input PDF for the SFP, QUESO internally sets fp_mc_qseq_size = 20000, as can be seen in the file display_env_sub0.txt. The file display_env_sub0.txt contains information from the subenvironment '0' that was generated during the run of the application code. Statistical Inverse Problem There are a few Matlab-ready commands that are very helpful tools for post-processing the data generated by QUESO when solving statistical inverse problems. This section discusses the results computed by QUESO with the code of Section 6.3.4, and shows how to use Matlab for the post-processing of such results. Chain Plots It is quite simple to plot, using Matlab, the chain of positions used in the DRAM algorithm implemented within QUESO. The sequence of Matlab commands presented in Listing 6.34 generates the graphic depicted in Figure 6 Histogram Plots In order to plot histograms of the parameter using either the raw chain or the filtered chain, you simply have to use the pre-defined Matlab function hist. KDE Plots Matlab function ksdensity (Kernel smoothing density estimate) together with the option 'pdf' may be used for plotting the KDE of the parameter. Listing 6.36: Matlab code for the KDE plot. CDF Plots Matlab function ksdensity (Kernel smoothing density estimate) with 'cdf' option may also be used for plotting the Cumulative Distribution Function of the parameter. Listing 6.37: Matlab code for the CDF plot. Autocorrelation Plots The code presented in Listing 6.38 uses matlab function autocorr to generate Figure 6.3.7 which presents the autocorrelation of the parameter g in both cases: raw and filtered chain. Listing 6.38: Matlab code for the autocorrelation plots. Covariance and Correlation Matrices Matlab function cov calculates the covariance matrix for a data matrix (where each column represents a separate quantity), and corr calculates the correlation matrix. Since our statistical inverse problem has only one parameter (the acceleration g due to gravity), both covariance and correlation matrices have dimension 1 × 1, i.e., they are scalars. Statistical Forward Problem Chain Plots It is quite simple to plot, using Matlab, the chain of positions generated by the Monte Carlo algorithm implemented within QUESO and called during the solution of the statistical forward problem. The sequence of Matlab commands presented bellow generates the graphic depicted in Figure 6 Histogram Plots In order to plot a histogram of the QoI, you may use the pre-defined Matlab function hist. The Matlab code presented in below shows how to create the Figure 6 KDE Plots Matlab function ksdensity (Kernel smoothing density estimate) together with the option 'pdf' may be used for plotting the KDE of the he QoI, displayed in Figure 6 Autocorrelation Plots The code presented in Listing 6.44 uses Matlab function autocorr to generate Figure 6.3.10, which presents the autocorrelation of the QoI d. Covariance and Correlation Matrices For a matrix input X, where each row is an observation, and each column is a variable, the Matlab function cov(X) may be used to calculate the covariance matrix. Thus, in order to calculated the covariance matrix between the parameter and the quantity of interest sequences generated by Monte Carlo sampler with QUESO, one may simply define X=[fp_mc_ParamSeq_unified fp_mc_QoiSeq_unified]. The code presented in Listing 6.45 shows the usage of Matlab commands for finding such the matrix. Analogously, the Matlab function corrcoef(X) returns a matrix of correlation coefficients calculated from an input matrix X whose rows are observations and whose columns are variables. In order to calculated the correlation matrix between the parameter and the QoI sequences, one may simply define X=[fp_mc_ParamSeq_unified fp_mc_QoiSeq_unified]. validationCycle This is the last and more complex of all QUESO examples. In this example, we numerically solve a statistical inverse problem related to a thermogravimetric experiment, where a material sample has its mass measured while loosing it through a controlled heating process. Given a simple mass evolution model that has a temperature profile and material properties as input parameters, and given thermogravimetric measurements with prescribed variances, the statistical inverse problems ask for the specification of the random variables that represent the unknown material properties. We compute probability density functions with the Bayesian approach and compute sets of realizations through the Metropolis-Hastings algorithm with delayed rejection. We qualitatively analyze the sensitivity of the solutions with respect to problem characteristics, namely "amount" of data and "quality" of data, and also with respect to algorithm options, namely chain initial position, number of delayed rejections and chain size. Thermogravimetric Experiments and a Simple Model Suppose a given material sample of initial mass m 0 and at initial temperature T 0 is heated with constant heating rate β (K/min). Heating is maintained until the sample fully ablates (decomposes). The sample mass m(T ) is measured at temperatures T > T 0 . Let w(T ) = m(T )/m 0 denote the mass fraction. It is convenient to transform the kinetic equation to a per unit temperature form by dividing through by β. Thus, a simple approach to the simulation of a thermogravimetric phenomenon consists on modeling the sample as a homogeneous material of scalar properties A > 0 and E > 0 whose relative mass w obeys the following initial value ordinal differential equation        dw dt = − Aw β exp − E RT , t 0, w(0) = 1, (6.4.1) where the kinetic parameters A and E are referred to, respectively, as pre-exponential factor (min −1 ) and activation energy (J/mol). In this combined SIP-SFP, we calibrate both model parameters A and E given the mathematical model in Equation (6.4.1) and experimental data (Section 6.4.2.3). Then the inferred values for A and E are then used for uncertainty propagation on the remaining mass fraction at time t = 3.9 s when β = 250 K/min, i.e., our quantity of interest is w(t = 3.9). Statistical Inverse Problem Let m = (A, E) be the vector of model parameters and M = R 2 + be the space of model parameters. Let V T denote the space of functions f : R + → R + that are weakly differentiable. V T will be the space of temperature profiles. Finally, let V w denote the space of functions f : R + → [0, 1] that are weakly differentiable. V w will be the space of relative mass evolutions. We will denote by w(m, T ) ∈ V w the solution of Equation (6.4.1) for given m ∈ M and T ∈ V T . Misfit Functions F(m) Let V S denote the space of all functions f : R + → R + that are square-Lebesgue-integrable over any finite interval. V S will be the space of misfit weight functions. Let V σ denote the space of all functions f : R + → R * + such that 1/f is square-Lebesgue-integrable over any finite interval. V σ will be the space of variance functions. Given a reference relative mass evolution function d ∈ V w , a temperature profile T ∈ V T , and some t F > 0, let F : M → R be the functional defined by F(m) = t F 0 {[w(m, T )](t) − d(t)} 2 · S(t) dt, or simply F(m) = t F 0 (w − d) 2 · S dt. (6.4.2) The functional (6.4.2) is general enough for our studies, since it can properly describe not only the case where one has continuous measurements d, but also the case of a finite set of N meas discrete measurements 0 d j 1, 1 i N meas at instants 0 t 1 < t 2 < . . . < t Nmeas . In the case of continuous measurements, for instance, one can set F 1 (m) = t F 0 {[w(m, T )](t) − d(t)} 2 · 1 σ 2 (t) dt, for some given variance function σ 2 ∈ V S satisfying σ(t) > 0 for all t 0. On the other hand, when measurements are discrete and a corresponding finite set of variances σ 2 j > 0, j = 1, 2, . . . , N meas is given, one can set F 2 (m) = t F 0 {[w(m, T )](t) −d(t)} 2 · Nmeas j=1 δ(t − t j ) σ 2 (t) dt, whered ∈ V w andσ ∈ V σ are any functions satisfyingd(t j ) = d j andσ(t j ) = σ j , j = 1, 2, . . . , N meas , in which case the functional simply becomes F 2 (m) = Nmeas j=1 {[w(m, T )](t j ) − d j } 2 σ 2 j , assuming, without loss of generality, that t F t Nmeas . Bayesian Approach: Prior RV, Likelihood and Posterior RV In deterministic inverse problems treat the unknown parameters as scalars or vectors and the goal is the calculation of their best values according to a given criteria, usually least squares, e.g. solving the unconstrained optimization problem min m∈M F(m). In statistical inverse problems, the unknown parameters are treated as random variables (RVs) and the goal is the specification of their probability density functions (PDFs) [33]. Applying the Bayesian approach π posterior (m) ∝ π prior (m) · π likelihood (m) we have that for the TGA SIP, the prior distribution and the likelihood are, respectively: π prior (m) ∝ e − 1 2 V (m) and π likelihood (m) ∝ e − 1 2 F (m) . Thus, we chose parameters (A, E) to have joint uniform prior PDF over the open square domain, i.e.: π prior = U((1.0 × 10 10 , 5.0 × 10 11 ) × (4.0 × 10 5 , 6.0 × 10 5 )). Data from experiments Statistical Forward Problem In spacecraft design, ablation is used to both cool and protect mechanical parts and/or payloads that would otherwise be damaged by extremely high temperatures. Two principal applications are heat shields for spacecraft entering a planetary atmosphere from space and cooling of rocket engine nozzles [60]. Suppose that an object about to re-enter the Earth atmosphere has a thermal protection layer (shield) of composition of the same sample material described in Section 6.4.2. Also, as the object re-enters the atmosphere, its shield loses mass according to Equation (6.4.1). The initial sample temperature is T 0 = 0.1 K and it is then heated with constant rate β = 5 K/m. We are interested in answering the following question: at scenario β = 250 K/min, what is the remaining mass fraction at time t = 3.9 s? In other words, the quantity of interest is w(t = 3.9s). The Input RV, QoI Function and Output RV The input random variables for this SFP are the inferred parameters A and E which are the solution (posterior PDF) of the inverse problem described in Section 6.4.2. The output random variable for this example is the remaining mass fraction at 3.9 s, i.e. w(t = 3.9). Note that, since there is uncertainty in the parameters A and E (both given as PDFs), one can expect that this uncertainty will be propagated to w(t = 3.9), which will also be given as a PDF. Finally, the QoI function for w is the solution of the Equation (6.4.1) evaluated when t = 3.9 s, which is calculated using numerical integration with adjustable and acceptable time-stepping using GSL function gsl_odeiv_evolve_apply 1 . Running the Example To run the executable provided (available after QUESO installation), and generate figures for the chains, PDFs, CDFs, etc., enter the following commands: TGA Example Code The program example given in this paper is compatible with version 0.47.1 of QUESO. The source code for the example is composed of 5 files: exTgaValidationCycle gsl.C (Listing 6.47), exTgaValidationCycle appl.h (Listing 6.48), exTgaValidationCycle likelihood.h (Listing 6.49) and exTgaValidationCycle qoi.C (Listing 6.50). / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Run application / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * uqAppl < QUESO :: GslVector , // type for parameter vectors QUESO :: GslMatrix , // type for parameter matrices QUESO :: GslVector , // type for qoi vectors QUESO :: GslMatrix // type for qoi matrices >(* env ) ; / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Finalize environment / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * delete env ; / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The driving routine " uqAppl () " : called by main () // There are 5 main tasks : // 1) initilization // 2) the ' calibration stage ' // 3) the ' validation stage ' // 4) the ' comparison stage ' // 5) memory release // Tasks 2 , 3 and 4 constitute the actual validation cycle . / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * template < class P_V , class P_M , class Q_V , class Q_M > void uqAppl ( const QUESO :: B as eE n vi ro nm e nt & env ) { if ( env . fullRank () == 0) { std :: cout << " Beginning run of ' uqTgaExample ' example \ n " << std :: endl ; } int iRC ; struct timeval timevalRef ; struct timeval timevalNow ; / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Task 1 of 5: instantiation of basic classes / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Instantiate the parameter space std :: vector < std :: string > paramNames (2 , " " ) ; paramNames tv_sec << " seconds \ n " << std :: endl ; } / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Task 3 of 5: validation stage / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * tv_sec << " seconds \ n " << std :: endl ; } / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Task 4 of 5: comparison stage / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The ( user defined ) data class that carries the data // needed by the ( user defined ) likelihood routine / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * template < / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The actual ( user defined ) likelihood routine / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * template < class P_V , class P_M > / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The ( user defined ) data class that carries the data // needed by the ( user defined ) qoi routine / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * template < class P_V , class P_M , class Q_V , class Q_M > struct q oi Ro ut i ne _D at a { double m_beta ; double m_cri ticalMas s ; double m_cri ticalTim e ; }; Input File The input file used with this TGA SIP-SFP QUESO provides QUESO with options for its environments, and for both MCMC and Monte-Carlo algorithms. It is displayed in Listing 6.51. Data Post-Processing and Visualization According to the specifications of the input file in Listing 6.51, both a folder named outputData and a the following files should be generated: The sequence of Matlab commands is identical to the ones presented in Sections 6.1.5, 6.2.5 and 6.3.8; therefore, are omitted here. The reader is invited to explore the Matlab file tga cycle plot.m for details of how the figures have been generated. KDE Plots of Parameters Matlab function ksdensity (Kernel smoothing density estimate) together with the option 'pdf' may be used to estimate the KDE of the parameters, as illustrated in Figure 6 CDF Plots of Parameters Matlab function ksdensity with 'cdf' option may also be used for plotting the Cumulative Distribution Function of each one of the parameters, as illustrated in Figure 6 Autocorrelation Plots of Parameters KDE, CDF and Autocorrelation Plots of QoI modal This example presents a combination of two statistical inverse problems in one. It presents the capability of the Multilevel method in sampling from a target distribution that has either one or two modes (distinct peaks). The random variable of interest has three parameters, i.e., θ = (θ 1 , θ 2 , σ 2 ) ∈ R 3 , where the third parameter may be seen as variation. The example also it gives the user the opportunity to chose either one single type of prior distribution, uniform, for the three components of the random variable, or two different priors: a uniform and a beta distribution. Choosing between a one-mode or a two-mode target distribution is done at execution level, as presented in the following code line: $ cd $HOME / LIBRARIES / QUESO -0.51.0/ $ cd examples / modal $ rm outputData /* $ ./ modal_gsl example . inp < num_of_nodes > where <num_of_nodes> is either 1 or 2. One-mode distribution In this case, the target distribution is assumed to have only one mode. Suppose also that the random variable θ can either have a uniform prior distribution for all its components, i.e.: π prior = U([0, 3]) × U([0, 3]) × U([0, 0.3]). or, the prior distribution is defined as a combination of uniform prior for θ 1 and θ 2 , with a beta prior for σ 2 : (6.5.1) Running the One-Mode Example To run the executable provided considering a one-mode distribution, enter the following commands: Two-mode distribution In this case, the target distribution is assumed to have two modes. Suppose that θ has a either uniform distribution for all its components, i.e.: Running the Two-Mode Example To run the executable provided considering a two-modes distribution, enter the following commands: $ Example Code The source code for the example is composed of 5 files: example main.C (Listing 6.54), example likelihood.h and example likelihood.C (Listings 6.55 and 6.56), example compute.h and example compute.C (Listings 6.57 and 6.58). Note that in line 12 of Listings 6.58 the #define directive creates the macro APPLS_MODAL_USES_CONCATENATION. Such macro, together with the directives #ifdef, #else, and #endif, tells the compiler that the application will use concatenated priors, by controlling compilation of portions of file example compute.C. Commenting line 12 of Listings 6.58 will make the application to use uniform priors only: Input File QUESO reads an input file for solving statistical problems, which provides options for the Multilevel or MCMC method. In this example, the Multilevel method is chosen to sample from the distribution. Many variables are common to both MCMC and Multilevel method, especially because the Multilevel method also has the option of delaying the rejection of a candidate. The names of the variables have been designed to be informative in this case as well: env: refers to QUESO environment; ip: refers to inverse problem; ml: refers to Multilevel; dr: refers to delayed rejection; rawChain: refers to the raw, entire chain; filteredChain: refers to a filtered chain (related to a specified lag); last: refers to instructions specific for the last level of the Multilevel algorithm. The user may select options for a specific level by naming its number, i.e., in case the user wants to write the raw chain of the level 3 in a separate file, say 'rawChain_ml_level3.m', he/she may include the line: Create your own Makefile Makefiles are special format files that together with the make utility will help one to compile and automatically build and manage projects (programs). Listing 6.60 presents a Makefile, named 'Makefile modal example violeta', that may be used to compile the code and create the executable modal_gsl. Naturally, it must be adapted to the user's settings, i.e., it has to have the correct paths for the user's libraries that have actually been used to compile and install QUESO (see Sections 2.1-2.4). The 'export' instruction above is only necessary if the user has not saved it in his/her .bashrc file. Data Post-Processing and Visualization According to the specifications of the input file in Listing 6.59, both a folder named outputData and a the following files should be generated: rawChain_ml.m display_sub0.txt The sequence of Matlab commands is identical to the ones presented in Sections 6.1.5, 6.2.5, 6.3.8 and 6.4.7; therefore, are omitted here. The reader is invited to explore the Matlab files plot modal all levels 1mode.m and/or plot modal all levels 2modes.m for details of how the figures have been generated. Scatter Plots The code presented in Listing 6.61 uses Matlab function plotmatrix to generate Figures 6.5.1 and 6.5.2 which presents the scatter plots and histograms of the parameters θ 1 and θ 2 , based on the generated raw chains. fprintf (1 , ' Scatter plots and histograms of raw chains -Level 1 < press any key >\ n ') ; plotmatrix ( ip_ml_1_rawChain_unified , '+b ') set ( gca , ' fontsize ' ,20) ; xlabel ( '\ theta_1 \ theta_2 \ theta_3 ' , ' fontsize ' ,16) ; ylabel ( '\ theta_3 \ theta_2 \ theta_1 ' , ' fontsize ' ,16) ; title ( ' Scatter plots and histograms , Level 1 -1 mode ') Listing 6.61: Matlab code for the scatter plots depicted in Figures 6.5.1 and 6.5.2. Figure 6.5.1: Scatter plots for θ 1 , θ 2 and θ 3 = σ 2 , levels 1, 3, 5 and 7 (last). One mode distribution. KDE Plots bimodal This example replicates the problem in "Section 4.1 A 1D Problem" of [8]: it presents how to use QUESO and the Multilevel method for sampling from a posterior PDF composed of the sum of two Gaussian distributions. Let's define D = [−250, 250] and the three distributions π prior : D → R + , f 1 : R → R + and f 2 : R → R + by: π prior = 1 |D| = 1 500 , ∀ θ ∈ D f 1 (θ) = 1 (2π) 1/2 |V 1 | exp − 1 2 (θ − µ 1 ) T V −1 1 (θ − µ 1 ) , ∀ θ ∈ R f 2 (θ) = 1 (2π) 1/2 |V 2 | exp − 1 2 (θ − µ 2 ) T V −1 2 (θ − µ 2 ) , ∀ θ ∈ R, (6.6.1) where µ 1 = 10, V 1 = 1 2 , µ 2 = 100, V 2 = 5 2 . In this example, we want to sample the posterior PDF given by: π posterior (θ) ∝ 1 2 f 1 (θ) + 1 2 f 2 (θ) · π prior = f (θ) · π prior (6.6.2) where f (θ) = 1 2 f 1 (θ) + 1 2 f 2 (θ) is the likelihood function, which is depicted in Figure 6.6.1. Running the Example To run the executable provided (available after QUESO installation), and generate figures for the chains, PDFs, CDFs, etc., enter the following commands: Example Code The source code for the example is composed of 5 files: bimodal main.C (Listing 6.62), bimodal likelihood.h and bimodal likelihood.C (Listings 6.63 and 6.64), bimodal compute.h and bimodal compute.C (Listings 6.65 and 6.66). 148 } Listing 6.66: File bimodal compute.C. Input File QUESO reads an input file for solving statistical problems, which provides options for the Multilevel or MCMC method. In this example, the Multilevel method is chosen to sample from the distribution. Many variables are common to both MCMC and Multilevel method, especially because the Multilevel method also has the option of delaying the rejection of a candidate. The names of the variables have been designed to be informative in this case as well: env: refers to QUESO environment; ip: refers to inverse problem; ml: refers to Multilevel; dr: refers to delayed rejection; rawChain: refers to the raw, entire chain; filteredChain: refers to a filtered chain (related to a specified lag); last: refers to instructions specific for the last level of the Multilevel algorithm. The user may select options for a specific level by naming its number, i.e., in case the user wants to define a different number of extra stages together with the scales for each stage (in the DRAM part of the ML algorithm) for the level 3, he/she may include the following instructions: i p _ m l _ 3 _ d r _ m a x N u m E x t r a S t a g e s = 1 i p _ m l _ 3 _ d r _ l i s t O f S c a l e s F o r E x t r a S t a g e s = 3.333 in the input file. The options used for solving this example are displayed in Listing 6.67. Create your own Makefile Similarly to the other examples presented in this user's manual and also available with QUESO distribution, a user-created makefile is available: 'Makefile bimodal violeta'. When adapted to the user's settings, namely paths for QUESO required libraries, it may be used to compile the code and create the executable bimodal_gsl. Thus, to compile, build and execute the code, the user just needs to run the following commands in the same directory where the files are: Again, the 'export' instruction above is only necessary if the user has not saved it in his/her .bashrc file. Data Post-Processing and Visualization According to the specifications of the input file in Listing 6.67, both a folder named outputData and a the following files should be generated: rawChain_ml.m display_sub0.txt The sequence of Matlab commands is identical to the ones presented in Sections 6.1.5, 6.2.5, 6.3.8 and 6.4.7; therefore, are omitted here. The reader is invited to explore the Matlab files plot likelihood normalized taus.m, plot likelihood unnormalized taus.m and/or plot all.m, for details of how the figures have been generated. KDE and CDF Plots Intermediary Likelihood Plots hysteretic This example replicates the problem in "Section 4.3 A Hysteretic Model Class " of [8], and which is also discussed in [6]. In this example we consider the nonlinear seismic response of a four-story building. This response is modeled with an inelastic shear building model with some linear viscous damping and hysteretic bilinear interstory restoring forces [6]. More specifically, let t ≥ 0 denote time, let a g (t) be a given total acceleration at the base (Fig. 13), and for the i-th floor [degree of freedom (dof)], 1 ≤ i ≤ N o ≡ 4, let us denote: m i = lumped mass, q i (t) = horizontal displacement, F i (t) = hysteretic restoring force (6.7.1) The hysteretic restoring force is illustrated in Figure 6.7.1 and the horizontal base (ground) acceleration (input data) used in [8] is illustrated in 6.7.2. We also define the mass matrix M and the stiffness matrix K: M =     m 1 0 0 0 0 m 2 0 0 0 0 m 3 0 0 0 0 m 4     and K =     k 1 + k 2 −k 2 0 0 −k 2 k 2 + k 3 −k 3 0 0 −k 3 k 3 + k 4 −k4 0 0 −k 4 k 4     and the Rayleigh damping matrix C = ρM + γK for given positive scalar parameters ρ and γ. The response q(t) ≡ [q 1 (t), q 2 (t), q 3 (t), q 4 (t)] is modeled as satisfying the equation of motion: Mq(t) + Cq(t) + F (t) = −M ·    1 . . . 1    4×1 · a g (t), (6.7.2) Figure 6.7.2: Horizontal base acceleration (input data) used in the hysteretic test problem [8]. where F (t) ≡ [F 1 (t), F 2 (t), F 3 (t), F 4 (t) ]. In this model, the hysteretic restoring force F (t) depends on the whole time history [q(t),q(t)] of responses from the initial instant until time t. The (noisy) measured data y = (y 1 , y 2 , y 3 , y 4 ) available for model calibration consists of 4 s of accelerometer data at each floor (refer to Fig. 6.7.3), with a sample interval ∆t = 0.01 s. The simulated dynamic data was obtained by adding Gaussian white noise to the output simulation of the hysteretic model with the following input values: m 1 = m 2 = m 3 = m 4 = 2 × 10 4 kg, k 1 = 2.2 × 10 7 N m −1 , k 2 = 2.0 × 10 7 N m −1 , k 3 = 1.7 × 10 7 N m −1 , k 4 = 1.45 × 10 7 N m −1 , r 1 = r 2 = r 3 = r 4 = 0.1, u 1 = u 2 = 8 × 10 −3 m, u 3 = u 4 = 7 × 10 −3 m, ρ = 7.959 × 10 −1 , γ = 2.5 × 10 −3 , σ 2 = 0.6 2 , where for i = 1, 2, 3, 4, k i is the initial stiffness, r i is the post-yield stiffness reduction factor, and u i is yield displacement. According to Cheung and Prudencio [8], these input values were chosen deliberately so that the excitation a g did not cause some of the upper floors to enter the nonlinear regime; that is, so that our test inverse problem did not become globally identifiable. In this section, 400 time-steps are used, as the data is available at instants t n = (n − 1) × ∆t, 1 ≤ n ≤ N T ≡ 401, ∆t = 0.01 however, Cheung and Prudencio used only 250 time steps [8]. An additive noise is assumed to be present in the measurements; i.e., y i (n) = q i (n) + ε i (n), 1 ≤ i ≤ N o , 1 ≤ n ≤ N T ≡ 401, where q i (n; θ 2 , ..., θ 15 ) denotes the output at time t n = n∆t (∆t = 0.01s) at the i-th observed degree of freedom predicted by the proposed structural model, and y i (n) denotes the corresponding measured output. They considered a total of 15 unknown parameters θ = (θ 1 , ..., θ 15 ) and modeled the variables ε i as independently and identically distributed Gaussian variables with mean zero and some unknown prediction-error variance σ 2 . The variance σ 2 is assumed to be the same for all N o = 4 floors. The first component θ 1 is equal to the prediction error variance σ 2 and the other 14 parameters are related to the four triples (k i , r i , u i ), 1 ≤ i ≤ N o (see Fig. 6.7.1), to ρ, and to γ. The likelihood function is given by: f (y|θ) = 1 (2πσ 2 ) NoN T /2 exp − 1 2σ 2 No i=1 N T n=1 [y i (t n ) − q i (t n ; θ 2 , . . . , θ 15 )] 2 . (6.7.3) An inverse gamma prior was used for θ 1 = σ 2 , and a 14-dimensional Gaussian prior was used for θ 2 , ..., θ 15 with zero mean and diagonal covariance matrix equal to a scaled identity matrix. Running the Example To run the executable provided (available after QUESO installation), and generate figures for the chains, KDEs, CDFs, autocorrelation and scatter plots, enter the following commands: Additional figures may be generated if the user allows the procedure debug hyst( be called by the compiler in Line 11 of file example main.C; in that case, call the function cpp gen.m inside Matlab/Octave. Example Code The source code for the example is composed of 5 files: example main.C (Listing 6.68), example likelihood.h and example likelihood.C (Listings 6.69 and 6.70), example compute.h and example compute.C (Listings 6.71 and 6.72), and finally example hyst.h and example hyst.C, which contain the Hysteretic model properly said. Note that in line 11 of Listings 6.68 the '#if 1' directive tells the compiler that the application will call compute(), which internally uses QUESO and the Multilevel algorithm. On the contrary, the user may calculate the hysteretic force without uncertainty by changing the directive to '#if 0', which can assist the analysis of the resulting data. Input File The options used for solving this example are displayed in Listing 6.73. Listing 6.73: Options for QUESO library used in application code (Listings 6.68-6.72). Create your own Makefile Similarly to the other examples presented in this user's manual and also available with QUESO distribution, a user-created makefile is available: 'Makefile hysteretic violeta' which may personalized to each user's computer settings and used to compile the code and create the executable hysteretic_gsl. Thus to compile, build and execute the code, commands similar to the following should be entered: Again, the 'export' instruction above is only necessary if the user has not saved the path for the libraries used during QUESO installation in his/her .bashrc file. Data Post-Processing and Visualization According to the specifications of the input file in Listing 6.73, both a folder named outputData and a the following files should be generated: rawChain_ml.m display_sub0.txt Note that in this hysteretic problem a total of 13 levels are required for the Multilevel method (e.g. see the contents of file rawChain ml.m). The sequence of Matlab commands is identical to the ones presented in Sections 6.1.5, 6.2.5, 6.3.8 and 6.4.7; therefore, are omitted here. The reader is invited to explore the Matlab files plot all.m and/or cpp gen.m, for details of how the figures have been generated. Permission for modification of the technical content is crucial too. When people modify the software, adding or changing features, if they are conscientious they will change the manual too−so they can provide accurate and clear documentation for the modified program. A manual that leaves you no choice but to write a new manual to document a changed version of the program is not really available to our community. KDE Plots Some kinds of limits on the way modification is handled are acceptable. For example, requirements to preserve the original author's copyright notice, the distribution terms, or the list of authors, are ok. It is also no problem to require modified versions to include notice that they were modified. Even entire sections that may not be deleted or changed are acceptable, as long as they deal with nontechnical topics (like this one). These kinds of restrictions are acceptable because they don't obstruct the community's normal use of the manual. However, it must be possible to modify all the technical content of the manual, and then distribute the result in all the usual media, through all the usual channels. Otherwise, the restrictions obstruct the use of the manual, it is not free, and we need another manual to replace it. Please spread the word about this issue. Our community continues to lose manuals to proprietary publishing. If we spread the word that free software needs free reference manuals and free tutorials, perhaps the next person who wants to contribute by writing documentation will realize, before it is too late, that only free manuals contribute to the free software community. If you are writing documentation, please insist on publishing it under the GNU Free Documentation License or another free documentation license. Remember that this decision requires your approval−you don't have to let the publisher decide. Some commercial publishers will use a free license if you insist, but they will not propose the option; it is up to you to raise the issue and say firmly that this is what you want. If the publisher you are dealing with refuses, please try other publishers. If you're not sure whether a proposed license is free, write to lice give the recipients all the rights that we gave you. You must make sure that they, too, receive or can get the source code. If you link other code with the library, you must provide complete object files to the recipients, so that they can relink them with the library after making changes to the library and recompiling it. And you must show them these terms so they know their rights. We protect your rights with a two-step method: (1) we copyright the library, and (2) we offer you this license, which gives you legal permission to copy, distribute and/or modify the library. To protect each distributor, we want to make it very clear that there is no warranty for the free library. Also, if the library is modified by someone else and passed on, the recipients should know that what they have is not the original version, so that the original author's reputation will not be affected by problems that might be introduced by others. Finally, software patents pose a constant threat to the existence of any free program. We wish to make sure that a company cannot effectively restrict the users of a free program by obtaining a restrictive license from a patent holder. Therefore, we insist that any patent license obtained for a version of the library must be consistent with the full freedom of use specified in this license. Most GNU software, including some libraries, is covered by the ordinary GNU General Public License. This license, the GNU Lesser General Public License, applies to certain designated libraries, and is quite different from the ordinary General Public License. We use this license for certain libraries in order to permit linking those libraries into non-free programs. When a program is linked with a library, whether statically or using a shared library, the combination of the two is legally speaking a combined work, a derivative of the original library. The ordinary General Public License therefore permits such linking only if the entire combination fits its criteria of freedom. The Lesser General Public License permits more lax criteria for linking other code with the library. We call this license the "Lesser" General Public License because it does Less to protect the user's freedom than the ordinary General Public License. It also provides other free software developers Less of an advantage over competing non-free programs. These disadvantages are the reason we use the ordinary General Public License for many libraries. However, the Lesser license provides advantages in certain special circumstances. For example, on rare occasions, there may be a special need to encourage the widest possible use of a certain library, so that it becomes a de-facto standard. To achieve this, non-free programs must be allowed to use the library. A more frequent case is that a free library does the same job as widely used non-free libraries. In this case, there is little to gain by limiting the free library to free software only, so we use the Lesser General Public License. In other cases, permission to use a particular library in non-free programs enables a greater number of people to use a large body of free software. For example, permission to use the GNU C Library in non-free programs enables many more people to use the whole GNU operating system, as well as its variant, the GNU/Linux operating system. Although the Lesser General Public License is Less protective of the users' freedom, it does ensure that the user of a program that is linked with the Library has the freedom and the wherewithal to run that program using a modified version of the Library. The precise terms and conditions for copying, distribution and modification follow. Pay close attention to the difference between a "work based on the library" and a "work that uses the library". The former contains code derived from the library, whereas the latter must be combined with the library in order to run. TERMS AND CONDITIONS FOR COPYING, DISTRIBU-TION AND MODIFICATION 0. This License Agreement applies to any software library or other program which contains a notice placed by the copyright holder or other authorized party saying it may be distributed under the terms of this Lesser General Public License (also called "this License"). Each licensee is addressed as "you". A "library" means a collection of software functions and/or data prepared so as to be conveniently linked with application programs (which use some of those functions and data) to form executables. The "Library", below, refers to any such software library or work which has been distributed under these terms. A "work based on the Library" means either the Library or any derivative work under copyright law: that is to say, a work containing the Library or a portion of it, either verbatim or with modifications and/or translated straightforwardly into another language. (Hereinafter, translation is included without limitation in the term "modification".) "Source code" for a work means the preferred form of the work for making modifications to it. For a library, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the library. Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running a program using the Library is not restricted, and output from such a program is covered only if its contents constitute a work based on the Library (independent of the use of the Library in a tool for writing it). Whether that is true depends on what the Library does and what the program that uses the Library does. 1. You may copy and distribute verbatim copies of the Library's complete source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and distribute a copy of this License along with the Library. You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee. 2. You may modify your copy or copies of the Library or any portion of it, thus forming a work based on the Library, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions: a) The modified work must itself be a software library. b) You must cause the files modified to carry prominent notices stating that you changed the files and the date of any change. c) You must cause the whole of the work to be licensed at no charge to all third parties under the terms of this License. d) If a facility in the modified Library refers to a function or a table of data to be supplied by an application program that uses the facility, other than as an argument passed when the facility is invoked, then you must make a good faith effort to ensure that, in the event an application does not supply such function or table, the facility still operates, and performs whatever part of its purpose remains meaningful. (For example, a function in a library to compute square roots has a purpose that is entirely well-defined independent of the application. Therefore, Subsection 2d requires that any application-supplied function or table used by this function must be optional: if the application does not supply it, the square root function must still compute square roots.) These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Library, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Library, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Library. In addition, mere aggregation of another work not based on the Library with the Library (or with a work based on the Library) on a volume of a storage or distribution medium does not bring the other work under the scope of this License. 3. You may opt to apply the terms of the ordinary GNU General Public License instead of this License to a given copy of the Library. To do this, you must alter all the notices that refer to this License, so that they refer to the ordinary GNU General Public License, version 2, instead of to this License. (If a newer version than version 2 of the ordinary GNU General Public License has appeared, then you can specify that version instead if you wish.) Do not make any other change in these notices. Once this change is made in a given copy, it is irreversible for that copy, so the ordinary GNU General Public License applies to all subsequent copies and derivative works made from that copy. This option is useful when you wish to copy part of the code of the Library into a program that is not a library. 4. You may copy and distribute the Library (or a portion or derivative of it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange. If distribution of object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place satisfies the requirement to distribute the source code, even though third parties are not compelled to copy the source along with the object code. 5. A program that contains no derivative of any portion of the Library, but is designed to work with the Library by being compiled or linked with it, is called a "work that uses the Library". Such a work, in isolation, is not a derivative work of the Library, and therefore falls outside the scope of this License. However, linking a "work that uses the Library" with the Library creates an executable that is a derivative of the Library (because it contains portions of the Library), rather than a "work that uses the library". The executable is therefore covered by this License. Section 6 states terms for distribution of such executables. When a "work that uses the Library" uses material from a header file that is part of the Library, the object code for the work may be a derivative work of the Library even though the source code is not. Whether this is true is especially significant if the work can be linked without the Library, or if the work is itself a library. The threshold for this to be true is not precisely defined by law. If such an object file uses only numerical parameters, data structure layouts and accessors, and small macros and small inline functions (ten lines or less in length), then the use of the object file is unrestricted, regardless of whether it is legally a derivative work. (Executables containing this object code plus portions of the Library will still fall under Section 6.) Otherwise, if the work is a derivative of the Library, you may distribute the object code for the work under the terms of Section 6. Any executables containing that work also fall under Section 6, whether or not they are linked directly with the Library itself. 6. As an exception to the Sections above, you may also combine or link a "work that uses the Library" with the Library to produce a work containing portions of the Library, and distribute that work under terms of your choice, provided that the terms permit modification of the work for the customer's own use and reverse engineering for debugging such modifications. You must give prominent notice with each copy of the work that the Library is used in it and that the Library and its use are covered by this License. You must supply a copy of this License. If the work during execution displays copyright notices, you must include the copyright notice for the Library among them, as well as a reference directing the user to the copy of this License. Also, you must do one of these things: a) Accompany the work with the complete corresponding machine-readable source code for the Library including whatever changes were used in the work (which must be distributed under Sections 1 and 2 above); and, if the work is an executable linked with the Library, with the complete machine-readable "work that uses the Library", as object code and/or source code, so that the user can modify the Library and then relink to produce a modified executable containing the modified Library. (It is understood that the user who changes the contents of definitions files in the Library will not necessarily be able to recompile the application to use the modified definitions.) b) Use a suitable shared library mechanism for linking with the Library. A suitable mechanism is one that (1) uses at run time a copy of the library already present on the user's computer system, rather than copying library functions into the executable, and (2) will op-erate properly with a modified version of the library, if the user installs one, as long as the modified version is interface-compatible with the version that the work was made with. c) Accompany the work with a written offer, valid for at least three years, to give the same user the materials specified in Subsection 6a, above, for a charge no more than the cost of performing this distribution. d) If distribution of the work is made by offering access to copy from a designated place, offer equivalent access to copy the above specified materials from the same place. e) Verify that the user has already received a copy of these materials or that you have already sent this user a copy. For an executable, the required form of the "work that uses the Library" must include any data and utility programs needed for reproducing the executable from it. However, as a special exception, the materials to be distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable. It may happen that this requirement contradicts the license restrictions of other proprietary libraries that do not normally accompany the operating system. Such a contradiction means you cannot use both them and the Library together in an executable that you distribute. 7. You may place library facilities that are a work based on the Library side-by-side in a single library together with other library facilities not covered by this License, and distribute such a combined library, provided that the separate distribution of the work based on the Library and of the other library facilities is otherwise permitted, and provided that you do these two things: a) Accompany the combined library with a copy of the same work based on the Library, uncombined with any other library facilities. This must be distributed under the terms of the Sections above. b) Give prominent notice with the combined library of the fact that part of it is a work based on the Library, and explaining where to find the accompanying uncombined form of the same work. 8. You may not copy, modify, sublicense, link with, or distribute the Library except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, link with, or distribute the Library is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance. 9. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Library or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Library (or any work based on the Library), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Library or works based on it. 10. Each time you redistribute the Library (or any work based on the Library), the recipient automatically receives a license from the original licensor to copy, distribute, link with or modify the Library subject to these terms and conditions. You may not impose any further restrictions on the recipients' exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties with this License. 11. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Library at all. For example, if a patent license would not permit royalty-free redistribution of the Library by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Library. If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply, and the section as a whole is intended to apply in other circumstances. It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice. This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License. 12. If the distribution and/or use of the Library is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Library under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License. 13. The Free Software Foundation may publish revised and/or new versions of the Lesser General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Library specifies a version number of this License which applies to it and "any later version", you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Library does not specify a license version number, you may choose any version ever published by the Free Software Foundation. 14. If you wish to incorporate parts of the Library into other free programs whose distribution conditions are incompatible with these, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally. THE LIBRARY IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE LIBRARY, TO THE EXTENT PERMITTED BY NO WARRANTY BECAUSE END OF TERMS AND CONDITIONS How to Apply These Terms to Your New Programs If you develop a new library, and you want it to be of the greatest possible use to the public, we recommend making it free software that everyone can redistribute and change. You can do so by permitting redistribution under these terms (or, alternatively, under the terms of the ordinary General Public License). To apply these terms, attach the following notices to the library. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this library; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA Also add information on how to contact you by electronic and paper mail. You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the library, if necessary. Here is a sample; alter the names: Yoyodyne, Inc., hereby disclaims all copyright interest in the library 'Frob' (a library for tweaking knobs) written by James Random Hacker. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. Applicability and Definitions This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The "Invariant Sections" are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words. A "Transparent" copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not "Transparent" is called "Opaque". Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only. The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text. The "publisher" means any person or entity that distributes copies of the Document to the public. A section "Entitled XYZ" means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as "Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" of such a section when you modify the Document means that it remains a section "Entitled XYZ" according to this definition. The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. Verbatim Copying You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3. You may also lend copies, under the same conditions stated above, and you may publicly display copies. Copying in Quantity If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects. If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages. If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public. It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document. Modifications You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version: A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement. C. State on the Title page the name of the publisher of the Modified Version, as the publisher. D. Preserve all the copyright notices of the Document. E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices. F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below. G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice. H. Include an unaltered copy of this License. I. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence. J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission. K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein. L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles. M. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version. N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section. O. Preserve any Warranty Disclaimers. If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles. You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties-for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version. Combining Documents You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements". Collections of Documents You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document. Aggregation with Independent Works A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate. Translation Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title. Termination You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License. However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. Future Revisions of This License The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http: //www.gnu.org/copyleft/. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document. Relicensing "Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site. "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization. "Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document. An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008. The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing. ADDENDUM: How to use this License for your documents To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright (C) YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ''GNU Free Documentation License''. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with ... Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Key Statistical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The Software Stack of an Application Using QUESO . . . . . . . . . . . . . .51.4 Algorithms for solving Statistical Inverse Problems . . . . . . . . . . . . . . . 6 1.4.1 DRAM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.2 Adaptive Multilevel Stochastic Simulation Algorithm . . . . . . . . . . 10 1.5 Algorithms for solving the Statistical Forward Problem . . . . . . . . . . . . . 16 2 Installation 18 2.1 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.1 Obtain and Install QUESO Dependencies . . . . . . . . . . . . . . . . . 19 2.1.2 Prepare your LINUX Environment . . . . . . . . . . . . . . . . . . . . 20 2.2 Obtaining a Copy of QUESO . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Recommended Build Directory Structure . . . . . . . . . . . . . . . . . 21 2.3 Configure QUESO Building Environment . . . . . . . . . . . . . . . . . . . . . 21 2.4 Compile, Check and Install QUESO . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 QUESO Developer's Documentation . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Summary of Installation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 The Build Directory Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 vii viii 2.8 The Installed Directory Structure . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.9 Create your Application with the Installed QUESO . . . . . . . . . . . . . . . 26 3 C++ Classes in the Library 28 3.1 Core Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Environment Class (and Options) . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.3 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Templated Basic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Vector Set, Subset and Vector Space Classes . . . . . . . . . . . . . . . 34 3.2.2 Scalar Function and Vector Function Classes . . . . . . . . . . . . . . . 34 3.2.3 Scalar Sequence and Vector Sequence Classes . . . . . . . . . . . . . . 37 3.3 Templated Statistical Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.1 Vector Realizer Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.2 Vector Random Variable Class . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.3 Statistical Inverse Problem (and Options) . . . . . . . . . . . . . . . . 41 3.3.4 Metropolis-Hastings Solver (and Options) . . . . . . . . . . . . . . . . 43 3.3.5 Multilevel Solver (and Options) . . . . . . . . . . . . . . . . . . . . . . 43 3.3.6 Statistical Forward Problem (and Options) . . . . . . . . . . . . . . . . 49 3.3.7 Monte Carlo Solver (and Options) . . . . . . . . . . . . . . . . . . . . . 49 3.4 Miscellaneous Classes and Routines . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Important Remarks 52 4.1 Revisiting Input Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Revisiting Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Running with Multiple Chains or Monte Carlo Sequences . . . . . . . . . . . . 53 4.4 Running with Models that Require Parallel Computing . . . . . . . . . . . . . 54 4.5 A Requirement for the DRAM Algorithm . . . . . . . . . . . . . . . . . . . . . 54 5 Global Sensitivity Analysis 55 5.1 Sensitivity Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1.1 First Order Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1.2 Total Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1.3 Estimation of S(θ i ) and T (θ i ) . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 QUESO Examples 63 6.1 simpleStatisticalInverseProblem . . . . . . . . . . . . . . . . . . . . . . . 64 6.1.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.1.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.1.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Figure 1 . 12.1 represents general inverse and forward problems respectively. There are many possible sources of uncertainty on a computational model. First, d need not be equal to the actual values of observables because of errors in the measurement process. Second, the values of the input parameters to the phenomenon might not be precisely known. Third, the appropriate set of equations governing the phenomenon might not be well understood. representation of (a) a generic forward problem and (b) a generic inverse problem. Figure 1 .2. 2 : 12The representation of a statistical forward problem. Θ denotes a random variable related to parameters, θ denotes a realization of Θ and Q denotes a random variable related to quantities of interest. Figure 1 . 12.3: The representation of a statistical inverse problem. Θ denotes a random variable related to parameters, θ denotes a realization of Θ and r denotes model equations, y denotes some model output data and d denotes experimental data. a) multiple instances of a problem where the physical model requires a single processor, or (b) multiple instances of a problem where the physical model requires multiple processors, or (c) independent sets of types (a) and (b). Figure 1 1Figure 1.3.1: An application software stack. QUESO requires the input of a likelihood routine π like : R n → R + for IPs and of a QoI routine q : R n → R m for FPs. These application level routines provide the bridge between the statistical algorithms in QUESO, physics knowledge in the model library, and relevant experimental (calibration and validation) data. • Recursively define α i : R n × . . . θ|D) = f (θ|D, M j ) τ · π prior (θ|M j ), = 0, 1, . . . , L,(1.4.4) rtest ) : PASSED : Test 1 ( TGA Validation Cycle ) -------------------------------------------------------------------- keme lli@ma rgarid a :~/ LIBRARIES / QUESO -0.51.0/ bin$ ./ queso_version ---------------------------------------------------------------QUESO Library : Version = 0.47.1 (47.1) Development Build example_main . o ex amp le _l ik el ih oo d . o example_qoi . o example_compute . o $ ( CXX ) example_main . o e xam pl e_ li ke li ho od . o example_qoi . o \ example_compute . o -o example_gsl $ ( LIBS ) %. o : %. C $ ( CXX ) $ ( INC_PATHS ) $ ( CXXFLAGS ) $ < CHAPTER 3 Figure 3 . 31.1 depicts class diagram for the environment class and Figure 3.1.2 display its collaboration graph; and Figure 3.1.3 displays environment options class. Figure 3 . 31.1: The class diagram for the Environment class described in Section 3.1.1. Figure 3 .1. 2 : 32Collaboration graph for the environment class described in Section 3.1.1. Figure 3 .1. 3 : 33The environment options class with its attributes and methods. Figure 3 . 31.4: The class diagram for the vector class described in Section 3.1.2. Joint PDF, marginal PDF, and CDF are all examples of scalar functions present in statistical problems. QUESO currently supports basic PDFs such as uniform and Gaussian and also more complex PDFs, such as the ones coming from a Bayesian analysis. They are implemented in the classes UniformJointPdf, GaussianJointPdf, and BayesianJointPdf, respectively. The posterior PDF may be represented within QUESO by GenericJointPdf. See Diagram 3.2.2 for the scalar function class. The handling of vector functions within QUESO is also quite straightforward. Indeed, the definition of a vector function q : B ⊂ R n → R m requires only the extra specifi-() and 10 more... + mpiSum() + matlabLinearInterpExtrap() + print() + subReadContents() + subWriteContents() -copy() -resetLU() -multiply() -internalSvd() Figure 3 . 31.5: The class diagram for the matrix class. Figure 3 . 32.1: The class diagram for vector set, vector subset and vector space classes, described in Section 3.2.1. cation of the image vector space R m . The classes representing the vector function class GenericVectorFunction and ConstantVectorFunction are derived from BaseVectorFunction and are presented in Diagram 3 Figure 3 . 32.2: The class diagram for the scalar function class. Figures 3.2.4 and 3.2.5 display the class diagram for the scalar sequence and vector sequence classes, respectively. Figure 3 . 32.3: The class diagram for the vector function class described in Section 3.2.2. Figure 3 . 32.4: The class diagram for the scalar sequence class. Figure 3 .2. 5 : 35The class diagram for the vector sequence class. Figure 3 . 33.1: The class diagram for the vector random variable class. .3.2a. One important characteristic of the QUESO design is that it separates 'what the problem is' from 'how the problem is solved'. The prior and the posterior RV are instances of the BaseVectorRv<V,M> class, while the likelihood function is an instance of the BaseScalarFunction<V,M> class. The solution of a SIP is computed by calling either solveWithBayesMetropolisHastings() or solveWithBayesMLSampling(), which are member functions of the class StatisticalInverseProblem<P_V,P_M> class. Upon return from a solution operation, the posterior RV is available through the postRv() member function. More details are provided about solveWithBayesMetropolisHastings() and solveWithBayesMLSampling() in Sections 3.3.4 and 3.3.5, respectively. Figure 3 . 33.2b displays the StatisticalInverseProblemOptions class, i.e. that class that handles a variety of options for solving the SIP. Such options may be provided to QUESO by the user's input file; and they are listed in Figure 3 .3. 2 : 32The statistical inverse problem class, which implements the representation inFigure 1.2.3, and statistical inverse problem options class. Figure 3 .3. 3 : 33The Metropolis-Hastings sequence generator class and the Metropolis-Hastings sequence generator options class. Figure 3 .3. 4 : 34MLSampling<P_V,P_M>. This class implements the Adaptive Multilevel Stochastic Simulation Algorithm of Cheung and Prudencio [8]. The Multilevel sequence generator class is assisted by two extra classes, MLSamplingOptions and MLSamplingLevelOptions, for handling the options to be used. The Multilevel class, the Multilevel options and level options classes are depicted in Figure 3.3.5. A collaboration graph for the Multilevel class is presented in Figure 3.3.6; whereas its associated options are presented in Table 3.3.3. Collaboration graph of the Metropolis-Hastings sequence generator class. Figure 3 .3. 5 :Figure 3 . 353The Multilevel sequence generator options class (3.3.5a) and its associated classes for handling options. 3.6: Collaboration graph of the Multilevel sampling class. Figure 3 .3. 7 : 37The statistical forward problem class, which implements the representation inFigure 1.2.2, and the statistical forward problem options class. QUESO::MonteCarloSG< P_V, P_M, Q_V, Figure 3 . 33.8: The Monte Carlo sequence generator class and the Monte Carlo sequence generator options class. 6 :A 6Use the two sets of data, A and B, to generate intermediate data files comprising samples, A for both m and c. 7: Repeat steps 2-4 four times to generate the derived files: m qoi samplesAi.txt, m qoi samplesBi.txt, c qoi samplesAi.txt and c qoi samplesAi.txt. (Instead of estimating the QoI for pseudorandom samples from the prior, the QoI is now computed for the set of samples in corresponding intermediate data files generated in the previous step.) . 1 : 1class P_V = QUESO :: GslVector , class P_M = QUESO :: GslMatrix , class Q_V = QUESO :: GslVector , class Q_M = QUESO :: GslMatrix > class Qoi_mc : public QUESO :: BaseVectorFunction < P_V , P_M , Q_V , Q_M > { public : Qoi_mc ( const char * prefix , const QUESO :: VectorSet < P_V , P_M > & domainSet , const QUESO :: VectorSet < Q_V , Q_M > & imageSet ) ; virtual~Qoi_mc () ; virtual void compute ( const P_V & domainVector , const P_V * domainDirection , Q_V & imageVector , QUESO :: DistArray < P_V * > * gradVectors , QUESO :: DistArray < P_M * > * hessianMatrices , QUESO :: DistArray < P_V * > * hess ianEffec ts ) const ; private : double x_loc ; mutable double m ,c , mf , cf , count ; mutable std :: fstream qoi_samples ; mutable std :: fstream samples ; File sensitivity mc.h. 1 #i n c l u d e <cmath> 2 #i n c l u d e <f s t r e a m > 3 #i n c l u d e <i o m a n i p > // f o r s e t p r e c i s i o n 4 5 #i n c l u d e <q u e s o / G s l V e c t o r . h> 6 #i n c l u d e <q u e s o / G s l M a t r i x . h> 7 #i n c l u d e < s e n s i t i v i t y m c . h> 8 9 template < class P_V , class P_M , class Q_V , class Q_M > 10 Qoi_mc < P_V , P_M , Q_V , Q_M >:: Qoi_mc ( const char * prefix , 11 const QUESO :: VectorSet < P_V , P_M > & domainSet , 12 const QUESO :: VectorSet < Q_V , Q_M > & imageSet ) 13 : QUESO :: BaseVectorFunction < P_V , P_M , Q_V , Q_M >( prefix , domainSet , imageSet ) , 14 x_loc (3) . open ( " c_ qo i_ s am pl es A i . txt " , std :: fstream :: in | std :: fstream :: out | std :: fstream :: app ) ;40 41 // ----Generate Qoi using samples from a text files -------------42 count ++; 43 samples . open ( " ./ files_sense / c_samples_Ai . txt " , std :: fstream :: in | std :: fstream :: out | std :: // -------Generate Qoi using pseudo -random MC samples ------------57 // std :: cout << " m = " << domainVector [0] << std :: endl ; 58 // m = domainVector [0]; // Sample of the RV ' line slope ' 59 // c = domainVector [1]; // Sample of the RV 'y -intercept ' std :: setprecision (4) << y_obs << " \ t \ t " << m << " \ t \ t " << c << std :: endl ; template class Qoi_mc < QUESO :: GslVector , QUESO :: GslMatrix , QUESO :: GslVector , Listing 5.2: File sensitivity mc.C. Figure 5 . 53.1: (a) Analysis of convergence for the first order sensitivity indices for slope, m and y-intercept, c with increasing sample size. (b) Bar-graph representation of estimates for S(θ i ) based on estimators suggested by Sobol[47] and Saltelli et al.[44] using 25000 pseudo-random samples.This chapter assumes that the user has successfully installed QUESO and its dependencies. It presents a variety of examples of how to use QUESO in order to develop applications.There are examples that solve a statistical inverse problem (Sections 6.1, 6.5, 6.6 and 6.7), a statistical forward problem (Section 6.2) or a combination of both, where the solution of the former serves as input to the later (Sections 6.3 and 6.4). Three of the first four examples (Sections 6.1, 6.3 -6.4) use the DRAM algorithm for solving the SIP, and the last three examples use the Multilevel algorithm. Each section presents: result, the user should have created several of PNG figures containing marginal posterior PDF, chain positions, histograms, cumulative density distributions and autocorrelation of both parameters. The name of the figure files have been chosen to be informative, as shown in the Listing above. file must be specified in command line as argv[1] , just after executable argv [0] " ) ; QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( MPI_COMM_WORLD , argv[1] , " " , NULL ) ; #e l s e QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( argv[1] , " " , NULL ) ; checking : the user , at the application level , expects // vector ' paramValues ' to have size 2.U Q _ F A T A L _ T E S T _ M A C R O (paramValues . sizeGlobal () code exemplifies multiple Metropolis -Hastings solvers , each calling this likelihood // routine . In this simple example , only node 0 in each subenvi ronment does the job even // though there might be more than one node per sub -environment . In a more realistic // situation , if the user is asking for multiple nodes per subenvironment , then the model // code in the likelihood routines might really demand more than one node . Here we use // ' env . subRank () ' only . A realistic application might want to use either ' env . subComm () ' // or ' env . subComm () . Comm () '. double result = 0.; const QUESO :: B as eE n vi ro nm e nt & env = paramValues . env () ; if ( env . subRank () == 0) { const QUESO :: GslVector & meanVector = *(( l i k e l i h o o d R o u t i n e _ D a t a T y p e *) fu nc t io nD at a Pt r ) -> meanVector ; const QUESO :: GslMatrix & covMatrix = *(( l i k e l i h o o d R o u t i n e _ D a t a T y p e *) fu nc t io nD at a Pt r ) -> covMatrix ; QUESO :: GslVector diffVec ( paramValues -meanVector ) ; result = scalarProduct ( diffVec , covMatrix . inver tMultip ly ( diffVec ) compute ( const QUESO :: Fu l lE nv ir o nm en t & env ) { // Step 1 of 5: Instantiate the parameter space QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpace ( env , " param_ " , 2 , NULL ) ; 13 // Step 2 of 5: Instantiate the parameter domain QUESO :: GslVector paramMins ( paramSpace . zeroVector () ) ; paramMins . cwSet ( -INFINITY ) ; QUESO :: GslVector paramMaxs ( paramSpace . zeroVector () ) ; paramMaxs . cwSet ( INFINITY ) ; 18 QUESO :: BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomain ( " param_ " , paramSpace , paramMins , paramMaxs ) ; // Step 3 of 5: Instantiate the likelihood function object QUESO :: GslVector meanVector ( paramSpace . zeroVector (Instantiate the inverse problem QUESO :: UniformVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > 43 priorRv ( " prior_ " , paramDomain ) ; QUESO :: GenericVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > postRv ( " post_ " , paramSpace ) ; QUESO :: StatisticalInverseProblem < QUESO :: GslVector , QUESO :: GslMatrix > ip ( " " , NULL , priorRv , likelihoodFunctionObj , postRv ) ; 48 // Step 5 of 5: Solve the inverse problem QUESO :: GslVector paramInitials ( paramSpace . zeroVector () ) ; paramInitials [0] = 0.example compute.C. # ' i p ' : i n f o r m a t i o n f o r M e t r o p o l i s −H a s t i n g s a l g o r i t h m # ############################################## #i p m h h e l p= a n y t h i n g i p _ m h _ d a t a O u t p u t F i l e N a for QUESO library used in application code (Listings 6.1-6.5). CXX ) $ ( INC_PATHS ) $ ( CXXFLAGS ) $ < Listing 6.7: Makefile for the application code in Listings 6.1-6.5 Thus, to compile, build and execute the code, the user just needs to run the following commands in the same directory where the files are: $ cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / s i m p l e S t a t i s t i c a l I n v e r s e P r o b l e m / $ export L D_ L IB RA RY _ PA TH = $ L D _ L I B R A R Y _ P A T H :\ $HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib $ make -f M a k e f i l e _ e x a m p l e _ m a r g a r i d a $ ./ s i m p l e _ s i p _ e x a m p l e example . inp plot ( lags , ACF_raw , 'b --* ' , lags , ACF_filt , ' b * -' , lags2 , ACF_raw2 , 'g --* ' , lags2 , ACF_filt2 , ' g * -' , ' linewidth ' ,3) ; >> h = legend ( '\ theta_1 , raw chain ' , '\ theta_1 , filtered chain ' , '\ theta_2 , raw chain ' , '\ theta_2 , filtered chain ' , ' location ' , ' northeast ') ; Listing 6.9: Matlab code for the autocorrelation plots depicted inFigure 6.1.1. Figure 6 .1. 1 : 61Autocorrelation plots obtained with QUESO for the SIP. : Matlab code for the KDE plots displayed in the left of Figure 6.1.2. Figure 6 .1. 2 : 62Kernel Density Estimation. QUESO results for estimation of the KDE of θ 1 and θ 2 are plotted against the analytical expressions π post (θ o r r _ m a t r i x _ t h e t a 1 _ t h e t a 2 = corr ( i p _ m h _ r a w C h a i n _ u n i f i e d ) result, the user should have created several of PNG figures containing marginal posterior PDF, chain positions of the parameters and the QoI, histogram, cumulative density distribution and autocorrelation. The name of the figure files have been chosen to be informative, as shown in the Listing above. argc != 2 , QUESO :: UQ_UNAVAILABLE_RANK , " main () " , " input file must be specified in command line as argv[1] , just after executable argv [0] " ) ; QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( MPI_COMM_WORLD , argv[1] , " " , NULL ) ; #e l s e QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( argv[1] , " " , NULL ) ; .12: File simple sfp example main.C.:: DistArray < QUESO :: GslVector * >* gradVectors , QUESO :: DistArray < QUESO :: GslMatrix * >* hessianMatrices , QUESO :: DistArray < QUESO :: GslVector * >* hessi anEffect s ) { // Logic just to avoid warnings from INTEL compiler const QUESO :: GslVector * aux1 = p aramDire ction ; if ( aux1 ) {}; QUESO :: DistArray < QUESO :: GslVector * >* aux2 = gradVectors ; if ( aux2 ) {}; aux2 = he ssianEff ects ; QUESO :: DistArray < QUESO :: GslMatrix * >* aux3 = h e ss ia nM a tr ic es ; if ( aux3 ) {};// Just checking : the user , at the application level , expects // vector ' paramValues ' to have size 2 and // vector ' qoiValues ' to have size 1. :: UQ_UNAVAILABLE_RANK , " qoiRoutine () " , " qoiValues vector does not have size 1 " ) ; code exemplifies multiple Monte Carlo solvers , each calling this qoi routine . // In this simple example , only node 0 in each sub -environment does the job even though // there might be more than one node per sub -environment . // In a more realistic situation , if the user is asking for multiple nodes per sub -// environment , then the model code in the qoi routine might really demand more than one // node . Here we use ' env . subRank () ' only . A realistic application might want to use // either ' env . subComm () ' or ' env . subComm () . Comm () '.const QUESO :: B as eE n vi ro nm e nt & env = paramValues . env () ; if ( env . subRank () compute ( const QUESO :: Fu l lE nv ir o nm en t & env ) ; #e n d i f Listing 6.15: File simple sfp example compute.h. compute ( const QUESO :: Fu l lE nv ir o nm en t & env ) { // Step 1 of 6: Instantiate the parameter space QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpace ( env , " param_ " , 2 , NULL ) ; 13 // Step 2 of 6: Instantiate the parameter domain QUESO :: GslVector paramMins ( paramSpace . zeroVector () ) ; paramMins . cwSet ( -INFINITY ) ; QUESO :: GslVector paramMaxs ( paramSpace . zeroVector () ) ; 18 paramMaxs . cwSet ( INFINITY ) ; QUESO :: BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomain ( " param_ " , paramSpace , paramMins , paramMaxs ) ; // Step 3 of 6: Instantiate the qoi space 23 QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > qoiSpace ( env , " qoi_ " , 1 , NULL ) ; < QUESO :: GslVector , QUESO :: GslMatrix > paramRv ( " param_ " , paramDomain , meanVector , covMatrix ) ; 53 QUESO :: GenericVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > qoiRv ( " qoi_ " , qoiSpace ) ; QUESO :: StatisticalForwardProblem < QUESO :: GslVector , QUESO :: GslMatrix , QUESO :: GslVector , QUESO :: GslMatrix > 58 fp ( " " , NULL , paramRv , qoiFunctionObj , qoiRv ) ; File simple sfp example compute.C. File name simple sfp example.inp with options for QUESO library used in application code (Listings 6.12-6.16). $ ( CXX ) $ ( INC_PATHS ) $ ( CXXFLAGS ) $ < Listing 6.18: Makefile for the application code in Listings 6.12-6.16 $ cd HOME / LIBRARIES / QUESO -0.51.0/ examples / s i m p l e S t a t i s t i c a l F o r w a r d P r o b l e m $ export L D_ L IB RA RY _ PA TH = $ L D _ L I B R A R Y _ P A T H :\ $HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib $ make -f M a k e f i l e _ s f p _ e x a m p l e _ m a r g a r i d a $ ./ s i m p l e _ s f p _ e x a m p l e s i m p l e _ s f p _ e x a m p l e . inp >> fp_q_seq % if commands of Listings 3.19/3.20 have not been called >> nbins =20; >> hist ( fp_mc_QoiSeq_unified , nbins ) ; >> title ( ' QoI Histogram ' , ' fontsize ' ,20) ; >> xlabel ( ' QoI =\ theta_1 +\ theta_2 ' , ' fontname ' , ' Times ' , ' fontsize ' ,20) >> ylabel ( ' Frequency ' , ' fontsize ' ,20) ; Listing 6.21: Matlab code for the QoI histogram plot. fp_q_seq % if commands of Listing 5.19 have not been called >> [ fi , xi ] = ksdensity ( fp_mc_QoiSeq_unified , ' function ' , ' pdf ') ; >> x = sort ( f p _ m c _ Q o i S e q _ u n i f i e d ) ; >> mu =1; >> sigma2 =5; >> f =( exp ( -(x -mu ) .*( x -mu ) / sigma2 /2) ) / sqrt (2* pi * sigma2 ) ; >> plot ( xi , fi , ' -m ' , ' linewidth ' ,4) ; >> hold ; >> plot (x ,f , ' --k ' , ' linewidth ' ,2) ; >> h = legend ( ' QoI = \ theta_1 +\ theta_2 ' , ' analytical ' , ' location ' , ' northwest ') ; Listing 6.22: Matlab code for the KDE displayed in Figure 6.2.2 fp_q_seq % if commands of Listing 5.19 have not been called >> [f , xi ] = ksdensity ( fp_mc_QoiSeq_unified , ' function ' , ' cdf ') ; >> plot ( xi ,f , ' -m ' , ' linewidth ' ,3) Listing 6.23: Matlab code for the QoI CDF plot displayed in Figure 6.2.3. Figure 6 .2. 1 : 61QoI histogram. Figure 6 .2. 2 : 62Kernel Density Estimation. QUESO results are plotted against the PDF of a Gaussian distribution Q(x) = 1 √ 10π exp − 1 10 (x − 1) 2 , where µ = 1 and σ 2 = 5. Figure 6 .2. 3 :Figure 6 .3. 1 : 6361Cumulative Distribution Function. An object falls from altitude h 0 with zero initial velocity (v 0 = 0). Figure 6 . 63.2: Object traveling with projectile motion. $g r a v i t y _ p l o t s _ ip # i n s i d e m a t l a b $ g r a v i t y _ p l o t s _ fp # i n s i d e m a t l a b $ exit # i n s i d e m a t l a b $ ls -l outputData /*. png s f p _ g r a v i t y _ a u t o c o r r e l a t i o n . png sf p _g ra vi t y_ cd f . png s f p _ g r a v i t y _ c h a i n _ p o s . png s f p _ g r a v i t y _ h i s t cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / gravity $ rm outputData /* $ ./ gravity_gsl g ra vi t y_ in v_ f wd . inp The console output of the program is: k em el li @ vi ol et a :~/ LIBRARIES / QUESO -0.51.0/ examples / gravity$ ./ gravity_gsl g ra vi t y_ in v_ f wd . inp ------------------------------------------------------------------home / kemelli / LIBRARIES / hdf5 -1.8.10 --------------------------------------------------------------------------------------------------------Beginning run at Mon Apr 29 17:27:32 2013 MPI node of worldRank 0 has fullRank 0 , belongs to s ubEnviro nment of id 0 , and has subRank 0 MPI node of worldRank 0 belongs to sub communicator with full ranks 0 MPI node of worldRank 0 also belongs to inter0 communicator with full ranks 0 SIP -> Gravity estimation ' at Mon Apr 29 17:27:32 2013 Solving the SIP with Metropolis Hastings Beginning ' SFP -> Projectile motion ' at Mon Apr 29 17:27:33 2013 Solving the SFP with Monte Carlo Ending run of ' Gravity + Projectile motion ' example at Mon Apr 29 17:27:33 2013 Ending run at Mon Apr 29 17:27:33 2013 Total run time = 1 seconds k em el li @ vi ol et a :~/ LIBRARIES / QUESO -0.51.0/ examples / gravity$ Listing 6.24: Console output of program gravity gsl In order to generate chain plots, histograms, KDEs, etc., the user may use Matlab/GNU Octave and call the following command lines: $ matlab $ result, the user should have created several of PNG figures containing marginal posterior PDF, chain positions of the parameters and the QoI, histogram, cumulative density distribution and autocorrelation. The name of the figure files have been chosen to be infor-mative, as shown in the listing above. / 162 fp . s o l v e W i t h M o n t e C a r l o 162* ------------------------------------------------------------------* Brief description of this file :* * This is an example of how to use QUESO classes and algorithms in order to define and solve * a statistical inverse problem ( SIP ) and / or a statistical forward problem ( SFP ) . * The SIP consists on calibrating the magnitude 'g ' of acceleration gravity using * measurements of the time that it takes for an object in free fall to reach the ground from * a given height and zero initial velocity . The solution of the SIP is the posterior * probability density function ( PDF ) of 'g '. * The SFP consists of calculating the maximum distance traveled by an object in projectile * motion . The posterior PDF of 'g ' from the SIP might be used as input to the SFP . * * The code consists of 7 files : * -' gravity_main .C ' ( this file ) * -' g ra vi t y_ co mp u te .C ' ( the driving application code ) * -' g ra vi t y_ co mp u te .h ' * -' g r a v i t y _ l i k e l i h o o d .C ' ( necessary for the SIP ) * -' g r a v i t y _ l i k e l i h o o d .h ' * -' gravity_qoi .C ' ( necessary for the SFP ) * -' gravity_qoi .h ' * -----------------------------------------------------------------*/ #i n c l u d e <g r a v i t y c o m p u t e . h> int main ( int argc , char * argv []) { // Initialize QUESO environment #i f d e f QUESO HAS MPI MPI_Init (& argc ,& argv ) ; QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( MPI_COMM_WORLD , argv [1] , " " , NULL ) ; #e l s e QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( argv [1] , " " , NULL ) ; File gravity compute.h. * This file is divided in two parts : 2 * -the first one handles the statistical inverse problem ( SIP ) for estimating * the magnitude 'g ' of gravity acceleration ; and * -the second part handles the statistical forward problem ( SFP ) for * predicting the maximum distance traveled by a projectile . * 7 * The SIP definition requires a user defined likelihood function ; refer to * files ' g r a v i t y _ l i k e l i h o o d .h ' and ' g r a v i t y _ l i k e l i h o o d .C '. The SFP definition * requires a user defined qoi function ; refer to files ' gravity_qoi .h ' and * ' gravity_qoi .e ( const QUESO :: F u ll En vi r on me nt & env ) { struct timeval timevalNow ; 32 gettimeofday (& timevalNow , NULL ) ; if ( env . fullRank () == 0) { std :: cout << " \ nBeginning run of ' Gravity + Projectile motion ' example at " << ctime (& timevalNow . tv_sec ) << " \ n my fullRank = " << env . fullRank () 37 << " \ n my s u b E n v i r o n m e n t I d = " << env . subId () << " \ n my subRank = " << env . subRank () << " \ n my interRank = " << env . inter0Rank () << std :: endl << std :: endl ; } 42 // Just examples of possible calls if (( env . subDisp layFile () ) && ( env . d i s p l a y V e r b os i t y () >= 2) ) { * env . sub DisplayF ile () << " Beginning run of ' Gravity + Projectile motion ' example at " 47 << ctime (& timevalNow . tv_sec ) env . fullRank () == 0) { std :: cout << " Beginning ' SIP -> Gravity estimation ' at " << ctime (& timevalNow . tv_sec ) << std :: endl ; } 62 // ------------------------------------------------------// SIP Step 1 of 6: Instantiate the parameter space // ------------------------------------------------------QUESO :: VectorSpace < > paramSpace ( env , " param_ " , 1 , NULL ) ; 67 // ------------------------------------------------------// SIP Step 2 of 6: Instantiate the parameter domain // ------------------------------------------------------QUESO :: GslVector paramMin Values ( paramSpace . zeroVector () ) ; 72 QUESO :: GslVector paramMax Values ( paramSpace . zeroVector () ) ; para mMinValu es [0] = 8.; para mMaxValu es [0] = 11.; 77 QUESO :: BoxSubset < > paramDomain ( " param_ " , paramSpace , paramMinValues , param MaxValu es ) ; // ------------------------------------------------------// SIP Step 3 of 6: Instantiate the likelihood function 82 // object to be used by QUESO . // ------------------------------------------------------Likelihood < > lhood ( " like_ " , paramDomain ) ; // ------------------------------------------------------87 // SIP Step 4 of 6: Define the prior RV // ------------------------------------------------------QUESO :: UniformVectorRV < > priorRv ( " prior_ " , paramDomain ) ; // ------------------------------------------------------92 // SIP Step 5 of 6: Instantiate the inverse problem // ------------------------------------------------------// Extra prefix before the default " rv_ " prefix QUESO :: GenericVectorRV < > postRv ( " post_ " , paramSpace ) ; 97 // No extra prefix before the default " ip_ " prefix QUESO :: StatisticalInverseProblem < > ip ( " " , NULL , priorRv , lhood , postRv ) ; // ------------------------------------------------------// SIP Step 6 of 6: Solve the inverse problem , that is , 102 // set the 'pdf ' and the ' realizer ' of the posterior RV // ------------------------------------------------------QUESO :: GslVector paramInitials ( paramSpace . zeroVector () ) ; priorRv . realizer () . realization ( paramInitials ) ; 107 QUESO :: GslMatrix p r o p o s a l C o v M a t r i x ( paramSpace . zeroVector () ) ; p r o p o s a l C o v M a t r i x (0 ,0) = std :: pow ( std :: abs ( paramInitials [0]) / 20.0 , 2.0) ; ip . s o l v e W i t h B a y e s M e t r o p o l i s H a s t i n g s ( NULL , paramInitials , & p r o p o s a l C o v M a t r i x ) ; & timevalNow , NULL ) ; std :: cout << " Beginning ' SFP -> Projectile motion ' at " << ctime (& timevalNow . tv_sec ) << std :: endl ; 122 // ------------------------------------------------------// SFP Step 1 of 6: Instantiate the parameter * and * qoi spaces . // SFP input RV = FIP posterior RV , so SFP parameter space // has been already defined . 127 // ------------------------------------------------------QUESO :: VectorSpace < > qoiSpace ( env , " qoi_ " , 1 , NULL ) ; // ------------------------------------------------------// SFP Step 2 of 6: Instantiate the parameter domain 132 // ------------------------------------------------------// Not necessary because input RV of the SFP = output RV of SIP . // Thus , the parameter domain has been already defined . 137 // ------------------------------------------------------// SFP Step 3 of 6: Instantiate the qoi object // to be used by QUESO . // ------------------------------------------------------Qoi < > qoi ( " qoi_ " , paramDomain , qoiSpace ) ; 142 // ------------------------------------------------------// SFP Step 4 of 6: Define the input RV // ------------------------------------------------------147 // Not necessary because input RV of SFP = output RV of SIP // ( postRv ) . // ------------------------------------------------------// SFP Step 5 of 6: Instantiate the forward problem 152 // ------------------------------------------------------QUESO :: GenericVectorRV < > qoiRv ( " qoi_ " , qoiSpace ) ; QUESO :: StatisticalForwardProblem < > fp ( " " , NULL , postRv , qoi , qoiRv ) ; 157 // ------------------------------------------------------// SFP Step 6 of 6: Solve the forward problem // -----------------------------------------------------std :: cout << " Solving the SFP with Monte Carlo " << std :: endl << std :: endl ; . sub DisplayF ile () << " Ending run of ' Gravity + Projectile motion ' example at " << ctime (& timevalNow . tv_sec ) << std :: endl ; } if ( env . fullRank () == 0) { 172 std :: cout << " Ending run of ' Gravity + Projectile motion ' example at " << ctime (& timevalNow . tv_sec ) << std :: endl ; } } Listing 6.27: File gravity compute.C. The first part of the code (lines 60-150) handles the statistical forward problem, whereas the second part of the code (lines 151-216) handles the statistical forward problem. <V , M >:: Likelihood ( const char * prefix , const QUESO :: VectorSet <V , M > & domain ) : QUESO :: BaseScalarFunction <V , M >( prefix , domain ) , m_heights (0) , m_times Listing 6. 29 :imageVector [ 0 ] = d i s t a n c e T r av e l e 290File gravity likelihood.C. #d e f i n e QUESO EXAMPLE GRAVITY QOI H #i n c l u d e <q u e s o / V e c t o r F u n c t i o n . h> #i n c l u d e <q u e s o / D i s t A r r a y . h> template < class P_V = QUESO :: GslVector , class P_M = QUESO :: GslMatrix , class Q_V = QUESO :: GslVector , class Q_M = QUESO :: GslMatrix > class Qoi : public QUESO :: BaseVectorFunction < P_V , P_M , Q_V , Q_M > { public : Qoi ( const char * prefix , const QUESO :: VectorSet < P_V , P_M > & domainSet , const QUESO :: VectorSet < Q_V , Q_M > & imageSet ) ; virtual~Qoi () ; virtual void compute ( const P_V & domainVector , const P_V * domainDirection , Q_V & imageVector , QUESO :: DistArray < P_V * > * gradVectors , QUESO :: DistArray < P_M * > * hessianMatrices , QUESO :: DistArray < P_V * > * hess ianEffe cts ) const ; void setAngle ( double angle ) ; void s e t I n i t i a l V e l o c i t y ( double velocity ) ; void s e t I n i t i a l H e i g h t ( double height ) ; private : double m_angle ; double m _ i n i t i a l V e l o c i t y ; double m _i n it ia lH e ig ht ; }; class P_V , class P_M , class Q_V , class Q_M > Qoi < P_V , P_M , Q_V , Q_M >:: Qoi ( const char * prefix , const QUESO :: VectorSet < P_V , P_M > & domainSet , const QUESO :: VectorSet < Q_V , Q_M > & imageSet ) : QUESO :: BaseVectorFunction < P_V , P_M , Q_V , Q_M >( prefix , domainSet , imageSet ) class P_V , class P_M , class Q_V , class Q_M > Qoi < P_V , P_M , Q_V , Q_M >::~Qoi () { // Deconstruct here } template < class P_V , class P_M , class Q_V , class Q_M > void Qoi < P_V , P_M , Q_V , Q_M >:: compute ( const P_V & domainVector , const P_V * domainDirection , Q_V & imageVector , QUESO :: DistArray < P_V * > * gradVectors , QUESO :: DistArray < P_M * > * hessianMatrices , QUESO :: DistArray < P_V * > * hess ianEffec ts ) const { if ( domainVector . sizeLocal () != 1) { q ue so _e r ro r_ ms g ( " domainVector does not have size 1 " ) ; } if ( imageVector . sizeLocal () != 1) { q ue so _e r ro r_ ms g ( " imageVector does not have size 1 " ) ; } double g = domainVector [0]; // Sample of the RV ' gravity acceleration ' double d i s t a nc e T r a v e l e d = 0.0; double aux = m _ i n i t i a l V e l o c i t y * std :: sin ( m_angle ) ; d i s t a n c e T r a v e l ed = ( m _ i n i t i a l V e l o c i t y * std :: cos ( m_angle ) / g ) *( aux + std :: sqrt ( std :: pow ( aux , 2) + 2.0 * g * m _i ni t ia lH ei g ht ) ) ; an input file for solving statistical problems. In the case of a SIP, it expects a list of options for MCMC, while in case of SFP it expects a list of options for Monte Carlo. The input file 'gravity inv fwd.inp used in this example is presented in Listing 6.32. _ c o m p u t e C o r r e l a t i o n s = 1 f p _ d a t a O u t p u t F i l e N a m e = outputData / sfp_gravity f p _ d a t a O u t p u t A l l o w e d S e t = Some options for QUESO library used in application code (Listings 6.25-6.29). Listing 6 . 633 presents a Makefile, named Makefile example violeta, that may be used to compile the code and create the executable gravity_gsl. Naturally, it must be adapted to the user's settings, i.e., it has to have the correct paths for the user's libraries that were actually used to compile and install QUESO (see Sections 2.( CXX ) $ ( INC_PATHS ) $ ( CXXFLAGS ) $ < Listing 6.33: Makefile for the application code in Listings 6.25-6.29 ylabel ( '\ theta =g ' , ' fontsize ' ,20) ; >> xlabel ( ' Number of positions ' , ' fontsize ' ,20) ; >> title ( ' DRAM Chain Positions ( raw ) ',' fontsize ' ,20) ; Listing 6.34: Matlab code for the chain plot. % inside Matlab >> s i p _ g r a v i t y _ r a w _ c h a i n >> nbins =100; >> hist ( ip_mh_rawChain_unified , nbins ) code for the histogram plot. f , xi ] = ksdensity ( ip_mh_rawChain_unified , ' function ' , ' pdf ') ; >> plot ( xi ,f , ' -b ' , ' linewidth ' ,3) >> title ( ' Parameter Kernel Density Estimation ' , ' fontsize ' ,20) ; >> xlabel ( ' Gravity ( m / s^2) ',' fontsize ' ,20) ; >> ylabel ( ' KDE ' , ' fontsize ' ,20) ; >> grid on ; f , xi ] = ksdensity ( ip_mh_rawChain_unified , ' function ' , ' cdf ') ; >> plot ( xi ,f , ' -b ' , ' linewidth ' ,3) >> title ( ' Parameter Cumulative Distribution Function ' , ' fontsize ' ,20) ; >> xlabel ( ' Gravity ( m / s^2) ',' fontsize ' ,20) ; >> ylabel ( ' CDF ' , ' fontsize ' ,20) ; >> grid on ; ACF_raw , lags , bounds ]= autocorr ( ip_mh_rawChain_unified , nlags , 0) ; >> [ ACF_filt , lags , bounds ]= autocorr ( ip_mh_filtChain_unified , nlags , 0) ; >> plot ( lags , ACF_raw , ' bo -' , lags , ACF_filt , ' r * -' , ' linewidth ' ,3) ; >> ylabel ( ' A ut o co rr el a ti on for \ theta =g ' , ' fontsize ' ,20) ; >> xlabel ( ' Lag ' , ' fontsize ' ,20) ; >> title ( ' Parameter Autocorrelation ' , ' fontsize ' ,20) ; Figure 6 . 63.3: MCMC raw chain with 20000 positions and a filtered chain with lag of 20 positions. Figure 6 .3. 4 : 64Histograms of parameter θ = g.(a) Raw chain (b) Filtered chain Figure 6 . 63.5: Kernel Density Estimation.>> grid on ; >> h = legend ( ' raw chain ' , ' filtered chain ' , ' location ' , ' northeast ') ; >> set (h , ' fontsize ' ,16) ; Matlab code for finding the covariance matrix. Figure 6 .3. 6 : 66Cumulative Distribution Function. Figure 6 .3. 7 : 67Autocorrelation plots. ( ' QoI ' , ' fontsize ' ,20) ; >> xlabel ( ' Number of positions ' , ' fontsize ' ,20) ; >> title ( ' MC Chain Positions ' , ' fontsize ' ,20) ; Listing 6.40: Matlab code for the chain plot. title ( ' QoI Histogram ' , ' fontsize ' ,20) ; >> xlabel ( ' Distance traveled ( m ) ',' fontsize ' ,20) ; >> ylabel ( ' Frequency ' , ' fontsize ' ,20) ; >> grid on ; Listing 6.41: Matlab code for the QoI histogram plot. Figure 6 . 63.8: MC chain positions and histogram of QoI = d. f , xi ] = ksdensity ( fp_mc_QoiSeq_unified , ' function ' , ' pdf ') ; >> plot ( xi ,f , ' -b ' , ' linewidth ' ,3) >> title ( ' QoI Kernel Density Estimation ',' fontsize ' ,20) ; >> xlabel ( ' Distance traveled ( m ) ',' fontsize ' ,20) ; >> ylabel ( ' KDE ' , ' fontsize ' ,20) ; >> grid on ; Listing 6.42: Matlab code for the QoI KDE plot. 6.3.8.2.4 CDF Plots Matlab function ksdensity (Kernel smoothing density estimate) with 'cdf' option may also be used for plotting the Cumulative Distribution Function of the QoI, displayed in Figure 6.3.9b. % inside Matlab >> s f p _ g r a v i t y _ q o i _ s e q . m >> [f , xi ] = ksdensity ( fp_mc_QoiSeq_unified , ' function ' , ' cdf ') ; >> plot ( xi ,f , ' -b ' , ' linewidth ' ,3) >> title ( ' QoI Cumulative Distribution Function ',' fontsize ' ,20) ; >> xlabel ( ' Distance traveled ( m ) ',' fontsize ' ,20) ; >> ylabel ( ' CDF ' , ' fontsize ' ,20) ; >> grid on ; Listing 6.43: Matlab code for the QoI CDF plot. Figure 6 .3. 9 : 69Kernel Density Estimation and Cumulative Distribution Function of QoI. ACF , lags , bounds ] = autocorr ( fp_mc_QoiSeq_unified , nlags , 0) ; >> plot ( lags , ACF , ' bo -' , ' linewidth ' ,3) ; >> ylabel ( ' A ut o co rr el a ti on for QoI = d ' , ' fontsize ' ,20) ; >> xlabel ( ' Lag ' , ' fontsize ' ,20) ; >> title ( ' QoI Autocorrelation ' , ' fontsize ' ,20) ; >> grid on ; Listing 6.44: Matlab code for the QoI autocorrelation plot. Figure 6 . 63.10: Autocorrelation plot. Listing 6.45: Matlab code for the matrix of covariance between parameter g and QoI d. code for the matrix of correlation between parameter g and quantity of interest d. Figure 6 . 64.1: Mass fraction decay over temperature given different heating rates. Data from J. A. Conesa, R. F. Marcilla and J. A. Caballero, "Thermogravimetric studies on the thermal decomposition of polyethylene", J. Anal. Appl. Pyrolysis, $ cd $HOME / LIBRARIES / QUESO -0.50.0/ examples / va l id at io n Cy cl e $ rm outputData /* $ ./ e x T g a V a l i d a t i o n C y c l e _ g s l tagCycle . inp $ matlab $ tga_ cycle_pl ot . m _v al _ Qo I_ CD F . png ca l _v al _Q o I_ PD F . png c a l _ v a l _ Q o I _ a u t o c o r r e l a t i o n . png As a result, the user should have created several of PNG figures containing marginal posterior PDF, chain positions of the parameters and the QoI, histogram, cumulative density distribution and autocorrelation. The name of the figure files have been chosen to be informative, as shown in the Listing above. , e n v O p t i o n s V a l u e QUESO classes and algorithms . The code itself is in the templated * routine ' uqAppl (* env ) '. This routine is called right after the init ializati on * of the MPI environment and of the QUESO environment and is available in * file ' focuses on the templated class StatisticalInverseProblem < P_V , P_M >. * Example (2) focuses on the templated class StatisticalForwardProblem < P_V , P_M , Q_V , Q_M >. * Example (3) focuses on both classes , since it uses the solution of the inverse problem * as input to the forward problem . * * This example with TGA uses both statistical problem templated classes , twice each in * fact , but it also : * -uses the templated class ValidationCycle < P_V , P_M , Q_V , Q_M > , * -reads experimental data from files in order to compute the likelihood function * at candidate parameter vectors generated by the Metropolis -Hastings algorithm , and * -compares the qoi cdfs computed from the two statistical forward problems at * the prediction scenario .* --------------------------------------------------------------------------* -------------------------------------------------------------------------#i f d e f UQ EXAMPLES USES QUESO INPUT FILE U Q _ F A T A L _ T E S T _ MA C R O ( argc != 2 , QUESO :: UQ_UNAVAILABLE_RANK , " main () " , " input file must be specified in command line as argv [1] , just after executable argv [0] " ) ; #i f d e f QUESO HAS MPI QUESO :: Fu ll En v ir on me n t * env = new QUESO :: F ul lE nv i ro nm en t ( MPI_COMM_WORLD , argv [1] , " " , e n v O p t i o n s V a l u e s ) ; #e l s e QUESO :: Fu ll En v ir on me n t * env = new QUESO :: F ul lE nv i ro nm en t ( argv [1] , " " -> m _ d i s p l a y V e r b o s i t y = 2; envOptionsValues -> m_seed = 0; #i f d e f QUESO HAS MPI QUESO :: Fu ll En v ir on me n t * env = new QUESO :: F ul lE nv i ro nm en t ( MPI_COMM_WORLD , " declaration : actual code is below template < class P_V , class P_M , class Q_V , class Q_M > void u q A p p l _ L o c a l C o m p a r i s o n S t a g e ( QUESO :: ValidationCycle < P_V , P_M , Q_V , Q_M >& cycle ) ; template < class P_V , class P_M , class Q_V , class Q_M > void u q A p p l _ U n i f i e d C o m p a r i s o n S t a g e ( QUESO :: ValidationCycle < P_V , P_M , Q_V , Q_M >& cycle ) ; :: VectorSpace < P_V , P_M > paramSpace ( env , " param_ " , paramNames . size () ,& paramNames ) ; // Instantiate the parameter domain P_V paramMi nValues ( paramSpace . zeroVector () ) ; para mMinValu es [0] = 2.40 e +11; para mMinValu es [1] = 1.80 e +05; P_V paramMa xValues ( paramSpace . zeroVector () ) ; para mMaxValu es [0] = 2.80 e +11; para mMaxValu es [1] = 2.20 e +05; QUESO :: BoxSubset < P_V , P_M > paramDomain ( " param_ " , paramSpace , paramMinValues , param MaxValu es ) ; // Instantiate the qoi space std :: vector < std :: string > qoiNames (1 , " " ) ; qoiNames [0] = " T i m e F o r 2 5 P e r c e n t O f M a s s " ; QUESO :: VectorSpace < Q_V , Q_M > qoiSpace ( env , " qoi_ " , qoiNames . size () ,& qoiNames ) ; // Instantiate the validation cycle QUESO :: ValidationCycle < P_V , P_M , Q_V , Q_M > cycle ( env , gettimeofday (& timevalRef , NULL ) ; if ( iRC ) {}; // just to remove compiler warning if ( env . fullRank () == 0) { std :: cout << " Beginning ' calibration stage ' at " << ctime (& timevalRef . tv_sec ) << std :: endl ; } // Inverse problem : instantiate the prior rv QUESO :: UniformVectorRV < P_V , P_M > calPriorRv ( " cal_prior_ " , // Extra prefix before the default " rv_ " prefix paramDomain ) ; // Inverse problem : instantiate the likelihood function object ( data + routine ) likelihoodRoutine_Data < P_V , P_M > c a l L i k e l i h o o d R o u t i n e _ D a t a ( env , " s c e n a r i o _ 5 _ K _ m i n . dat " , problem : solve it , that is , set 'pdf ' and ' realizer ' of the posterior rv P_V p a r a m I n i t i a l V a l u e s ( paramSpace . zeroVector () ) ; if ( env . cycle . calIP () . postRv () . imageSet () . vectorSpace () . n e w P r o p o s a l M a t r i x ( NULL ,& p a r a m I n i t i a l V a l u e s ) ; #i f d e f UQ EXAMPLES USES QUESO INPUT FILE #e l s e QUESO :: Ss Op ti o ns Va lu e s s s O p ti o n s V a l u e s 1 ; QUESO :: Ss Op ti o ns Va lu e s s s O p ti o n s O p t i o n s V a l u e s1 . m _ a u t o C o r r D i s p l a y = true ; s s O p t i o n s V a l u e s1 . m _a u to Co rr W ri te = true ; s s O p t i o n s V a l u e s1 . m_kdeCompute = false ; s s O p t i o n s V a l u e s1 . m _ c o v M a t r i x C o m p u t e = true ; s s O p t i o n s V a l u e s1 . m _ c o r r M a t r i x C o m p u problem : instantiate it ( parameter rv = posterior rv of inverse problem ; qoi rv is instantiated internally ) double b et a _p re di c ti on = 250.; double c r i t i c a l M a s s _ p r e d i c t i o n = 0.; double c r i t i c a l T i m e _ p r e d i c t i o n = 3.9; qoiRoutine_Data < P_V , P_M , Q_V , Q_M > c a l Q o i R o u t i n e _ D a t a ; c a l Q o i R o u t i n e _ D a t a . m_beta = be ta _ pr ed ic t io n ; c a l Q o i R o u t i n e _ D a t a . m_criti calMass = c r i t i c a l M a s s _ p r e d i c t i o n ; problem : solve it , that is , set ' realizer ' and ' cdf ' of the qoi rv QUESO :: Mc Op ti o ns Va lu e s * c a l F p M c O p t i o n s V a l u e s = NULL ; #i f d e f UQ EXAMPLES USES QUESO INPUT FILE #e l s e QUESO :: Ss Op ti o ns Va lu e s s s O p ti o n s V a l u e s 3 ; QUESO :: Ss Op ti o ns Va lu e s s s O p ti o n s -> m _ q s e q D i s p l a y P e r i o d = 20000; calFpMcOptionsValues -> m _ q s e q M e a s u r e R u n T i m e s = true ; calFpMcOptionsValues -> m _ q s e q D a t a O u t p u t F i l e N a m e = " outputData / f i l e _ c a l _ f p _ q o i 2 " ; calFpMcOptionsValues -> m _ q s e q D a t a O u t p u t A l l o w e d S e t . insert (0) ; calFpMcOptionsValues -> m _ q s e q D a t a O u t p u t A l l o w e d S e t . insert (1gettimeofday (& timevalNow , NULL ) ; if ( env . fullRank () == 0) { std :: cout << " Ending ' calibration stage ' at " << ctime (& timevalNow . tv_sec ) << " Total ' calibration stage ' run time = " << timevalNow . tv_sec -timevalRef . O p t i o n s V a l u eV a l u e s iRC = gettimeofday (& timevalRef , NULL ) ; if ( env . fullRank () == 0) { std :: cout << " Beginning ' validation stage ' at " << ctime (& timevalRef . tv_sec ) << std :: endl ; } // Inverse problem : no need to instantiate the prior rv (= posterior rv of calibration inverse problem ) // Inverse problem : instantiate the likelihood function object ( data + routine ) likelihoodRoutine_Data < P_V , P_M > v problem : instantiate it ( posterior rv is instantiated internally ) QUESO :: S i p O p t i o n s V al u e s * v a l I p O p t i o n s V a l u e s = NULL ; #i f d e f UQ EXAMPLES USES QUESO INPUT FILE #e l s e v a l I p O p t i o n s V a l u e s = new QUESO :: S i p problem : solve it , that is , set 'pdf ' and ' realizer ' of the posterior rv QUESO :: Mh Op ti o ns Va lu e s * v a l I p M h O p t i o n s V a l u e s = NULL ; const QUESO :: SequentialVectorRealizer < P_V , P_M >* tmpRealizer = dynamic_cast < const QUESO :: SequentialVectorRealizer < P_V , P_M >* >(&( cycle . calIP () . postRv () . realizer () ) ) ; // Use ' realizer () ' because the post . rv was computed with Metr . Hast . > m _ d i s p l a y C a n d i d a t e s = false ; valIpMhOptionsValues -> m _ p u t O u t O f B o u n d s I n C h a i n = true ; valIpMhOptionsValues -> m _ t k U s e L o c a l H e s s i a n = false ; valIpMhOptionsValues -> m _ t k U s e N e w t o n C o m p o n e n t = true ; valIpMhOptionsValues -> m _ d r M a x N u m E x t r a S t a g problem : instantiate it ( parameter rv = posterior rv of inverse problem ; // qoi rv is instantiated internally ) qoiRoutine_Data < P_V , P_M , Q_V , Q_M > v problem : solve it , that is , set ' realizer ' and ' cdf ' of the qoi rv QUESO :: Mc Op ti o ns Va lu e s * v a l F p M c O p t i o n s V a l u e s = NULL ; #i f d e f UQ EXAMPLES USES QUESO INPUT FILE #e l s e QUESO :: Ss Op ti o ns Va lu e s s s O p ti o n s V a l u e s 7 ; QUESO :: Ss Op ti o ns Va lu e s s s O p ti o n s O p t i o n s V a l u e s8 . m _ a u t o C o r r C o m p u t e V i a F f t = true ; s s O p t i o n s V a l u e s8 . m _ a u t o C o r r S e c o n d L a g = 2; s s O p t i o n s V a l u e s8 . m _ a u t o C o r r L a g S p a c i n g = 1; s s O p t i o n s V a l u e s8 . m _ a u t o C o r r N u m L a g s = 15; s s O p t i o n s V a l u e s8 . m _ a u t o C o r r D i s p l gettimeofday (& timevalNow , NULL ) ; if ( env . fullRank () == 0) { std :: cout << " Ending ' validation stage ' at " << ctime (& timevalNow . tv_sec ) << " Total ' validation stage ' run time = " << timevalNow . tv_sec -timevalRef . iRC = gettimeofday (& timevalRef , NULL ) ; if ( env . fullRank () == 0) { std :: cout << " Beginning ' comparison stage ' at " << ctime (& timevalRef . tv_sec ) << std :gettimeofday (& timevalNow , NULL ) ; if ( env . fullRank () == 0) { std :: cout << " Ending ' comparison stage ' at " << ctime (& timevalNow . tv_sec ) << " Total ' comparison stage ' run time = " << timevalNow . tv_sec -timevalRef . tv_sec << " seconds \ n " << std :: endl ; :: vector < double > m_Te3 ; // temperatures std :: vector < double > m_Me3 ; // relative masses const QUESO :: B as eE n vi ro nm e nt * m_env ; }; template < class P_V , class P_M > likelihoodRoutine_Data < P_V , P_M >:: l i k e l i h o o d R o u t i n e _ D a t a ( const QUESO :: B as eE n vi ro nm e nt & env , kinetic parameters and convert heating rate to K / s int aux1 = fscanf ( inp , " % lf % lf " ,& m_beta1 ,& m_variance1 ) ; m_beta1 /= 60.; if ( aux1 ) {}; // just to eliminate warnings unsigned int nu mO bs e rv at io n s = 0; double tmpTe ; double tmpMe ; while ( fscanf ( inp , " % lf % lf " ,& tmpTe ,& tmpMe ) != EOF ) { U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns >= m_Te1 . size () ) , env . fullRank () , " uqAppl () , in uqTgaEx4 . h " , " input file 1 has too many observations " ) ; m_Te1 [ n um O bs er v at io ns ] = tmpTe ; m_Me1 [ n um O bs er v at io ns ] = tmpMe ; n um Ob se r va ti on s ++; } U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns != m_Te1 . size () ) , env . fullRank () , " uqAppl () , in uqTgaEx4 . h " , " input file 1 has a smaller number of observations than expected " ) ; // Close input file on experimental data fclose ( inp ) ; } // Read experimental data if ( inpName2 ) { m_Te2 . resize (11 ,0.) ; m_Me2 . resize (11 ,0.) ; // Open input file on experimental data FILE * inp ; inp = fopen ( inpName2 , " r " ) ; // Read kinetic parameters and convert heating rate to K / s int aux2 = fscanf ( inp , " % lf % lf " ,& m_beta2 ,& m_variance2 ) ; m_beta2 /= 60.; if ( aux2 ) {}; // just to eliminate warnings unsigned int nu mO bs e rv at io n s = 0; double tmpTe ; double tmpMe ; while ( fscanf ( inp , " % lf % lf " ,& tmpTe ,& tmpMe ) != EOF ) { U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns >= m_Te2 . size () ) , env . fullRank () , " uqAppl () , in uqTgaEx4 . h " , " input file 2 has too many observations " ) ; m_Te2 [ n um O bs er v at io ns ] = tmpTe ; m_Me2 [ n um O bs er v at io ns ] = tmpMe ; n um Ob se r va ti on s ++; } U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns != m_Te2 . size () ) , env . fullRank () , " uqAppl () , in uqTgaEx4 . h " , " input file 2 has a smaller number of observations than expected " ) ; kinetic parameters and convert heating rate to K / s int aux3 = fscanf ( inp , " % lf % lf " ,& m_beta3 ,& m_variance3 ) ; m_beta3 /= 60.; if ( aux3 ) {}; // just to eliminate warnings unsigned int nu mO bs e rv at io n s = 0; double tmpTe ; double tmpMe ; while ( fscanf ( inp , " % lf % lf " ,& tmpTe ,& tmpMe ) != EOF ) { U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns >= m_Te3 . size () ) , env . fullRank () , " uqAppl () , in uqTgaEx4 . h " , " input file 3 has too many observations " ) ; m_Te3 [ n um O bs er v at io ns ] = tmpTe ; m_Me3 [ n um O bs er v at io ns ] = tmpMe ; n um Ob se r va ti on s ++; } U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns != m_Te3 . size () ) , env . fullRank () , " uqAppl () , in uqTgaEx4 . h " , " input file 3 has a smaller number of observations than expected " resultValue = 0.; const QUESO :: B as eE n vi ro nm e nt & env = *((( likelihoodRoutine_Data < P_V , P_M > *) fu n ct io n Da ta Pt r ) -> m_env ) ; likelihoodRoutine_Data < P_V , P_M > *) fu nc ti o nD at aP t r ) -> m_variance1 ; const std :: vector < double >& Te = (( likelihoodRoutine_Data < P_V , P_M > *) f un ct i on Da ta P tr ) -> m_Te1 ; const std :: vector < double >& Me = (( likelihoodRoutine_Data < P_V , P_M > *) f un ct i on Da ta P tr ) Me . size () ) && ( t_old <= Te [ i ]) && ( Te [ i ] <= t ) ) { Mt [ i ] = ( Te [ i ] -t_old ) *( Mass [0] -M_old [0]) /( t -t_old ) + M_old [0]; misfit += ( Me [ i ] -Mt [ i ]) *( Me [ i ] -Mt [ i ]) ; // printf ( " % i % lf % lf % lf % lf \ n " ,i , Te [ i ] , Me [ i ] , Mt [ i ] , misfit ) printf ( " loopSize = % d \ n " , loopSize ) ; if (( paramValues . env () . d i s p l a y V e r b o s i t y () >= 10) && ( paramValues . env () . fullRank () == 0likelihoodRoutine_Data < P_V , P_M > *) fu nc ti o nD at aP t r ) -> m_variance2 ; const std :: vector < double >& Te = (( likelihoodRoutine_Data < P_V , P_M > *) f un ct i on Da ta P tr ) -> m_Te2 ; const std :: vector < double >& Me = (( likelihoodRoutine_Data < P_V , P_M > *) f un ct i on Da ta P tr ) -> m_Me2 ; std :: vector < double > Mt ( Me . size () Me . size () ) && ( t_old <= Te [ i ]) && ( Te [ i ] <= t ) ) { Mt [ i ] = ( Te [ i ] -t_old ) *( Mass [0] -M_old [0]) /( t -t_old ) + M_old [0]; misfit += ( Me [ i ] -Mt [ i ]) *( Me [ i ] -Mt [ i ]) ; // printf ( " % i % lf % lf % lf % lf \ n " ,i , Te [ i ] , Me [ i ] , Mt [ i ] likelihoodRoutine_Data < P_V , P_M > *) fu nc ti o nD at aP t r ) -> m_variance3 ; const std :: vector < double >& Te = (( likelihoodRoutine_Data < P_V , P_M > *) f un ct i on Da ta P tr ) -> m_Te3 ; const std :: vector < double >& Me = (( likelihoodRoutine_Data < P_V , P_M > *) f un ct i on Da ta P tr ) -> m_Me3 ; std :: vector < double > Mt ( Me . size () Me . size () ) && ( t_old <= Te [ i ]) && ( Te [ i ] <= t ) ) { Mt [ i ] = ( Te [ i ] -t_old ) *( Mass [0] -M_old [0]) /( t -t_old ) + M_old [0]; misfit += ( Me [ i ] -Mt [ i ]) *( Me [ i ] -Mt [ i ]) ; // printf ( " % i % lf % lf % lf % lf \ n " ,i , Te [ i ] , Me [ i ] , Mt [ i ] , misfit ) ; / / The actual ( user defined ) qoi routine template < class P_V , class P_M , class Q_V , class Q_M > void qoiRoutine ( const P_V = (( qoiRoutine_Data < P_V , P_M , Q_V , Q_M > *) fu nc t io nD a ta Pt r ) -> m_beta ; double criticalMass = (( qoiRoutine_Data < P_V , P_M , Q_V , Q_M > *) f u nc ti on D at aP tr ) -> m_cri ticalMa ss ; double criticalTime = (( qoiRoutine_Data < P_V , P_M , Q_V , Q_M > *) f u nc ti on D at aP tr ) pe ra t ur e_ ol d + ( temperature -t em p er at ur e _o ld ) * ( M_old [printf ( " loopSize = % d \ n " , loopSize ) ; if (( paramValues . env () . d i s p l a y V e r b o s i t y () >= 3) && ( paramValues . env () . fullRank () == 0) ) { printf ( " In qoiRoutine () , A = %g , E = %g , beta = %.3 lf , criticalTime = %.3 lf , criticalMass = %.3 lf : qoi = % lf .\ n " ,A ,E , beta , criticalTime , criticalMass , qoiValues [d i f // EX TGA VALIDATION CYCLE QOI H Listing 6.50: File exTgaValidationCycle qoi.h. c l e _ c a l _ i p _ m h _ a m _ e t a = 1.92 c y c l e _ c a l _ i p _ m h _ a m _ e p s i l o n = 1. e -5 c y c l e _ c a l _ i p _ m h _ f i l t e r e d C h a i n _ g e n e r a c l e _ v a l _ i p _ m h _ d i s p l a y C a n d i d a t e s = 0 c y c l e _ v a l _ i p _ m h _ p u t O u t O f B o u n d s I n C h a i n = 1 c y c l e _ v a l _ i p _ m h _ t k _ u s e L o c a l H e s s i a n = 0 c y c l e _ v a l _ i p _ m h _ t k _ u s e N e w t o n C o m p o n e File name tgaCycle.inp with options for QUESO library used in application code (Listings 6.47-6.49). Figure 6 . 64.2: Posterior distributions of parameters A and E. Figure 6 . 64.3: Cumulative density functions of parameters A and E. Figure 6 . 64.4 presents the autocorrelation of the parameters A and E in both cases: calibration and validation stages. Figures 6.4.5a and 6.4.5b present PDF and CDF of QoI, respectively and Figure 6.4.6 presents its autocorrelation. π prior = U([0, 3]) × U([0, 3]) × B(α,β), with α = 3 and β = 0.09709133373799. Figure 6 . 64.4: Autocorrelation of parameters A and E (filtered chain). Figure 6 .4. 5 : 65QoI PDF and CDF, during calibration and validation stages. Figure 6 .4. 6 : 66QoI $ cd $HOME / LIBRARIES / QUESO -0.51.0/ $ cd examples / modal $ rm outputData /* $ ./ modal_gsl example . Running the example with a one-mode distribution. As a result, the user should have created several of PNG figures scatter plots of each one of the levels and the kernel density estimation of the parameters, for each level in the Multilevel method. The name of the figure files have been chosen to be informative, as shown in the Listing above. π prior = U([0, 3]) × U([0, 3]) × U([0, 0.3]).or, the prior distribution is defined as a combination of uniform prior for the θ 1 , cd $HOME / LIBRARIES / QUESO -0.51.0/ $ cd examples / modal $ rm outputData /* Running the example with a two-mode distribution. As a result, the user should have created several of PNG figures scatter plots of each one of the levels and the kernel density estimation of the parameters, for each level in the Multilevel method. The name of the figure files have been chosen to be informative, as shown in the Listing above. executable argv [0] , input file must be specified in command line as argv[1] , then numModes (1 or 2) must be specified as argv[2] " ) ; QUESO :: Fu ll En v ir on me n t * env = new QUESO :: Fu ll En v ir on me n t ( MPI_COMM_WORLD , argv [1] , " " , NULL ) ; // Compute unsigned int numModes = ( unsigned int ) atoi ( argv [: cout << std :: endl << " FIM ! " << std :: endl << std :: endl ; return 0; } Listing 6.54: File example main.C. numModes = (( l i k e l i h o o d R o u t i n e _ D a t a T y p e *) fu n ct io n Da ta Pt r ) -> numModes ; double aux1 = theta1 + 2.* theta2 ; double aux2 = sqrt ( theta1 * theta1 +4.* theta2 * theta2 ) ; double w1 = 10.* sqrt (10.*( aux1 + aux2 ) ) ; double w2 = 10.* sqrt (10.*( aux1 -aux2 ) ) ; double sum1 = 0.; double sum2 = 0.; e <q u e s o / E n v i r o n m e n t . h> void compute ( const QUESO :: Fu l lE nv ir o nm en t & env , unsigned int numModes ) ; #e n d i f Listing 6.57: File example compute.h. :: InverseGammaVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRvB ( " priorB_ " , paramDomainB , alpha , beta ) ; 96 QUESO :: ConcatenatedVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRv ( " prior_ " , priorRvA , priorRvB , paramDomain ) ; #e l s e QUESO :: UniformVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > 101 priorRv ( " prior_ " , paramDomain ) ; #e n d i f QUESO :: GenericVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > postRv ( " post_ " , paramSpace ) ; 106 QUESO :: StatisticalInverseProblem < QUESO :: GslVector , QUESO :: GslMatrix > ip ( " " , NULL , priorRv , likelihoodFunctionObj , postRv ) ; numPosTotal = postRv . realizer () . subPeriod () ; if ( env . s ubDispla yFile () ) { * env . sub DisplayF ile () << " numPosTotal = " << numPosTotal 121 << std :: endl ; } QUESO :: GslVector auxVec ( paramSpace . zeroVector () .58: File example compute.C. : input file. The options used for solving this example are displayed in Listing 6.59. p m l h e l p = a n y t h i n g i p _ m l _ d a t a O u t p u t F i l e N a m e = outputData / sipOutput_ml i p _ m l _ d a t a O u t p u t A l l o w e d S e t Options for QUESO library used in application code (Listings 6.54-6.58). mp le _ co mp u te . o \ -o modal_gsl $ ( LIBS ) %. o : %. C $ ( CXX ) $ ( INC_PATHS ) $ ( CXXFLAGS ) $ < Listing 6.60: Makefile for the application code in Listings 6.54-6.58Thus, to compile, build and execute the code, the user just needs to run the following commands in the same directory where the files are:$ cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / modal / $ export L D_ L IB RA RY _ PA TH = $ L D _ L I B R A R Y _ P A T H :\ $HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib $ make -f M a k e f i l e _ m o d a l _ v i o l e t a $ ./ modal_gsl example . inp < num_modes > Figures 6 . 65.3 and 6.5.4 present the KDE plots of the parameters θ 1 , θ 2 , θ 3 and target PDF in both cases: one-mode and two-modes distribution. the autocorrelation of the parameters θ 1 , θ 2 and θ 3 in both cases: one-mode and two-modes distribution. Figure 6 . 65.2: Scatter plots for θ 1 , θ 2 and θ 3 = σ 2 , levels 1, 3, 6 and 9 (last). Two-mode distribution. Figure 6 . 65.3: KDE plots for θ 1 , θ 2 , θ 3 = σ 2 , and the target PDF. One mode distribution. Figure 6 . 65.4: KDE plots for θ 1 , θ 2 , θ 3 = σ 2 , and the target PDF. Two-mode distribution. Figure 6 .5. 5 : 65Autocorrelation plots for θ 1 , θ 2 and θ 3 = σ 2 . One-mode distribution. Figure 6 . 65.6: Autocorrelation plots for θ 1 , θ 2 and θ 3 = σ 2 . Two-mode distribution. Figure 6 .6. 1 : 61Likelihood function given by f = f 1 /2 + f 2 /2, where f 1 and f 2 are defined in Equation (6.6.1). result, the user should have created several of PNG figures containing marginal posterior PDF, cumulative density distribution and autocorrelation. The name of the figure files have been chosen to be informative, as shown in the Listing above. #i n c l u d e <b i m o d a l c o m p u t e . h> int main ( int argc , char * argv []) { // Initialize environment #i f d e f QUESO HAS MPI MPI_Init (& argc ,& argv ) ; QUESO :: Fu ll En v ir on me n t * env = new QUESO :: F ul lE nv i ro nm en t ( MPI_COMM_WORLD , argv [1] , " " , NULL ) ; #e l s e QUESO :: Fu ll En v ir on me n t * env = new QUESO :: F ul lE nv i ro nm en t ( argv [1] , " " , NULL ) ; c l u d e <q u e s o / G s l M a t r i x . h> struct l i k e l i h o o d R o u t i n e _ D a t a T y ( paramValues . env () . subDispl ayFile () ) && ( paramValues . env () . d i s p l a y V e r b o si t y () > 0) ) { // detailed output debug * paramValues . env () . subD isplayF ile () << " Leaving likelihood function " << " : paramValues = " << paramValues << " , returnValue = " << returnValue << std :e <q u e s o / E n v i r o n m e n t . h> void compute ( const QUESO :: Fu l lE nv ir o nm en t & env ) ; #e n d i f Listing 6.65: File bimodal compute.h. Note that in line 57 of Listings 6.66 the '#if 0' directive tells the compiler that the application will not use DRAM algorithm, but rather the Multilevel solver (line 65). Naturally, the user may chose to use the DRAM algorithm by changing the directive in line 57 to '#if 1'. = seq1 . subMeanExtra (0 , seq1 . s ub Se qu e nc eS iz e () ) ; 103 if ( env . s ubDispla yFile () ) { * env . sub DisplayF ile () << " seq1 . size () = " << seq1 . su bS eq u en ce Si z e () << " \ n seq1 . mean () = " << mean1 << " \ n seq1 . std () = " << sqrt ( seq1 . s u b S a m p l e V a r i a n c e E x t r a (0 , seq1 . s ub Se qu e nc eS iz e () , mean1 ) ) << std :: endl ; 108 } double mean2 = seq2 . subMeanExtra (0 , seq2 . s ub Se qu e nc eS iz e () ) ; if ( env . s ubDispla yFile () ) { * env . sub DisplayF ile () << " seq2 . size () = " << seq2 . su bS eq u en ce Si z e () 113 << " \ n seq2 . mean () = " << mean2 << " \ n seq2 . std () = " << sqrt ( seq2 . s u b S a m p l e V a r i a n c e E x t r a (0 , seq2 . s ub Se qu e nc eS iz e () , mean2 ) ) << std :: endl ; } 118 double meanAll = seqAll . subMeanExtra (0 , seqAll . s ub S eq ue nc e Si ze () ) ; if ( env . s ubDispla yFile () ) { * env . sub DisplayF ile () << " seqAll . size () = " << seqAll . s u bS eq ue n ce Si ze () << " \ n seqAll . mean () = " << meanAll << " \ n seqAll . std () = " << sqrt ( seqAll . s u b S a m p l e V a r i a n c e E x t r a (0 , seqAll . s ub Se q ue nc eS i ze () , meanAll ) ) 123 << std :: endl ; } // ------------------------------------------------------// Test if likelihood is normalized 128 // -----------------------------------------------------unsigned int numGridPoints = 1000001; double xMin = paramDomain . minValues () [0]; double xMax = paramDomain . maxValues () [0]; double intervalSize = ( xMax -xMin ) /(( double ) numGridPoints -1) ; 133 double integral = 0.; for ( unsigned int i = 0; i < numGridPoints ; ++ i ) { auxVec [0] = xMin + i * intervalSize ; integral += l i k e l i h o o d F u n c t i o n O b j . actualValue ( auxVec , NULL , NULL , NULL , NULL ) ; } 138 integral *= intervalSize ; if ( env . s ubDispla yFile () ) { * env . sub DisplayF ile () << " integral = " << integral << std : i p _ c o m p u t e S o l u t i o n = 1 i p _ d a t a O u t p u t F i l e N a m e = outputData / sipOutput i p _ d a t a O u t p u t A l l o w e d S e t Listing 6.67: Options for QUESO library used in application code (Listings 6.62-6.66). $ cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / bimodal / $ export L D_ L IB RA RY _ PA TH = $ L D _ L I B R A R Y _ P A T H :\ $HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib $ make -f M a k e f i l e _ b i m o d a l _ v i o l e t a $ ./ bimodal_gsl example . inp Figure 6 . 66.2 presents the KDE and CDF plots of the parameter θ. Figure 6 . 66.3 presents the autocorrelation of the parameter θ, in each one of the intermediate levels. Figure 6 . 66.2: KDE and CDF plots of parameter θ, for all fours levels. Figure 6 .6. 3 : 63Autocorrelation plots for θ, all four levels. Figure 6 .6. 4 : 64Intermediary likelihood functions f (θ) τ , where τ i is the exponent computed at the i-th level of Multilevel algorithm. In this simple problem, only four levels are needed, i.e. i = 1 . . . 4. The cyan-colored curve (exponent τ = 1) is the same curve as inFigure 6.6.1. Figure 6 . 67.1: Illustration of the hysteretic restoring force [see Eq. (6.7.2)] used in our hysteretic test problem. The terms r i , k i , and u i denote model parameters. Figure 6 . 67.3: Horizontal acceleration of each of the four floors (measured data aimed for calibration) used in our hysteretic test problem. $ cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / example $ rm outputData /* $ ./ example_gsl example _1chain . inp $ matlab $ plot_all . m result, the user should have created several of PNG figures containing kernel density estimate of the 15 parameters, cumulative density distribution, autocorrelation and scatter plots. The name of the figure files have been chosen to be informative, as shown in the Listing above. main ( int argc , char * argv []) { 5 // Initialize environment #i f d e f QUESO HAS MPI MPI_Init (& argc ,& argv ) ; e _ D a t a . floor [ i ] = new std :: vector < double >( numTimeSteps ,0.) ; } l i k e l i h o o d R o u t i n e _ D a t a . accel . resize ( numTimeSteps ,0.) ; FILE * inp ; inp = fopen ( " an . txt " ," r " ) ; unsigned int nu mO bs e rv at io n s = 0; double tmpA ; while ( fscanf ( inp , " % lf " ,& tmpA ) != EOF ) { l i k e l i h o o d R o u t i n e _ D a t a . accel [ n um Ob s er va t io ns ] = tmpA ; n um Ob se r va ti on s ++; } n um Ob se r va ti on s =0; FILE * inp1_1 ; inp1_1 = fopen ( " m e a s u r e d _ d a ta 1 _ 1 . txt " ," r " ) ; a . floor [2]) [ n um O bs er va t io ns ]= tmpA ; n um Ob se r va ti on s ++; } n um Ob se r va ti on s =0; FILE * inp1_4 ; inp1_4 = fopen ( " m e a s u r e d _ d a ta 1 _ 4 . txt " ," r " ) ; while ( fscanf ( inp1_4 , " % lf " ,& tmpA ) != EOF ) { (* l i k e l i h o o d R o u t i n e _ D a t a . floor [3]) [ n um O bs er va t io ns ]= tmpA ; n um Ob se r va ti on s ++; } QUESO :: GenericScalarFunction < QUESO :: GslVector , QUESO :<< " udd_cpp = zeros ( " << numFloors << " ," << numTimeSteps << " ) ;\ n " << " udd_cpp = [ " ; for ( unsigned int i = 0; i < numFloors ; ++ i ) { for ( unsigned int j = 0; j < numTimeSteps ; ++ j ) { udd . g e t P o s i t i o n V a l u e s (j , auxVec ) ; myFile << auxVec [ i ] << " " ; } myFile << std :: endl ; } myFile << " ]; " << std :: endl ; // Write ' resfor_cpp ' myFile << " resfor_cpp = zeros ( " << numFloors << " ," << numTimeSteps << " ) ;\ n " << " resfor_cpp = [ " ; for ( unsigned int i = 0; i < numFloors ; ++ i ) { for ( unsigned int j = 0; j < numTimeSteps ; ++ j ) { resfor . g e t P o s i t i o n V a l u e s (j , auxVec ) ; myFile << auxVec [ i ] << " " ; } myFile << std :: endl ; } myFile << " ]; " << std :: endl ; // Write ' ru_cpp ' myFile << " ru_cpp = zeros ( " << numFloors << " ," << numTimeSteps << " ) ;\ n " << " ru_cpp = [ " ; for ( unsigned int i = 0; i < numFloors ; ++ i ) { for ( unsigned int j = 0; j < numTimeSteps ; ++ j ) { ru . g e t P o s i t i o n V a l u e s (j , auxVec ) ; myFile << auxVec [ i ] << " " ; } myFile << std :: endl ; } myFile << " ]; " << std :: endl ; myFile . close () ; return ; } Listing 6.72: File example compute.C. $ cd $HOME / LIBRARIES / QUESO -0.51.0/ examples / hysteretic / $ export L D_ L IB RA RY _ PA TH = $ L D _ L I B R A R Y _ P A T H :\ $HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib $ make -f M a k e f i l e _ h y s t e r e t i c _ v i o l e t a $ ./ hyster etic_gs l example . inp Figure 6 . 67.4 presents the KDE plots of each parameter θ i , i = 1, . . . , 15. The Multilevel method also provides data about the logarithm of the likelihood function as well as of the target PDF. Figure 6 . 67.5 presents the KDE plots of both the likelihood function and of its logarithm.6.7.5.2 Autocorrelation and CDF Plots Figure 6 . 67.6a combines the CDF of all parameters θ i , i = 1, . . . , 15 into a single plot. Figure 6.7.6b presents their autocorrelations. Figure 6 . 67.4: KDE plots of parameter θ at the last level. Figure 6 . 67.5: KDE plots of the likelihood function, given by Eq. (6.7.3), and of its logarithm, at the last level. Figure 6 . 67.6: CDF and autocorrelation plots of parameter θ at the last level. APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLD-ERS AND/OR OTHER PARTIES PROVIDE THE LIBRARY "AS IS" WITHOUT WAR-RANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE LIBRARY IS WITH YOU. SHOULD THE LIBRARY PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 16. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE LIBRARY AS PERMITTED ABOVE, BE LI-ABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE LIBRARY (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE LIBRARY TO OPERATE WITH ANY OTHER SOFTWARE), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. one line to give the library's name and an idea of what it does. Copyright (C) year name of author This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. 2001, 2002, 2007, 2008 Free Software Foundation, Inc. http://fsf.org/ Equation (1.4.4). Compute τ so that Equation (1.4.14) is satisfied if τ > 1 then 41:τ ← 1 Recompute w ( )[k] andw ( )[k]Compute an estimate for the sample covariance matrix for π Select, from previous level, the initial positions for the Markov chains int (θ) are generated by doing the following for k = 1, . . . , n Generate chains: draw a number k from a discrete distribution P ( ) (k) in Equation(1.4.13) via Metropolis-Hastings i.e., obtain θ ( )[k] = P (l)[k]37: Set ← + 1 begin next level 38: Compute plausibility weights w ( )[k] via Equation (1.4.9) Compute normalized weights w ( )[k] via Equation (1.4.10) Compute n ( ) eff via Equation (1.4.11) 39: 40: 42: end if 43: ( ) int via Equation (1.4.12) 44: 45: Compute sizes of the chains the sum of the sizes = n ( ) total 46: Redistribute chain initial positions among processors Then the n ( ) total samples θ ( )[k] , from π ( ) ( ) total : 47: 48: Input: Starting point x 0 , step size δ, number of trials M , number of steps per trial N , unnormalized density or probability function P (x) for the target distribution. Output: Random number sequence x i , i = 0, 1, 2, . . . 49: for i = 0 . . . M do50: Monte Carlo is implemented in QUESO and it is the chosen algorithm to compute a sample of the output RV (the QoI) of the SFP for each given sample of the input RV.60: accepts ← accepts+1 61: end if 62: end if 63: end for 64: end for export LD_LIBRARY_PATH = $LD_LIBRARY_PATH :\ which can be placed in the user's .bashrc or other startup file.In addition, the user must set the following environmental variables:The latest supported public release of QUESO is available in the form of a tarball (tar format compressed with gzip) from QUESO Home Page: https://github.com/libqueso. you have copied the file 'queso-0.47.1.tar.gz' into $HOME/queso download/. Then just follow these commands to expand the tarball:$ cd $HOME / queso_download / $ tar xvf queso -0.47.1. tar . gz$HOME / LIBRARIES / gsl -1.15/ lib /:\ $HOME / LIBRARIES / boost -1.53.0/ lib /:\ $HOME / LIBRARIES / hdf5 -1.8.10/ lib /:\ $HOME / LIBRARIES / trilinos -11.2.4/ lib : $ export CC = gcc $ export CXX = g ++ $ export MPICC = mpicc $ export MPICXX = mpic ++ $ export F77 = fort77 $ export FC = gfortran 2.2 Obtaining a Copy of QUESO Suppose $ cd queso -0.47.1 #e n t e r t h e d i r e c t o r y r >> bin include lib examples 2.7 The Build Directory Structure The QUESO build directory contains three main directories, src, examples and test. They are listed below and more specific information about them can be obtained with the Developer's documentation from Section 2.5 above. 1. src: this directory contains the QUESO library itself, and its main subdirectories are: (a) basic/: contain classes for dealing with vector sets, subsets and spaces, scalar and vector functions and scalar and vector sequences (b) core/: contain classes that handle QUESO environment, and vector/matrix operations (c) stats/: contain classes that implement vector realizers, vector random variables, statistical inverse and forward problems; and the Monte Carlo and the Metropolis-Hasting solvers Details of QUESO classes are presented in Chapter 3. 2. examples: examples of different applications, with distinct levels of difficulty, using QUESO. The following examples have been thoroughly documented and are included in Chapter 6: The build directory contains only the source files. The executables are available under the QUESO installation directory, together with example of Makefiles that may be used to re-build the examples without the need of re-building the library.These tests can optionally be called during QUESO installation steps by entering the instruction: make check.3. test: a set of tests used as part of the periodic QUESO regression tests, conduct to ensure that more recent program/code changes have not adversely affected existing features of the library, as described in Section 2.4. (a) gsl tests (b) t01 valid cycle/ (c) t02 sip sfp/ (d) t03 sequence/ (e) t04 bimodal/ (f) test Environment/ (g) test GaussianVectorRVClass/ (h) test GslMatrix/ (i) test GslVector/ (j) test uqEnvironmentOptions/ lli@ma rgarid a :~/ LIBRARIES / QUESO -0.51.0/ bin$ 2. lib: contains the static and dynamic versions of the library. The full to path to this directory, e.g., $HOME/LIBRARIES/QUESO-0.51.0/lib should be added to the user's LD_LIBRARY_PATH environmental variable in order to use QUESO library with his/her application code: Note that due to QUESO being compiled/built with other libraries (GSL, Boost, Trilinos and HDF5), LD_LIBRARY_PATH had already some values set in Section 2.1.2. 3. include: contains the library .h files. 4. examples: contains the same examples of QUESO build directory, and listed in Section 2.7, together with their executables and Matlab files that may be used for visualization purposes. A selection of examples are described in details in Chapter 6; the user is invited understand their formulation, to run them and understand their purpose.$ export LD_LIBRARY_PATH = $LD_LIBRARY_PATH :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib 1 . 1world: MPI WORLD COMM; 2. full: communicator passed to the environment constructor, of size F and usually equal to the world communicator;3. sub: communicator of size F/S that contains the number of MPI nodes necessary to solve a statistical IP or a statistical FP;4. self: MPI SELF COMM, of size 1; and 5. inter0: communicator of size S formed by all MPI nodes that have subrank 0 in their respective subcommunicators. Table 3 . 31.1: Input file options for a QUESO environment. Option name Default value Description 〈PREFIX〉env help Produces help message for environment class 〈PREFIX〉env numSubEnvironments 1 Number of subenvironments 〈PREFIX〉env subDisplayFileName "." Output filename for sub-screen writing 〈PREFIX〉env subDisplayAllowAll 0 Allows all subenvironments to write to output file 〈PREFIX〉env subDisplayAllowedSet "" Subenvironments that will write to output file 〈PREFIX〉env displayVerbosity 0 Sets verbosity 〈PREFIX〉env syncVerbosity 0 Sets syncronized verbosity 〈PREFIX〉env seed 0 Set seed Table 3 . 33.1.QUESO::StatisticalInverseProblem< P_V, P_M > -m_env -m_priorRv -m_likelihoodFunction -m_postRv -m_solutionDomain -m_solutionPdf -m_subSolutionMdf -m_subSolutionCdf -m_solutionRealizer -m_mhSeqGenerator and 5 more... + StatisticalInverseProblem() + ~StatisticalInverseProblem() + computeSolutionFlag() + solveWithBayesMetropolisHastings() + solveWithBayesMLSampling() + priorRv() + postRv() + logEvidence() + meanLogLikelihood() + eig() + print() (a) StatisticalInverseProblem QUESO::StatisticalInverseProblemOptions + m_ov + m_prefix -m_env -m_optionsDesc -m_option_help -m_option_computeSolution -m_option_dataOutputFileName -m_option_dataOutputAllowedSet + StatisticalInverseProblemOptions() + StatisticalInverseProblemOptions() + ~StatisticalInverseProblemOptions() + scanOptionsValues() + print() -defineMyOptions() -getMyOptionValues() (b) StatisticalInverseProblemOptions Table 3 . 33.1: Input file options for a QUESO statistical inverse problem. Option name Default Value Description 〈PREFIX〉ip help Produces help message for statistical inverse problem 〈PREFIX〉ip computeSolution 1 Computes solution process 〈PREFIX〉ip dataOutputFileName "." Name of data output file 〈PREFIX〉ip dataOutputAllowedSet "" Subenvironments that will write to data out- put file Table 3 . 33.2: Input file options for a QUESO Metropolis-Hastings solver. Table 3 . 33.3: Input file options for a QUESO Multilevel solver (to be continued). Table 3 . 33.4: Input file options for a QUESO statistical forward problem.Option Name Default Value Description 〈PREFIX〉fp computeSolution 1 Computes the solution process 〈PREFIX〉fp computeCovariances 1 Compute p-q covariances 〈PREFIX〉fp computeCorrelations 1 Table 3 . 33.5: Input file options for a QUESO statistical forward problem solved via Monte Carlo algorithm. Table 5 . 51.1: Commonly used estimators and corresponding references for Alternatively, the user may call the file simple fp plots.m, which contains the above commands, together with a variety of others, for data visualization: Listing 6.20: Matlab code for loading the data in both parameter and QoI chains of the SFP, by calling the file simple fp plots.m.% inside Matlab >> clear all >> fp_p_seq . m >> fp_q_seq . m Listing 6.19: Matlab code for loading the data in both parameter and QoI chains of the SFP. % inside Matlab >> clear all >> si mp le _ fp _p lo t s Table 6 . 63.1: Measurement data d of size n d = 14. The object falls from altitude h 0 in t seconds, with standard deviation of σ seconds in the time measurement [1]. altitude [m] time [s] Std. Dev. σ [s] 10 1.41 0.02 20 2.14 0.12 30 2.49 0.02 40 2.87 0.01 50 3.22 0.03 60 3.49 0.01 70 3.81 0.03 80 4.07 0.03 90 4.32 0.03 100 4.47 0.05 110 4.75 0.01 120 4.99 0.04 130 5.16 0.01 140 5.26 0.09 6.3.1.2 The Prior RV, Likelihood and Posterior RV Listing 6.28: File gravity likelihood.h.class Likelihood : public QUESO :: BaseScalarFunction <V , M > { public : Likelihood ( const char * prefix , const QUESO :: VectorSet <V , M > & domain ) ; virtual~Likelihood () ; virtual double lnValue ( const V & domainVector , const V * domainDirection , V * gradVector , M * hessianMatrix , V * hessianEffect ) const ; virtual double actualValue ( const V & domainVector , const V * domainDirection , V * gradVector , M * hessianMatrix , V * hessianEffect ) const ; private : std :: vector < double > m_heights ; // heights std :: vector < double > m_times ; // times std :: vector < double > m_stdDevs ; // uncertainties in time measurements }; #e n d i f template class Likelihood < QUESO :: GslVector , QUESO :: GslMatrix >;, m_stdDevs (0) { // Data available in / inputData / data02 . dat double const heights [] = {10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , 110 , 120 , 130 , 140}; double const times [] = {1.41 , 2.14 , 2.49 , 2.87 , 3.22 , 3.49 , 3.81 , 4.07 , 4.32 , 4.47 , 4.75 , 4.99 , 5.16 , 5.26}; double const stdDevs [] = {0.020 , 0.120 , 0.020 , 0.010 , 0.030 , 0.010 , 0.030 , 0.030 , 0.030 , 0.050 , 0.010 , 0.040 , 0.010 , 0.09}; std :: size_t const n = sizeof ( heights ) / sizeof (* heights ) ; m_heights . assign ( heights , heights + n ) ; m_times . assign ( times , times + n ) ; m_stdDevs . assign ( stdDevs , stdDevs + n ) ; } template < class V , class M > Likelihood <V , M >::~Likelihood () { // Deconstruct here } template < class V , class M > double Likelihood <V , M >:: lnValue ( const V & domainVector , const V * domainDirection , V * gradVector , M * hessianMatrix , V * hessianEffect ) const { double g = domainVector [0]; double misfitValue = 0.0; for ( unsigned int i = 0; i < m_heights . size () ; ++ i ) { double modelTime = std :: sqrt (2.0 * m_heights [ i ] / g ) ; double ratio = ( modelTime -m_times [ i ]) / m_stdDevs [ i ]; misfitValue += ratio * ratio ; } return -0.5 * misfitValue ; } template < class V , class M > double Likelihood <V , M >:: actualValue ( const V & domainVector , const V * domainDirection , V * gradVector , M * hessianMatrix , V * hessianEffect ) const { return std :: exp ( this -> lnValue ( domainVector , domainDirection , gradVector , hessianMatrix , hessianEffect ) ) ; } Table 6 . 64.1 presents the data collected in the TGA experiment.Observation Temperature Relative mass Variance index "i" T i (K) m * obs,i (%) V i 1 673.034 96.5855 0.1 2 682.003 95.1549 0.1 3 690.985 92.5048 0.1 4 699.979 88.6353 0.1 5 708.989 83.0585 0.1 6 718.02 75.5306 0.1 7 727.089 64.1003 0.1 8 735.96 47.5474 0.1 9 744.904 23.6777 0.1 10 754.062 03.2234 0.1 11 763.049 00.0855448 0.1 Table 6.4.1: Experimental data. The parallel capabilities of QUESO have been exercised on the Ranger system of the TACC[2] with up to 16k processors. https://github.com/libqueso $ export LD_LIBRARY_PATH = $LD_LIBRARY_PATH :\ $HOME / LIBRARIES / QUESO -0.51.0/ lib http://www.gnu.org/software/gsl/manual/html_node/Evolution.html#index-gsl_005fodeiv2_ 005fevolve_005fapply #i n c l u d e <e x a m p l e c o m p u t e . h> #i n c l u d e < e x a m p l e l i k e l i h o o d . h> #i n c l u d e <q u e s o / G s l M a t r i x . h> #i n c l u d e <q u e s o / S t a t i s t i c a l I n v e r s e P r o b l e m . h> #i n c l u d e <q u e s o / G e n e r i c S c a l a r F u n c t i o n . h> 6 #i n c l u d e <q u e s o / G e n e r i c V e c t o r R V . h> #i n c l u d e <q u e s o / U n i f o r m V e c t o r R V . h> #i n c l u d e <q u e s o / C o n c a t e n a t e d V e c t o r R V . h> #i n c l u d e <q u e s o / InverseGammaVectorRV . h> #i n c l u d e <q u e s o / C o n c a t e n a t i o n S u b s e t . h> 11 #d e f i n e APPLS MODAL USES CONCATENATION void compute ( const QUESO :: Fu l lE nv ir o nm en t & env , unsigned int numModes ) { / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / #i n c l u d e <q u e s o / G s l M a t r i x . h> #i n c l u d e <q u e s o / S t a t i s t i c a l I n v e r s e P r o b l e m . h> #i n c l u d e <q u e s o /1 D1DFunction . h> #i n c l u d e <q u e s o / G e n e r i c S c a l a r F u n c t i o n . h> #i n c l u d e <q u e s o / G e n e r i c V e c t o r R V . h> } / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // Task 5 of 5: release memory before leaving routine . / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * if ( env . fullRank () == 0) { std :: cout << " Finishing run of ' uqTgaExample ' example " << std :: endl ; } return ; } / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The ' local comparison stage ' of the driving routine " uqAppl () " / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * template < class P_V , class P_M , class Q_V , class Q_M > / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The ' unified comparison stage ' of the driving routine " uqAppl () " / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * template < class P_V , class P_M , class Q_V , class Q_M > / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // The ODE ( state dot ) function / / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * int func(The following article was written by Richard Stallman, founder of the GNU Project.The biggest deficiency in the free software community today is not in the software−it is the lack of good free documentation that we can include with the free software. Many of our most important programs do not come with free reference manuals and free introductory texts. Documentation is an essential part of any software package; when an important free software package does not come with a free manual and a free tutorial, that is a major gap. We have many such gaps today.Consider Perl, for instance. The tutorial manuals that people normally use are nonfree. How did this come about? Because the authors of those manuals published them with restrictive terms−no copying, no modification, source files not available−which exclude them from the free software world.That wasn't the first time this sort of thing happened, and it was far from the last. Many times we have heard a GNU user eagerly describe a manual that he is writing, his intended contribution to the community, only to learn that he had ruined everything by signing a publication contract to make it non-free.Free documentation, like free software, is a matter of freedom, not price. The problem with the non-free manual is not that publishers charge a price for printed copies−that in itself is fine. (The Free Software Foundation sells printed copies of manuals, too.) The problem is the restrictions on the use of the manual. Free manuals are available in source code form, and give you permission to copy and modify. Non-free manuals do not allow this.The criteria of freedom for a free manual are roughly the same as for free software. Redistribution (including the normal kinds of commercial redistribution) must be permitted, so that the manual can accompany every copy of the program, both on-line and on paper.PreambleThe licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public Licenses are intended to guarantee your freedom to share and change free software-to make sure the software is free for all its users.This license, the Lesser General Public License, applies to some specially designated software packages-typically libraries-of the Free Software Foundation and other authors who decide to use it. You can use it too, but we suggest you first think carefully about whether this license or the ordinary General Public License is the better strategy to use in any particular case, based on the explanations below.When we speak of free software, we are referring to freedom of use, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish); that you receive source code or can get it if you want it; that you can change the software and use pieces of it in new free programs; and that you are informed that you can do these things.To protect your rights, we need to make restrictions that forbid distributors to deny you these rights or to ask you to surrender these rights. These restrictions translate to certain responsibilities for you if you distribute copies of the library or if you modify it.For example, if you distribute copies of the library, whether gratis or for a fee, you must . . . . Create Your Own Makefile, . . . . . . . . . . . . . . . . . . . . . 70Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 70 . Data Post-Processing, . . 71 6.2 simpleStatisticalForwardProblem . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.5 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 80 6.3 gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.1 Statistical Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.2 Statistical Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3.3 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3.4 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.5 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3.6 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . ; Visualization . . . . . . . . . . . . . . . . . . 97. . . . . . . . . . . . . Visualization, . . 71 6.2 simpleStatisticalForwardProblem . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.5 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 80 6.3 gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.1 Statistical Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.2 Statistical Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3.3 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3.4 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.5 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3.6 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . ; Visualization . . . . . . . . . . . . . . . . . . 9796 6.3.7 Running the Gravity Example with Several Processors . . . . . . . . . 97 6.3.8 Data Post-ProcessingData Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 71 6.2 simpleStatisticalForwardProblem . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.5 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 80 6.3 gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.1 Statistical Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.3.2 Statistical Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3.3 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3.4 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.5 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3.6 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.7 Running the Gravity Example with Several Processors . . . . . . . . . 97 6.3.8 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 97 106 6.4.1 Thermogravimetric Experiments and a Simple Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validationcycle, . . . . . . . . . . 106 6.4.2 Statistical Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4.3 Statistical Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.4 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4.5 TGA Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4.6 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.4.7 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 132validationCycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.4.1 Thermogravimetric Experiments and a Simple Model . . . . . . . . . . 106 6.4.2 Statistical Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4.3 Statistical Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.4 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4.5 TGA Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4.6 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.4.7 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 132 134 6.5.1 One-mode distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal, . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.5.2 Two-mode distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.5.3 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5.4 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.5.5 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.5.6 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 145modal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.5.1 One-mode distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.5.2 Two-mode distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.5.3 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5.4 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.5.5 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.5.6 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bimodal, . . 149 6.6.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.6.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.6.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.6.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.6.5 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 157bimodal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.6.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.6.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.6.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.6.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.6.5 Data Post-Processing and Visualization . . . . . . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hysteretic, . . 159 6.7.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.7.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.7.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.7.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 172 r R V// --------------------------------------------------h> void compute ( const QUESOhysteretic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.7.1 Running the Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.7.2 Example Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.7.3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.7.4 Create your own Makefile . . . . . . . . . . . . . . . . . . . . . . . . . 172 r R V . h> void compute ( const QUESO :: Fu l lE nv ir o nm en t & env ) { // ------------------------------------------------------ QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpace. // --------------------------------------------------env , " param_ " , 1 , NULL// ------------------------------------------------------ QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpace ( env , " param_ " , 1 , NULL ) ; GslVector paramMaxs ( paramSpace . zeroVector (. QUESO :: GslVector paramMaxs ( paramSpace . zeroVector () ) ; BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > QUESO :: ScalarSequence < double > seq1 ( env , numPosSmallerThan40. QUESO :: BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > QUESO :: ScalarSequence < double > seq1 ( env , numPosSmallerThan40 , " " ) ; QUESO :: ScalarSequence < double > seq2 ( env , numPosTotal -numPosSmallerThan40. QUESO :: ScalarSequence < double > seq2 ( env , numPosTotal -numPosSmallerThan40 , " " ) ; QUESO :: ScalarSequence < double > seqAll ( env , numPosTotal. QUESO :: ScalarSequence < double > seqAll ( env , numPosTotal , " " ) ; const QUESO :: B as eE n vi ro nm e nt & env = paramValues. env (const QUESO :: B as eE n vi ro nm e nt & env = paramValues . env () ; . U Q _ F A T A L _ T E S T _ M A C R O, paramValues . sizeLocal () != 15 , env . fullRank (. o d (. invalid parameter size " )U Q _ F A T A L _ T E S T _ M A C R O ( paramValues . sizeLocal () != 15 , env . fullRank () , o d () " , " invalid parameter size " ) ; const std :: vector < std :: vector < double >* >& floor =. > floor ; const std :: vector < double >& accel =. Pt r ) -> accelconst std :: vector < std :: vector < double >* >& floor = (( l i k e l i h o o d R o u t i n e _ D a t a T y p e *) fu n ct io nD a ta Pt r ) -> floor ; const std :: vector < double >& accel = (( l i k e l i h o o d R o u t i n e _ D a t a T y p e *) fu n ct io nD a ta Pt r ) -> accel ; . U Q _ F A T A L _ T E S T _ M A C R O, env . fullRank (o d (. invalid ' numFloors ' " )U Q _ F A T A L _ T E S T _ M A C R O (( numFloors != 4) , env . fullRank () , o d () " , " invalid ' numFloors ' " ) ; . U Q _ F A T A L _ T E S T _ M A C R O, env . fullRank (o d (. invalid ' numTimeSteps ' " )U Q _ F A T A L _ T E S T _ M A C R O (( numTimeSteps != 401) , env . fullRank () , o d () " , " invalid ' numTimeSteps ' " ) ; . U Q _ F A T A L _ T E S T _ M A C R O, floor [ i ] -> size () != numTimeSteps , env . fullRank (. o d (. invalid number of steps " )U Q _ F A T A L _ T E S T _ M A C R O ( floor [ i ] -> size () != numTimeSteps , env . fullRank () , o d () " , " invalid number of steps " ) ; } Queso :: Vectorspace, &lt; Queso, GslVector , QUESO :: GslMatrix > floorSpace ( env , " floor. } QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > floorSpace ( env , " floor_ " , numFloors , NULL ) ; GslVector kVec ( floorSpace . zeroVector (. QUESO :: GslVector kVec ( floorSpace . zeroVector () ) ; . Exp, e +7 * exp ( paramValues [4]) ; GslVector rVec ( floorSpace . zeroVector (. QUESO :: GslVector rVec ( floorSpace . zeroVector () ) ; GslVector uVec ( floorSpace . zeroVector (. -1 * exp ( paramValues [13QUESO :: GslVector uVec ( floorSpace . zeroVector () ) ; -1 * exp ( paramValues [13]) ; SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > u ( floorSpace , numTimeSteps. QUESO :: SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > u ( floorSpace , numTimeSteps , " " ) ; / Queso, SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > ud ( floorSpace , numTimeSteps. // absolute displacement QUESO :: SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > ud ( floorSpace , numTimeSteps , " " ) ; // Velocity Queso, SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > udd. floorSpace , numTimeSteps , " " ) ; ( const QUESO :: Fu l lE nv ir o nm en t & env ) { struct timeval timevalNow// velocity QUESO :: SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > udd ( floorSpace , numTimeSteps , " " ) ; ( const QUESO :: Fu l lE nv ir o nm en t & env ) { struct timeval timevalNow ; Beginning run of ' Hysteretic ' example at " << ctime (& timevalNow . tv_sec ). cout << std :: endl <<. std :: cout << std :: endl << " Beginning run of ' Hysteretic ' example at " << ctime (& timevalNow . tv_sec ) ; VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpaceA. // ------------------------------------------------------QUESOenv , " paramA_ " , 1 , NULL// Step 1 of 5: Instantiate the parameter space // ------------------------------------------------------ QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpaceA ( env , " paramA_ " , 1 , NULL ) ; VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpaceB. 14env , " paramB_QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpaceB ( env , " paramB_ " , 14 , NULL ) ; VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpace. 15env , " param_QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > paramSpace ( env , " param_ " , 15 , NULL ) ; . paramMinsA . cwSet. 0paramMinsA . cwSet (0) ; . paramMaxsA . cwSet. 5paramMaxsA . cwSet (5) ; BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomainA ( " paramA. paramSpaceA , paramMinsA , paramMaxsAQUESO :: BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomainA ( " paramA_ " , paramSpaceA , paramMinsA , paramMaxsA ) ; GslVector paramMinsB ( paramSpaceB . zeroVector (. QUESO :: GslVector paramMinsB ( paramSpaceB . zeroVector () ) ; . paramMinsB . cwSet ( -INFINITY. paramMinsB . cwSet ( -INFINITY ) ; GslVector paramMaxsB ( paramSpaceB . zeroVector (. QUESO :: GslVector paramMaxsB ( paramSpaceB . zeroVector () ) ; . paramMaxsB . cwSet ( INFINITY. paramMaxsB . cwSet ( INFINITY ) ; BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomainB ( " paramB. paramSpaceB , paramMinsB , paramMaxsBQUESO :: BoxSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomainB ( " paramB_ " , paramSpaceB , paramMinsB , paramMaxsB ) ; ConcatenationSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomain. paramSpace , paramDomainA , paramDomainBQUESO :: ConcatenationSubset < QUESO :: GslVector , QUESO :: GslMatrix > paramDomain ( " " , paramSpace , paramDomainA , paramDomainB ) ; std :: cout << " \ tI nstantia ting the Likelihood ; calling internally the hysteretic model. << std :: endl. object // ------------------------------------------------------// Step 3 of 5: Instantiate the likelihood function object // ------------------------------------------------------ std :: cout << " \ tI nstantia ting the Likelihood ; calling internally the hysteretic model " << std :: endl ; UniformVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRvA ( " priorA. paramDomainAQUESO :: UniformVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRvA ( " priorA_ " , paramDomainA ) ; GslVector meanVec ( paramSpaceB . zeroVector (. QUESO :: GslVector meanVec ( paramSpaceB . zeroVector () ) ; GslVector diagVec ( paramSpaceB . zeroVector (. QUESO :: GslVector diagVec ( paramSpaceB . zeroVector () ) ; GslMatrix covMatrix. diagVecQUESO :: GslMatrix covMatrix ( diagVec ) ; GaussianVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRvB ( " priorB. paramDomainB , meanVec , covMatrixQUESO :: GaussianVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRvB ( " priorB_ " , paramDomainB , meanVec , covMatrix ) ; ConcatenatedVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRv ( " prior. priorRvA , priorRvB , paramDomainQUESO :: ConcatenatedVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > priorRv ( " prior_ " , priorRvA , priorRvB , paramDomain ) ; GenericVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > postRv ( " post. paramSpaceQUESO :: GenericVectorRV < QUESO :: GslVector , QUESO :: GslMatrix > postRv ( " post_ " , paramSpace ) ; StatisticalInverseProblem < QUESO :: GslVector , QUESO :: GslMatrix > ip. NULL , priorRv , likelihoodFunctionObj , postRvQUESO :: StatisticalInverseProblem < QUESO :: GslVector , QUESO :: GslMatrix > ip ( " " , NULL , priorRv , likelihoodFunctionObj , postRv ) ; cout << " \ tSolving the SIP with Multilevel method. // ------------------------------------------------------std<< std :: endl ; g (// Step 5 of 5: Solve the inverse problem // ------------------------------------------------------ std :: cout << " \ tSolving the SIP with Multilevel method " << std :: endl ; g () ; Ending run of ' Hysteretic ' example at " << ctime (& timevalNow . tv_sec ) << std :: endl ; return. cout <<. std :: cout << " Ending run of ' Hysteretic ' example at " << ctime (& timevalNow . tv_sec ) << std :: endl ; return ; vector < double > accel. std :: vector < double > accel ( numTimeSteps ,0.) ; . File * Inp, FILE * inp ; while ( fscanf ( inp. % lf. & tmpA ) != EOFwhile ( fscanf ( inp , " % lf " ,& tmpA ) != EOF ) { um Ob s er va ti o ns >= accel . size () ) , env . fullRank (. U Q _ F A T A L _ T E S T _ M A C R O, n. debug_hyst (. input file has too many lines " ) ; ++;U Q _ F A T A L _ T E S T _ M A C R O (( n um Ob s er va ti o ns >= accel . size () ) , env . fullRank () , " debug_hyst () " , " input file has too many lines " ) ; ++; } U Q _ F A T A L _ T E S T _ M A C R O, um O bs er va t io ns != accel . size () ) , env . fullRank (. n. debug_hyst (. input file has a smaller number of observations than expected " )} U Q _ F A T A L _ T E S T _ M A C R O (( n um O bs er va t io ns != accel . size () ) , env . fullRank () , " debug_hyst () " , " input file has a smaller number of observations than expected " ) ; VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > floorSpace. env , " floor_QUESO :: VectorSpace < QUESO :: GslVector , QUESO :: GslMatrix > floorSpace ( env , " floor_ " , numFloors , NULL ) ; GslVector kVec ( floorSpace . zeroVector (. QUESO :: GslVector kVec ( floorSpace . zeroVector () ) ; GslVector rVec ( floorSpace . zeroVector (. QUESO :: GslVector rVec ( floorSpace . zeroVector () ) ; GslVector uVec ( floorSpace . zeroVector (. QUESO :: GslVector uVec ( floorSpace . zeroVector () ) ; SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > u ( floorSpace , numTimeSteps. QUESO :: SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > u ( floorSpace , numTimeSteps , " " ) ; / Queso, SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > ud ( floorSpace , numTimeSteps. // absolute displacement QUESO :: SequenceOfVectors < QUESO :: GslVector , QUESO :: GslMatrix > ud ( floorSpace , numTimeSteps , " " ) ; . // Velocity, // velocity Diagrams for Interactive Learning. Diagrams for Interactive Learning. http://interactagram.com/. Accessed on August 12th, 2012. TACC: Texas Advanced Computing Center. TACC: Texas Advanced Computing Center. http://www.tacc.utexas.edu/, 2001-2013. Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation. J L Beck, S.-K Au, Journal Of Engineering Mechanics. 128J. L. Beck and S.-K. Au. Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation. Journal Of Engineering Mechanics, 128:380-391, 2002. Introduction to Bayesian Scientific Computing. D Calvetti, E Somersalo, Surveys and Tutorials in the Applied Mathematical Sciences. 2SpringerD. Calvetti and E. Somersalo. Introduction to Bayesian Scientific Computing, volume 2 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, 2007. Bayesian Methods for Data Analysis. Texts in Statistical Science. B P Carlin, T A Louis, CRS Press3rd editionB. P. Carlin and T. A. Louis. Bayesian Methods for Data Analysis. Texts in Statistical Science. CRS Press, 3rd edition, 2006. Stochastic analysis, model and reliability updating of complex systems with applications to structural dynamics. S H Cheung, California Institute of Technology.PhD thesisS. H. Cheung. Stochastic analysis, model and reliability updating of complex systems with applications to structural dynamics. PhD thesis. Jan 2009, California Institute of Technology., 2009. . S H Cheung, Todd A Oliver, E E Prudencio, Serge Prudhomme, Robert D , S. H. Cheung, Todd A. Oliver, E. E. Prudencio, Serge Prudhomme, and Robert D. Bayesian uncertainty analysis with applications to turbulence modeling. Moser, Reliability Engineering & System Safety. 969Moser. Bayesian uncertainty analysis with applications to turbulence modeling. Reliability Engineering & System Safety, 96(9, SI):1137-1149, 2011. Parallel adaptive multilevel sampling algorithms for the Bayesian analysis of mathematical models. S H Cheung, E E Prudencio, International Journal for Uncertainty Quantification. 23215237S. H. Cheung and E. E. Prudencio. Parallel adaptive multilevel sampling algorithms for the Bayesian analysis of mathematical models. International Journal for Uncertainty Quantification, 2(3):215237, 2012. Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J Ching, Y Chen, Journal Of Engineering Mechanics. 1337816832J. Ching and Y. Chen. Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. Journal Of Engineering Mechanics, 133(7):816832, 2007. Division on Engineering and Physical Sciences, National Research Council. Assessing the Reliability of Complex Models: Mathematical and Statistical Foundations of Verification, Validation, and Uncertainty Quantification. Committee on Mathematical Foundations of Verification, Validation, and Uncertainty Quantification; Board on Mathematical Sciences and Their Applications. The National Academies PressCommittee on Mathematical Foundations of Verification, Validation, and Uncertainty Quantification; Board on Mathematical Sciences and Their Applications, Division on Engineering and Physical Sciences, National Research Council. Assessing the Reliability of Complex Models: Mathematical and Statistical Foundations of Verification, Validation, and Uncertainty Quantification. The National Academies Press, 2012. http://www.nap. edu/openbook.php?record_id=13395. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I theory. R I Cukier, C M Fortuin, K E Shuler, A G Petschek, J H Schaibly, The Journal of chemical physics. 598R.I. Cukier, C.M. Fortuin, K.E. Shuler, A.G. Petschek, and J.H. Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I theory. The Journal of chemical physics, 59(8):3873-3878, 1973. Asymptotic Theory of Statistics and Probability. A Dasgupta, Springer Texts in Statistics. SpringerA. DasGupta. Asymptotic Theory of Statistics and Probability. Springer Texts in Statis- tics. Springer, 2008. Probability: Theory and Examples. R Durret, Duxbury Advanced Series. Thomson Brooks/Cole3rd editionR. Durret. Probability: Theory and Examples. Duxbury Advanced Series. Thomson Brooks/Cole, 3rd edition, 2005. . Mark Galassi, James Theiler, Brian Gough, Gerard Jungman, Many Others, GNU Scientific Library. Mark Galassi, James Theiler, Brian Gough, Gerard Jungman, and many others. GNU Scientific Library. http://www.gnu.org/software/gsl/, 1996-2011. Bayesian Data Analysis. Texts in Statistical Science. A Gelman, J B Carlin, H S Stern, D B Rubin, Chapman & Hall/CRC2nd editionA. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis. Texts in Statistical Science. Chapman & Hall/CRC, 2nd edition, 2004. Stochastic finite elements: a spectral approach. R G Ghanem, P D Spanos, Courier Corporation. R.G. Ghanem and P.D. Spanos. Stochastic finite elements: a spectral approach. Courier Corporation, 2003. Delayed rejection in reversible jump Metropolis-Hastings. P J Green, A Mira, BIOMETRIKA. 884P. J. Green and A. Mira. Delayed rejection in reversible jump Metropolis-Hastings. BIOMETRIKA, 88(4):1035-1053, 2001. HDF5 (Hierarchical Data Format 5). Hdf The, Group, The HDF Group. HDF5 (Hierarchical Data Format 5). http://www.hdfgroup.org/ HDF5/, 2000-2012. . The MathWorks Group. MATLAB. The MathWorks Group. MATLAB . http://www.mathworks.com/products/matlab/, 2013. A radial basis function method for global optimization. H-M Gutmann, Journal of Global Optimization. 193H-M Gutmann. A radial basis function method for global optimization. Journal of Global Optimization, 19(3):201-227, 2001. DRAM: Efficient adaptive MCMC. H Haario, M Laine, A Mira, E Saksman, Stat. Comput. 16H. Haario, M. Laine, A. Mira, and E. Saksman. DRAM: Efficient adaptive MCMC. Stat. Comput., 16:339-354, 2006. An adaptive Metropolis algorithm. H Haario, E Saksman, J Tamminen, Bernoulli. 72H. Haario, E. Saksman, and J. Tamminen. An adaptive Metropolis algorithm. Bernoulli, 7(2):223-242, 2001. Monte Carlo sampling methods using Markov chains and their applications. W K Hastings, BIOMETRIKA. 571W. K. Hastings. Monte Carlo sampling methods using Markov chains and their applica- tions. BIOMETRIKA, 57(1):97-109, 1970. . Michael Heroux, Trilinos, Michael Heroux. Trilinos. http://trilinos.sandia.gov/, 2013. An overview of the Trilinos project. Michael A Heroux, Roscoe A Bartlett, Vicki E Howle, Robert J Hoekstra, Jonathan J Hu, Tamara G Kolda, Richard B Lehoucq, Kevin R Long, Roger P Pawlowski, Eric T Phipps, Andrew G Salinger, Heidi K Thornquist, Ray S Tuminaro, James M Willenbring, Alan Williams, Kendall S Stanley, ACM Trans. Math. Softw. 313Michael A. Heroux, Roscoe A. Bartlett, Vicki E. Howle, Robert J. Hoekstra, Jonathan J. Hu, Tamara G. Kolda, Richard B. Lehoucq, Kevin R. Long, Roger P. Pawlowski, Eric T. Phipps, Andrew G. Salinger, Heidi K. Thornquist, Ray S. Tuminaro, James M. Willen- bring, Alan Williams, and Kendall S. Stanley. An overview of the Trilinos project. ACM Trans. Math. Softw., 31(3):397-423, 2005. Importance measures in global sensitivity analysis of nonlinear models. T Homma, A Saltelli, Reliability Engineering & System Safety. 521T. Homma and A. Saltelli. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering & System Safety, 52(1):1-17, 1996. Optimal Bayesian experimental design for combustion kinetics. X Huan, Y Marzouk, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, number AIAA 2011-513. Orlando, FloridaX. Huan and Y. Marzouk. Optimal Bayesian experimental design for combustion kinet- ics. In 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, number AIAA 2011-513, Orlando, Florida, 2011. . Silicon Graphics International. STL: Standard Template Library. Silicon Graphics International. STL: Standard Template Library. http://www.sgi.com/ tech/stl/, 2000-2011. . J Jacod, P Protter, Probability Essentials. Springer2nd editionJ. Jacod and P. Protter. Probability Essentials. Springer, 2nd edition, 2004. Analysis of variance designs for model output. M J W Jansen, Computer Physics Communications. 1171M.J.W. Jansen. Analysis of variance designs for model output. Computer Physics Com- munications, 117(1):35-43, 1999. Probability Theory: The Logic of Science. E T Jaynes, Cambridge University PressE. T. Jaynes. Probability Theory: The Logic of Science. Cambridge University Press, 2003. The C++ Standard Library, A Tutorial and Reference. N M Josuttis, Addison Wesley1st editionN. M. Josuttis. The C++ Standard Library, A Tutorial and Reference. Addison Wesley, 1st edition, 1999. Statistical and Computational Inverse Problems. J Kaipio, E Somersalo, Applied Mathematical Sciences. 160SpringerJ. Kaipio and E. Somersalo. Statistical and Computational Inverse Problems, volume 160 of Applied Mathematical Sciences. Springer, 2005. M Laine, DRAM -Delayed Rejection Adaptive Metropolis. M. Laine. DRAM -Delayed Rejection Adaptive Metropolis. http://helios.fmi.fi/ lainema/dram/, 2006-2008. Accessed on October 8th, 2013. MCMC Toolbox for Matlab. helios.fmi.fi/~lainema/mcmc. M Laine, M. Laine. MCMC Toolbox for Matlab. helios.fmi.fi/~lainema/mcmc/, 2006-2008. Accessed on March 28th, 2013. Adaptive MCMC methods with applications in environmental and geophysical models. M Laine, Lappeenranta, FinlandLappeenranta University of TechnologyPhD thesisM. Laine. Adaptive MCMC methods with applications in environmental and geophysical models. PhD thesis, Lappeenranta University of Technology, Lappeenranta, Finland, 2008. . S B Lippman, J Lajoie, B E Moo, Primer, Addison Wesley4th editionS. B. Lippman, J. Lajoie, and B. E. Moo. C++ Primer. Addison Wesley, 4th edition, 2005. . Andrew Makhorin, Glkp Development Team, Glpk, GNU Linear Programming KitAndrew Makhorin and GLKP Development Team. GLPK (GNU Linear Programming Kit). http://www.gnu.org/software/glpk/, 2000-2012. Equations of state calculations by fast computing machines. N Metropolis, A W Rosenbluth, M N Rosenbluth, A H Teller, E Teller, Journal of Chemical Physics. 216N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equa- tions of state calculations by fast computing machines. Journal of Chemical Physics, 21(6):1087-1092, 1953. On Metropolis-Hastings algorithms with delayed rejection. Metron -International. A Mira, Journal of Statistics. 593-4A. Mira. On Metropolis-Hastings algorithms with delayed rejection. Metron -Interna- tional Journal of Statistics, 59(3-4):231-241, 2001. The Parallel C++ Statistical Library 'QUESO': Quantification of Uncertainty for Estimation, Simulation and Optimization. E E Prudencio, K W Schulz, Euro-Par 2011: Parallel Processing Workshops. Berlin / HeidelbergSpringer7155E. E. Prudencio and K. W. Schulz. The Parallel C++ Statistical Library 'QUESO': Quantification of Uncertainty for Estimation, Simulation and Optimization. In Euro-Par 2011: Parallel Processing Workshops, volume 7155 of Lecture Notes in Computer Science, pages 398-407. Springer Berlin / Heidelberg, 2012. The Bayesian Choice. C P Robert, Springer Verlag2nd editionC. P. Robert. The Bayesian Choice. Springer Verlag, 2nd edition, 2004. Monte Carlo Statistical Methods. C P Robert, G Casella, Springer Verlag2nd editionC. P. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Verlag, 2nd edition, 2005. Variance based sensitivity analysis of model output. design and estimator for the total sensitivity index. A Saltelli, P Annoni, I Azzini, F Campolongo, M Ratto, S Tarantola, Computer Physics Communications. 1812A. Saltelli, P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola. Variance based sensitivity analysis of model output. design and estimator for the total sensitivity index. Computer Physics Communications, 181(2):259-270, 2010. Global sensitivity analysis: the primer. A Saltelli, M Ratto, T Andres, F Campolongo, J Cariboni, D Gatelli, M Saisana, S Tarantola, John Wiley & SonsA. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola. Global sensitivity analysis: the primer. John Wiley & Sons, 2008. Density Estimation for Statistics and Data Analysis. B W Silverman, Number 26 in Monographs on Statistics & Applied Probability. Chapman & Hall/CRCB. W. Silverman. Density Estimation for Statistics and Data Analysis. Number 26 in Monographs on Statistics & Applied Probability. Chapman & Hall/CRC, 1986. On sensitivity estimation for nonlinear mathematical models. I M Sobol, Matematicheskoe Modelirovanie. 2I.M. Sobol'. On sensitivity estimation for nonlinear mathematical models. Matematich- eskoe Modelirovanie, 2(1):112-118, 1990. Global sensitivity indices for the investigation of nonlinear mathematical models. I M Sobol, Matematicheskoe Modelirovanie. 19I.M. Sobol'. Global sensitivity indices for the investigation of nonlinear mathematical models. Matematicheskoe Modelirovanie, 19(11):23-24, 2007. . Boost Development Team. Boost C++ Libraries. Boost Development Team. Boost C++ Libraries. http://www.boost.org/, 1998-2013. GCC Team. GCC, the GNU compiler collection. GCC Team. GCC, the GNU compiler collection. http://gcc.gnu.org/, 1987-2013. GNU Development Team. GNU build system (autotools). GNU Development Team. GNU build system (autotools). http://www.gnu.org/ software/software.html, ?-2013. Retrieved August 4th, 2013. The groovy toolkit (GRVY). Grvy Development Team, GRVY Development Team. The groovy toolkit (GRVY). https://red.ices.utexas. edu/projects/software/wiki/GRVY, 1996-2009. . Intel Team. Intel c++ compiler. Intel Team. Intel c++ compiler. http://software.intel.com/en-us/c-compilers, 1987-2013. Retrieved August 4th, 2013. MPICH Development Team. MPICH: High-Performance Portable MPI. MPICH Development Team. MPICH: High-Performance Portable MPI. http://www. mpich.org/, 2001-2013. . Octave Development Team. GNU Octave. Octave Development Team. GNU Octave. http://www.gnu.org/software/octave/, 1988-2013. Open MPI: Open source high performance computing. Mpi Development Open, Team, Open MPI Development Team. Open MPI: Open source high performance computing. http://www.open-mpi.org/, 2008-2012. . Dimitri Van Heesch, Dimitri van Heesch. Doxygen. http://www.doxygen.org/, 2009-2013. Design analysis for optimal calibration of diffusivity in reactive multilayers. M Vohra, X Huan, T P Weihs, O M Knio, arXiv:1610.02558arXiv preprintM. Vohra, X. Huan, T.P. Weihs, and O.M. Knio. Design analysis for optimal calibration of diffusivity in reactive multilayers. arXiv preprint arXiv:1610.02558, 2016. Development of a reduced model of formation reactions in zr-al nanolaminates. M Vohra, J Winokur, K R Overdeep, P Marcello, T P Weihs, O M Knio, Journal of Applied Physics. 11623233501M. Vohra, J. Winokur, K.R. Overdeep, P. Marcello, T.P. Weihs, and O.M. Knio. De- velopment of a reduced model of formation reactions in zr-al nanolaminates. Journal of Applied Physics, 116(23):233501, 2014. The wiener-askey polynomial chaos for stochastic differential equations. D Xiu, G E Karniadakis, SIAM journal on scientific computing. 242D. Xiu and G.E. Karniadakis. The wiener-askey polynomial chaos for stochastic differ- ential equations. SIAM journal on scientific computing, 24(2):619-644, 2002.
[ "https://github.com/libqueso.", "https://github.com/libqueso" ]
[ "INSPIRED: Intention-based Privacy-preserving Permission Model", "INSPIRED: Intention-based Privacy-preserving Permission Model" ]
[ "Hao Fu ", "Zizhan Zheng \nDepartment of Computer Science\nTulane University\nNew OrleansUSA\n\nDepartment of Computer Science\nPennsylvania State University\nPennsylvaniaUSA\n", "Sencun Zhu ", "Prasant Mohapatra ", "\nDepartment of Computer Science\nUniversity of California\nDavisUSA\n" ]
[ "Department of Computer Science\nTulane University\nNew OrleansUSA", "Department of Computer Science\nPennsylvania State University\nPennsylvaniaUSA", "Department of Computer Science\nUniversity of California\nDavisUSA" ]
[]
Mobile operating systems adopt permission systems to protect system integrity and user privacy. In this work, we propose INSPIRED, an intention-aware dynamic mediation system for mobile operating systems with privacy preserving capability. When a security or privacy sensitive behavior is triggered, INSPIRED automatically infers the underlying program intention by examining its runtime environment and justifies whether to grant the relevant permission by matching with user intention. We stress on runtime contextual-integrity by answering the following three questions: who initiated the behavior, when was the sensitive action triggered and under what kind of environment was it triggered? Specifically, observing that mobile applications intensively leverage user interface (UI) to reflect the underlying application functionality, we propose a machine learning based permission model using foreground information obtained from multiple sources. To precisely capture user intention, our permission model evolves over time and it can be user-customized by continuously learning from user decisions. Moreover, by keeping and processing all user's behavioral data inside her own device (i.e., without sharing with a third-party cloud for learning), INSPIRED is also privacy-preserving. Our evaluation shows that our model achieves both high precision and recall (95%) based on 6,560 permission requests from both benign apps and malware. Further, it is capable of capturing users' specific privacy preferences with an acceptable median fmeasure (84.7%) for 1,272 decisions from users. Finally, we show INSPIRED can be deployed on real Android devices to provide real-time protection with a low overhead.
null
[ "https://arxiv.org/pdf/1709.06654v1.pdf" ]
9,879,035
1709.06654
0339584e6c0b073e2f62383a7a76d448766143f1
INSPIRED: Intention-based Privacy-preserving Permission Model Hao Fu Zizhan Zheng Department of Computer Science Tulane University New OrleansUSA Department of Computer Science Pennsylvania State University PennsylvaniaUSA Sencun Zhu Prasant Mohapatra Department of Computer Science University of California DavisUSA INSPIRED: Intention-based Privacy-preserving Permission Model Mobile operating systems adopt permission systems to protect system integrity and user privacy. In this work, we propose INSPIRED, an intention-aware dynamic mediation system for mobile operating systems with privacy preserving capability. When a security or privacy sensitive behavior is triggered, INSPIRED automatically infers the underlying program intention by examining its runtime environment and justifies whether to grant the relevant permission by matching with user intention. We stress on runtime contextual-integrity by answering the following three questions: who initiated the behavior, when was the sensitive action triggered and under what kind of environment was it triggered? Specifically, observing that mobile applications intensively leverage user interface (UI) to reflect the underlying application functionality, we propose a machine learning based permission model using foreground information obtained from multiple sources. To precisely capture user intention, our permission model evolves over time and it can be user-customized by continuously learning from user decisions. Moreover, by keeping and processing all user's behavioral data inside her own device (i.e., without sharing with a third-party cloud for learning), INSPIRED is also privacy-preserving. Our evaluation shows that our model achieves both high precision and recall (95%) based on 6,560 permission requests from both benign apps and malware. Further, it is capable of capturing users' specific privacy preferences with an acceptable median fmeasure (84.7%) for 1,272 decisions from users. Finally, we show INSPIRED can be deployed on real Android devices to provide real-time protection with a low overhead. I. INTRODUCTION Millions of mobile applications (or apps for short) are available to users due to the fast penetration of smart devices. On the one hand, these apps access device resources to support various functionalities. For example, a weather app queries user locations to provide precise humidity information; the referral page of a utility app uses SMS to invite friends. On the other hand, they may also abuse the resources, e.g., by transmitting sensitive data to a third party that is unintended by the user or sending premium SMS stealthily to introduce extra cost to the user. To this end, mobile operating systems such as Android and iOS adopt permission systems as an important line of defense for protecting the security and privacy of users. In particular, early versions of Android present the list of permissions requested by an app when it is installed, where the users can only make an all-or-nothing decision. More recently, Android 6.0 implements an opt-in system similar to iOS, where users are allowed to grant or deny a permission to an app when it is needed by the app for the first time. But even this approach does not provide sufficient protection as an adversary can easily induce users to grant the permission first, and then exploit the same resource for malicious purposes. A recent study [56] showed that at least 80% users would have preferred to prevent at least one permission request involved in the study, and suggested the necessity of more fine-grained control of permissions. However, simply querying users for every sensitive resource access is annoying and causing dialog fatigue. Ideally, a permission system should be able to identify suspicious permission requests on the fly and automatically by taking user preferences into account, and notify users only when necessary. To enable effective run-time permission control, it is crucial to account for the context pertinent to sensitive permission requests, as shown in several recent user studies [40,56,57]. They observed that the user's preference is strongly correlated with the foreground app and the visibility of the permission requesting app (i.e., whether the app is currently visible to the user or not). The intuition is that users often rely on display to infer the purpose of a permission request, and they tend to block resource requests that are considered to be irrelevant to app's functionalities [56]. Thus, a permission system that can properly identify and utilize foreground data may significantly improve accuracy and reduce user involvement. We posit that to fully achieve contextual integrity [14], it is important to capture more detailed foreground information beyond visibility, in order to detect the precise context surrounding a permission request. In this paper, we propose a run-time permission system that can automatically infer user's expectation using detailed foreground information. The main idea is to determine whether a permission request is expected by inspecting who is requesting the permission, when the request is initiated, and under what circumstances it is initiated, so that an appropriate action can be taken regarding the request (accept, deny, or notify the user). We present the design and implementation of a lightweight run-time permission control system, called INSPIRED (InteNtion-baSed PrIvacy-preseRving pErmission moDel). INSPIRED continuously identifies mismatches between app intentions and user intentions almost instantaneously with a low overhead. Moreover, it adapts to users' privacy preferences on the fly, without worrying about disclos-ing the personal decisions. In addition, INSPIRED is designed to be resilient to code obfuscation and name manipulation, encouraging adoption for monitoring commercial apps. These distinguishing features are achieved through the following key ideas. First, INSPIRED detects unexpected permission requests through examination of contextual foreground data. As a critical part that a user interacts with, the foreground user interface (UI) of an app fulfills and reflects the underlying app functionality. For instance, a user interacting with an SMS composing page would expect the app to ask for the SEND_SMS permission once the sending button is clicked. But an SMS message sent by a flashlight instance should be considered suspicious or malicious. Even under a message composing scenario, no message should be actually sent without proper user interactions, e.g., clicking the send button. We observe that a single widget in a window often cannot provide accurate information on app functionality. Consider the button for sending messages in an SMS app shown in Figure 1 (left). The button alone does not provide enough information on its purpose. A user needs to observe the whole page to understand its semantics. Without considering the relationships among widgets, one cannot tell if the sending behavior is legitimate or not. To this end, our approach leverages the semantic similarities at the window-level, even for windows crawled from different apps. The intuition is that benign apps tend to have a clear and informative UI to guide users so that foreground features such as words appeared on the screen reflect the underlying program logic. For instance, Figure 1 (right) shows the user interface of a different SMS app, which has similar widgets to specify message recipient, content, and the transmission behavior. Given the large number of apps with similar functionalities and UIs, it is possible to learn the correspondence between UI patterns and their semantics using foreground data crawled from popular apps, which remain valid for new apps encountered at runtime. Second, INSPIRED adopts a two-level framework to strike a balance between usability and control. We observe that to reduce user involvement, it is important to understand app intentions for requesting a permission, so that one can tell if a permission request is necessary to fulfill app's functionality. For instance, accessing CAMERA is necessary for scanning bar code, while requesting the SEND_SMS permission is very suspicious for a weather UI. As app intentions are independent of individual users, it is feasible and desirable to build a tool to understand app intentions automatically. However, such a one-fit-all approach is often insufficient as different users may have very different preferences on the same permission request even in a similar context [40,57]. For instance, one user may think attaching the current location while taking a picture is appropriate, whereas another user may feel uncomfortable about potential location leakage. Previous studies suggested that predicting the decisions of one user using data collected from others has inherent limitations [40,57]. Therefore, it is crucial to adapt the permission system by incorporating individual user's preferences at runtime. These observations lead us to a two-phase solution: (1) in the offline phase, we apply program analysis techniques to analyze a large corpus of sensitive apps and extract foreground data surrounding sensitive permission checks. Our approach can automatically identify the relationship between widgets. The foreground data along with the corresponding permission requests are then used to build a one-fit-all model using machine learning. (2) in the lightweight, online phase, we improve the one-fit-all model by incorporating individual users' privacy preferences using self-adaptive learning, which can be implemented completely on the local device or assisted by remote servers (e.g., cloud-based training). Since the latter approach requires the device to transfer security and privacy related data of a user to an untrusted third party, it may introduce additional privacy concerns. Therefore, INSPIRED chooses to implement the self-adaptive learning module completely on the device so that no sensitive data would be leaked. INSPIRED can also be combined with other runtime mediation techniques to provide protection at different levels. For instance, we can combine INSPIRED with TaintDroid [22] to offer information-flow level protection. In summary, this paper makes the following contributions: • We propose a novel intention-aware permission system based on app foreground information to enforce runtime contextual integrity. Our approach adopts a two-layer framework where (1) program analysis together with offline learning are used to identify UI patterns (layout and beyond) for stable app intentions that remain unchanged across users, and (2) runtime permission control and adaptive learning are used to incorporate user preferences on the fly. • As a proof of concept, we implement a prototype of the II. PROBLEM STATEMENT In this paper, we target threats from third-party apps who may improperly access device resources. Such threats come from either intended malicious logic embedded in an app or vulnerable components of an app that can be exploited by the attackers. We assume that the underlying operating system is trustworthy and uncompromised. We assume that apps are Fig. 1: Two message-composing window pages in different SMS apps. Texts shown on the windows such as New message, Compose, Type message and Enter recipients indicate the underlying app purpose. Also, two windows share a similar UI structure even their underlying implementations are different. isolated from each other through sandboxing and their system calls can be intervened by the permission system. Our ultimate goal is to design a run-time permission system that enforces contextual integrity with minimum user involvement. Contextual Integrity: The current permission systems of popular mobile operation systems defy user expectations over half the time since they do not consider the varying contexts of the requests [56]. We envision that to enforce contextual integrity in mobile platforms, one need to ask the following three questions regarding a permission request: Who initiated the request? An app may request the same permission for different purposes. For instance, a map app may request user's locations for updating the map as well as for advertisement. Although it can be difficult to know the exact purpose of a permission request, it is important and feasible to distinguish the different purposes by tracing the sources of permission requests as we show in this work. When did it happen? Ideally, a permission should be requested only when it is needed. This implies that the temporal pattern of permission requests is an important piece of contextual data. For instance, it is helpful to know if a permission is requested at the beginning or at the termination of the current app activity, and if it is triggered by proper user interactions, such as clicking, long clicking, checking, etc. What kind of environment? A proper understanding of the overall theme or scenario when a permission is requested is critical for proper permission control. For instance, it is expected that different scenarios such as entertainment, navigation, or message composing may request very different permissions. In contrast to who and when that focus on detailed behavioral patterns, what focuses on a high level understanding of the context. They are complementary to each other. We propose to answer the above three questions using the foreground data surrounding a permission request. Recent studies have shown the significance of foreground visibility in understanding user's expectation [40,57]. We propose to go one step further to build a run-time permission system that can capture and exploit more comprehensive foreground data from the above three perspectives. Minimum User Effort: Recent studies on run-time permission control focus on characterizing users' behavioral habit and attempt to mimic users' decisions whenever possible [40,56,57]. Although this approach caters to individual user's privacy preference, it also raises some concerns. First, users could be less cautious and the potential poor decisions made by users could lead to poor access control [57]. Second, many malicious resource accesses are user independent (although they may still be context dependent), which should be rejected by the run-time permission system without notifying the user. Furthermore, the permission system should automatically grant the permissions required for the core functional logic indicated by the context of the running app to reduce user intervention. Note that the core logic here is defined for the current dynamic context, which may not be a core functionality mentioned in the static description. For instance, an SMS message sent under "Invite friends" page after proper user interaction (i.e. clicking "Invite" button) is used to fulfill the core logic in the current context (i.e. friend referral), which may not be a main functionality of the app. Third, a user may be concerned with the liability of the system if it is being continuously monitored and analyzed. To improve usability and reduce incautious decisions, a user friendly permission system should involve user decisions only when necessary. We propose INSPIRED, a new permission system that continuously captures semantically-meaningful information about app behaviors. INSPIRED enforces contextual integrity through comprehensive inspection of the foreground from three distinct perspectives. In particular, it answers the questions of "who", "when" and "what" by examining the follow-ing foreground elements: • Activation widgets: INSPIRED models "who" by identifying the widget that triggers a sensitive resource request. A widget is a UI element shown on a foreground window, which is normally implemented using android.view.View and can be a button, a checkbox, etc. Users may install apps with harmful widgets injected, which leads to a severe consequence since the permissions granted by users to functional widgets can be abused by unintended ones. As shown in Figure 2, an advertisement widget that parasitizes on a weather app can stealthily collect users' location information using location-related permissions granted to the app. Therefore, INSPIRED focuses on widget-level and considers the permissions requested by an improper widget as suspicious. • Trigger events: INSPIRED discovers the set of events that lead to sensitive requests. A sensitive method call invoked without any prior visible event should be suspended. For example, sending a short message by clicking the send button in the message composing page of an SMS app should be considered legitimate, but no messages should be transmitted without actual user click. To verify the correctness of this temporal property, INSPIRED tracks back to the trigger event of a permission request. • Windows: INSPIRED infers the overall theme of the environment through a full inspection of the windows. Consider the screenshots taken from the SMS apps (see Figure 1). The title New message together with the text Type message inside the window indicate a message composing environment. We further observe that windows from different apps often share a similar layout when fulfilling a similar functionality. To capture the structural properties of windows, INSPIRED maps the absolute positions of the elements in windows to their relative positions. Moreover, INSPIRED adopts a two-layer design to protect users from malicious logic with minimum user intervention, while catering to individual user's privacy preferences. The offline module of INSPIRED uses features collected from program analysis to train a one-fit-all model to capture the app intentions, through a proper modeling of benign and malicious permission requests. With on-device deployment, this model is improved by incorporating personal privacy preferences to capture user intentions at runtime. The unique features provided by INSPIRED can be summarized as follows: • Automatically grant necessary permission requests and reject improper ones with minimum user involvement. • When needed, notify users to improve decision accuracy. • Keep users' decisions and behavioral data on local devices. Overall, we achieve the following design goals: • Intention-based detection: Our approach detects mismatches between app intentions and user intentions. It infers the purpose of a sensitive permission request through inspection of the foreground context. It stresses on contextual integrity by conducting analysis from three distinct perspectives. Our approach is able to meet users' personal expectations through continuous updates of the on-device learning modules. • Limited user involvement: Our system notifies a user only when the decision is user dependent and the current scenario is new to the user. In other cases, it automatically accepts or denies an app request based on the latest model with the user's previous decisions incorporated. • High scalability and adaptivity: Our approach is scalable to a large number of diverse permission requests. It is transparent to app source code and requires no additional developer efforts. Its accuracy and usability can be continuously improved with more apps available in the app stores and more user decisions incorporated. • Obfuscation resilience: Previous research utilized namespace at the code level to build context-aware permission models [40,53,54]. However, commercial apps and malwares often modify their classes, methods and variable names to prevent reverse engineering, as shown in Figure 3. Malicious apps may further simulate the name space of official Android packages to evade detection. In contrast, our foreground-based design is resilient to code obfuscation and name space manipulation. • Privacy-preserving: Our solution not only protects users from privacy threats caused by third-party apps, but also eliminates the potential privacy risk due to sharing user data with a third-party cloud by keeping and processing all user sensitive data on the devices. Figure 4 depicts the overall architecture of INSPIRED, which contains two main phases. III. SYSTEM ARCHITECTURE • Offline Phase: The offline phase is responsible for building a one-fit-all model that can be customized in the online phase. To build the model, we collect a large number of benign apps and malicious apps, and develop a lightweight static analysis technique to extract the set of sensitive API calls and the corresponding foreground windows. Subsequently, the windows are dynamically rendered to extract their layouts as well as the information of their embedded widgets. The detail of contextual data collection is given in Section IV-A. The system calls, widgets and layouts are then used to extract features to build a learning model that classifies each sensitive API call of third-party apps as either legitimate, illegal or user-dependent. Section IV-B2 describes this offline classification procedure. • Online Phase: In the online phase, the one-fit-all model trained in the offline phase is customized as follows. For each sensitive API call invoked by a third-party app, our mediation system will intercept the call and leverage the online learning model to identify its nature (initially, the online model is the same as the offline model). The sensitive API call is allowed if it is classified as legal, and is blocked (optionally with a pop-up warning window) if it is classified as illegal. Otherwise, the API call is considered as undetermined and the user will be notified for decision making. User's decisions are then fed back to the online learning model so that automatic decisions can be made for similar scenarios in the future. To better assist user's decisions, detailed contextual information is provided in addition to the sensitive API call itself. Moreover, we provide special mechanisms to handle background requests without foreground contexts. We will discuss the implementation of our online permission system in Section V. IV. OFFLINE ANALYSIS AND LEARNING In this section, we discuss our approach for building a onefit-all model using program analysis and machine learning. A. Foreground Data Extraction INSPIRED models the context of a sensitive request using the foreground data associated with the request. Although one can manually interact with an app and record the foreground data, it is infeasible to build a faithful model by analyzing a large number of apps manually. An alternative approach is using existing random fuzzing techniques such as Monkey [7], which generates random inputs in order to trigger as many sensitive behaviors as possible. However, random fuzzing is inefficient, as it may generate many inputs with similar program behavior. More importantly, without prior knowledge of app behaviors, random testing wastes time on exploiting code paths that are irrelevant to sensitive resource accesses. In this work, we propose a hybrid approach to collect relevant foreground data, including the set of widgets, the triggering events and the windows associated with sensitive API calls. Our approach has two phases, a static analysis phase and a dynamic rendering phase. In particular, we adopt static program analysis attempts to accurately locate the foreground components that would trigger a permission request. Compared with random fuzzing, our approach achieves better coverage and eliminates redundant traces. The identified foreground components are then rendered dynamically with actual execution, which provides more complete and precise information compared to a pure static approach. Pure static analysis, as an over-approximation approach, is criticized by generating false relationships between UI elements [16]. To illustrate our hybrid approach, we use the code in Listing 1 as an example throughout this section. The code presents the underlying logic of the open-source SMS app QKSMS [5], shown on the right side of Figure 1. 1) Static Analysis: Our static analysis takes the entire app package as input, and outputs its security or privacy sensitive program behaviors, with the corresponding foreground components identified. We detect sensitive behaviors by performing analysis over constructed call graphs. The foreground components that would trigger the sensitive behaviors are then located through data flow analysis. For each target app, we first identify its permissionprotected API calls through method signatures. We construct a call graph for the given app with the help of FlowDroid [13] and iterate over the graph to locate the target calls. The list of permission-protected API methods is provided in PScout [23]. For instance, in QKSMS, sendTextMessage() at line 14 is marked as a sensitive API call that requests the SEND_SMS permission. The set of call graph entry points of the sensitive API calls are then identified by traversing through the call graph. For in- Further, the set of widgets that invoke the entry points are extracted by locating the event handlers of the entry points. By modeling the call relationship inside the ComposeView, we get to know that the handler of onClick() is setOnClickListener() at Line 9, which is initialized by the widget mButton. We then conduct a data flow analysis to track the source of mButton. After knowing where the widget mButton is initialized, we are able to get its unique resource id (compose_button) within the app by inspecting the initialization procedure (Line 8). As the foreground windows set contexts, our analysis goes beyond individual widgets by further identifying the windows that the widgets belong to. A window is represented by an Activity in Android. In our case, we aim to identify the Activity that includes mButton. Since mButton is initialized inside ComposeView, we search for the usage of ComposeView within the app. ComposeView is declared in ComposeFragment, from which we can finally identify ComposeActivity as the window for the widget mButton. We notice that the over-approximation of the static analysis phase may introduce some misidentified UI elements that do not actually correlate with the indicated permission request. We manually filter the misidentified samples before building the learning model to lower the impact of false alarms as much as possible. However, we remark that it can be beneficial to keep some contextual instances that do not really request a permission and label them as illegal since they simulate more scenarios that should not use the permission. 2) Dynamic Rendering: For each target Activity such as ComposeActivity recognized by our static analysis, we then render it with actual execution to precisely extract its layout and widget information. Actual execution enables us to extract data of interest loaded at runtime. Capturing rendering information specified by source code is intractable for static rendering approaches such as SUPOR [31], which solely leverage app resource files to uncover the layout hierarchies and identify sensitive inputs. For instance, the title of the crafting page ("Compose") of QKSMS, a critical piece of context while using the app, is declared in the Java code (line 25 in Listing 1) instead of in the resource files. Losing this kind of dynamically generated information may hinder the progress of our upcoming task to precisely infer the purpose of the underlying program behavior. Moreover, our dynamic rendering avoids further counting the falsely recognized elements introduced by the over-approximation nature of static analysis. Most Activities cannot be directly called by default. Hence, for each app, we automatically instrument the app configuration file manifest.xml with tag <android: exported> and then repackage it into a legal apk file. After installation of the new package, we wake up the interested Activities one by one with the adb commands provided by Android. Once an Activity is awaken, the contextual foreground app data, including the layout and widget information, is then extracted and stored into XML files with UiAutomator [6]. We found that some Activities cannot be correctly started by this way and we ignored them for now. If necessary, we can manually interact with those cases to extract the user interfaces we need. B. Learning Using the extracted foreground data, we are able to build a machine learning model to detect both user-intended and userunintended behaviors. Given a permission request, we consider it as : Legitimate: if the permission is necessary to fulfill the core functionality indicated by the corresponding foreground context. The requests in this category would be directly allowed by our runtime mediation system to eliminate unnecessary user intervention. We emphasize that the core functionality here is with respect to the running foreground context, not the app as a whole. For example, some utility apps may include a referral feature for inviting friends to try this app through SMS messages. This is typically not a core functionality of the app and the developers normally do not mention this feature on the app description page. However, the SMS messages sent under the "Invite friends" page after proper user interactions (e.g., clicking "Invite" button) should be considered as user intended. In contrast, description-based approaches [42,43] would unnecessarily raise alarms. Illegitimate: if the permission neither serves the core functionality indicated by the foreground context, nor provides any utility gain to the user. An illegitimate request can be triggered by either malicious code snippet or false program logic. The latter can happen as developers sometimes require needless permissions due to the misunderstanding of the official development documents [23]. User-dependent: if the request does not confidently fall into the above two categories; that is, it is not required by the core functionality suggested by the foreground context, but the user may obtain certain utility by allowing it. Intuitively, in addition to the core functionality, the foreground context may also indicate several minor features that require sensitive permissions. Whether these additional features are desirable can be user dependent. For example, besides the CAMERA permission, a picture shooting instance may also ask unnecessary permissions such as ACCESS_LOCATION to add a geotag to photos. Although some users may be open to embed their location information into their photos which may be shared online later, those who are more sensitive to location privacy may consider this a bad practice. In this case, we treat ACCESS_LOCATION as a user-dependent request and leave the decision to individual users. 1) Features: Before extracting features from the collected foreground contextual data, we pre-process the crawled layouts to better retrieve their structural properties. Mobile devices have various resolutions. With absolution positions, the solution derived from one device may not scale to another device. Therefore, we divide a window into nine grids and map the absolute positions to the relative positions. As shown in Figure 2, the advertisement widget is mapped to the bottom three grids, while the main frame of the app occupies the central grids. The processed layouts are then used to extract features. As we discussed in Section II, we construct three feature sets to enforce contextual integrity. More specifically, we derive the following features from a sensitive request: Who: The static phase of our foreground data collection described in Section IV-A allows us to identify the widgets leading to sensitive API calls. We then collect the feature values of the target widgets using the dynamically extracted layout files. In particular, the feature set of "who" includes the following attributes of the target widgets: text: The text shown on the widget. class: The Java class of the widget instance. position: Its relative position in the layout. size: The percentage of screen area occupied by the widget. isPassword: Whether the widget is a password. isClickable, isLongClickable, isCheckable, isScrollable: Whether the widget can be clicked, long clicked, checked and scrolled. It is possible that the permission request is triggered by an Activity rather then a widget. In this case, we would leave the value of this feature set as empty and rely on the "what" feature set to handle windows. When: The call graph traversal gives us entries of sensitive API calls. An entry point can be either a lifecycle callback or an event listener. The lifecycle of an app models the transition between states such as creation, pause, resume and termination. The event listeners of an app monitor and respond to runtime events. Both lifecycle callbacks and event listeners are prior events happened before an API call and serve as useful temporal context to the call. We therefore use the class names and method names of entry methods as the "when" feature set. What: The text shown on target widgets could be too generic (such as "Ok" and "Yes") to convey any meaningful context. Therefore, we also derive features from the windows to help infer the overall theme of the requesting environment. We iterate over the view hierarchy of the window layout and extract all the related widgets that have text labels. For each obtained widget, we save the text displayed on the widget and its relative position in the window as features. Including both textual and structural attributes provides better scalability to capture semantical and structural similarities across millions of pages. Although developers may adopt various design styles for the same functionality, their implementations usually share a similar foreground characterization. For instance, we do not need to know whether a window is implemented with Material design. Instead, learning the title shown at the top of the window, such as "Compose" and "New message", is crucial. These form our "what" feature set. By focusing on features directly visible to users, our approach is resilient to code level obfuscation and name manipulation. Note that the entry methods are overridden of the existing official SDK APIs and cannot be renamed by the third parties. For each of the three feature sets mentioned above, we generate a separate feature vector. Note that attributes of a widget leading to sensitive API calls appear in both the "who" feature set and the "what" feature set. However, they are treated separately to stress the triggering widget. For the "what" set, text and position from all the widgets shown on the window are included, while for the "who" set, only those related to the triggering widget are included. All textual features are pre-processed using NLP techniques before subjecting to learning algorithms. In particular, we perform identifier splitting, stop-word filtering, stemming and leverage bag-ofwords model to convert them into feature vectors. The process is similar to other text-based learning methods [28,53]. It is certainly possible to further raise the bar of potential attacks by considering more types of feature. We will discuss the feasible extensions in Section VII. 2) Learning: Using the three sets of features discussed above, we train a one-fit-all learning model as follows. More specifically, one classifier is trained for each permission type with a data mining tool Weka [9] using the manually labeled sensitive API calls related to that permission. The classifiers are trained separately for different permissions to eliminate potential interference. Each instance is labeled as either legal or illegal based on the foreground contextual data we collect, including: the entry point method signature, the screenshot of the window, and the highlighted widget invoking the API call (if there is such a widget). We ensure contextual integrity by checking whether they altogether imply the sensitive API call. The behavior is marked as illegal if it is not supported by any type of the foreground data. For instance, SEND_SMS requested under the "Compose" page without user interactions, or required by an advertisement view, is categorized as illegal. As we mentioned in Section III, our one-fit-all models will be continuously updated at runtime to incorporate individual user's preferences. One option is to keep sending data to a remote cloud for pruning the models. However, since the content shown on the device is often deeply personal, transmitting this kind of sensitive contextual and behavioral data out of the device would raise serious concerns on potential leaks [26]. Consider the SMS composing example again, the window may contain private information typed by the user, which is inappropriate to share with a third-party service. But the limited computational power of mobile devices makes it infeasible to repeatedly train complicated models from scratch inside the devices. To meet both the privacy and performance requirements, we apply light-weight incremental classifiers that can be updated instantaneously using new instances with a low performance overhead, which matches the memory and computing constraints of smart phones [60]. One key question is which incremental learning technique to use. To this end, we have evaluated popular incremental learning algorithms. The detailed results are given in Section VI. V. ONLINE PERMISSION SYSTEM In this section, we provide the details about the implementation of our online permission system. With the help of the pre-trained model, INSPIRED automatically grants legitimate permission requests, denies illegitimate requests, and customizes the model according to user preferences. A. Mediation and Data Extraction To implement run-time access control, INSPIRED dynamically intercepts sensitive calls, collects features for them, and finally classifies them using an online learning model. The online model is initialized as the one-fit-all model trained offline and is customized dynamically to model user preference as discussed below. Android does not include official APIs that allows a thirdparty app to mediate other apps' requests. Instead of modifying the OS and flashing the new firmware, INSPIRED is written in Java as a standalone Android app and can be easily installed on Android devices with root access. The implementation of INSPRIED is based on XPosed [11], an open-source method hooking framework for Android. XPosed provides native support to intercept method calls, which enables us to execute our own code before and after execution of the hooked method. To detect improper permission requests at runtime, IN-SPIRED dynamically extracts information from the UI elements associated with sensitive calls. Consider the example shown in Figure 5, the sendTextMessage() is triggered after clicking the mButton widget shown on the MainActivity window. INSPIRED needs to retrieve the memory references of the interested UI elements, including the running instances of mButton and MainActivity. However, simply intercepting the target sensitive call is insufficient. The problem is that although we can extract the values of the variables appeared in the current call (e.g., sendTextMessage(...)), retrieving the values from the prior calls (e.g., onClick(...)) is currently infeasible in XPosed, which makes it difficult to retrieve the trigger UI instances by only hooking the sensitive API call. To address the above problem, INSPIRED intercepts the invocations of both Activity lifecycle callbacks (e.g., performCreate(Activity) for Activity.onCreate()) and event listeners (e.g., performClickView for onClick(View))) in addition to sensitive API calls. For each of these methods, it records the references of the method parameters. For instance, in the above example, the references to mButton and the Activity are stored when processing onClick(mButton). When it encounters a sensitive API call, INSPIRED retrieves the latest widget and activity it saved, and extracts the same features from them as in the offline model. In particular, "who" features are collected from the widget and "what" features are extracted from the activity by iterating over all its widgets. Moreover, INSPIRED examines call stack traces to determine the entry point methods leading to sensitive calls, which are used to derive the "when" features. Other method signatures available in the call stack can be used to build the "program namespace" features. It is possible that the latest saved widget is not the one that really triggered the sensitive request due to multi-threading. However, this rarely happens in reality and we will further discuss it in Section VII. After converting the features into numerical values, IN-SPIRED uses the online learning model to predict the type of the sensitive request. It automatically grants the permission if it is classified to be legitimate with high confidence and rejects it if it is classified to be illegitimate with high confidence. For a rejected request, INSPIRED further pops up a warning to the user including the details of the request. A request that is neither legal or illegal with high confidence would be treated as user-dependent, which is handled by the user preference module as discussed below. As users can switch between Activities, a request may be initiated by a background Activity. By tracking the memory references of the associated UI elements, INSPIRED is able to reason about the background requests even if the associated UI elements are currently invisible. B. User Preference Modeling To incorporate user preferences, INSPIRED notifies the user if the online model identifies a request as user-dependent. Consider the example shown in Figure 6. The UI shows a product review page and a location permission is requested once the "Upload" button is clicked. On the one hand, the user may be beneficial from sharing location if the seller provides subsequent services to promote customer experience based on the user's review and location. On the other hand, the sharing behavior could put the user at risk since there is no guarantee how exactly the location information would be used by the app developer. As the page does not provide enough evidences whether location sharing is necessary, INSPIRED treats the instance as user-dependent, and then creates a prompt to accept user decision. Our prompt not only alarms the user about the existence of the permission request, but also highlights the widget that triggered the request and the activation event. The user decision, along with the features of the instance, is then used to update our learning model. Discussed in Section IV-B2, our classifiers are built though incremental learning in order to take care of both privacy concern and performance overhead. The incremental learning model im-mediately accepts the new instance and adjusts the decision strategy to better match user criteria next time. C. Background Services In an Android app, an Activity can start a background Service through inter-component communication. When a sensitive call is initiated by a Service, its call stack does not contain the information of the starting Activity. In this case, INSPIRED monitors the calls of Activity.startService(Intent) to track the relationship between running Activities and Services. INSPIRED can then use the information available from the Activity to infer the purpose of a Service request. One problem with this approach is that a Service may still be alive even when the foreground Activity has finished. In this case, INSPIRED simply notifies the user about the background request and lets the user decide whether to allow or deny the request. Alternatively, we can always reject such requests. We argue that sensitive services should not exist unless they provide sufficient foreground clues to indicate their purposes. Users tend to reject requests without foreground as suggested by three recent important user studies [40,56,57]. Google also further restricts background services in the most recent Android O [2]. D. Defense Against GUI Spoofing To ensure that the foreground data is indeed associated with the background request, INSPIRED ignores the widgets that are not owned by the permission requesting app. Thus, INSPIRED is resilient to GUI spoofing that tries to evade detection by hiding behind the interfaces of other apps. More advanced GUI spoofing attacks have also been proposed in the literature [15]. For example, when a benign app running in the foreground expects a sensitive permission to be granted, a malware may replicate and replace the window of the benign app to elicit the user. An adversary may also programmatically simulate user behaviors to interact with other apps. However, such attacks can be hard to implement in practice as they require Accessibility feature [1] enabled to the malware by the user. It is worth noting that using Accessibility may play against the malware itself, since Android repeatedly warns the user about the threats caused by Accessibility. If needed, INSPIRED can also intercept the method calls initiated from Accessibility to further alarm users. E. Handling of False Automatic Decisions Achieving 100% precision and recall is intractable for any machine learning algorithm. To provide better usability, INSPIRED notifies the user of each rejection and provides rich contextual information, including the activation event, the triggering widget, and the screenshot, to help the user perceive the cause. For any false automatic decision made by the system, the user can override it at the backend and our incremental learning models will incorporate the user's decision immediately. VI. EVALUATION We evaluate the effectiveness of INSPIRED by answering the following questions: • RQ1: Can INSPIRED effectively identify misbehaviors (i.e., inconsistencies between context and behavior) in mobile apps? How do the feature sets of who, when and what contribute to the effectiveness of misbehavior identification? • RQ2: Can INSPIRED be applied to capture personal privacy preferences of users? • RQ3: Can INSPIRED be deployed in real mobile devices with a low performance overhead? We note that RQ1 measures the effectiveness of the one-fit-all models where individual user preferences are not involved. A request that cannot be confidently labelled as either legal or illegal is considered as user-dependent, which is not counted in RQ1. We let RQ2 capture these scenarios that rely more on user preferences. Machine learning can still help in this case using data collected from individual users. A. RQ1: Accuracy in Identifying Misbehaviors We crawled more than 10,000 apps from Google Play in November 2016, all of which were top-ranked apps across 25 categories. We also used a VirusShare data set [8], which contains more than 5,000 malware samples. From these datasets, we manually labeled 6,560 identified permission requests that belong to 1,844 different apps. Each request was marked as legitimate or illegitimate through the associated foreground contextual data, including the widget (if any), the events and the window. In particular, we determined whether a request (e.g., "RECORD AUDIO") was initiated by an appropriate widget (e.g., a "microphone" button) after a proper interaction (e.g., clicking) and under a correct environment (e.g. voice assistant). The sample sizes of some datasets are imbalanced. For example, the number of legal usage of CAMERA is much higher than the illegal ones. It is well known that imbalanced data can severely hinder the learning performance of classification algorithms [52]. We therefore leveraged SMOTE [18] to oversample the heavily skewed datasets before feeding them into the classifiers. 1) Overall Effectiveness: For each permission type, we leveraged the labeled requests both as training and test data in a five-fold cross validation. Specifically, we randomly divided all instances of the same permission into 5 equally sized buckets, training the classifier on 4 of the buckets, and using the remaining bucket for testing. We repeated the process 5 times and every bucket was used exactly once as the testing data. We applied cross validation on every permission type and measured the results in terms of precision, recall and fmeasure [3]. As our online learning approach is a continuous training process that adapts to user decisions, a classifier that can process one example at a time is desired. To determine which machine learning technique to use, we evaluated the SVM. Compared to non-updatable classifiers, all these methods can iteratively incorporate new user feedback to update their knowledge and do not assume the availability of a sufficiently large training set before the learning process can start [48]. A summary of the results is given in Table I, where the mean values are calculated over all permission types. As we can see, logistic regression achieved the best result among all four classifiers. Table II further provides detailed results of logistic regression on each permission type. We considered 7 common permissions as for now and will investigate more permissions in the future. We observe that among all the permission types, differentiating requests of DEVICE_ID is more challenging since developers normally do not provide sufficient information in apps to indicate why the permission is requested. More human intervention could be beneficial regarding DEVICE_ID. 2) Feature Comparison: To measure how each feature set contributes to the effectiveness of behavior classification, we used the same learning technique (e.g., Logistic Regression) with different feature sets under "who", "when" and "what" and some combinations of them, respectively. The cross validation results of RECORD_AUDIO are presented in Table III. Since the comparison results of other permissions share the similar trend, we omit them here. For each feature set, we evaluated its effectiveness by comparing the accuracy of our learning models when the feature set is used and when it is not. We found that the "what" features contributed the most among the three feature sets. As we mentioned in Section I, benign instances often share similar themes that can be inferred from window content and layout. For example, an audio recorder instance typically has a title Recorder, a timer frame 00:00 at the center and two buttons with words start and stop, respectively.From these keywords and their positions in the page, INSPIRED is often able to tell whether the user is under a recording theme. Although the "what" features successfully predicted most audio recorder instances, it may be of limited use in other cases where RECORD_AUDIO permission is used. For instances, developers tend to integrate voice search into their apps to better serve users. However, as the searching scenarios differ greatly from each other, it is hard to classify their intentions using "what" features only. The "who" features help alleviate the above problem by further examining the meta data of the corresponding widget. For instance, co.uk.samsnyder.pa:id/speakButton is an image button for speech recognition, which does not provide useful "what" features as the image button does not contain any extractable textual information. However, the word "speak" in the resource-id clearly indicates the purpose of the button. In addition to the meta data, the relative position and the class attribute of a widget can help locate non-functional components, e.g., the widgets for advertisement. We observed that for RECORD_AUDIO, the "who" features and the "when" features are highly correlated in most cases, this is because most sensitive method calls initiated by widgets are bound with the event onClick(). However, there are exceptions. For instance, com.webstar.walkies is an Internet-based walkie talkie app [10] that transfers users' audio information to each other. The tips "Press & Hold " shown in its main window indicate that the recording should start only after user clicking. However, it actually starts recording once the app is open. This misbehavior can be effectively identified using the "when" features, which emphasizes that apps should request a permission only after proper user interactions. In summary, "what" features work well in differentiating between most legitimate and illegitimate instances at the current stage. However, as malware continues to evolve, we expect that collecting more comprehensive contextual data including "who", "when" and "what" can provide better protection. The last row in Table III shows that the combination of all the three feature sets provides the best results. Other types of features, such as the keywords extracted from hostnames, could potentially further increase the accuracy of INSPIRED. We will investigate them in the future. B. RQ2: Effectiveness of Capturing Personal Preferences We conducted a lab-based survey to measure the effectiveness of our models to capture individual user's preferences, where we asked participants to classify a set of requests Fig. 7: The precision and recall of each user that were not faithfully labelled as legal or illegal in RQ1. The survey was composed and spread through Google Forms. Among the 24 participants, 3 were professors, 6 were undergraduate students and 15 were graduate students. Each user is asked to classify 50 location accessing requests collected from 40 real apps, covering several user-dependent scenarios such as shopping, photo geo-tagging, news, personal assistant and product rating. We collected 1,272 user decisions from the 24 users. To simulate the real decision making on device, for each request, the following information is displayed to the participants: 1) Screenshot: the screenshot taken from the app right after the request was initiated, with the triggering widget highlighted. 2) Prior event: the event led to the request, such as app start and user clicking. 3) Meta-information: the app name and a Google Play link are included, whereby the participants can find more information about the app. We evaluated the effectiveness of our user preference modeling by updating the pre-trained model constructed during the evaluation phase of RQ1 with the decisions collected from each individual user. For each user's decisions, we randomly partitioned them into three sets and used two of the three sets as the training set to update the pre-trained model, and the rest set as the testing set. The updated model was then used to predict the decisions in the testing set. Our model yielded a median f-measure of 84.7% among the 24 users, which is reasonably good due to the limited number of samples. We expect our model to be more accurate with more user feedback. Figure 7 presents the detailed result of each individual. A quarter of users' results have more than 90% precision and 90% recall. Our model performed surprisingly well for one individual, with 100% precision and 100% recall. We found that some users shared very similar preferences, which leads to several small clusters. One individual tends to behave conservatively by rejecting nearly all requests, giving a sharp outlier in the lower right corner with a perfect precision but a terrible recall. We also observe that some users made inconsistent decisions under a similar context. For instance, one user allowed a request from a product rating page but rejected another with a closely related context. The root cause of the conflicting behaviors is unclear to us, which leaves room for further improvement of our model. One possible explanation is that sometimes users are less cautious and make random decisions as suggested in [57]. Fortunately, our system can greatly help protect users from malicious behaviors caused by malware even if users make random decisions. This is because in offline training, our model has already learned many misbehaviors by malware and accordingly, it is able to block them at runtime automatically. We also conducted a controlled experiment to test whether the finer-grained contextual info shown in our prompts can help users make better decisions. We used the screenshots similar to Figure 6 with location-based functionality at the center and a behavioral advertisement at the bottom. Without prompts, 79.2% of the participants chose to grant the permission. After being alerted that the location requests were actually initiated by advertisements, 73.9% of the users changed their minds to reject the requests. These results encourage the deployment of INSPIRED to better assist users against unintended requests. C. RQ3: Usability on Real Devices In this subsection, we measured the overhead incurred by INSPIRED. We installed the online module of INSPIRED on a Google Nexus 5 running Android 5.1.1 with 2.26 GHz quadcore CPU and 2GB RAM. 1) CPU Time: We installed some popular apps from different categories on the phone, interacted with them as in common daily use, and monitored the performance overhead introduced by INSPIRED. The performance data were collected using the runtime profiling tool Traceview [4], which is officially supported for debugging Android apps through tracking the performance information of each method call. We modified the device firmware to let Traceview monitor the released commercial apps without requiring their debuggable installation packages. The overhead introduced by INSPIRED is measured within the target monitored app. Table IV shows the average CPU overhead of INSPIRED when interacting with 5 representative apps installed on the phone. Each of these apps has at least 10 million installations according to Google Play. The first column gives the average number of sensitive requests made by each app per minute. The second column shows the average total time that INSPIRED spent on inspecting a request, excluding the time waiting for user's decisions. The third column gives the average CPU time that INSPIRED spent on a request, excluding the waiting time on I/O. The last column gives the percentage of the CPU time used by INSPIRED within an app over the total CPU time that app used during execution. Note that the value was measured within each target app, not the total CPU time used by the entire device. We observe that INSPIRED consumed less than 5% total CPU time for all the five apps and the values vary a lot across apps. In particular, INSPIRED incurred the highest overhead on Wechat, which can be explained by two main reasons. First, Wechat intensively requests permissions when used. As a complicated communication and social app, it needs to access several sensors to provide functionalities such as voice input, location sharing, video call, etc. It also periodically reads the device ID for analytical purpose. Second, Wechat adopts its own GUI library, which takes INSPIRED longer time for analysis.Yelp and Yahoo Weather also frequently initiate sensitive requests. They continuously update locations to provide nearby services and weather information, respectively. Compared to Wechat, their UI structures are simpler and hence cost less time to analyze. Amazon asks to access microphone and location for embedded voice assistance, which has limited foreground information and was triggered only after proper user interactions. During the experiment, Paypal only initiated sensitive requests when the app was first started. The lower frequency of permission requests and the simpler UI together led to the least overhead for Amazon and Paypal. 2) Memory Usage: As the method profiling provided by TraceView did not include the memory cost. we estimated the rough memory usage of INSPIRED by dumping the runtime objects into files. We serialized the running INSPIRED objects and the related referenced objects such as Weka instances at the decision points, in which the memory use should reach the peak value. The average memory use was 5,712 KB over 50 separate decision points. Among them, over 95% memory can be attributed to the Weka machine learning module. 3) Storage: The size of the installation package of our run-time control system is 8.7 MB. After installation, the total storage occupied is 19.86 MB, including the INSPIRED classes, Xposed library, Weka library, Android support library and the resource files. We can reduce the size by discarding the unused classes files inside the libraries, and further reduction is possible by compressing some resource files. 4) Network bandwidth: INSPIRED does not generate any network traffic on its normal use. This is a significant overhead reduction compared to cloud-based systems that continuously consume bandwidth to upload user data. VII. DISCUSSION In this section, we discuss the limitations of our approach and make suggestions on future directions. Features: Similar to existing detection methods based on machine learning [12,29,41,43,53,58,62], INSPIRED could be bypassed with feature engineering through carefully designed evasion logic. An adversary may deliberately make an app (or repackage an existing app) that contains some valid user interfaces to justify certain permission requests while piggybacking his illegitimate sensitive information flow in the same contexts. For example, he can modify an SMS app to send out user intended SMS, while at the same time, deliver messages to a malicious receiver. However, we argue that the design philosophy of INSPIRED makes such attacks more difficult to succeed. First, the adversary can only target apps that are legitimate to use the target permissions. For example, Although INSPIRED provides a more detailed characterization of user interface than existing approaches [28,33,35] to better detect improper permission requests, it leaves room to consider more advanced features. Moreover, an adversary who knows the precise list of features we use can potentially obfuscate the user interface to match our criteria. For example, one may put human invisible text labels (e.g. using white text on a white background) on the screen to deceive our system. Although such an attack is possible, it cannot easily bypass the current version of INSPIRED, as INSPIRED considers multiple types of UI features. As we mentioned before, our system would warn the user if it encounters confused scenarios that do not lead to a confident decision. We envision that it is a long-term battle to fight against increasingly more advanced adversary. Our approach is flexible to incorporate more UI-related features (e.g., colors and images) to cope with emerging new attacks. Implementation: As mentioned in Section V, our run-time system stores the references of encountered UI elements and leverages the information available in the call stack to match sensitive API calls to the corresponding UI elements. However, the mapping could be imprecise due to multithreading. One reason is that the call stack does not contain the caller's information of a child thread. Although we can track the initiation procedure of certain threads, there is no universal solution yet to track all possible threads inside Android apps. Even if the call stack contains the caller's information, we may still incorrectly identify the relationship between sensitive calls and UI elements. For example, a user may click two buttons in a short time period, where only the first click leads to a sensitive call, but the time of actual invocation is later than the second click. In this case, our current implementation matches the API call to the most recently used button, which may not be the one that triggers the sensitive call. The problem could be alleviated by modifying the base code of Xposed to log the values we need inside the runtime environment. We currently focus on the apps designed in English. However, our design could be easily extended to add multilanguage support. Beyond Lab-based User Study: We so far did preliminary lab-based user study in evaluating our proof-of-concept system. The demographic distribution of participants is not comprehensive and the data set is small. Once our system is ready for daily use, we will release it to popular app stores and get feedback from actual deployments beyond the controlled lab environment. VIII. RELATED WORK Several previous studies have documented the limitations of mobile permission systems [24,34,51,55,64]. In particular, enforcing contextual integrity in mobile permission systems is considered as an important research direction. Early studies on building context-aware systems mainly depend on manually crafted policies specific to certain behaviors [17,19,21,38,50,63]. More recently, researchers began to investigate methods that can automatically infer context-aware policies from users' behavioral traits [40,56,57]. They observe that the visibility of apps is the crucial factor that contributes to users' decisions on permission control. However, these approaches do not capture more fine-grained foreground information beyond visibility and package names. Some recent efforts have also been made to detect unexpected app behavior from UI data. For instance, AppIntent [59] uses symbolic execution to extract a sequence of GUI manipulations leading to data transmissions. PERUIM [35] relates user interface with permission requests through program analysis. Both approaches require user efforts to locate suspicious program behaviors. AsDroid [33] identifies the mismatch between user interface and program behavior with heuristic rules. DroidJust [20] tracks the sensitive data flows to see whether they are eventually consumed by any human sensible API calls. Rubin et al. [49] detect covert communications inside mobile apps that do not trigger UI changes with control flow analysis. Roesner et al. [47] propose to regulate resource access initiated by UI elements. Ringer et al. [46] extend the idea and design a GUI library for Android. As these approaches rely on a small set of human crafted policies, they can only recognize certain misbehaviors within the domains. Most recently, FlowIntent [28] examines all textual information shown on the foreground windows with machine learning. Though similar in spirit, it only touches upon a subset of the challenges that INSPIRED tries to address. More specifically, we extended this line of research in several ways. First, we proposed to protect contextual integrity through analyzing UI data from three distinctive perspectives: who, when and what. Second, we provided a two-layer machine learning framework that can automatically grant the necessary permission requests and reject the improper requests without requiring user involvement, as well as improving the decision accuracy based on user feedback. Third, we implemented our permission system on real devices and conducted comprehensive evaluations. Our system can be easily installed on actual devices and incurs limited overhead. In addition to UI centric approaches, many different approaches have been proposed to detect unexpected behaviors targeting mobile platforms. Examples include WHYPER [42], CHABADA [29] and AutoCog [43], which assess descriptionto-permission fidelity; DroidSift [62], AppContext [58] and HSOMINER [41], which identify malwares by training on conditional API calls; SUPOR [31], UIPicker [39] and Bid-Text [32], which detect sensitive leakage from user input; LeakSemantic [27] and Recon [45], which performs privacy protection at network layer. Moreover, Wang et.al [53] attempt to infer the mapping from permission to app functionality using class, method and variable names. Many other studies have been done to combat UI deception and spoofing [15,25,30,44]. These works are orthogonal to our work and can be combined with INSPIRED to further protect users. IX. CONCLUSION We propose INSPIRED, an intention-aware privacypreserving permission system for Android. INSPIRED automatically infers the underlying program intention by examining its runtime foregroud and justifies whether to grant the relevant permission by matching with user intention. It can be user-customized by continuously learning from user decisions to precisely capture user intention, It is also privacy-preserving by keeping and processing all user's behavioral data inside her own device (i.e., without sending to a third-party cloud for training or learning). Experiments show that our model achieves both high precision and high recall (95%) based on 6,560 requests from both benign apps and malware. Further, it is capable of capturing users' specific privacy preferences with an acceptable median f-measure (84.7%) for 1,272 decisions collected from 24 users. Finally, we show that INSPIRED can be deployed on real Android devices to provide real-time protection with a low overhead. Fig. 2 : 2Advertisement in a Weather App. The advertisement widget located at the bottom may stealthily collect user location by exploiting the location permission granted to the weather functionality. Fig. 3 : 3The code obfuscation adopted by a commercial app (left) and the name manipulation leveraged by a DroidKungfu malware (right). Fig. 4: System Architecture Fig. 5: Online Extraction Fig. 6 : 6An example user prompt shown by INSPIRED. In the top right corner, the "Upload" button that is accessing the device location is highlighted. INSPIRED permission system. INSPIRED is implemented as a standalone app and can be easily installed on Android devices with root access, without requiring OS modification. Moreover, INSPIRED is designed to be completely transparent to third-party apps. Hence, it requires no modification for apps to run under INSPIRED control. The experimental installation package can be found at https://sites.google.com/view/inspired-mobile.• We show that INSPIRED achieves both high precision and high recall (95%) for 6,560 requests from both authentic apps and malware. Further, it is able to capture users' specific privacy preferences with an acceptable median f-measure (84.7%) for 1,272 decisions collected from users. We further show that INSPIRED can be deployed on real Android devices to provide real-time protection with a low overhead. stance, the onClick() method inside ComposeView (line 20) is found as an entry method of sendTextMessage().Listing 1: Code Example 1 public class ComposeView extends LinearLayout implements View.OnClickListener { 2 private FrameLayout mButton; 3 4 @Override 5 public void onFinishInflate() { 6 ... 7 // Get references to the views 8 mButton = (FrameLayout) findViewById(R.id.compose_button); 9 mButton.setOnClickListener(this); 10 } 11 12 private void handleComposeButtonClick() { 13 switch (mButtonState) { 14 case SEND: sendTextMessage(...); break; 15 ... 16 } 17 } 18 19 @Override 20 public void onClick(View v) { 21 ... 22 handleComposeButtonClick(); 23 ... 24 } 25 } 26 27 public class ComposeFragment extends QKFragment implements ComposeView.OnSendListener{ 28 public View onCreateView(LayoutInflater inflater ...) { 29 mComposeView = (ComposeView) view.findViewById(R.id.compose_view); 30 mComposeView.setLabel("Compose"); 31 ... 32 } 33 } TABLE I : IResults for Different ClassifierAlgorithm Median F-measure Average Precision Average Recall Hoeffding Tree 77.9% 81.7% 78.3% Naive Bayes 93.9% 93.3% 92.9% SVM 95.5% 95.4% 95.4% Logistic Regression 96.1% 95.8% 95.5% TABLE II : IIResults for Different PermissionPermission Precision Recall F-Measure DEVICE_ID 89.8% 89.3% 89.3% LOCATION 93.8% 93.9% 93.8% CAMERA 95.0% 95.0% 95.0% RECORD_AUDIO 96.0% 96.1% 96.1% BLUETOOTH 97.9% 97.9% 97.9% NFC 96.7% 96.6% 96.6% SEND_SMS 99.8% 99.8% 99.8% effectiveness of four commonly used learning methods that support incremental classification, including Hoeffding Tree, (Multinomial) Naive Bayes, Logistic Regression and (linear) TABLE III : IIIClassification with Different Feature SetsFeature Type Precision Recall F-Measure Who 81.9% 78.8% 75.7% When 69.7% 70.7% 70.0% What 95.4% 95.3% 95.3% Who & When 80.0% 79.1% 76.9% Who & What 95.6% 95.6% 95.6% When & What 95.6% 95.6% 95.6% Who & When & What 96.0% 96.1% 96.1% TABLE IV Target IVApp Requests/min Time/Request(ms) CPU Time/Request(ms) CPU Time (%)he could only manipulate a limited number of communication or utility apps to access SMS related permissions. Second, the adversary is restricted to exploit the target permissions under proper scenes only. For example, even if he successfully elicited an end user to install the malicious SMS app, he could only send out a message under the composing page, and when to access such pages is fully controlled by the user. Thus, by enforcing contextual integrality, INSPIRED is more robust than approaches that only check description-to-permission fidelity[29,36,37,42,43,61]. Third, as INSPIRED examines the trigger event and the activation widget, the adversary should carefully plug the payload into the correct position of the targeted app source code. In the example above, he cannot simply introduce a malicious background service. Instead, he should place the malicious logic inside the clicking handler of the send button to succeed. Moreover, INSPIRED can be integrated with other techniques based upon different feature sets to provide more comprehensive protection. A promising direction is to add runtime data-flow tracking support, which enables INSPIRED to better understand the semantic relationships among the widgets. In that case, an SMS is restricted to the recipient specified by the To: widget.Wechat 12.6 174.7 76.7 4.4% Yelp 5.8 56.4 21.8 2.2% Yahoo Weather 2.5 42.3 11.3 1.4% Amazon 0.8 23.0 8.7 0.6% Paypal 0.4 27 11.8 0.2% ACKNOWLEDGMENT Hidden for double blind. . Accessibility, Accessibility. https://developer.android.com/guide/topics/ ui/accessibility/index.html. Precision and recall. Precision and recall. https://en.wikipedia.org/wiki/ Precision and recall. . Qksms, Qksms. https://github.com/moezbhatti/qksms. Ui/application exerciser monkey. Ui/application exerciser monkey. https://developer. android.com/studio/test/monkey.html. . Virusshare, Virusshare. https://virusshare.com/. Weka. Weka. http://www.cs.waikato.ac.nz/ml/weka/. . Wi-Fi Walkie Talkie, Wi-fi walkie talkie. https://play.google.com/store/apps/ details?id=com.webstar.walkies. Drebin: Effective and explainable detection of android malware in your pocket. D Arp, M Spreitzenbarth, M Hubner, H Gascon, K Rieck, Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS). the ISOC Network and Distributed System Security Symposium (NDSS)D. Arp, M. Spreitzenbarth, M. Hubner, H. Gascon, and K. Rieck. Drebin: Effective and explainable detection of android malware in your pocket. In Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS), 2014. Flowdroid: Precise context, flow, field, object-sensitive and lifecycle-aware taint analysis for android apps. S Arzt, S Rasthofer, C Fritz, E Bodden, A Bartel, J Klein, Y Le Traon, D Octeau, P Mcdaniel, PLDI. S. Arzt, S. Rasthofer, C. Fritz, E. Bodden, A. Bartel, J. Klein, Y. Le Traon, D. Octeau, and P. McDaniel. Flowdroid: Precise context, flow, field, object-sensitive and lifecycle-aware taint analysis for android apps. In PLDI, 2014. Privacy and contextual integrity: Framework and applications. A Barth, A Datta, J C Mitchell, H Nissenbaum, IEEE Symposium on Security and Privacy (SP). A. Barth, A. Datta, J. C. Mitchell, and H. Nissenbaum. Privacy and contextual integrity: Framework and appli- cations. In IEEE Symposium on Security and Privacy (SP), 2006. What the app is that? deception and countermeasures in the android user interface. A Bianchi, J Corbetta, L Invernizzi, Y Fratantonio, C Kruegel, G Vigna, IEEE Symposium on Security and Privacy (SP). A. Bianchi, J. Corbetta, L. Invernizzi, Y. Fratantonio, C. Kruegel, and G. Vigna. What the app is that? decep- tion and countermeasures in the android user interface. In IEEE Symposium on Security and Privacy (SP), 2015. Data flow oriented ui testing: exploiting data flows and ui elements to test android applications. N P BorgesJr, N. P. Borges Jr. Data flow oriented ui testing: exploiting data flows and ui elements to test android applications. In ISSTA, 2017. ipshield: A framework for enforcing context-aware privacy. S Chakraborty, C Shen, K R Raghavan, Y Shoukry, M Millar, M B Srivastava, NSDI. S. Chakraborty, C. Shen, K. R. Raghavan, Y. Shoukry, M. Millar, and M. B. Srivastava. ipshield: A framework for enforcing context-aware privacy. In NSDI, 2014. Smote: synthetic minority over-sampling technique. N V Chawla, K W Bowyer, L O Hall, W P Kegelmeyer, Journal of artificial intelligence research. 16N. V. Chawla, K. W. Bowyer, L. O. Hall, and W. P. Kegelmeyer. Smote: synthetic minority over-sampling technique. Journal of artificial intelligence research, 16:321-357, 2002. . K Z Chen, N M Johnson, S Dai, K Macnamara, T R Magrino, E X Wu, M Rinard, D X Song, K. Z. Chen, N. M. Johnson, S. Dai, K. MacNamara, T. R. Magrino, E. X. Wu, M. Rinard, and D. X. Song. Contextual policy enforcement in android applications with permission event graphs. Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS). the ISOC Network and Distributed System Security Symposium (NDSS)Contextual policy enforcement in android applications with permission event graphs. In Proceedings of the ISOC Network and Distributed System Security Sympo- sium (NDSS), 2013. Droidjust: automated functionalityaware privacy leakage analysis for android applications. X Chen, S Zhu, WiSecX. Chen and S. Zhu. Droidjust: automated functionality- aware privacy leakage analysis for android applications. In WiSec, 2015. Crepe: Context-related policy enforcement for android. M Conti, V T N Nguyen, B Crispo, ternational Conference on Information Security. SpringerM. Conti, V. T. N. Nguyen, and B. Crispo. Crepe: Context-related policy enforcement for android. In In- ternational Conference on Information Security, pages 331-345. Springer, 2010. Taintdroid: an information-flow tracking system for realtime privacy monitoring on smartphones. W Enck, P Gilbert, S Han, V Tendulkar, B.-G Chun, L P Cox, J Jung, P Mcdaniel, A N Sheth, OSDI. W. Enck, P. Gilbert, S. Han, V. Tendulkar, B.-G. Chun, L. P. Cox, J. Jung, P. McDaniel, and A. N. Sheth. Taint- droid: an information-flow tracking system for realtime privacy monitoring on smartphones. In OSDI, 2010. Android permissions demystified. A P Felt, E Chin, S Hanna, D Song, D Wagner, Proceedings of the ACM SIGSAC conference on Computer & communications security (CCS). the ACM SIGSAC conference on Computer & communications security (CCS)A. P. Felt, E. Chin, S. Hanna, D. Song, and D. Wagner. Android permissions demystified. In Proceedings of the ACM SIGSAC conference on Computer & communica- tions security (CCS), 2011. Android permissions: User attention, comprehension, and behavior. A P Felt, E Ha, S Egelman, A Haney, E Chin, D Wagner, SOUPS. A. P. Felt, E. Ha, S. Egelman, A. Haney, E. Chin, and D. Wagner. Android permissions: User attention, comprehension, and behavior. In SOUPS, 2012. Android ui deception revisited: Attacks and defenses. E Fernandes, Q A Chen, J Paupore, G Essl, J A Halderman, Z M Mao, A Prakash, Proceedings of the 20th International Conference on Financial Cryptography and Data Security. the 20th International Conference on Financial Cryptography and Data SecurityE. Fernandes, Q. A. Chen, J. Paupore, G. Essl, J. A. Halderman, Z. M. Mao, and A. Prakash. Android ui deception revisited: Attacks and defenses. In Proceed- ings of the 20th International Conference on Financial Cryptography and Data Security, 2016. Appstract: on-the-fly app content semantics with better privacy. E Fernandes, O Riva, S Nath, MobiCom. E. Fernandes, O. Riva, and S. Nath. Appstract: on-the-fly app content semantics with better privacy. In MobiCom, pages 361-374, 2016. Leaksemantic: Identifying abnormal sensitive network transmissions in mobile applications. H Fu, Z Zheng, S Bose, M Bishop, P Mohapatra, Computer Communications (INFOCOM). H. Fu, Z. Zheng, S. Bose, M. Bishop, and P. Mohapatra. Leaksemantic: Identifying abnormal sensitive network transmissions in mobile applications. In Computer Com- munications (INFOCOM), IEEE Proceedings on, 2017. Flowintent: Detecting privacy leakage from user intention to network traffic mapping. H Fu, Z Zheng, A K Das, P H Pathak, P Hu, P Mohapatra, Annual IEEE International Conference on Sensing, Communication, and Networking (SECON). H. Fu, Z. Zheng, A. K. Das, P. H. Pathak, P. Hu, and P. Mohapatra. Flowintent: Detecting privacy leakage from user intention to network traffic mapping. In Annual IEEE International Conference on Sensing, Communica- tion, and Networking (SECON), 2016. Checking app behavior against app descriptions. A Gorla, I Tavecchia, F Gross, A Zeller, IEEE/ACM International Conference on Software engineering (ICSE). A. Gorla, I. Tavecchia, F. Gross, and A. Zeller. Checking app behavior against app descriptions. In IEEE/ACM In- ternational Conference on Software engineering (ICSE), 2014. A taxonomy of attacks and a survey of defence mechanisms for semantic social engineering attacks. R Heartfield, G Loukas, ACM Computing Surveys (CSUR). 48337R. Heartfield and G. Loukas. A taxonomy of attacks and a survey of defence mechanisms for semantic social engineering attacks. ACM Computing Surveys (CSUR), 48(3):37, 2016. Supor: precise and scalable sensitive user input detection for android apps. J Huang, Z Li, X Xiao, Z Wu, K Lu, X Zhang, G Jiang, USENIX Security. J. Huang, Z. Li, X. Xiao, Z. Wu, K. Lu, X. Zhang, and G. Jiang. Supor: precise and scalable sensitive user input detection for android apps. In USENIX Security, 2015. Detecting sensitive data disclosure via bi-directional text correlation analysis. J Huang, X Zhang, L Tan, Proceedings of the ACM SIGSOFT International Symposium on Foundations of Software Engineering (FSE). the ACM SIGSOFT International Symposium on Foundations of Software Engineering (FSE)J. Huang, X. Zhang, and L. Tan. Detecting sensitive data disclosure via bi-directional text correlation analysis. In Proceedings of the ACM SIGSOFT International Sym- posium on Foundations of Software Engineering (FSE), 2016. Asdroid: detecting stealthy behaviors in android applications by user interface and program behavior contradiction. J Huang, X Zhang, L Tan, P Wang, B Liang, IEEE/ACM International Conference on Software engineering (ICSE). J. Huang, X. Zhang, L. Tan, P. Wang, and B. Liang. Asdroid: detecting stealthy behaviors in android applica- tions by user interface and program behavior contradic- tion. In IEEE/ACM International Conference on Software engineering (ICSE), 2014. Privacy as part of the app decision-making process. P G Kelley, L F Cranor, N Sadeh, CHI. P. G. Kelley, L. F. Cranor, and N. Sadeh. Privacy as part of the app decision-making process. In CHI, 2013. Peruim: understanding mobile application privacy with permission-ui mapping. Y Li, Y Guo, X Chen, Proceedings of the ACM International Joint Conference on Pervasive and Ubiquitous Computing (Ubicomp). the ACM International Joint Conference on Pervasive and Ubiquitous Computing (Ubicomp)Y. Li, Y. Guo, and X. Chen. Peruim: understanding mo- bile application privacy with permission-ui mapping. In Proceedings of the ACM International Joint Conference on Pervasive and Ubiquitous Computing (Ubicomp), 2016. Checking more and alerting less: Detecting privacy leakages via enhanced data-flow analysis and peer voting. K Lu, Z Li, V P Kemerlis, Z Wu, L Lu, C Zheng, Z Qian, W Lee, G Jiang, Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS). the ISOC Network and Distributed System Security Symposium (NDSS)K. Lu, Z. Li, V. P. Kemerlis, Z. Wu, L. Lu, C. Zheng, Z. Qian, W. Lee, and G. Jiang. Checking more and alerting less: Detecting privacy leakages via enhanced data-flow analysis and peer voting. In Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS), 2015. A survey of app store analysis for software engineering. W Martin, F Sarro, Y Jia, Y Zhang, M Harman, IEEE Transactions on Software Engineering. W. Martin, F. Sarro, Y. Jia, Y. Zhang, and M. Harman. A survey of app store analysis for software engineering. IEEE Transactions on Software Engineering, 2016. Conxsense: automated context classification for context-aware access control. M Miettinen, S Heuser, W Kronz, A.-R Sadeghi, N Asokan, Proceedings of the 9th ACM symposium on Information, computer and communications security (Asia CCS). the 9th ACM symposium on Information, computer and communications security (Asia CCS)M. Miettinen, S. Heuser, W. Kronz, A.-R. Sadeghi, and N. Asokan. Conxsense: automated context classification for context-aware access control. In Proceedings of the 9th ACM symposium on Information, computer and communications security (Asia CCS), 2014. Uipicker: User-input privacy identification in mobile applications. Y Nan, M Yang, Z Yang, S Zhou, G Gu, X Wang, USENIX Security. Y. Nan, M. Yang, Z. Yang, S. Zhou, G. Gu, and X. Wang. Uipicker: User-input privacy identification in mobile ap- plications. In USENIX Security, 2015. Smarper: Context-aware and automatic runtime-permissions for mobile devices. K Olejnik, I I Dacosta Petrocelli, J C Soares Machado, K Huguenin, M E Khan, J.-P Hubaux, IEEE Symposium on Security and Privacy (SP). K. Olejnik, I. I. Dacosta Petrocelli, J. C. Soares Machado, K. Huguenin, M. E. Khan, and J.-P. Hubaux. Smarper: Context-aware and automatic runtime-permissions for mobile devices. In IEEE Symposium on Security and Privacy (SP), 2017. Dark hazard: Learning-based, large-scale discovery of hidden sensitive operations in android apps. X Pan, X Wang, Y Duan, X Wang, H Yin, Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS. the ISOC Network and Distributed System Security Symposium (NDSSX. Pan, X. Wang, Y. Duan, X. Wang, and H. Yin. Dark hazard: Learning-based, large-scale discovery of hidden sensitive operations in android apps. In Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS), 2017. Whyper: Towards automating risk assessment of mobile applications. R Pandita, X Xiao, W Yang, W Enck, T Xie, USENIX Security. R. Pandita, X. Xiao, W. Yang, W. Enck, and T. Xie. Whyper: Towards automating risk assessment of mobile applications. In USENIX Security, 2013. Autocog: Measuring the description-topermission fidelity in android applications. Z Qu, V Rastogi, X Zhang, Y Chen, T Zhu, Z Chen, Proceedings of the ACM SIGSAC conference on Computer & communications security (CCS). the ACM SIGSAC conference on Computer & communications security (CCS)Z. Qu, V. Rastogi, X. Zhang, Y. Chen, T. Zhu, and Z. Chen. Autocog: Measuring the description-to- permission fidelity in android applications. In Proceed- ings of the ACM SIGSAC conference on Computer & communications security (CCS), 2014. Windowguard: Systematic protection of gui security in android. C Ren, P Liu, S Zhu, Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS). the ISOC Network and Distributed System Security Symposium (NDSS)C. Ren, P. Liu, and S. Zhu. Windowguard: Systematic protection of gui security in android. In Proceedings of the ISOC Network and Distributed System Security Symposium (NDSS), 2017. Recon: Revealing and controlling pii leaks in mobile network traffic. J Ren, A Rao, M Lindorfer, A Legout, D Choffnes, MobiSys. J. Ren, A. Rao, M. Lindorfer, A. Legout, and D. Choffnes. Recon: Revealing and controlling pii leaks in mobile network traffic. In MobiSys, 2016. Audacious: User-driven access control with unmodified operating systems. T Ringer, D Grossman, F Roesner, Proceedings of the ACM SIGSAC conference on Computer & communications security (CCS). the ACM SIGSAC conference on Computer & communications security (CCS)T. Ringer, D. Grossman, and F. Roesner. Audacious: User-driven access control with unmodified operating systems. In Proceedings of the ACM SIGSAC conference on Computer & communications security (CCS), 2016. User-driven access control: Rethinking permission granting in modern operating systems. F Roesner, T Kohno, A Moshchuk, B Parno, H J Wang, C Cowan, IEEE Symposium on Security and privacy (SP). F. Roesner, T. Kohno, A. Moshchuk, B. Parno, H. J. Wang, and C. Cowan. User-driven access control: Re- thinking permission granting in modern operating sys- tems. In IEEE Symposium on Security and privacy (SP), 2012. Incremental learning for robust visual tracking. D A Ross, J Lim, R.-S Lin, M.-H Yang, International journal of computer vision. 771D. A. Ross, J. Lim, R.-S. Lin, and M.-H. Yang. Incre- mental learning for robust visual tracking. International journal of computer vision, 77(1):125-141, 2008. Covert communication in mobile applications. J Rubin, M I Gordon, N Nguyen, M Rinard, ASE. J. Rubin, M. I. Gordon, N. Nguyen, and M. Rinard. Covert communication in mobile applications. In ASE, 2015. Understanding and capturing peoples privacy policies in a mobile social networking application. N Sadeh, J Hong, L Cranor, I Fette, P Kelley, M Prabaker, J Rao, Personal and Ubiquitous Computing. 136N. Sadeh, J. Hong, L. Cranor, I. Fette, P. Kelley, M. Prabaker, and J. Rao. Understanding and capturing peoples privacy policies in a mobile social network- ing application. Personal and Ubiquitous Computing, 13(6):401-412, 2009. Asking for (and about) permissions used by android apps. R Stevens, J Ganz, V Filkov, P Devanbu, H Chen, MSR. R. Stevens, J. Ganz, V. Filkov, P. Devanbu, and H. Chen. Asking for (and about) permissions used by android apps. In MSR, 2013. Classification of imbalanced data: A review. Y Sun, A K Wong, M S Kamel, International Journal of Pattern Recognition and Artificial Intelligence. 2304Y. Sun, A. K. Wong, and M. S. Kamel. Classification of imbalanced data: A review. International Journal of Pat- tern Recognition and Artificial Intelligence, 23(04):687- 719, 2009. Using text mining to infer the purpose of permission use in mobile apps. H Wang, J Hong, Y Guo, Proceedings of the ACM International Joint Conference on Pervasive and Ubiquitous Computing (Ubicomp). the ACM International Joint Conference on Pervasive and Ubiquitous Computing (Ubicomp)H. Wang, J. Hong, and Y. Guo. Using text mining to infer the purpose of permission use in mobile apps. In Proceedings of the ACM International Joint Conference on Pervasive and Ubiquitous Computing (Ubicomp), 2015. Understanding the purpose of permission use in mobile apps. H Wang, Y Li, Y Guo, Y Agarwal, J I Hong, ACM Transactions on Information Systems (TOIS). 35443H. Wang, Y. Li, Y. Guo, Y. Agarwal, and J. I. Hong. Understanding the purpose of permission use in mobile apps. ACM Transactions on Information Systems (TOIS), 35(4):43, 2017. Permission evolution in the android ecosystem. X Wei, L Gomez, I Neamtiu, M Faloutsos, ACSAC. X. Wei, L. Gomez, I. Neamtiu, and M. Faloutsos. Per- mission evolution in the android ecosystem. In ACSAC, 2012. Android permissions remystified: A field study on contextual integrity. P Wijesekera, A Baokar, A Hosseini, S Egelman, D Wagner, K Beznosov, USENIX Security. P. Wijesekera, A. Baokar, A. Hosseini, S. Egelman, D. Wagner, and K. Beznosov. Android permissions remystified: A field study on contextual integrity. In USENIX Security, 2015. The feasibility of dynamically granted permissions: Aligning mobile privacy with user preferences. P Wijesekera, A Baokar, L Tsai, J Reardon, S Egelman, D Wagner, K Beznosov, IEEE Symposium on Security and Privacy (SP). P. Wijesekera, A. Baokar, L. Tsai, J. Reardon, S. Egel- man, D. Wagner, and K. Beznosov. The feasibility of dynamically granted permissions: Aligning mobile privacy with user preferences. In IEEE Symposium on Security and Privacy (SP), 2017. Appcontext: Differentiating malicious and benign mobile app behaviors using context. W Yang, X Xiao, B Andow, S Li, T Xie, W Enck, IEEE/ACM International Conference on Software engineering (ICSE). W. Yang, X. Xiao, B. Andow, S. Li, T. Xie, and W. Enck. Appcontext: Differentiating malicious and benign mobile app behaviors using context. In IEEE/ACM International Conference on Software engineering (ICSE), 2015. Appintent: Analyzing sensitive data transmission in android for privacy leakage detection. Z Yang, M Yang, Y Zhang, G Gu, P Ning, X S Wang, Proceedings of the ACM SIGSAC conference on Computer & communications security (CCS). the ACM SIGSAC conference on Computer & communications security (CCS)Z. Yang, M. Yang, Y. Zhang, G. Gu, P. Ning, and X. S. Wang. Appintent: Analyzing sensitive data transmission in android for privacy leakage detection. In Proceedings of the ACM SIGSAC conference on Computer & commu- nications security (CCS), 2013. Incremental clustering for human activity detection based on phone sensor data. X Yin, W Shen, X Wang, IEEE International Conference on Computer Supported Cooperative Work in Design (CSCWD). X. Yin, W. Shen, and X. Wang. Incremental clustering for human activity detection based on phone sensor data. In IEEE International Conference on Computer Supported Cooperative Work in Design (CSCWD), 2016. Revisiting the description-to-behavior fidelity in android applications. L Yu, X Luo, C Qian, S Wang, IEEE International Conference on Software Analysis, Evolution, and Reengineering (SANER). L. Yu, X. Luo, C. Qian, and S. Wang. Revisiting the description-to-behavior fidelity in android applications. In IEEE International Conference on Software Analysis, Evolution, and Reengineering (SANER), 2016. Semanticsaware android malware classification using weighted contextual api dependency graphs. M Zhang, Y Duan, H Yin, Z Zhao, Proceedings of the ACM SIGSAC conference on Computer & communications security (CCS). the ACM SIGSAC conference on Computer & communications security (CCS)M. Zhang, Y. Duan, H. Yin, and Z. Zhao. Semantics- aware android malware classification using weighted contextual api dependency graphs. In Proceedings of the ACM SIGSAC conference on Computer & communica- tions security (CCS), 2014. Rethinking permission enforcement mechanism on mobile systems. Y Zhang, M Yang, G Gu, H Chen, IEEE Transactions on Information Forensics and Security. 1110Y. Zhang, M. Yang, G. Gu, and H. Chen. Rethinking permission enforcement mechanism on mobile systems. IEEE Transactions on Information Forensics and Secu- rity, 11(10):2227-2240, 2016. Permission use analysis for vetting undesirable behaviors in android apps. Y Zhang, M Yang, Z Yang, G Gu, P Ning, B Zang, IEEE transactions on information forensics and security. 9Y. Zhang, M. Yang, Z. Yang, G. Gu, P. Ning, and B. Zang. Permission use analysis for vetting undesirable behaviors in android apps. IEEE transactions on infor- mation forensics and security, 9(11):1828-1842, 2014.
[ "https://github.com/moezbhatti/qksms." ]
[ "Auditable Register Emulations", "Auditable Register Emulations" ]
[ "Vinicius V Cogo \nFaculdade de Ciências\nLASIGE\nUniversidade de Lisboa\nLisboaPortugal\n", "Alysson Bessani \nFaculdade de Ciências\nLASIGE\nUniversidade de Lisboa\nLisboaPortugal\n" ]
[ "Faculdade de Ciências\nLASIGE\nUniversidade de Lisboa\nLisboaPortugal", "Faculdade de Ciências\nLASIGE\nUniversidade de Lisboa\nLisboaPortugal" ]
[]
The widespread prevalence of data breaches amplifies the importance of auditing storage systems. Here we initiate the study of auditable storage emulations, which provides the capability for an auditor to discover the previously executed reads in a register. We precisely define the notion of auditable register and its properties, and establish tight bounds and impossibility results for auditable storage emulations in a Byzantine setting. Our formulation considers readwrite registers that securely store data using information dispersal and support fast reads. In such scenario, given a maximum number f of faulty storage objects and a minimum number τ of data blocks required to recover a written value, we prove that (1) auditability is impossible if τ ≤ 2f ; (2) τ ≥ 3f + 1 is required for implementing a weak form of auditability; and (3) a stronger form of auditability is impossible. We also show that totally ordering operations or using non-fast reads enables such strong auditability, albeit requiring more replicas.
null
[ "https://arxiv.org/pdf/1905.08637v1.pdf" ]
160,009,932
1905.08637
e921cec9f47a5a481583cc139a7bccdf61b5e319
Auditable Register Emulations Vinicius V Cogo Faculdade de Ciências LASIGE Universidade de Lisboa LisboaPortugal Alysson Bessani Faculdade de Ciências LASIGE Universidade de Lisboa LisboaPortugal Auditable Register Emulations The widespread prevalence of data breaches amplifies the importance of auditing storage systems. Here we initiate the study of auditable storage emulations, which provides the capability for an auditor to discover the previously executed reads in a register. We precisely define the notion of auditable register and its properties, and establish tight bounds and impossibility results for auditable storage emulations in a Byzantine setting. Our formulation considers readwrite registers that securely store data using information dispersal and support fast reads. In such scenario, given a maximum number f of faulty storage objects and a minimum number τ of data blocks required to recover a written value, we prove that (1) auditability is impossible if τ ≤ 2f ; (2) τ ≥ 3f + 1 is required for implementing a weak form of auditability; and (3) a stronger form of auditability is impossible. We also show that totally ordering operations or using non-fast reads enables such strong auditability, albeit requiring more replicas. Introduction Motivation. Characteristics like cost-effectiveness, high scalability, and ease of use promoted the migration from private storage infrastructures to public multi-tenant clouds in the last decade. Security and privacy concerns were the main deterrents for this migration since the beginning [33]. Numerous secure storage systems have been proposing the use of advanced cryptographic primitives to securely disperse data across multiple clouds (i.e., independent administrative domains) to reduce the risk of data breaches [30]. Given a secure storage system composed of n storage objects, these information dispersal techniques split and convert the original data item into n coded blocks [21,26,30]. Each coded block is stored in a different object and clients need to obtain only τ out of n coded blocks to effectively recover the original data. In this type of solution, no object contains the whole data item, which differentiates information dispersal from fully-replicated storage systems (e.g., [1,4,24])-where each object stores a full copy of the data. Despite the advances in secure storage systems, the increasing severity of data breaches (e.g., [28]) and the tightening of privacy-related regulations (e.g., GDPR [11]) have been driving the demand for further improvements on this topic. For instance, auditability [19] may enable the systematic verification of who has effectively read data in secure storage systems. Notably, such verification will allow one to separate these users from the whole set of clients that are authorized to read data but have never done so. It is an important step to detect data breaches (including those caused by authorized users, e.g., Snowden's case [15]), analyze leakages, and sanction misuses. Problem. In this paper, we address the following question: How to extend Byzantine-resilient storage emulations with the capability of auditing who has effectively read data? The answer must encompass the techniques used in these emulations, such as information dispersal [21,26,30] and (available) Byzantine quorum systems [24], for providing a R/W register abstraction [22]. More specifically, we address the problem of protecting storage systems from readers trying to obtain data without being detected (i.e., audit completeness) and protecting correct readers from faulty storage objects trying to incriminate them (i.e., audit accuracy). Related Work. Several auditing schemes were proposed to verify the integrity of data stored in multi-tenant external infrastructures [20,25,32]. They mainly focus on cryptographic techniques to produce retrievability proofs without the need to fetch all data from the system (e.g., [18,36]) or on providing public integrity verification (e.g., [34,35]). However, to the best of our knowledge, there is no previous work on auditing who has effectively read data in a dispersed storage. Another topic related to our work is the accountability, which focuses on making systems' components accountable in a way their actions become non-repudiable [13,37]. Works in the accountability literature have discussed generic scenarios for networked systems [14], described the necessary cryptographic building blocks [13,37], or how evidences should be stored and protected [14]. Several other works have explored the space complexity of fault-tolerant register emulations (e.g., [2,7,9,10]), including disintegrated storage [5] (e.g., [3,6,31]). However, none of these works focuses on the quorum requirements for auditing read accesses in a storage system despite the existence of Byzantine objects. Contributions. This paper initiates the study of auditing in Byzantine-resilient storage by presenting lower bounds and impossibility results related with the implementation of an auditable register on top of n base objects despite the existence of f faulty ones. Our results show that, given a minimum number τ of data blocks required to recover a data item from information dispersal schemes, (1) auditability is impossible with τ ≤ 2f (or n ≤ 4f ); (2) when fast reads (reads executed in a single communication round-trip [12]) are supported, τ ≥ 3f + 1 (or n ≥ 5f + 1) is required for implementing a weak form of auditability, while a stronger form of auditability is impossible; and (3) totally ordering operations and using non-fast reads can still provide such strong auditability. Preliminaries System Model. Our system is composed of an arbitrary number of client processes Π = {p 1 , p 2 , ...}, which interact with a set of n storage objects O = {o 1 , o 2 , ..., o n }. Clients can be subdivided into three main classes: writers Π W = {p w1 , p w2 , ...}, readers Π R = {p r1 , p r2 , ...}, and auditors Π A = {p a1 , p a2 , ...}. These different roles do not necessarily mean they have to be performed by different processes. A configuration C is a vector of the states of all entities (i.e., processes and objects) in the system. An initial configuration is a specific configuration where all entities of the system are in their initial states. An algorithm A defines the behavior of processes in Π as deterministic state machines, which can modify the system's states through actions (e.g., invoke and response). An execution segment is a (finite or infinite) sequence of alternated configurations and actions. An execution ρ is an execution segment that begins in an initial configuration. An event is the occurrence of an action in an execution. Clients invoke operations in the storage objects and wait for responses. Low-level operations in objects are instantaneous, atomic actions. A sequence of invocations and responses compose a history σ of the system. We consider the following relationships between operations within a history. First, if a history σ contains the invocation of an operation op 1 and its response, then op 1 is complete in σ. Otherwise, if σ contains only the former, then operation op 1 is pending. Second, if the response of an operation op 1 precedes the invocation of another operation op 2 in σ, then op 1 precedes op 2 . Third, if operations op 1 and op 2 do not precede each other in σ, then they are concurrent. Fourth, a history σ that does not contain concurrent operations is considered sequential. Fault Model. Clients (i.e., writers, readers, and auditors) and storage objects that obey their specifications are said to be correct. Faulty readers can be either malicious or honest. Malicious (i.e., Byzantine) faulty readers may behave arbitrarily, i.e., combining omissive and assertive behaviors. For instance, they may deviate from the protocol specification by invoking contradictory operations in all or only some storage objects. Alternatively, honest faulty readers may only crash. Faulty writers and faulty auditors are honest and can only fail by crashing. Writers are trusted because they are the data owners, who are the most interested part in the auditable register we are proposing in this work. This is a common assumption in the BFT storage literature (e.g., [6,24]) since malicious writers could write invalid values anyway, compromising the application's state. Furthermore, auditors are also trusted because they are controlled either by the same entity as writers or third-party entities writers trust. A faulty storage object may deviate from its specification arbitrarily-i.e., data can be created, corrupted, deleted, or leaked to unauthorized parties [23]. They can also present omissive behavior, intermittent or not. Although our system tolerates the aforementioned faults, it operates correctly only if no more than f storage objects are faulty. R/W Register Specification. A read-write (R/W) register is an object that stores a data value v from any domain V and provides two low-level operations [22]: • rw-write(v): writes the data value v ∈ V, passed as argument, in the data object and returns an ack to confirm the operation succeed. • rw-read(): returns the data value v ∈ V currently stored in the data object or ⊥ if no value has been written on it. The behavior of a R/W register is given by its sequential specification, where every low-level rw-read operation on it returns the value written by the last preceding rw-write operation on this object, or the special value ⊥ / ∈ V if no such operation exists. Emulated Registers. We emulate a shared register that stores a value v from domain V using R/W registers. It exposes high-level s-read() and s-write(v) operations to processes in Π. We consider multi-writer multi-reader (MWMR) registers where multiple writers from Π w invoke swrite operations and multiple readers from Π R invoke s-read operations. Our safety requirement is the safe semantics: a read can return any value if it is concurrent with a write operation, and returns the value written by the most recent write operation if there is no write operation concurrent with the read [22]. Our liveness requirement is wait-freedom, where every operation invoked by a process p returns within a finite number of events [17]. High-level read operations (e.g., s-read ) in our system are fast reads. Definition 1 (Fast read [12]). An s-read operation is fast if it completes in a single communication round-trip between the reader and the storage objects. Information Dispersal. Storage objects in O are untrusted independent entities. We assume they store data in blocks generated using information dispersal schemes (i.e., a special case of disintegrated storage [5]) such as erasure codes [26,27] and secret sharing [21,29]. In these schemes, a high-level s-write(v) operation w v converts a value v ∈ V into n coded blocks b v 1 , b v 2 , ..., b vn from a domain B. Each coded block b v k produced in w v is marked with a unique write label l wv and sent to the k th storage object in the system. These techniques guarantee that no object o k stores an entire copy of value v and no client process recovers v by accessing less than a certain fraction of these blocks. For simplicity, we assume every distinct value is written only once. A high-level s-read() operation recovers the original value v from any subset of a specific number (τ ) of correct blocks b v k written in w v . It means readers do not need to fetch blocks from all (n) storage objects. More specifically, we introduce the notion of an effective read as described in Definition 2. Definition 2 (Effective read). A value v ∈ V, written in w v , is effectively read by a reader p r , in history σ, when p r has read τ correct blocks b v k of w v from different objects o k . Note that an effective read depends only on the number of correct blocks (written in the same write w v ) that a reader p r has already obtained from different objects. Reader p r does not necessarily obtain all these blocks on a single s-read operation. There might exist cases where it is accomplished only after receiving responses from many subsequent s-read operations. We also consider information dispersal schemes have a maximum number of Byzantine faults f tolerated at the same time. As a consequence, the minimum number of blocks to recover the original data must comply with Proposition 1, which guarantees that f malicious objects cannot create nonexistent correct values. Proposition 1. Any information dispersal scheme requires τ > f to tolerate up to f Byzantine faults [16]. Available f -Threshold Byzantine Quorum Systems. An available (f -threshold) quorum system guarantees the consistency and availability of stored data by executing operations in only a subset (i.e., a quorum Q) of storage objects, instead of requiring clients to communicate with all of them to execute high-level operations. We consider all protocols in this paper require Byzantine quorum systems, which contain at least f + 1 objects at the intersection of the quorums of any two operations [24]. To provide wait-free register emulations using quorum systems in asynchronous environments, these systems need to be available. Definition 3 (Available f -threshold quorums [24]). Any available f -threshold quorum Q is composed of q = n − f replicas to tolerate up to f faults. We assume available quorums are composed of q = n − f replicas (instead of q ≤ n − f [24]) because we are interested in the limit case where quorums of maximum size are used. Using available quorums guarantees that any complete s-write operation w v stores correctly labeled blocks in at least a quorum of storage objects. At any point in history σ, an object that participated in the quorum of the last preceding s-write is considered an up-to-date object, whereas an object that did not participate on it is considered stale. Finally, we assume that information dispersal schemes (e.g., erasure codes) are implemented using available quorums and comply with Proposition 2 to tolerate up to f Byzantine objects. This proposition defines the required minimum number (n) of objects to be present in the whole system. Proposition 2. Any available Byzantine fault-tolerant τ -of-n information dispersal scheme requires at least n ≥ τ + 2f replicas to tolerate up to f faulty objects [16]. This bound comes from the fact that available quorums require the response from only n − f replicas, with up to f of these responses from malicious objects. Note that malicious readers can obtain τ correct blocks without accessing a quorum of storage objects. Auditable Register Emulations We extend the aforementioned register emulation by adding an operation getLog() to the storage objects (i.e., R/W registers). Invoking this operation on object o k returns the log L k containing records of every preceding read executed in this object. A record p r , l wv k ∈ L k contains the identifier of the reader p r and the write label l wv associated with the value v whose the block b v k was read by p r . Based on these individual fail-prone logs, we define an auditable register emulation that has access to a virtual log L k∈{1..n} L k from which we can infer who has effectively read a value from the register. This emulation provides three high-level operations: a-write(v), a-read(), and a-audit(). An a-write(v) is an unmodified s-write(v) described in the previous section. An a-read() is an extended s-read() that reads blocks b v k from a quorum of objects, causing p r , l wv k to be added to L k on each accessed correct storage object o k . Finally, the third operation is an a-audit(). It obtains records from L and produces a set of evidences E about the values read. Each evidence E pr,v contains a set of at least records from different storage objects, which proves that v was effectively read by each reader p r in history σ. We are interested in auditing effective reads because we intend to audit who has actually read a data value v-including malicious readers that do not follow the read protocol and leave operations pending. A correct auditor receives E and reports all evidenced reads. An auditable register provides an a-audit() operation that guarantees completeness and at least one form of accuracy, as stated in Definitions 4-6. Definition 4 (Completeness). Every value v ∈ V effectively read by a reader p r before the invocation of an a-audit in history σ is reported in E resulting from this audit operation, i.e., E pr,v ∈ E. Definition 5 (Weak Accuracy). A correct reader p r that has never invoked an a-read before the invocation of an a-audit in history σ will not be reported in E resulting from this audit operation, i.e., ∀v ∈ V, E pr,v / ∈ E. Definition 6 (Strong Accuracy). A correct reader p r that has never effectively read a value v ∈ V before the invocation of an a-audit in history σ will not be reported in E resulting from this audit operation as having read v, i.e., E pr,v / ∈ E. While completeness is intended to protect the storage system from readers trying to obtain data without being detected, accuracy focuses on protecting correct readers from malicious storage objects incriminating them. The difference between the weak and the strong variants of the accuracy property is in the precision of the auditor's report. With weak accuracy, the audit reports only readers that have invoked read operations, which already separates them from the whole set of clients that are authorized to read a specific register but have never done so. With strong accuracy, the audit reports only the values readers have effectively read. Strong accuracy implies weak accuracy. Complementing the other side of the accuracy property, incorrect (honest or malicious) readers may be reported by the audit in pending, incomplete, or partial reads. It means they had the intention to read the data and aborted it or crashed while doing so. Preliminary Results Auditing fully-replicated systems is impossible in the presence of Byzantine objects because each object stores a full copy of data and can alone give it to readers without logging the operations. Therefore, we consider information dispersal techniques as the basic form of disintegrated storage [5] for auditable register emulations. Records Available for Auditing Registers. In this section, we identify the minimum number of records from every effective read that will be available for a-audit operations. This number is then used to define the required number of records to produce an evidence E pr,v . The three operations (write, read, and audit) of auditable registers are possibly executed on different available quorums that need to intersect. The worst-case scenario is presented in Figure 1a, where we illustrate the minimum intersections between each of these three quorums, identified as groups G 1−4 . In this figure, every storage object belongs to a group and all objects within the same group contain the same state. Lemma 1. Any available a-audit() operation obtains at least τ − 2f records of every preceding effective read in history σ. Proof. Let us assume any available quorum system composed of n storage objects, where clients wait for replies from q = n−f of them (Definition 3). Let us assume that, in the worst-case scenario, the quorums Q w for an a-write(x), Q r for an a-read(), and Q a for an a-audit() operation differ as depicted in Figure 1a. In this scenario, the size of the intersection |Q w ∩ Q r ∩ Q a | is n − 3f . From these n − 3f objects, f of them are malicious and n − 4f are correct, up-to-date objects. Let us assume a malicious reader that effectively reads a value v after accessing only τ objects instead of contacting the whole quorum Q r . In the worst-case scenario, the malicious reader obtains f correct blocks from malicious storage objects (that do not log the read); f blocks from correct, up-to-date objects in G 1 (that log the read but do not participate in the audit quorum Q a ); and n − 4f blocks from correct, up-to-date objects (that log the records and participate in Q a ). Since these n − 4f correct, up-to-date objects are the only ones that have both logged the records and participate in Q a , by applying Proposition 2, we have that τ − 2f is the minimum number of available records in L from Q a for detecting every value v effective read in the system. Evidences are created in a-audit() only after finding, in L, at least records from different objects attesting the same read value and reader. Corollary 1. The required number ( ) of records to produce an evidence E pr,v is ≤ τ − 2f . Resilience Lower Bounds We present impossibility results for the three properties of auditable registers: completeness (Definition 4), weak accuracy (Definition 5), and strong accuracy (Definition 6). From this point on, correct storage objects start a history σ with the initial configuration composed of blocks of a value v ∈ V and an empty log L = ∅. For simplicity, we depict the log records p r , l wv as v r in the figures. Lemma 2. It is impossible to satisfy the completeness of auditable registers with τ ≤ 2f . Proof. Without loss of generality, let us assume that τ = 2f . According to Proposition 2, a system with τ = 2f must have at least 4f nodes. Consider an auditable register implemented using four subset groups of objects G 1−4 , as depicted in Figure 1b. Each group is composed of f objects. Group G 2 contains malicious objects that may behave arbitrarily. A malicious reader p r1 obtains f coded blocks from the correct nodes in G 3 (that log the operation) and f blocks from the malicious nodes in G 2 (which do not log the operation). This malicious reader can decode the original value v after receiving the τ = 2f correct blocks, performing thus an effective read. However, any audit quorum Q a that does not include at least one object from group G 3 will not receive any record for the read operation from reader p r1 . This history violates the completeness property since an evidence is never created if there is no record of this read to do so. |G 3 | = f |G 1 | = f |G 2 | = n-3f |G 4 | = f ack a-write(x) x a-read( ) {x r } a-audit( ) Q a Q w Q r (a) |G 4 |=f |G 1 |=f { } b v { } b v { } b v { } b v malicious |G 2 |=f { } b m { } b m |G 3 |=f { } b v {v r1 } b v σ 1 time block log v a-read( ) { } a-audit( ) (b) |G 1 |≥τ { } b v malicious |G 2 |=f {v r1 } b m |G 3 |=f { } b v time log {v Lemma 3. It is impossible to satisfy the weak accuracy of auditable registers with ≤ f . Proof. Without loss of generality, let us assume that = f . Consider an auditable register implemented using three subset groups of objects G 1−3 , as depicted in Figure 1c, where the group G 1 is composed of at least τ objects and the remaining groups are composed of exactly f objects each. Group G 2 contains malicious objects that may behave arbitrarily and log a nonexistent read operation. Any audit quorum Q a that includes the f objects from group G 2 will receive f records for a read operation that has never been invoked. If = f is enough to create an evidence and report an effective read, then a correct auditor must report it, violating the weak accuracy property. The next theorem presents a scenario where these two properties are intended to be supported simultaneously. Theorem 1. It is impossible to satisfy both completeness and weak accuracy of auditable registers with τ ≤ 3f . Proof. Without loss of generality, let us assume that τ = 3f is enough to satisfy both the weak accuracy and completeness properties. According to Proposition 2, a system with τ = 3f must have at least 5f nodes. Consider an auditable register implemented using five subset groups of objects G 1−5 , as depicted in Figure 2, where each group is composed of f objects and group G 2 contains only malicious objects. In history σ 1 (Figure 2a), a malicious reader p r1 obtains 2f coded blocks from correct objects in G 3 and G 4 (that log the operation) and other f correct blocks from the malicious objects in G 2 (which do not log the operation). Any audit quorum Q a will receive at least f records either from the objects in group G 3 or G 4 . Evidences are created using these ≥ f records to report the read, satisfying completeness. However, as proved in Lemma 3 and illustrated in history σ 2 (Figure 2b), it is impossible to guarantee weak accuracy with ≤ f . Now, we turn our attention to strong accuracy, i.e., the capability of an auditor to report exactly which value v each reader p r has read. Figure 2: A scenario where τ = 3f does not guarantee both completeness and weak accuracy. |G 4 |=f |G 5 |=f |G 1 |=f { } b v { } b v { } b v {v r1 } b v { } b v { } b v malicious |G 2 |=f { } b m { } b m |G 3 |=f { } b v {v r1 } b v time log v a-read( ) {v r1 } a-audit( ) σ 1 block (a) |G 4 |=f |G 5 |=f |G 1 |=f { } b v { } b v { } b v malicious |G 2 |=f {v r1 } b m |G 3 |=f { } b v time log {v r1 } a-audit( ) σ 2 block (b) Lemma 4. It is impossible to satisfy the strong accuracy of auditable registers with < τ + f . Proof. Let us assume without loss of generality that = τ + f − 1. Consider an auditable register implemented using n ≥ 2f + τ objects divided in four subset groups of objects G 1−4 , as depicted in Figure 3a, where group G 3 is composed of τ − 1 objects, group G 4 is composed of at least one object, and the remaining groups are composed of exactly f objects each. Group G 2 contains only malicious objects. A writer p w1 starts a write operation concurrently with a read operation from a correct reader p r1 . At a first moment, only objects from group G 4 have received the write request and indeed wrote the blocks for value x on their registers. Then, the read request arrives in objects from groups G 2−4 , where: the f malicious objects from G 2 return blocks for the wrong value m (and log the read of v by p r1 ); τ − 1 correct objects from group G 3 return blocks for the value v (and log it); and all objects from group G 4 correctly return blocks for the value x (and log the read of x by p r1 ). In this history σ 1 , the reader p r1 has not received τ correct data blocks for value v, which means it is unable to recover v. However, any audit quorum Q a that includes objects from groups G 2 and G 3 will return τ +f −1 records p r1 , l wv k . If = τ + f − 1 is enough to produce an evidence of an effective read, then the auditor will report this non-effective read, violating the strong accuracy property. Theorem 2. It is impossible to satisfy both completeness and strong accuracy of auditable registers. Proof. Without loss of generality, let us assume any number of blocks τ ≥ 2f + 1 required to effectively read a value (Lemma 2). Consider an auditable register implemented using n ≥ 2f + τ objects (Proposition 2) divided in seven subset groups of objects G 1−7 , as depicted in Figure 3b. Group G 1 is composed of f − 1 objects; G 2 and G 7 are composed of 1 object each; G 3 , G 5 , and G 6 are composed of f objects each; and G 4 is composed of τ − 2f − 1 objects. Group G 3 contains only malicious objects. A writer p w1 executes a complete write operation in objects from groups G 3−7 . Then, a correct reader p r1 executes a read operation in groups G 1−5 and G 7 . Malicious objects from G 3 return their correct blocks for value x (but do not log the read operation). Reader p r1 receives exactly τ correct data blocks and obtains the original value x (i.e., an effective read). A second correct reader p r2 executes a read operation in groups G 2−6 . Malicious objects from G 3 do not return their correct blocks for value x (but they do log the read of x by p r2 ). Reader p r2 receives τ − 1 correct data blocks for value x, which is insufficient to recover the original value x (i.e., a non-effective read). After these two reads, any audit quorum Q a receives the same number or more records p r2 , l wx k of a non-effective read than records p r1 , l wx k of an effective read. For instance, an audit quorum composed of objects G 2−6 receives τ + f − 1 records p r2 , l wx k and τ − f − 1 records p r1 , l wx k . Defining l = τ − f − 1 is enough to report the effective read from p r1 , but it violates the strong accuracy by also reporting the non-effective read from p r2 . As a consequence, it is impossible to define a single reporting threshold to be used to produce evidences without violating either completeness or strong accuracy. |G 4 |≥1 |G 1 |=f { } b v { } b v { } b x { } b v {x r1 } b x { } b v malicious |G 2 |=f { } b m { } b m {v r1 } b m |G 3 |=τ-1 { } b v { } b v {v r1 } b v time … a-write(x) ? a-read( ) log {vr1} a-audit( ) σ 1 block (a) malicious |G 4 |=τ-2f-1 |G 2 |=1 |G 3 |=f |G 5 |=f |G 6 |=f |G 7 |=1 |G 1 |=f-1 { } b v { } b m { } b v { } b v { } b v { } b v { } b v { } b v { } b m { } b x { } b x { } b x { } b x { } b v {v r1 } b v { } b m {x r1 } b x {x r1 } b x { } b x {x r1 } b x {v r1 } b v {v r1 , v r2 } b v {x r2 } b m {x r1 , x r2 } b x {x r1 , x r2 } b x {x r2 } b x {x r1 } b x {v r1 } b v time ack a-write(x) x a-read( ) Remark: Adding any number of objects to the scenario of Figure 3b does not change the impossibility result of Theorem 2. The reason is that with the unlimited concurrency in our model, each additional object may have a value v i different from all values stored in the other objects. Audit Algorithm We present a generic auditability algorithm for registers implemented on top of available f -threshold quorum systems to prove that all bounds from the previous section (Lemmata 2-4 and Theorem 1) are tight in our system model. In the present section, we prove that this algorithm satisfies the completeness property with τ ≥ 2f +1 (Lemma 5) and the weak accuracy with ≥ f +1 (Lemma 6). Then, we prove that it supports both completeness and weak accuracy with τ ≥ 3f +1 (Theorem 3). Finally, we prove it supports the strong accuracy property alone with ≥ τ + f (Lemma 7). The implementation for a-audit() is presented in Algorithm 1. It starts with an empty set E that will be used to store the evidences attesting the effective reads (Line 2). It then queries n storage objects to obtain the list of records on their logs (i.e., o k .getLog()), waits the response from at least q of them, and stores these responses in the array of logs L (Lines 4-7). For each previously seen record, it verifies if the number of different logs L[k] containing records for the same reader p r and value v is at least . In doing so, it adds the evidence E pr,v to the reporting set of evidences E (Lines 8-11). After verifying all records, the audit operation returns the set E (Line 14), which is used by auditors to report that the detected readers have effectively read the mentioned data values. Lemma 5. Algorithm 1 satisfies the completeness property of auditable registers with τ ≥ 2f + 1. Proof. Let us assume τ ≥ 2f + 1. Then in this case, based on Corollary 1 (i.e., ≤ τ − 2f ), any audit quorum Q a receives at least one record from a correct, up-to-date storage object that has participated on each effective read (i.e., ≥ 1). Consequently, Algorithm 1 satisfies completeness by obtaining the records from any audit quorum Q a if τ ≥ 2f + 1. Lemma 6. Algorithm 1 satisfies the weak accuracy of auditable registers with ≥ f + 1. Proof. To support the weak accuracy of auditable registers, Algorithm 1 simply needs to make the f records from malicious objects insufficient to create an evidence reporting an effective read if a correct reader has never invoked a read on any correct object. Therefore, ≥ f + 1 ensures that at least one correct object has also received a read request from this client. Theorem 3. Algorithm 1 satisfies both completeness and weak accuracy of auditable registers with τ ≥ 3f + 1. Proof. Assuming τ ≥ 3f + 1 directly satisfies completeness (Lemma 5). Based on Corollary 1 (i.e., ≤ τ − 2f ), any audit quorum Q a receives at least f + 1 records from correct, up-to-date storage objects that have participated on each effective read. As proved in Lemma 6, ≥ f + 1 is enough for Algorithm 1 to satisfy weak accuracy. A consequence of Theorem 3 and Proposition 2 is that existing information dispersal schemes that support fast reads, such as Hendricks et. al [16], require n ≥ 5f + 1 to support auditability. Lemma 7. Algorithm 1 satisfies the strong accuracy of auditable registers with ≥ τ + f . Proof. To support the strong accuracy property of auditable registers, evidences must be created using at least τ records for the same value from correct, up-to-date storage objects that have participate in the read-where τ > f (Proposition 1). Since f malicious servers can also participate in any audit quorum Q a , this number must be added to the minimum number of records required to create an evidence. These two requirements make strong accuracy satisfiable with ≥ τ + f by accessing any audit quorum Q a because, in this case, the f malicious objects make no difference when reporting a value effectively read by a correct reader. Alternative Models for the Algorithm As already proved in Theorem 2, it is impossible to satisfy both completeness and strong accuracy of auditable registers in our model. In this section, we present two modifications on our assumptions that allow Algorithm 1 to overcome the lower bound in Lemma 4 and satisfy both completeness and strong accuracy. Sequential Operations Serializing operations using total order broadcast [8] allows our system to execute operations sequentially. By doing so, we limit the number of different values present in storage objects since high-level read and write operations are executed in total order. In the worst case, the system will have f objects with a correct but stale value and f malicious objects with arbitrary values. With this limitation, strong accuracy becomes easier, and then it can be satisfied together with completeness. Lemma 8. Totally ordering operations in our model allows Algorithm 1 to satisfy strong accuracy with ≥ 2f + 1. Proof. If there is no concurrent operations in the system, there will be always at most f correct stale storage objects and f malicious ones. The worst case is when the f stale objects participate in a read quorum Q r and the f malicious objects mimic them, resulting in 2f records for the read of a stale value. Requiring ≥ 2f + 1 to create an evidence guarantees that it is achieved only when a reader obtains at least 4f + 1 correct blocks from the same data value, which can only happen with the most up-to-date value. Theorem 4. Totally ordering operations in our model allows Algorithm 1 to satisfy both completeness and strong accuracy with τ ≥ 4f + 1. Proof. With τ ≥ 4f +1 and based on Lemma 1, any audit quorum Q a receives at least 2f +1 records from correct, up-to-date storage objects that have participated on each effectively read value v. As proved in Lemma 8, ≥ 2f + 1 is enough to satisfy strong accuracy in our model when there is no concurrency. Moreover, this limitation does not change the lower bound of Lemma 5, which means that τ ≥ 4f + 1 is enough for Algorithm 1 to also satisfy completeness in our model. Non-fast Reads There are read algorithms that use more than one communication round (i.e., a non-fast read [12]) to ensure correct readers will only fetch, from correct objects, the blocks of the most up-to-date value stored in the register. For instance, DepSky-CA [6] is a register emulation in which the first round of a read obtains the identifier of the most up-to-date value available in a quorum of objects, while the second round actually reads the coded blocks for that value. As a consequence, correct objects log reads from correct clients only if they hold the most up-to-date value available in a quorum of objects. Lemma 9. Applying Algorithm 1 to DepSky-CA protocol satisfies strong accuracy with ≥ f +1. Proof. Let us assume registers follow DepSky-CA protocol [6] with two communication rounds in read operations and τ ≥ 3f + 1. No read operation will result in correct records for more than one data value and every read operation results in the creation of records in at least a quorum of objects. In the worst-case scenario, the f malicious storage objects can only forge at most f records for another value. Requiring ≥ f + 1 to create an evidence and report an effective read guarantees that it is achieved only when the reported value was actually read in a quorum of objects. Theorem 5. Applying Algorithm 1 to DepSky-CA protocol satisfies both completeness and strong accuracy with τ ≥ 3f + 1. Proof. Let us assume registers follow DepSky-CA protocol [6] with two communication rounds in read operations. Assuming τ ≥ 3f + 1 and based on Lemma 1, any audit quorum Q a receives at least f + 1 records from correct, up-to-date storage objects that have participated on each effective read. As proved in Lemma 9, ≥ f + 1 is enough to satisfy strong accuracy when using DepSky-CA algorithm. Moreover, this limitation does not change the lower bound of Lemma 5, which means that τ ≥ 3f + 1 is also enough to satisfy completeness. Figure 1 : 1(a) The worst-case scenario of a triple quorum applied to auditable registers. (b) A scenario where τ = 2f does not guarantee completeness. (c) A scenario where = f does not guarantee weak accuracy. Figure 3 : 3(a) A scenario where < τ + f does not guarantee strong accuracy. (b) A generic scenario that proves it is impossible to satisfy both completeness and strong accuracy of auditable registers. (Algorithm 1) The a-audit() algorithm. Byzantine disk paxos: optimal resilience with Byzantine shared memory. Ittai Abraham, V Gregory, Idit Chockler, Dahlia Keidar, Malkhi, Proc. of the 23th Annual ACM Symposium on Principles of Distributed Computing (PODC). of the 23th Annual ACM Symposium on Principles of Distributed Computing (PODC)Ittai Abraham, Gregory V Chockler, Idit Keidar, and Dahlia Malkhi. Byzantine disk paxos: optimal resilience with Byzantine shared memory. In Proc. of the 23th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 226-235, 2004. On using network attached disks as shared memory. Burkhard Marcos K Aguilera, Eli Englert, Gafni, Proc. of the 22nd Annual ACM Symposium on Principles of Distributed Computing (PODC). of the 22nd Annual ACM Symposium on Principles of Distributed Computing (PODC)Marcos K Aguilera, Burkhard Englert, and Eli Gafni. On using network attached disks as shared memory. In Proc. of the 22nd Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 315-324, 2003. Erasure-coded Byzantine storage with separate metadata. Elli Androulaki, Christian Cachin, Dan Dobre, Marko Vukolić, Proc. of the 18th International Conference on Principles of Distributed Systems (OPODIS). of the 18th International Conference on Principles of Distributed Systems (OPODIS)Elli Androulaki, Christian Cachin, Dan Dobre, and Marko Vukolić. Erasure-coded Byzantine storage with separate metadata. In Proc. of the 18th International Conference on Principles of Distributed Systems (OPODIS), pages 76-90, 2014. Sharing memory robustly in message-passing systems. Hagit Attiya, Amotz Bar-Noy, Danny Dolev, Journal of the ACM (JACM). 421Hagit Attiya, Amotz Bar-Noy, and Danny Dolev. Sharing memory robustly in message-passing systems. Journal of the ACM (JACM), 42(1):124-142, 1995. Integrated bounds for disintegrated storage. Alon Berger, Idit Keidar, Alexander Spiegelman, Proc. of the 32nd International Symposium on Distributed Computing (DISC). of the 32nd International Symposium on Distributed Computing (DISC)11Alon Berger, Idit Keidar, and Alexander Spiegelman. Integrated bounds for disintegrated storage. In Proc. of the 32nd International Symposium on Distributed Computing (DISC), pages 11:1-11:18, 2018. Dep-Sky: Dependable and secure storage in cloud-of-clouds. Alysson Bessani, Miguel Correia, Bruno Quaresma, Fernando Andre, Paulo Sousa, ACM Transactions on Storage (TOS). 9433Alysson Bessani, Miguel Correia, Bruno Quaresma, Fernando Andre, and Paulo Sousa. Dep- Sky: Dependable and secure storage in cloud-of-clouds. ACM Transactions on Storage (TOS), 9(4):12:1-12:33, 2013. Information-theoretic lower bounds on the storage cost of shared memory emulation. Zhiying Viveck R Cadambe, Nancy Wang, Lynch, Proc. of the 35th ACM Symposium on Principles of Distributed Computing (PODC). of the 35th ACM Symposium on Principles of Distributed Computing (PODC)Viveck R Cadambe, Zhiying Wang, and Nancy Lynch. Information-theoretic lower bounds on the storage cost of shared memory emulation. In Proc. of the 35th ACM Symposium on Principles of Distributed Computing (PODC), pages 305-313, 2016. Practical Byzantine Fault-Tolerance and Proactive Recovery. M Castro, B Liskov, ACM Transactions on Computer Systems (TOCS). 204M. Castro and B. Liskov. Practical Byzantine Fault-Tolerance and Proactive Recovery. ACM Transactions on Computer Systems (TOCS), 20(4):398-461, 2002. Amnesic distributed storage. Gregory Chockler, Rachid Guerraoui, Idit Keidar, Proc. of the 21st International Symposium on Distributed Computing (DISC). of the 21st International Symposium on Distributed Computing (DISC)Gregory Chockler, Rachid Guerraoui, and Idit Keidar. Amnesic distributed storage. In Proc. of the 21st International Symposium on Distributed Computing (DISC), pages 139-151, 2007. Space complexity of fault-tolerant register emulations. Gregory Chockler, Alexander Spiegelman, Proc. of the 36th ACM Symposium on Principles of Distributed Computing (PODC). of the 36th ACM Symposium on Principles of Distributed Computing (PODC)Gregory Chockler and Alexander Spiegelman. Space complexity of fault-tolerant register emu- lations. In Proc. of the 36th ACM Symposium on Principles of Distributed Computing (PODC), pages 83-92, 2017. on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation). Official Journal of the European Union. 27Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation). Official Journal of the European Union, L119:1-88, 2016. How fast can a very robust read be?. Rachid Guerraoui, Marko Vukolić, Proc. of the 25th Annual ACM Symposium on Principles of Distributed Computing (PODC). of the 25th Annual ACM Symposium on Principles of Distributed Computing (PODC)Rachid Guerraoui and Marko Vukolić. How fast can a very robust read be? In Proc. of the 25th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 248-257, 2006. A case for the accountable cloud. Andreas Haeberlen, ACM SIGOPS Operating Systems Review. 442Andreas Haeberlen. A case for the accountable cloud. ACM SIGOPS Operating Systems Review, 44(2):52-57, 2010. PeerReview: Practical accountability for distributed systems. Andreas Haeberlen, Petr Kouznetsov, Peter Druschel, Proc. of the 21th ACM SIGOPS Symposium on Operating Systems Principles (SOSP). of the 21th ACM SIGOPS Symposium on Operating Systems Principles (SOSP)Andreas Haeberlen, Petr Kouznetsov, and Peter Druschel. PeerReview: Practical account- ability for distributed systems. In Proc. of the 21th ACM SIGOPS Symposium on Operating Systems Principles (SOSP), pages 175-188, 2007. The Snowden files: The inside story of the world's most wanted man. Luke Harding, Guardian Faber PublishingLuke Harding. The Snowden files: The inside story of the world's most wanted man. Guardian Faber Publishing, 2014. Low-overhead Byzantine faulttolerant storage. James Hendricks, Gregory R Ganger, Michael K Reiter, Proc. of the 21st ACM SIGOPS Symposium on Operating Systems Principles (SOSP). of the 21st ACM SIGOPS Symposium on Operating Systems Principles (SOSP)James Hendricks, Gregory R. Ganger, and Michael K. Reiter. Low-overhead Byzantine fault- tolerant storage. In Proc. of the 21st ACM SIGOPS Symposium on Operating Systems Princi- ples (SOSP), pages 73-86, 2007. Impossibility and universality results for wait-free synchronization. Maurice P Herlihy, Proc. of the 7th Annual ACM Symposium on Principles of Distributed Computing (PODC). of the 7th Annual ACM Symposium on Principles of Distributed Computing (PODC)Maurice P. Herlihy. Impossibility and universality results for wait-free synchronization. In Proc. of the 7th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 276-290, 1988. PORs: proofs of retrievability for large files. Ari Juels, S Burton, KaliskiJr, Proc. of the 14th ACM Conference on Computer and Communications Security (CCS). of the 14th ACM Conference on Computer and Communications Security (CCS)Ari Juels and Burton S Kaliski Jr. PORs: proofs of retrievability for large files. In Proc. of the 14th ACM Conference on Computer and Communications Security (CCS), pages 584-597, 2007. Towards achieving accountability, auditability and trust in cloud computing. K L Ryan, Bu Sung Ko, Siani Lee, Pearson, Proc. of the 1st International Conference on Advances in Computing and Communications (ACC). of the 1st International Conference on Advances in Computing and Communications (ACC)Ryan KL Ko, Bu Sung Lee, and Siani Pearson. Towards achieving accountability, auditability and trust in cloud computing. In Proc. of the 1st International Conference on Advances in Computing and Communications (ACC), pages 432-444, 2011. Cloud data auditing techniques with a focus on privacy and security. Manjur Kolhar, M Mosleh, Saied M Abd Abu-Alhaj, El-Atty, IEEE Security & Privacy. 151Manjur Kolhar, Mosleh M Abu-Alhaj, and Saied M Abd El-atty. Cloud data auditing tech- niques with a focus on privacy and security. IEEE Security & Privacy, 15(1):42-51, 2017. Secret sharing made short. Hugo Krawczyk, Proc. of the 13th Annual International Cryptology Conference on Advances in Cryptology (CRYPTO). of the 13th Annual International Cryptology Conference on Advances in Cryptology (CRYPTO)Hugo Krawczyk. Secret sharing made short. In Proc. of the 13th Annual International Cryp- tology Conference on Advances in Cryptology (CRYPTO), pages 136-146, 1993. . Leslie Lamport. On interprocess communication. Distributed Computing. 12Leslie Lamport. On interprocess communication. Distributed Computing, 1(2):86-101, 1986. The Byzantine generals problem. Leslie Lamport, Robert Shostak, Marshall Pease, ACM Transactions on Programming Languages and Systems (TOPLAS). 43Leslie Lamport, Robert Shostak, and Marshall Pease. The Byzantine generals problem. ACM Transactions on Programming Languages and Systems (TOPLAS), 4(3):382-401, 1982. Byzantine quorum systems. Dahlia Malkhi, Michael Reiter, Distributed Computing. 114Dahlia Malkhi and Michael Reiter. Byzantine quorum systems. Distributed Computing, 11(4):203-213, 1998. Enabling security in cloud storage SLAs with CloudProof. Ada Raluca, Jacob R Popa, David Lorch, Helen J Molnar, Li Wang, Zhuang, Proc. of the USENIX Annual Technical Conference (ATC). of the USENIX Annual Technical Conference (ATC)Raluca Ada Popa, Jacob R Lorch, David Molnar, Helen J Wang, and Li Zhuang. Enabling security in cloud storage SLAs with CloudProof. In Proc. of the USENIX Annual Technical Conference (ATC), pages 355-368, 2011. Efficient dispersal of information for security, load balancing, and fault tolerance. Michael Rabin, Journal of the ACM (JACM). 362Michael Rabin. Efficient dispersal of information for security, load balancing, and fault toler- ance. Journal of the ACM (JACM), 36(2):335-348, 1989. Effective erasure codes for reliable computer communication protocols. Luigi Rizzo, ACM SIGCOMM Computer Communication Review (CCR). 272Luigi Rizzo. Effective erasure codes for reliable computer communication protocols. ACM SIGCOMM Computer Communication Review (CCR), 27(2):24-36, 1997. The 21 biggest data breaches of the 21st century. Optimum Security, Optimum Security. The 21 biggest data breaches of the 21st century. https:// optimumsecurity.ca/21-biggest-data-breach-of-21-century, 2019. How to share a secret. Adi Shamir, Communications of the ACM (CACM). 2211Adi Shamir. How to share a secret. Communications of the ACM (CACM), 22(11):612-613, 1979. On cloud storage and the cloud of clouds approach. Daniel Slamanig, Christian Hanser, Proc. of the 7th International Conference for Internet Technology And Secured Transactions (ICITST). of the 7th International Conference for Internet Technology And Secured Transactions (ICITST)Daniel Slamanig and Christian Hanser. On cloud storage and the cloud of clouds approach. In Proc. of the 7th International Conference for Internet Technology And Secured Transactions (ICITST), pages 649-655, 2012. Space bounds for reliable storage: Fundamental limits of coding. Alexander Spiegelman, Yuval Cassuto, Gregory Chockler, Idit Keidar, Proc. of the 35th ACM Symposium on Principles of Distributed Computing (PODC). of the 35th ACM Symposium on Principles of Distributed Computing (PODC)Alexander Spiegelman, Yuval Cassuto, Gregory Chockler, and Idit Keidar. Space bounds for reliable storage: Fundamental limits of coding. In Proc. of the 35th ACM Symposium on Principles of Distributed Computing (PODC), pages 249-258, 2016. Iris: A scalable cloud file system with efficient integrity checks. Emil Stefanov, Marten Van Dijk, Ari Juels, Alina Oprea, Proc. of the 28th ACM Annual Computer Security Applications Conference (ACSAC). of the 28th ACM Annual Computer Security Applications Conference (ACSAC)Emil Stefanov, Marten van Dijk, Ari Juels, and Alina Oprea. Iris: A scalable cloud file system with efficient integrity checks. In Proc. of the 28th ACM Annual Computer Security Applications Conference (ACSAC), pages 229-238, 2012. Security and privacy challenges in cloud computing environments. Hassan Takabi, B D James, Gail-Joon Joshi, Ahn, IEEE Security & Privacy. 86Hassan Takabi, James BD Joshi, and Gail-Joon Ahn. Security and privacy challenges in cloud computing environments. IEEE Security & Privacy, 8(6):24-31, 2010. Privacy-preserving public auditing for secure cloud storage. Cong Wang, S M Sherman, Qian Chow, Kui Wang, Wenjing Ren, Lou, IEEE Transactions on Computers (TC). 622Cong Wang, Sherman SM Chow, Qian Wang, Kui Ren, and Wenjing Lou. Privacy-preserving public auditing for secure cloud storage. IEEE Transactions on Computers (TC), 62(2):362- 375, 2013. Privacy-preserving public auditing for data storage security in cloud computing. Cong Wang, Qian Wang, Kui Ren, Wenjing Lou, Proc. of the IEEE INFOCOM. of the IEEE INFOCOMCong Wang, Qian Wang, Kui Ren, and Wenjing Lou. Privacy-preserving public auditing for data storage security in cloud computing. In Proc. of the IEEE INFOCOM, pages 1-9, 2010. Enabling public auditability and data dynamics for storage security in cloud computing. Qian Wang, Cong Wang, Kui Ren, Wenjing Lou, Jin Li, IEEE Transactions on Parallel and Distributed Systems (TPDS). 22Qian Wang, Cong Wang, Kui Ren, Wenjing Lou, and Jin Li. Enabling public auditability and data dynamics for storage security in cloud computing. IEEE Transactions on Parallel and Distributed Systems (TPDS), 22(5):847-859, 2011. The role of accountability in dependable distributed systems. R Aydan, Jeffrey S Yumerefendi, Chase, 1st Workshop on Hot Topics in System Dependability (HotDep). Aydan R Yumerefendi and Jeffrey S Chase. The role of accountability in dependable distributed systems. In 1st Workshop on Hot Topics in System Dependability (HotDep), 2005.
[]
[ "A General Formula for Black Hole Gravitational Wave Kicks", "A General Formula for Black Hole Gravitational Wave Kicks" ]
[ "James R Van Meter [email protected] ", "M Coleman Miller ", "John G Baker ", "William D Boggs ", "Bernard J Kelly " ]
[]
[]
Although the gravitational wave kick velocity in the orbital plane of coalescing black holes has been understood for some time, apparently conflicting formulae have been proposed for the dominant out-of-plane kick, each a good fit to different data sets. This is important to resolve because it is only the out-of-plane kicks that can reach more than 500 km s −1 and can thus eject merged remnants from galaxies. Using a different ansatz for the out-of-plane kick, we show that we can fit almost all existing data to better than 5%. This is good enough for any astrophysical calculation, and shows that the previous apparent conflict was only because the two data sets explored different aspects of the kick parameter space.
10.1088/0004-637x/719/2/1427
[ "https://arxiv.org/pdf/1003.3865v1.pdf" ]
981,860
1003.3865
3c21593c95952cdfbb5036cbb96e505a9089228b
A General Formula for Black Hole Gravitational Wave Kicks 19 Mar 2010 James R Van Meter [email protected] M Coleman Miller John G Baker William D Boggs Bernard J Kelly A General Formula for Black Hole Gravitational Wave Kicks 19 Mar 2010Subject headings: black hole physics -galaxies: nuclei -gravitational waves Although the gravitational wave kick velocity in the orbital plane of coalescing black holes has been understood for some time, apparently conflicting formulae have been proposed for the dominant out-of-plane kick, each a good fit to different data sets. This is important to resolve because it is only the out-of-plane kicks that can reach more than 500 km s −1 and can thus eject merged remnants from galaxies. Using a different ansatz for the out-of-plane kick, we show that we can fit almost all existing data to better than 5%. This is good enough for any astrophysical calculation, and shows that the previous apparent conflict was only because the two data sets explored different aspects of the kick parameter space. (2004)). As discussed in Merritt et al. (2004), the kick magnitude and distribution are important for discussions of hierarchical merging, supermassive black hole formation, galactic nuclear dynamics, and the degree to which black holes influence galaxy formation. The community has converged on the formula for the kick speed in the original orbital plane (Baker et al. 2007;Campanelli et al. 2007b;Gonzalez et al. 2007), but apparently conflicting dependences for the out-of-plane kick (which dominates the total kick for most configurations) have been proposed. suggested that for a binary with component masses m 1 and m 2 ≥ m 1 , the kick scales as η 2 , where η ≡ m 1 m 2 /(m 1 + m 2 ) 2 is the symmetric mass ratio; however the data in Baker et al. (2008) were fit much better with an η 3 dependence. The conflict is only apparent, however, because there were no runs in common between the two data sets. This suggests that an analysis might be performed with a new ansatz that can fit all of the existing data. Here we perform such a fit, and demonstrate that there is a single formula for the outof-plane kick that fits almost all existing data to better than 5% accuracy. The new ansatz is similar to one recently suggested by , for which, however, a fit was not attempted. Its form is based straightforwardly on the post-Newtonian (PN) approximation and includes both the aforementioned η 2 and η 3 terms, as well as a slightly more complicated spin-angle dependence. In § 2 we describe our new runs and we describe the new ansatz in § 3. In § 4 we list all 95 runs we have fit, from different numerical relativity groups, and our fitting procedure and best-fit parameters. In § 5 we discuss the implications of our results, and indicate the fraction of kicks above 500 km s −1 and 1,000 km s −1 for representative spin and mass ratio distributions, comparing it with previous results. The goodness of our fit to the entire usable data set of out-of-plane kicks suggests that the full three-dimensional kick is now modelled well enough that it will not limit the accuracy of any astrophysical calculation. Numerical simulations We have performed new simulations representing 22 distinct physical cases. Defining q ≡ m 1 /m 2 and α i ≡ S i /m 2 i where S i is the spin angular momentum of the ith black hole, we used mass ratios of q = 0.674 or q = 0.515, and spins initially within the orbital plane of α 1 = α ⊥ 1 = 0.367 and α 2 = α ⊥ 1 = 0.177 or α 2 = α ⊥ 1 = 0.236, where the ⊥ superscript indicates orthogonality to the orbital axis. For each mass ratio we used one of eleven different spin orientations (given in Table 1), in order to probe the spin-angle dependence of the final recoil. Initial parameters were informed by a quasicircular PN approximation and initial data constructed using the spectral solver TwoPunctures (Ansorg et al. 2004). Evolutions were performed with the Einstein-solver Hahndol (Imbiriba et al. 2004;van Meter et al. 2006;Baker et al. 2007), with all finite differencing and interpolation at least fifth-order-accurate in computational grid spacing. Note for the purpose of characterizing the simulations, it is convenient to use units in which G = c = 1 and specify distance and time in terms of M ≡ m 1 + m 2 . The radiation field represented by the Weyl scalar Ψ 4 was interpolated to an extraction sphere at coordinate radius r = 50M and integrated to obtain the radiated momentum using the standard formula (Schnittman et al. 2008): P i = t −∞ dt r 2 16π dΩ x i r t −∞ dtΨ 4 2 . (1) For each of the 22 cases we ran two resolutions, with fine-grid spacings of h f = 3M/160 and h f = M/64. To give some indication of our numerical error, the total recoil from the two resolutions for each of the q = 0.674 cases agreed to within < ∼ 6%, and for each of the q = 0.515 cases to within 1% (the latter using a more optimal grid structure). In these simulations, variation in the magnitude of the final recoil within the x-y plane suggested non-negligible precession of the orbital plane. Indeed the coordinate trajectories of the black holes showed precession of up to ∼ 10 • . To calculate the component of the recoil velocity parallel to the "final" orbital axis, V || , we tried two different methods. In one method we simply took the dot product of the final, numerically computed recoil velocity V with the normalized orbital angular momentum L|L| −1 as calculated from the coordinate trajectories of the black holes, just when the common apparent horizon was found: V || ≡ V · L|L| −1 . In our second method we assumed that each black hole spin, which is initially orthogonal to the orbital angular momentum, remained approximately orthogonal throughout the simulation, i.e. L · S 1 ≈ L · S 2 ≈ 0. This assumption is consistent with PN calculations, to linear order in spin (Racine 2008;Kidder 1995), and is also supported numerically by the fact that the merger times in our simulations were independent of the initial spin, to within ∆t < 1M. In this case the in-plane recoil should depend only on the mass ratio (and not the spin), and we assume it is given by the formula found by Gonzalez et al. (2007), with coefficient values given by a previous fit (Baker et al. 2008). This implies V || ≡ V 2 − (V pred ⊥ ) 2 V z |V z | ,(3) where we have further assumed that the sign of V || should be the same as that of V z , given the modest amount of precession. These two definitions for V || were found to differ by a relative error of < ∼ 5% for all points except one for which V || ≪ V ⊥ . Relative to the maximum V || per mass ratio, they were found to differ only by < 2%. Note this lends further support to our assumption that the spins are orthogonal to the orbital angular momentum to good approximation throughout these simulations (Eq. (2)). Even V z was found to differ from the above definitions for V || by only < ∼ 2%, relative to the maximum. We will use Eq. (3) to give our canonical V || , for the purpose of analytic fits to the data. An additional assumption we will make about our data, important for fitting purposes, is that the amount of precession undergone by the spins in the orbital plane, i.e. the difference in spin-angles, between the initial data and the merger, is independent of the initial spin orientation. This assumption, valid to linear order in spin according to the PN approximation (Eq. (2.4) of (Kidder 1995)), was previously found to be the case through explicit computation of the spins in the simulations presented in Baker et al. (2008). We have not explicitly computed the spins in the new simulations presented here but the independence of the merger time with respect to initial spin orientation is consistent with the assumption of similar independence of spin precession. For the purpose of constructing an accurate phenomenological model, we added to this data set previously published data representing out-of-plane kicks. Our criteria for selecting relevant data are that the component of the final recoil parallel to the orbital plane is given and at least three different in-plane spin orientations are used. We arrived at a total of 95 data points: Table III, resulting from what they consider to be their best calculation of the final orbital plane. Ansatz An ansatz for the total recoil that was found to be very consistent with numerical results has the form (Baker et al. 2007;Campanelli et al. 2007b;Gonzalez et al. 2007) V = V ⊥m e 1 + V ⊥s (cos ξ e 1 + sin ξ e 2 ) + V e 3 ,(4)V ⊥m = Aη 2 1 − 4η(1 + Bη),(5)V ⊥s = H η 2 (1 + q) α 2 − qα 1 ,(6) where e 1 and e 3 are unit vectors in the directions of separation and the orbital axis just before merger, respectively, e 2 ≡ e 1 × e 3 , α 1 and α 2 represent the components of spins parallel to the orbital axis, ξ, A, B, and H are constant fitting parameters, and V is to be discussed. Every component of Eq. (4) has been modeled to some degree after PN expressions. The use of such PN-based formulae has proven very successful. For example, Eq. (5) for the in-plane, mass-ratio-determined recoil can be obtained from the corresponding PN expression (Eq. (23) of Blanchet et al. (2005)) simply by taking the leading-order terms and replacing instances of the PN expansion parameter (in this case frequency) with constant fitting parameters. Using this, Gonzalez et al. (2007) obtained very good agreement with a large set of numerical results. Le Tiec et al. (2010), who calculated the recoil using a combination of the PN method with a perturbative approximation of the ringdown, also found that Eq. (5) gave a good phenomenological fit to their analytic results. Why such a prescription for generating an ansatz from the PN approximation should apply so well to merger dynamics is not perfectly understood. The effective replacement of powers of the frequency with constants may be defensable because the majority of the recoil is generated within a narrow time-window near merger, perhaps within a narrow range of frequencies. This is particularly evident for the out-of-plane recoil speed, which rapidly and monotonically increases up to a constant value around merger. However, values of the constant coefficients that appear in the PN expansion cannot be expected to remain unchanged because, as merger is approached, high-order PN terms with the same functional dependence on mass and spin as leading order terms can become comparable in magnitude. So, it is just for this functional dependence that we look to the PN approximation for guidance. Following the prescription implied above, from the PN expression for out-of-plane recoil given by Racine et al. (2009), Eqs. (4.40-4.42) (neglecting spin-spin interaction), the following can be straightforwardly obtained: V || = K 2 η 2 + K 3 η 3 q + 1 qα ⊥ 1 cos(φ 1 − Φ 1 ) − α ⊥ 2 cos(φ 2 − Φ 2 ) + K S (q − 1)η 2 (q + 1) 3 q 2 α ⊥ 1 cos(φ 1 − Φ 1 ) + α ⊥ 2 cos(φ 2 − Φ 2 ) ,(7) where K 2 , K 3 , and K S are constants, α ⊥ i represents the magnitude of the projection of the ith black hole's spin (divided by the square of the black hole's mass) into the orbital plane, φ i represents the angle of the same projection, as measured at some point before merger, with respect to a reference angle representing the direction of separation of the black holes, and Φ i represents the amount by which this angle precesses before merger, which depends on the mass ratio and the initial separation. We have ignored terms quadratic in the spin because we assume they are subleading. Note that this ansatz is equivalent to one suggested by , provided the angular parameters are suitably interpreted. In terms of the notation of Racine et al. (2009), (q + 1) −1 qα ⊥ 1 cos(φ 1 − Φ 1 ) − α ⊥ 2 cos(φ 2 − Φ 2 ) = −M −1 ∆ · n = −M −1 ∆ ⊥ cos(Θ),(8) (q + 1) −2 q 2 α ⊥ 1 cos(φ 1 − Φ 1 ) + α ⊥ 2 cos(φ 2 − Φ 2 ) = M −2 S · n = M −2 S ⊥ cos(Ψ),(9) where n is a unit separation vector, ∆ ≡ S 2 /m 2 − S 1 /m 1 , S ≡ S 1 + S 2 , ∆ ⊥ ≡ M(q + 1) −1 |α ⊥ 2 − qα ⊥ 1 |, S ⊥ ≡ M 2 (q + 1) −2 |α ⊥ 2 + q 2 α ⊥ 1 |, Θ is the angle between ∆ and n, and Ψ is the angle between S and n, all measured, for our purposes, at some point arbitrarily close to merger. It may be interesting to note that the expression multiplying K S takes into account effects of orbital precession (because it is non-vanishing if and only if the orbit precesses) and therefore represents physical phenomena neglected by previous fits. Fitting procedure and results The fitting of the out-of-plane recoil is in principle complicated because in addition to the overall factors K 2 , K 3 , and K S (which are the same for any mass ratio or spins), any particular set of runs with the same initial separation and mass ratio (which we will term a "block") has idiosyncratic values of Φ 1 and Φ 2 that, although not fundamentally interesting, need to be fit to the data. Therefore, in the 17 blocks of data we fit, there are formally 3 + 2 × 17 = 37 fit parameters. In the eight data blocks for which α ⊥ 1 = 0, Φ 1 never enters, and in the two for which α ⊥ 1 = α ⊥ 2 , Φ 1 = Φ 2 . Therefore the actual number of fitting parameters is 27, but this is still large enough that a multiparameter fit would be challenging. Fortunately, each (Φ 1 , Φ 2 ) pair only affects a single data block. We can therefore speed up the fitting, and incidentally concentrate on only the interesting parameters, if we (1) pick some values of K 2 , K 3 , and K S that apply to all data blocks, then (2) for each data block, find the values of Φ 2 and possibly Φ 1 that optimize the fit, repeating this using new values of Φ 1 and Φ 2 for each block. This gives an overall fit for the assumed values of K 2 , K 3 , and K S , having optimized over the uninteresting Φ parameters. The fit itself needs to be performed assuming uncertainties on each of the numerical measurements of the kick. Each such calculation is computationally expensive and systematic errors are usually difficult to quantify, hence we do not have enough information to do a true fit. As a substitute, we assume that for each block of data, the uncertainty σ in each kick is either equal to a fraction (fixed for all blocks) of the maximum magnitude kick in the block, or to a fixed fraction of the individual kick itself. The former may be justified because some sources of error will be independent of the phase of the angles when the holes merge, but we note that the fit performed with the latter assumption (that the uncertainty equals a fractional error of each kick) yields very similar values for the fitting factors. As we do not know what the actual fractional error is, in either case we adjust it so that for our best fit we get a reduced χ 2 of roughly unity (given our 95 data points and 27 fitting variables, this means we need a total χ 2 of about 68). We then evaluate every (K 2 , K 3 , K S ) triplet using χ 2 = (pred − kick) 2 /σ 2 . Minimizing χ 2 as calculated with respect to the maximum kick per block, we find that σ 2 = 0.0005V 2 ,max (block) gives χ 2 /dof = 1.0. We note that this value for σ is comparable to the numerical error as measured by the difference in kicks computed at different resolutions, when available (e.g. for the new simulations presented here, or the q = 0.25 case presented in Table IV of ). It is also worth noting that for the data from the new simulations, the uncertainty due to orbital precession, discussed in § 2, is less than 0.02V ,max < σ, and therefore cannot significantly affect the fit. Our best fit is K 2 = 30, 540 km s −1 , K 3 = 115, 800 km s −1 , and K S = 17, 560 km s −1 , with 1σ ranges 28, 900 − 32, 550 km s −1 , 107, 300 − 121, 900 km s −1 , and 15, 900 − 19, 000 km s −1 , respectively. We emphasize that all three coefficients are indispensable in obtaining a good fit; e.g. using the same definition of χ 2 as above, if K 3 = 0 then the best fit gives χ 2 /dof = 2.0, or if K S = 0 then the best fit gives χ 2 /dof = 1.6. Minimizing χ 2 as calculated with respect to each individual kick, we obtain similar results. In this case, σ 2 = 0.0016V 2 gives χ 2 /dof = 1.0. Our best fit becomes K 2 = 32, 092 km s −1 , K 3 = 108, 897 km s −1 , and K S = 15, 375 km s −1 , in agreement with the above fit to within ∼ 5% for K 2 and K 3 and < ∼ 12% for K S . In Table 4 we compare the predicted out-of-plane kicks with the measured ones for the entire data set. For fit #1 minimizing the error with respect to the maximum kick per block, of the 95 points, only 4 agree to worse than 10%, and all of those occur for kicks with magnitudes much less than the maximum in their data block. Therefore these could represent small phase errors rather than relatively large fractional velocity errors. Only 15 of the 95 points agree to worse than 5%. Fit #2 minimizing the error with respect to individual kicks performs even better in this regard, with only 10 points differing by more than 5% and only 2 points differing by more than 10%. In either case, the fits with the new ansatz are significantly better than either the simpler η 2 fit proposed by or the η 3 fit proposed by Baker et al. (2008). Campanelli et al. (2007b) (set E). The spin angles φ 1 and φ 2 are in radians, and the recoil velocities V || are in km s −1 . The 8th column shows the result of a fit intended to minimize the error relative to the maximum recoil velocity per (q, α ⊥ 1 , α ⊥ 2 ) triplet and the 9th column shows the relative error from that fit. The 10th column shows the result of a fit intended to minimize the conventional relative error and the 11th column shows the relative error from that fit. In both cases, the vast majority of points agree to well within 10%. set q α ⊥ 1 α ⊥ 2 φ 1 φ 2 num. V || fit V || #1 |∆V /V | fit V || #2 |∆V /V | Aset q α ⊥ 1 α ⊥ 2 φ 1 φ 2 num. V || fit V || #1 |∆V /V | fit V || #2 |∆V /V | Aset q α ⊥ 1 α ⊥ 2 φ 1 φ 2 num. V || fit V || #1 |∆V /V | fit V || #2 |∆V /V | C Ejection probabilities and discussion One of the most important outputs of kick calculations and fits is the probability distribution of kicks given assumptions about the mass ratio, spin magnitudes, and spin directions. This distribution is critical to studies of hierarchical merging in the early universe (e.g., Volonteri (2007)) as well as to the gas within galaxies (Devecchi et al. 2009) and an evaluation of the prospects for growth of intermediate-mass black holes in globular clusters (Holley-Bockelmann et al. 2008). In Table 2 we show the results of our work (from fit #1), compared with the proposed fit formula of Campanelli et al. (2007a). It is clear that our work gives distributions very close to those of Campanelli et al. (2007a), with perhaps slightly smaller kicks because of the η 3 term we include. In summary, we have demonstrated that a modified formula fits all available out-of-plane kicks extremely well. The wide range of mass ratios, spin magnitudes, and angles explores all the major aspects of parameter space for the out-of-plane kicks, and thus we do not expect new results to deviate significantly from our formula. The excellence of these fits suggests that the kick distribution is known to an accuracy that is sufficient for any astrophysical purpose. New simulations used for this work were performed on Jaguar at Oak Ridge National Laboratory. MCM acknowledges partial support from the National Science Foundation under grant AST 06-07428 and NASA ATP grant NNX08AH29G. The work at Goddard was supported in part by NASA grant 06-BEFS06-19. We also wish to think S. McWilliams and A. Buonanno for helpful discussions. Mass ratio and spin Speed threshold CLZM This work 1/10 ≤ q ≤ 1, α 1 = α 2 = 0.9 v > 500km s −1 0.364±0.0048 0.342526±0.00019 v > 1000km s −1 0.127±0.0034 0.120974±0.00011 1/4 ≤ q ≤ 1, α 1 = α 2 = 0.9 v > 500km s −1 0.699 ± 0.0045 0.697818±0.00026 v > 1000km s −1 0.364±0.0046 0.353393±0.00019 1/4 ≤ q ≤ 1, 0 ≤ α 1 , α 2 ≤ 1 v > 500km s −1 0.428±0.0045 0.415915±0.00020 v > 1000km s −1 0.142±0.0034 0.134615±0.00012 the 22 new ones presented here plus 73 drawn from Baker et al. (2008); Campanelli et al. (2007b); Dain et al. (2008); Lousto & Zlochower (2009). Note that for the cases in Lousto & Zlochower (2009) we use the second version of their values listed in their Table 1 . 1Recoil data from Lousto & Zlochower (2009) (set A), Baker et al.(2008) (set B), Table 1 - 1Continued num. V || fit V || #1 |∆V /V | fit V || #2 |∆V /V |Table 2. Fraction of kick speeds above a given threshold, compared with the results ofCampanelli et al. (2007a) (CLZM). In all cases we assume an isotropic distribution of spin orientations.set q α ⊥ 1 α ⊥ 2 φ 1 φ 2 E 1.000 0.515 0.515 3.304 0.162 47.0 46.3 0.014 46.2 0.017 E 1.000 0.515 0.515 0.000 3.142 -351.0 -354.4 0.010 -353.3 0.007 . M Ansorg, B Brügmann, W Tichy, Phys. Rev. D. 7064011Ansorg, M., Brügmann, B., & Tichy, W. 2004, Phys. Rev. D, 70, 064011 . J G Baker, W D Boggs, J M Centrella, B J Kelly, S T Mcwilliams, M C Miller, J R Van Meter, Astrophys. J. 6681140Baker, J. G., Boggs, W. D., Centrella, J. M., Kelly, B. J., McWilliams, S. T., Miller, M. C., & van Meter, J. R. 2007, Astrophys. J., 668, 1140 . Astrophys. J. 68229-. 2008, Astrophys. J., 682, L29 . L Blanchet, M S S Qusailah, C M Will, Astrophys. J. 635508Blanchet, L., Qusailah, M. S. S., & Will, C. M. 2005, Astrophys. J., 635, 508 . M Campanelli, C O Lousto, Y Zlochower, D Merritt, Phys. Rev. Lett. 659231102Astrophys. J.Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007a, Astrophys. J., 659, L5 -. 2007b, Phys. Rev. Lett., 98, 231102 . S Dain, C O Lousto, Y Zlochower, Phys. Rev. D. 7824039Dain, S., Lousto, C. O., & Zlochower, Y. 2008, Phys. Rev. D, 78, 024039 . B Devecchi, E Rasia, M Dotti, M Volonteri, M Colpi, Mon. Not. R. Astron. Soc. 394633Devecchi, B., Rasia, E., Dotti, M., Volonteri, M., & Colpi, M. 2009, Mon. Not. R. Astron. Soc., 394, 633 . J A Gonzalez, U Sperhake, B Brügmann, M D Hannam, S Husa, Phys. Rev. Lett. 9891101Gonzalez, J. A., Sperhake, U., Brügmann, B., Hannam, M. D., & Husa, S. 2007, Phys. Rev. Lett., 98, 091101 . K Holley-Bockelmann, K Gültekin, D M Shoemaker, N Yunes, Astrophys. J. 686829Holley-Bockelmann, K., Gültekin, K., Shoemaker, D. M., & Yunes, N. 2008, Astrophys. J., 686, 829 . B Imbiriba, J G Baker, D.-I Choi, J M Centrella, D R Fiske, J D Brown, J R Van Meter, K Olson, Phys. Rev. D. 70124025Imbiriba, B., Baker, J. G., Choi, D.-I., Centrella, J. M., Fiske, D. R., Brown, J. D., van Meter, J. R., & Olson, K. 2004, Phys. Rev. D, 70, 124025 . L E Kidder, Phys. Rev. D. 52821Kidder, L. E. 1995, Phys. Rev. D, 52, 821 . Le Tiec, A Blanchet, L Will, C M , Class. Quantum Grav. 2712001Le Tiec, A., Blanchet, L., & Will, C. M. 2010, Class. Quantum Grav., 27, 012001 . C O Lousto, M Campanelli, Y Zlochower, arXiv:0904.3541gr-qcLousto, C. O., Campanelli, M., & Zlochower, Y. 2009, arXiv:0904.3541 [gr-qc] . C O Lousto, Y Zlochower, Phys. Rev. D. 7964018Lousto, C. O., & Zlochower, Y. 2009, Phys. Rev. D, 79, 064018 . D Merritt, M Milosavljevic, M Favata, S A Hughes, D E Holz, Astrophys. J. 6079Merritt, D., Milosavljevic, M., Favata, M., Hughes, S. A., & Holz, D. E. 2004, Astrophys. J., 607, L9 . E Racine, Phys. Rev. D. 7844021Racine, E. 2008, Phys. Rev. D, 78, 044021 . E Racine, A Buonanno, L E Kidder, Phys. Rev. D. 8044010Racine, E., Buonanno, A., & Kidder, L. E. 2009, Phys. Rev. D, 80, 044010 . J D Schnittman, A Buonanno, J R Van Meter, J G Baker, W D Boggs, J M Centrella, B J Kelly, S T Mcwilliams, Phys. Rev. D. 7744031Schnittman, J. D., Buonanno, A., van Meter, J. R., Baker, J. G., Boggs, W. D., Centrella, J. M., Kelly, B. J., & McWilliams, S. T. 2008, Phys. Rev. D, 77, 044031 . J R Van Meter, J G Baker, M Koppitz, D.-I Choi, Phys. Rev. D. 73124011van Meter, J. R., Baker, J. G., Koppitz, M., & Choi, D.-I. 2006, Phys. Rev. D, 73, 124011 . M Volonteri, Astrophys. J. 6635Volonteri, M. 2007, Astrophys. J., 663, L5
[]
[ "Effects of weather and policy intervention on COVID-19 infection in Ghana", "Effects of weather and policy intervention on COVID-19 infection in Ghana" ]
[ "Iddrisu Wahab Abdul \nDepartment of Mathematics and Statistics\nUniversity of Energy and Natural Resources\nSunyaniGhana\n", "Peter Appiahene \nDepartment of Computer Science and Informatics\nUniversity of Energy and Natural Resources\nSunyaniGhana\n", "Justice A Kessie \nDepartment of Mathematics and Statistics\nUniversity of Energy and Natural Resources\nSunyaniGhana\n" ]
[ "Department of Mathematics and Statistics\nUniversity of Energy and Natural Resources\nSunyaniGhana", "Department of Computer Science and Informatics\nUniversity of Energy and Natural Resources\nSunyaniGhana", "Department of Mathematics and Statistics\nUniversity of Energy and Natural Resources\nSunyaniGhana" ]
[]
As the number of COVID-19 cases continues to surge and the disease continues to wreak more havoc globally, new revelations concerning the spread and transmission of the virus continue to emerge from research every day. Even though laboratory and epidemiological studies have demonstrated the effects of ambient temperature on the transmission and survival of coronaviruses, not much has been done on the effects of weather on the spread of COVID-19. This study investigates the effects of temperature, humidity, precipitation, wind speed and the specific government policy intervention of partial lockdown on the new cases of Prediction of Worldwide Energy Resources (POWER) project. Considering the nature of the data and the objectives of the study, a time series generalized linear model which allows for regressing on past observations of the response variable and covariates was used for model fitting. The results indicate significant effects of maximum temperature, relative humidity and precipitation in predicting new cases of the disease. Also, results of the intervention analysis indicate that the null hypothesis of no significant effect of the specific policy intervention of partial lockdown should be rejected (p-value=0.0164) at a 5% level of significance. These findings provide useful insights for policymakers and the public.
null
[ "https://arxiv.org/pdf/2005.00106v1.pdf" ]
218,470,244
2005.00106
3bc8ab111a155637ba06781f1a0ef6c606d4f30f
Effects of weather and policy intervention on COVID-19 infection in Ghana April 28, 2020 Iddrisu Wahab Abdul Department of Mathematics and Statistics University of Energy and Natural Resources SunyaniGhana Peter Appiahene Department of Computer Science and Informatics University of Energy and Natural Resources SunyaniGhana Justice A Kessie Department of Mathematics and Statistics University of Energy and Natural Resources SunyaniGhana Effects of weather and policy intervention on COVID-19 infection in Ghana April 28, 2020*Correspondence to: [email protected] COVID-19 infection in Ghana. Daily data on confirmed cases of COVID-19 from March 13, 2020 to April 21, 2020 were obtained from the official website of Our World in Data (OWID) dedicated to COVID-19 while satellite climate data for the same period was obtained from the official website of the National Aeronautics and Space Administration's (NASA's) As the number of COVID-19 cases continues to surge and the disease continues to wreak more havoc globally, new revelations concerning the spread and transmission of the virus continue to emerge from research every day. Even though laboratory and epidemiological studies have demonstrated the effects of ambient temperature on the transmission and survival of coronaviruses, not much has been done on the effects of weather on the spread of COVID-19. This study investigates the effects of temperature, humidity, precipitation, wind speed and the specific government policy intervention of partial lockdown on the new cases of Prediction of Worldwide Energy Resources (POWER) project. Considering the nature of the data and the objectives of the study, a time series generalized linear model which allows for regressing on past observations of the response variable and covariates was used for model fitting. The results indicate significant effects of maximum temperature, relative humidity and precipitation in predicting new cases of the disease. Also, results of the intervention analysis indicate that the null hypothesis of no significant effect of the specific policy intervention of partial lockdown should be rejected (p-value=0.0164) at a 5% level of significance. These findings provide useful insights for policymakers and the public. Introduction COVID-19 continues to wreak more havoc globally [1], [2]. As the number of infections keep surging high, deaths continue to increase at no small rate, active cases continue to scare the world [3], and a possible vaccine invention seems to be several months (if not years) away from now (though we do not lack the expertise, the technology and funding for such an expedition), and its treatment the world is quite unsure, leaving us all in horrendous fear. This pandemic has touched everything and activity in this world and has sent the world experts in health and data science to work to find possible solutions to this pandemic, while the experts in economics and policy makers have concerned themselves with the alleviation of the impact of COVID-19 on their nations' economy, its people and their businesses. Researchers have investigated the role of pre-existing conditions such as aging, hypertension, etc. on COVID-19 deaths [4], [5], environmental factors that influences COVID-19 deaths [5], [6] and the role of climate on COVID-19 related deaths and new confirmed cases [7], [8]. The relationships between climate and infectious diseases have long been established as seasonal variations and climate sensitivities are well known to explain some variability in many infectious diseases. Statistical, process-based, and landscape models are the key categories of models for the estimation of potential climate effects on infectious diseases. These three model categories tackle very different problems. Statistical models for instance, require that the relationship between the current geographic spread of the disease and actual localized climatespecific conditions be inferred empirically [9]. Hitherto, there have been the confirmations that ultraviolet light has a sterilizing effect, because the radiation damages the virus's genetic material and its ability to replicate [10], [11]. This was first put out by [10], whose findings informed us that temperature and irradiance of COVID-19 will decide the trajectory of the pandemic at warmer regions as well as whether rising temperatures will change direction and have consequences on public health policy. They also showed that case and death counts at higher temperatures (>14 °C) when aligned for the epidemic stage had significantly lower rates of growth. However, according to [8], in warmweather locations such as Singapore, Malaysia, and Thailand, COVID-19 have also proven lethal and pose wider concerns on the environmental factors. [12] studied the association between temperature and number of new infections in 122 cities in China and made some important revelations including the fact that temperature less than 3 °C was positively linked to newly confirmed COVID-19 cases. Another work done by [13] on the "role of temperature and humidity in the modulation of the doubling time of COVID-19 cases" affirms what was revealed by [12] and [10]. Their conclusion was that, temperature and humidity explain a total of 18% variability in the disease doubling time while the rest (82%) may be attributed to containment initiatives, general health policies, population growth, and transport or cultural aspects. In this study, we investigate the role of climate and specific government policy of partial lockdown in determining the number of new infections or confirmed cases in Ghana and to predict the future occurrences of this outbreak in the country. [14]- [16] studied the effects of lockdown on the number of new confirmed cases and/or on deaths. All of which revealed that lockdown serves an important intervention for preventing and minimizing the impact of an outbreak. Searching through literature, it appears Ghana at the moment does not have a statistical model that could predict future occurrences of new confirmed cases of COVID-19. To that extent, this study is novel. Materials and methods Study area The study covers the whole of Ghana. Figure 1 is a map of the study area indicating the cumulative confirmed cases in each region as of April 25, 2020. Data Daily data on new cases of COVID-19 from March 13, 2020 to April 21, 2020 were obtained from the official website of Our World in Data (OWID) dedicated to COVID-19 (https://ourworldindata.org/coronavirus-source-data). On the other hand, daily satellite climate data covering the same study period on maximum temperature, humidity, precipitation and maximum wind speed were obtained from the official website of the National Aeronautics and Space Administration's (NASA's) Prediction of Worldwide Energy Resources (POWER) project (https://power.larc.nasa.gov/data-accessviewer/). Statistical analysis To model the daily confirmed cases of COVID-19 in Ghana, which is a count variable, the generalized linear model (GLM) framework for time series of counts by [17] was adopted. These are models whose conditional mean of the response variable depends on previous observations of covariates and on its own previous values. Denote the new cases of COVID-19 by { ∶ ∈ ℕ}, and denote by { ∶ ∈ ℕ} a timevarying -dimensional covariate vector, say = � ,1 , . . . , , � ⏉ . The conditional mean ( |ℱ −1 ) of the new cases is modeled by a process, say {⋋ : ∈ ℕ}, such that ( |ℱ −1 ) = ⋋ . Denote by ℱ the history of the joint process { ,⋋ , +1 ∶ ∈ ℕ} up to time including the covariate information at time + 1. The general form of the model of interest developed by [17] is given in equation (1). (⋋ ) = 0 + � =1 �� − � + � α ℓ �⋋ − ℓ � + ⏉ , ℓ=1(1) Where ∶ ℝ + → ℝ is a link function and � ∶ ℕ 0 → ℝ is a transformational function. The parameter vector = ( 1 , … , ) ⏉ corresponds to the effects of covariates. To allow for regression on arbitrary past observations of the response, define a set = � 1 , 2 , … , � and integers 0 < 1 < 2 … < < ∞, with ∈ ℕ 0 . This enables regression on the lagged observations − 1 , − 2 , … , − . Parameters of the autocorrelation terms are represented by α ℓ and . The effect of government policy intervention such as the partial lockdown of the two largest cities in Ghana (Accra and Kumasi) and their environs on the daily confirmed cases of COVID-19 was analyzed using methods to test for such intervention effects developed by [18], [19]. These methods employ an approximate score test which has asymptotically a 2 distribution, assuming some regularity conditions [19]. Data analysis and model fitting was done in R [20] using the "tscount" package [21]. Results Descriptives Descriptive statistics for the daily confirmed cases of COVID-19 and weather variables in Ghana are presented in Table 1. As of April 21, 2020 Ghana had recorded a total of 1,040 cases of COVID-19 and an average of about 26.72 new cases every day. Regarding the weather variables, average daily maximum temperature, precipitation, humidity and maximum wind speed were 34.4°C, 2.56mm, 68.99% and 3.33m/s respectively. Figure 2 shows time series of daily new cases of COVID-19, maximum temperature, precipitation, humidity and maximum wind speed. The plot is an excellent way to begin comprehending the sort of process that produced the data and the relationships that exists therein. It is observed from the plot that neither the daily confirmed cases of COVID-19 nor the weather variables show any trend. Relationship between weather variables and new cases of COVID-19 Correlations between the weather variables and confirmed cases of COVID-19 are shown in Figure 3. While the new cases of COVID-19 correlated negatively with maximum temperature and humidity, it rather correlated positively with precipitation and maximum wind. This means that increasing maximum temperature and humidity results in decreasing new cases of COVID-19 whereas increasing precipitation and maximum wind results in increasing new cases of COVID-19. Among the weather variables, weak correlations were observed. Figure 3: Correlations between confirmed cases of COVID-19 and the weather variables Estimates of the model parameters are presented in It is observed from Table 2 that the 95% confidence interval for the estimated coefficient corresponding to the first order autocorrelation (beta_1) does not contain zero, indicating a clear dependence of confirmed COVID-19 cases on the number of new cases of the preceding day. It is also observed that three of the weather variables considered (maximum temperature, precipitation and relative humidity) were significant predictors of the number of new cases of COVID-19. However, maximum wind speed was not a significant predictor of the confirmed cases of COVID-19. The linear trend was also found to be significant. The model diagnostics are presented in Figure 4. Autocorrelation function of residuals, shown in Figure 4 (top left), does not exhibit any serial correlation or seasonality which has not been taken into account by the model. Figure 4 (bottom left) points to a probability integral transform (PIT) histogram which appears to approach uniformity. Figure 4 (top right) indicates the overall behavior of the data set. Figure 4 Model validation Time series of the actual daily new cases of COVID-19 used for model fitting and predicted cases based on the fitted model are shown in Figure 5. It is observed that the overall trend of the two curves is similar, and the values themselves are very close in some cases. This is an indication of a good model. Figure 5: In-sample model validation Intervention analysis Here, we test whether there was an abrupt shift in the number of new cases of COVID-19 occurring when the partial lockdown was introduced on March 30, 2020. The approximate score test was applied. With a p-value of 0.0164 (Table 3), the null hypothesis of no significant effect of the specific policy intervention of partial lockdown is rejected at a 5% significance level. Hence, partial lockdown of the two largest cities and their environs in Ghana significantly influenced the daily confirmed cases of COVID-19. Discussion In this study, we explored the possible effects of weather and government policy intervention on the daily confirmed cases of COVID-19. Specifically, the study considered the policy of partial lockdown of the two major cities (Accra and Kumasi) and their environs. The generalized linear model (GLM) framework for time series of counts by [17] was adopted for model fitting and analysis, which allows for regressing on past observations of the response variable and covariates. The results indicated that daily confirmed cases of COVID-19 in Ghana correlated negatively with maximum temperature and relative humidity. Previous studies on SARS-CoV and MERS-CoV also highlighted temperature as an important factor in the survival and transmission of such coronaviruses [22]- [26]. Recent studies have also underscored the importance of weather on the transmission of COVID-19 in particular. For instance, using a Modified Susceptible-Exposed-Infectious-Recovered (M-SEIR) model, [27] confirmed that transmission rate decreased with the increase of temperature, leading to further decrease of infection rate and outbreak scale in 31 provincial-level regions in mainland China. [28] also in a study suggested that for northern hemisphere countries, the rate of transmission of COVID-19 cases would significantly decrease as a result of warmer weather. [29] in a study stated that local weather condition with low temperature, mild diurnal temperature range and low humidity likely favor the transmission of COVID-19 cases in China. [30] in a study, analyzed the association between COVID-19 and climate indicators in New York City, USA. The authors used a secondary published data from New York city health services and National weather service, USA. The climate indicators included in the study were average temperature, minimum temperature, maximum temperature, rainfall, average humidity, wind speed, and air quality. Using Kendall and Spearman rank correlation tests for the data analysis, [30] suggested that that average temperature, minimum temperature, and air quality were significantly associated with COVID-19. Finally, using a global data on COVID-19 reported cases until 29th February 2020 and climate temperature data, [31] suggested that higher average temperature was strongly associated with lower COVID-19 incidence for temperatures of 1°C and higher. Our results further indicated the significance of maximum temperature, relative humidity and precipitation in predicting daily confirmed cases of COVID-19 in Ghana. This is consistent with the results of [32], who implemented a restricted cubic spline function and generalized linear mixture model to analyze the relationships between temperature and COVID-19 transmission in 429 cities in the world and suggested that to a certain extent, temperature could significantly change COVID-19 transmission, and there might be a best temperature for the viral transmission, which may partly explain why it first broke out in Wuhan, China. [33] also investigated the influence of air temperature and relative humidity on the transmission of COVID-19 in 100 Chinese cities and suggested that, under a linear regression framework, high temperature and high humidity significantly reduce the transmission of COVID-19. Additionally, results of the intervention analysis conducted in this study indicated that the null hypothesis of no significant effect of the specific policy intervention of partial lockdown should be rejected (p-value=0.0164) at a 5% level of significance. This finding underscores the significance of the partial lockdown of the two largest cities (Accra and Kumasi) and their environs in controlling the spread of the disease in Ghana. A number of studies [14]- [16] have been conducted on the effects of lockdown on the number of new confirmed cases and/or on deaths. All of which reveal that lockdown serves an important intervention for preventing and minimizing the impact of an outbreak. Conclusion In this paper, we investigated the effects of weather and government policy intervention on the daily confirmed cases of COVID-19 in Ghana. The results indicated that while daily confirmed cases of COVID-19 correlated negatively with maximum temperature and humidity, it rather correlated positively with precipitation and maximum wind. Also, maximum temperature, humidity and precipitation were significant predictors of daily confirmed cases of COVID-19 in Ghana. Furthermore, results of the intervention analysis revealed that the specific government policy of partial lockdown of the two largest cities (Accra and Kumasi) and their surroundings significantly influenced the daily confirmed cases of COVID-19, which provides useful implications for policy makers and the public. For instance, knowing that precipitation and the daily confirmed cases of COVID-19 are positively correlated could help public health responses by informing key preparations as we move into the rainy season. Figure 1 : 1Map of study area (Source: https://www.ghanahealthservice.org/covid19/; accessed on April 28, 2020) Figure 2 : 2Time series of daily confirmed cases of COVID-19, and weather variables from March 13, 2020 to April 21, 2020 (bottom right) shows a marginal calibration plot with minor fluctuations about zero. Hence the probabilistic calibration of the Poisson model is sufficient. Figure 4 : 4Diagnostic plots after model fitting to the data Table 1 : 1Descriptive statistics of daily confirmed cases of COVID-19 and satellite climate data in GhanaVariables N Mean Sd Min Max Range skew Se New_Cases 39 26.72 51.55 0 208 208 2.38 8.25 Max_Temp (°C) 39 34.46 2.94 28.52 40.44 11.92 0.19 0.47 Precipitation (mm) 39 2.56 6.09 0.10 37.95 37.85 5.04 0.97 Humidity (%) 39 68.99 9.95 48.62 84.33 35.71 -0.29 1.59 Max_Wind (m/s) 39 3.33 0.74 1.56 4.79 3.23 -0.23 0.12 Table 2 . 2The log-linear model with the logarithmic link was chosen for model fitting because it allows for negative covariate effects. Also, in order to capture the short range serial dependence, a first order autoregressive term(beta_1) was included. Maximum temperature (Max_Temp), precipitation (Precipitation), relative humidity (Humidity), and maximum wind speed (Max_Wind) were included as explanatory variables. A deterministic covariate (linearTrend) was included to describe a linear trend. The fitted model for the number of new cases of COVID-19 (New_Cases ) in day is accordingly given by New_Cases |ℱ −1 ~ ( ) With log( ) = 13.8430 − 0.2279New_Cases −1 − 0.2267Max_Temp + 0.0407Precipitation − 0.0605Humidity − 0.0930Max_Wind + 28.7428 /365 Table 2 : 2Estimates of model parametersEstimate Std.Error 95% CI(lower) 95% CI(upper) (Intercept) 13.8430 2.8291 8.7491 19.9134 beta_1 -0.2279 0.0884 -0.4334 -0.0968 Max_Temp -0.2267 0.0519 -0.3377 -0.1326 Precipitation 0.0407 0.0056 0.0288 0.051 Humidity -0.0605 0.0161 -0.0948 -0.0288 Max_Wind -0.0930 0.0598 -0.2132 0.0178 linearTrend 28.7428 3.7136 23.1412 37.678 Link function: log Distribution family: Poisson Number of coefficients: 7 Log-likelihood: -735.4298 AIC: 1484.86 BIC: 1496.504 QIC: 1484.86 Table 3 : 3Chi-square test of the effect of partial lockdown on new COVID-19 cases in GhanaChi-Square Statistic Degrees of freedom p-value 8.2247 2 0.0164 Differential COVID-19-attributable mortality and BCG vaccine use in countries. A Shet, D Ray, N Malavige, M Santosham, N Bar-Zeev, 10.1101/2020.04.01.20049478medrxiv.org. A. Shet, D. Ray, N. Malavige, M. Santosham, and N. Bar-Zeev, "Differential COVID-19- attributable mortality and BCG vaccine use in countries," medrxiv.org, 2020, doi: 10.1101/2020.04.01.20049478. Response to COVID-19 in Chinese neurosurgery and beyond. Y Sun, Y M , .-J Of Neurosurgery, U 2020, J. Neurosurg. 1Y. Sun, Y. M.-J. of Neurosurgery, and U. 2020, "Response to COVID-19 in Chinese neurosurgery and beyond," J. Neurosurg., vol. 1(aop), pp. 1-2, 2020. Early forecasts of the evolution of the COVID-19 outbreaks and quantitative assessment of the effectiveness of countering measures. E Daddi, M Giavalisco, arXivE. Daddi and M. Giavalisco, "Early forecasts of the evolution of the COVID-19 outbreaks and quantitative assessment of the effectiveness of countering measures," arXiv, Apr. 2020. Covid-19: risk factors for severe disease and death. R E Jordan, P Adab, K K Cheng, 10.1136/bmj.m1091thebmj. R. E. Jordan, P. Adab, and K. K. Cheng, "Covid-19: risk factors for severe disease and death," thebmj, 2020, doi: 10.1136/bmj.m1091. Risk factors associated with acute respiratory distress syndrome and death in patients with coronavirus disease. C Wu, X Chen, Y Cai, X Zhou, … S X , U 2020, JAMA Intern. Med. C. Wu, X. Chen, Y. Cai, X. Zhou, … S. X.-J. internal, and U. 2020, "Risk factors associated with acute respiratory distress syndrome and death in patients with coronavirus disease 2019 pneumonia in Wuhan, China," JAMA Intern. Med., 2020. Understanding the heterogeneity of adverse COVID-19 outcomes: the role of poor quality of air and lockdown decisions *** PRELIMINARY AND INCOMPLETE ***. L Becchetti, F Salustri, L. Becchetti and F. Salustri, "Understanding the heterogeneity of adverse COVID-19 outcomes: the role of poor quality of air and lockdown decisions *** PRELIMINARY AND INCOMPLETE ***," 2020. Effects of temperature variation and humidity on the death of COVID-19 in Wuhan, China. Y Ma, 10.1016/j.scitotenv.2020.138226Sci. Total Environ. 138226Y. Ma et al., "Effects of temperature variation and humidity on the death of COVID-19 in Wuhan, China," Sci. Total Environ., p. 138226, 2020, doi: 10.1016/j.scitotenv.2020.138226. Predict the next moves of COVID-19: reveal the temperate and tropical countries scenario. N A Hasan, M M Haque, 10.1101/2020.04.04.20052928medrxiv.org. N. A. Hasan and M. M. Haque, "Predict the next moves of COVID-19: reveal the temperate and tropical countries scenario," medrxiv.org, 2020, doi: 10.1101/2020.04.04.20052928. Climate change and infectious diseases. J Patz, A Githeko, J Mccarty, … S H , U , J. Patz, A. Githeko, J. McCarty, … S. H.-C. change and, and U. 2003, "Climate change and infectious diseases," 2009. Follow the Sun : Slower COVID-19 morbidity and mortality growth at higher irradiances. A Bäcker, papers.ssrn.com. A. Bäcker, "Follow the Sun : Slower COVID-19 morbidity and mortality growth at higher irradiances," papers.ssrn.com, pp. 1-17, 2019. Climate affects global patterns of COVID-19 early outbreak dynamics. G F Ficetola, D Rubolini, 10.1101/2020.03.23.20040501medRxiv, p. 2020.03.23.G. F. Ficetola and D. Rubolini, "Climate affects global patterns of COVID-19 early outbreak dynamics," medRxiv, p. 2020.03.23.20040501, 2020, doi: 10.1101/2020.03.23.20040501. Association between ambient temperature and COVID-19 infection in 122 cities from China. J Xie, Y Zhu, 10.1016/j.scitotenv.2020.138201Sci. Total Environ. 724138201J. Xie and Y. Zhu, "Association between ambient temperature and COVID-19 infection in 122 cities from China," Sci. Total Environ., vol. 724, p. 138201, 2020, doi: 10.1016/j.scitotenv.2020.138201. Role of temperature and humidity in the modulation of the doubling time of COVID-19 cases. B Oliveiros, L Caramelo, N C Ferreira, F Caramelo, medRxivB. Oliveiros, L. Caramelo, N. C. Ferreira, and F. Caramelo, "Role of temperature and humidity in the modulation of the doubling time of COVID-19 cases," medRxiv, 2020. Assessment of 21 Days Lockdown Effect in Some States and Overall India: A Predictive Mathematical Study on COVID-19 Outbreak. T Sardar, S S Nadim, J Chattopadhyay, arXiv Prepr. arXiv. 200403487T. Sardar, S. S. Nadim, and J. Chattopadhyay, "Assessment of 21 Days Lockdown Effect in Some States and Overall India: A Predictive Mathematical Study on COVID-19 Outbreak," arXiv Prepr. arXiv, vol. 2004, no. 03487, pp. 1-36, 2020. Evaluation of the lockdowns for the SARS-CoV-2 epidemic in Italy and Spain after one month follow up. A Tobías, 10.1016/j.scitotenv.2020.138539Sci. Total Environ. 725138539PG-138539-138539A. Tobías, "Evaluation of the lockdowns for the SARS-CoV-2 epidemic in Italy and Spain after one month follow up," Sci. Total Environ., vol. 725, no. PG-138539-138539, p. 138539, 2020, doi: https://doi.org/10.1016/j.scitotenv.2020.138539. Can we contain the COVID-19 outbreak with the same measures as for SARS?. A Wilder-Smith, C Chiew, V L , .-T L I Diseases, U , Lancet Infect. Dis. A. Wilder-Smith, C. Chiew, V. L.-T. L. I. Diseases, and U. 2020, "Can we contain the COVID-19 outbreak with the same measures as for SARS?," Lancet Infect. Dis., 2020. Tscount: An R package for analysis of count time series following generalized linear models. T Liboschik, K Fokianos, R Fried, 10.18637/jss.v082.i05J. Stat. Softw. 82T. Liboschik, K. Fokianos, and R. Fried, "Tscount: An R package for analysis of count time series following generalized linear models," J. Stat. Softw., vol. 82, 2017, doi: 10.18637/jss.v082.i05. Interventions in ingarch processes. K Fokianos, R Fried, K. Fokianos and R. Fried, "Interventions in ingarch processes," 2009. Interventions in log-linear Poisson autoregression. K Fokianos, R Fried, 10.1177/1471082X1201200401Stat. Modelling. 124K. Fokianos and R. Fried, "Interventions in log-linear Poisson autoregression," Stat. Modelling, vol. 12, no. 4, pp. 299-322, 2012, doi: 10.1177/1471082X1201200401. R: a language and environment for statistical computing computer program. R T C -R, Team, Vienna, Austriaversion 3.6. 1," R Core TeamR. T.-R. C. Team, "R: a language and environment for statistical computing computer program, version 3.6. 1," R Core Team, Vienna, Austria, 2019. Package 'tscount' Title Analysis of Count Time Series. T Liboschik, R Fried, K Fokianos, P Probst, 10.1002/jtsaJ. Stat. Softw. T. Liboschik, R. Fried, K. Fokianos, and P. Probst, "Package 'tscount' Title Analysis of Count Time Series," J. Stat. Softw., 2020, doi: 10.1002/jtsa. Weather: driving force behind the transmission of severe acute respiratory syndrome in China?. P Bi, J Wang, J E Hiller, 10.1111/j.1445-5994.2007.01358.xIntern. Med. J. 378P. Bi, J. Wang, and J. E. Hiller, "Weather: driving force behind the transmission of severe acute respiratory syndrome in China?," Intern. Med. J., vol. 37, no. 8, pp. 550-554, Aug. 2007, doi: 10.1111/j.1445-5994.2007.01358.x. Effects of Air Temperature and Relative Humidity on Coronavirus Survival on Surfaces. L M Casanova, S Jeon, W A Rutala, D J Weber, M D Sobsey, 10.1128/AEM.02291-09Appl. Environ. Microbiol. 769L. M. Casanova, S. Jeon, W. A. Rutala, D. J. Weber, and M. D. Sobsey, "Effects of Air Temperature and Relative Humidity on Coronavirus Survival on Surfaces," Appl. Environ. Microbiol., vol. 76, no. 9, pp. 2712-2717, May 2010, doi: 10.1128/AEM.02291-09. The Effects of Temperature and Relative Humidity on the Viability of the SARS Coronavirus. K H Chan, J S M Peiris, S Y Lam, L L M Poon, K Y Yuen, W H Seto, 10.1155/2011/734690Adv. Virol. 2011K. H. Chan, J. S. M. Peiris, S. Y. Lam, L. L. M. Poon, K. Y. Yuen, and W. H. Seto, "The Effects of Temperature and Relative Humidity on the Viability of the SARS Coronavirus," Adv. Virol., vol. 2011, 2011, doi: 10.1155/2011/734690. An initial investigation of the association between the SARS outbreak and weather: with the view of the environmental temperature and its variation. J Tan, 10.1136/jech.2004.020180J Epidemiol Community Heal. 59J. Tan et al., "An initial investigation of the association between the SARS outbreak and weather: with the view of the environmental temperature and its variation," J Epidemiol Community Heal., vol. 59, pp. 186-192, 2005, doi: 10.1136/jech.2004.020180. Stability of middle east respiratory syndrome coronavirus (MERS-CoV) under different environmental conditions. N Van Doremalen, T Bushmaker, V J Munster, 10.2807/1560-7917.ES2013.18.38.20590Eurosurveillance. 1838N. van Doremalen, T. Bushmaker, and V. J. Munster, "Stability of middle east respiratory syndrome coronavirus (MERS-CoV) under different environmental conditions," Eurosurveillance, vol. 18, no. 38, 2013, doi: 10.2807/1560-7917.ES2013.18.38.20590. The impact of temperature and absolute humidity on the coronavirus disease 2019 (COVID-19) outbreak -evidence from China. P Shi, 10.1101/2020.03.22.20038919P. Shi et al., "The impact of temperature and absolute humidity on the coronavirus disease 2019 (COVID-19) outbreak -evidence from China," medRxiv, no. 77, p. 2020.03.22.20038919, 2020, doi: 10.1101/2020.03.22.20038919. Temperature dependence of COVID-19 transmission. A Notari, Prepr. arXiv. 12417A. Notari, "Temperature dependence of COVID-19 transmission," Prepr. arXiv, vol. 2003, no. 12417, pp. 1-8, 2020. Impact of meteorological factors on the COVID-19 transmission: A multicity study in China. J Liu, 10.1016/j.scitotenv.2020.138513Sci. Total Environ. 726138513J. Liu et al., "Impact of meteorological factors on the COVID-19 transmission: A multi- city study in China," Sci. Total Environ., vol. 726, p. 138513, 2020, doi: 10.1016/j.scitotenv.2020.138513. Correlation between climate indicators and COVID-19 pandemic. M F Bashir, 10.1016/j.scitotenv.2020.138835Sci. Total Environ. 728138835M. F. Bashir et al., "Correlation between climate indicators and COVID-19 pandemic in New York, USA," Sci. Total Environ., vol. 728, p. 138835, 2020, doi: 10.1016/j.scitotenv.2020.138835. Preliminary evidence that higher temperatures are associated with lower incidence of COVID-19, for cases reported globally up to. M Bannister-Tyrrell, A Meyer, C Faverjon, A Cameron, medRxivM. Bannister-Tyrrell, A. Meyer, C. Faverjon, and A. Cameron, "Preliminary evidence that higher temperatures are associated with lower incidence of COVID-19, for cases reported globally up to 29th February 2020," medRxiv, 2020. Temperature Significantly Change COVID-19 Transmission in 429 cities. M Wang, M. Wang et al., "Temperature Significantly Change COVID-19 Transmission in 429 cities," medRxiv, no. February 2019, pp. 1-13, 2020. High Temperature and High Humidity Reduce the Transmission of COVID-19. J Wang, K Tang, K Feng, W Lv, 10.2139/ssrn.3551767SSRN Electron. J. J. Wang, K. Tang, K. Feng, and W. Lv, "High Temperature and High Humidity Reduce the Transmission of COVID-19," SSRN Electron. J., 2020, doi: 10.2139/ssrn.3551767.
[]
[ "Collective mode reductions for populations of coupled noisy oscillators", "Collective mode reductions for populations of coupled noisy oscillators" ]
[ "Denis S Goldobin \nInstitute of Continuous Media Mechanics\nUB RAS\nAcademician Korolev Street 1614013PermRussia\n\nDepartment of Theoretical Physics\nPerm State University\nBukirev Street 15614990PermRussia\n", "Irina V Tyulkina \nDepartment of Theoretical Physics\nPerm State University\nBukirev Street 15614990PermRussia\n", "Lyudmila S Klimenko \nInstitute of Continuous Media Mechanics\nUB RAS\nAcademician Korolev Street 1614013PermRussia\n\nDepartment of Theoretical Physics\nPerm State University\nBukirev Street 15614990PermRussia\n", "Arkady Pikovsky \nInstitute for Physics and Astronomy\nUniversity of Potsdam\nKarl-Liebknecht-Strasse 24/2514476Potsdam-GolmGermany\n\nDepartment of Control Theory\nNizhny Novgorod State University\nGagarin Avenue 23606950Nizhny NovgorodRussia\n" ]
[ "Institute of Continuous Media Mechanics\nUB RAS\nAcademician Korolev Street 1614013PermRussia", "Department of Theoretical Physics\nPerm State University\nBukirev Street 15614990PermRussia", "Department of Theoretical Physics\nPerm State University\nBukirev Street 15614990PermRussia", "Institute of Continuous Media Mechanics\nUB RAS\nAcademician Korolev Street 1614013PermRussia", "Department of Theoretical Physics\nPerm State University\nBukirev Street 15614990PermRussia", "Institute for Physics and Astronomy\nUniversity of Potsdam\nKarl-Liebknecht-Strasse 24/2514476Potsdam-GolmGermany", "Department of Control Theory\nNizhny Novgorod State University\nGagarin Avenue 23606950Nizhny NovgorodRussia" ]
[]
We analyze accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise, and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling, are considered as examples to test validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations.
10.1063/1.5053576
[ "https://arxiv.org/pdf/1808.07833v3.pdf" ]
53,256,331
1808.07833
f36c6d3fcf62446d0e3d0417af36772028d8ee6d
Collective mode reductions for populations of coupled noisy oscillators 14 Oct 2018 Denis S Goldobin Institute of Continuous Media Mechanics UB RAS Academician Korolev Street 1614013PermRussia Department of Theoretical Physics Perm State University Bukirev Street 15614990PermRussia Irina V Tyulkina Department of Theoretical Physics Perm State University Bukirev Street 15614990PermRussia Lyudmila S Klimenko Institute of Continuous Media Mechanics UB RAS Academician Korolev Street 1614013PermRussia Department of Theoretical Physics Perm State University Bukirev Street 15614990PermRussia Arkady Pikovsky Institute for Physics and Astronomy University of Potsdam Karl-Liebknecht-Strasse 24/2514476Potsdam-GolmGermany Department of Control Theory Nizhny Novgorod State University Gagarin Avenue 23606950Nizhny NovgorodRussia Collective mode reductions for populations of coupled noisy oscillators 14 Oct 2018(Dated: 16 October 2018) We analyze accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise, and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling, are considered as examples to test validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations. Synchrony of large ensembles of coupled elements can be characterised by the order parametersthe mean fields. Quite often the evolution of these collective variables is surprisingly simple, what makes a description with only a few order parameters feasible. Thus, one tries to construct accurate closed low-dimensional mathematical models for the dynamics of the first few order parameters. These models represent useful tools for gaining insight into the underlaying mechanisms of some more sophisticated collective phenomena: for example, one describes coupled populations by virtue of coupled equations for the relevant order parameters. A regular approach to the construction of closed low-dimensional systems is also beneficial for dealing with phenomena, which are beyond the applicability scope of these models; for instance, with such an approach, one can determine constrains on clustering in populations. There are two prominent types of situations, where the low-dimensional models can be constructed: (i) for a certain class of ideal paradigmatic systems of coupled phase oscillators, the Ott-Antonsen ansatz yields an exact equation for the main order parameter; (ii) the Gaussian approximation for the probability density of the phases, also yielding a low-dimensional closure, is frequently quite accurate. In this paper, we compare applications of these two model reductions for situations, where neither of them is perfectly accurate. Furthermore, we construct a new reduction approach which practically works as a first-order correction to the best of the two basic approximations. I. INTRODUCTION Models of globally coupled oscillators are relevant for many applications in physics, engineering, living and social systems [1][2][3][4][5][6] . The main effect here is synchronization, i.e. appearance of a nontrivial mean field due to interactions. This effect can be viewed as a nonequilibrium phase transtion, where the appearing ordered synchronized state is described by a set of order parameters. The famous Kuramoto model of coupled phase oscillators is a paradigmatic example for the synchronization transition, it is completely solvable in the thermodynamic limit of an infinite population. The properties of the transition are also quite well understood if additionally to the coupling, the oscillators are subject to independent noise terms. A description of globally coupled noisy oscillators can be reduced, in the thermodynamic limit of large ensemble, to a nonlinear Fokker-Planck equation (or to a Liouville equation in the noiseless case), which is a system with an infinite number of degrees of freedom. If one wants not simply find the stationary solutions, but to follow the evolution of the distributions, the problem of the reduction of the infinite-dimensional system to several essential degrees of freedom arises. This closure problem is in the focus of this paper. We will discuss and compare three variants of the reduction to a few global modes: (i) the Ott-Antonsen ansatz 7 , (ii) the Gaussian ansatz, recently considered by Hannay et al. 8 on the basis of previous works [9][10][11] , and (iii) the circular cumulant approach suggested in Ref. 12 . Neither of these approaches is exact for a population of coupled noisy oscillators, but they provide quite good approximations of the observed regimes. We will compare their accuracy for different ranges of parameters. II. BASIC MODELS Our basic model is a population of phase oscillators ϕ k (t) with intrinsic noise: ϕ k = ω k + Im(2h(t)e −iϕ k ) + √ Dη k (t) .(1) Here natural frequencies ω k have a Lorentzian (Cauchy) distribution g(ω) = γ/[π((ω − ω 0 ) 2 + γ 2 )], and γ is the distribution half-width. Parameter D is the noise intensity, terms η k are independent normalized white Gaussian random forces: η k (t)η m (t ′ ) = 2δ km δ(t − t ′ ), η k = 0. The coupling is determined by the complex force h, common for all oscillators. For the Kuramoto setup, this force is proportional to the mean field which is just the Kuramoto order parameter of the population h = K 2 Z 1 , Z 1 = e iϕ = 1 N N j=1 e iϕj . The Kuramoto model for noisy oscillators thus readṡ ϕ k = ω k + K N N j=1 sin(ϕ j − ϕ k ) + √ Dη k (t) .(2) With a slight modification of the common force h, namely h = a 2 + K 2 Z 1 , one obtains the equations for a population of noisy active rotators with the Kuramoto-type coupling, treated in Ref. 9 : ϕ k = ω k − a sin ϕ k + K N N j=1 sin(ϕ j − ϕ k ) + √ Dη k (t) . (3) Models (2) and (3) are the basic systems we consider below. III. MODE EQUATIONS In the thermodynamic limit N → ∞, starting from the Langevin equations (1), one can write for the distribution density of the subpopulation of the oscillators with natural frequency ω the Fokker-Planck equation ∂w(ϕ, t|ω) ∂t = − ∂ ∂ϕ Im(2h(t)e −iϕ )w + D ∂ 2 ∂ϕ 2 w . (4) This equation can be rewritten as an infinite system for the complex amplitudes of the Fourier modes z m = π −π dϕ w(ϕ, t|ω)e imϕ of the density w(ϕ, t|ω) = (2π) −1 m z m (t, ω)e −imϕ : z n = niωz n + nhz n−1 − nh * z n+1 − n 2 Dz n .(5) The quantities z n are the local order parameters at a given frequency, the global Kuramoto-Daido order parameters are obtained by the additional averaging over the distribution of the natural frequencies: Z n = dω g(ω)z n .(6) The main order parameter Z 1 is employed in the definition of the forces h in the two models we study in this paper. Below we consider only the Lorentzian distribution g and adopt the assumption by Ott and Antonsen 7 on the analyticity of z n (t, ω) as a function of complex ω in the upper half-plane. This allows for calculating the global Kuramoto-Daido order parameters via residues as Z n = z n (ω 0 + iγ) . In this way, one obtains an infinite system of equations for Z n (which are in fact moments e iϕ n of the complex observable e iϕ ) with Z 0 ≡ 1: Z n = n(iω 0 − γ)Z n + nhZ n−1 − nh * Z n+1 − n 2 DZ n . (7) IV. FINITE-DIMENSIONAL REDUCTIONS As we discussed above, it is desirable to reduce, at least approximately, the infinite system (7) to a finitedimensional one, and, in what follows, we discuss three ways to accomplish this. A. Ott-Antonsen reduction Here one assumes, following Ref. 7 , that all the higher order parameters can be expressed via the first one according to Z n = (Z 1 ) n .(8) This reduces the system (7) to just one equatioṅ Z 1 = (iω 0 − γ)Z 1 + h − h * Z 2 1 − DZ 1 .(9) The Ott-Antonsen (OA) reduction works exactly for D = 0, where it defines the so-called OA invariant manifold. This manifold corresponds to the probability density being the wrapped Cauchy distribution of the phases. B. Gaussian reduction Recently, on the basis of the analysis of some experimental data, another representation of the higher order parameters through the first one was suggested 8 : Z m = |Z 1 | m 2 −m Z m 1 .(10) Equivalently, if we introduce the amplitude and the argument of the Kuramoto order parameter Z 1 = R 1 e iψ , with R 1 = exp[−s 2 /2], we can rewrite (10) as Z m = R m 2 1 e imψ = e − 1 2 m 2 s 2 e imψ .(11) This relation means that the corresponding probability density of the phases is the wrapped Gaussian distribution. Substitution of (10) into Eq. (7) for n = 1 yields 8 Z 1 = (iω 0 − γ)Z 1 + h − h * |Z 1 | 2 Z 2 1 − DZ 1 .(12) C. Cumulant reduction Recently, we suggested 12 a reformulation of the model in terms of the "circular cumulants" κ n , instead of the formulation in terms of moments (7). The cumulants are determined via the power series of the cumulant-generating function defined as Ψ(k) = k ∂ ∂k exp(ke iϕ ) ≡ ∞ n=1 κ n k n .(13) For example, the first three circular cumulants are: κ 1 = Z 1 , κ 2 = Z 2 − Z 2 1 , and κ 3 = (Z 3 − 3Z 2 Z 1 + 2Z 3 1 )/2. The merit of the reformulation in terms of the cumulants is two-fold. (i) In terms of the cumulants, the OA manifold (8) is a state with one non-vanishing cumulant only: κ 1 = Z 1 and κ n>1 = 0. This allows for a representation of the states close to the OA solution as those with small higher cumulants. The cumulants κ 2 , κ 3 , . . . describe deviations from the OA manifold (from the wrapped Cauchy distribution), see below Fig. 1 for a visualization of the perturbation due to κ 2 . (ii) For general states with high synchrony, where |Z 1 | ≈ 1, the moments Z n decay slowly with n, while in terms of cumulants one has |κ 1 | ≈ 1 and |κ n>1 | ≪ 1, which also allows for a nice representation in terms of cumulants. In particular, in the case of the wrapped Gaussian distribution (11), the cumulants obey the hierarchy of smallness for arbitrary degree of synchrony; this hierarchy has a simple analytical form for high and low synchrony: κ n = −(−n) n−2 (n−1)! e inψ s 2(n−1) 1 + O(s 2 ) for s 2 ≪ 1 , (−1) n−1 Z n 1 1 + O(Z 2 1 ) for s 2 1 .(14) Hence, the cumulant representation appears to be a proper framework for perturbations both of the OA solution and of a highly synchronous state, although the exact equation system for the cumulants 12 is more complex than (7): κ n = n(iω 0 − γ)κ n + hδ 1n −h * (n 2 κ n+1 + n n−1 m=0 κ n−m κ m+1 ) −D(n 2 κ n + n n−2 m=0 κ n−1−m κ m+1 ) .(15) Note, that Eqs. (7) and (15) describe nonidentical oscillators with Lorentzian distribution of frequencies; identical ensembles correspond to γ = 0. In Ref. 12 , the infinite system (15) was analysed in the case of small noise intensity D, and was shown to generate, as a perturbation of the OA solution, the hierarchy κ n ∼ D n−1 for n ≥ 2. A first-order correction to the OA ansatz requires the cumulant κ 2 to be taken into account; from the infinite system (15) only two equations remain: Z 1 = (iω 0 − γ)Z 1 + h − h * (Z 2 1 + κ 2 ) − DZ 1 , κ 2 = 2(iω 0 − γ)κ 2 − 4h * (κ 3 + Z 1 κ 2 ) − D(4κ 2 + 2Z 2 1 ) .(16) To close these equations, one needs to specify κ 3 . The representation of κ 3 with maintaining the first order accuracy can be performed in several ways. In Ref. 12 , this cumulant was just set to zero: κ 3 = 0 .(17) On the other hand, any substitution κ 3 = const κ 2 2 /Z 1 yields the same first order accuracy for system (16), since it obeys the hierarchy κ n ∼ D n−1 . To find a proper representation of κ 3 for a Gaussian distribution with high synchrony, let us write the first three cumulants for s ≪ 1: κ 1 = Z 1 ≈ e iψ , κ 2 ≈ −s 2 e i2ψ , κ 3 ≈ 3 2 s 4 e i3ψ . One can see that the closure κ 3 = 3 2 κ 2 2 Z 1(18) is consistent with this distribution, although it potentially includes non-Gaussian situations, because in (18) Z 1 and κ 2 are independent of each other. Summarizing, the closure (18) is consistent simultaneously both with the hierarchy κ n ∼ D n−1 and with the Gaussian distribution with high synchrony, but generally can describe also states away from these limiting cases. We stress that the closure (18) should not be used in situations, where Z 1 is close to zero while κ 2 is not small. For the systems, where Z 1 can vanish without κ 2 2 /Z 1 remaining finite, a modification to closure (18) can be suggested: κ 3 = 3 2 κ 2 2 Z * 1 ;(19) this modification is equivalent to Eq. (18) at s → 0, but less accurately corresponds to the wrapped Gaussian distribution for |Z 1 | < 1. It is also not less accurate than the first-order correction to the OA solution. It is instructive to visualize the perturbation of the OA probability density corresponding to one nonvanishing second circular cumulant κ 2 . With two nonvanishing cumulants, the moment-generating function is F (k) = ∞ m=0 Z m (t) k m m! = exp kZ 1 + κ 2 k 2 2 . Assuming smallness of κ 2 , we approximate it as F (k) ≈ (1 + κ 2 Z m 1 + m(m−1) 2 κ 2 Z m−2 1 . Summation of the Fourier series with these Fourier coefficients yields w (ϕ) = w OA (ϕ) + w C (ϕ), where w OA (ϕ) = 1 − |Z 1 | 2 2π|e iϕ − Z 1 | 2 is the wrapped Cauchy distribution corresponding to the OA ansatz, and w C (ϕ) = Re π −1 κ 2 e iϕ (e iϕ − Z 1 ) 3 . is the correction corresponding to a nonvanishing second cumulant. We illustrate the perturbation of the probability density in Fig. 1. We depict the OA-density relative to the argument of the order parameter, by using ϑ = ϕ − arg(Z 1 ). One can see that the perturbation is localized close to the maximum of the unperturbed density w OA ; its exact position depends on the difference of the arguments of the two cumulants involved Θ = arg(κ 2 ) − 2arg(Z 1 ). V. ACCURACY OF DIFFERENT APPROXIMATIONS Above we have outlined five possible finite-dimensional descriptions of the noisy interacting population: Eqs. (9), (12), (16,17), (16,18), and (16,19) (cf. Table I For the ensemble of identical oscillators (γ = 0) in a steady state, where h = const (in a rotating reference frame, if necessary), the stationary distribution of phases according to (4) is the von Mises distribution w = exp 2|h| D cos ϕ − arg(h) 2πI 0 (2|h|/D) , where I 0 (·) is the modified Bessel function of order 0. In the case |h| ≫ D, it is close to the wrapped Gaussian distribution; thus, one expects that the Gaussian approximation will provide an accurate steady state in this limit. Simultaneously, this is the case of high synchrony, where substitutions (18) and (19) are relevant. Below we compare the accuracy of the steady states according to the approximations outlined, for the Kuramoto model (2) and the active rotator model (3). A. Kuramoto model The Kuramoto model for noisy oscillators (2) contains three parameters: γ, D, and K. However, by virtue of a time normalization, one can get rid of one parameter. The critical coupling for the onset of synchronization is K cr = 2(γ + D). Thus, it is convenient to choose γ = 1 − D, so that the critical coupling is K cr = 2. First, we calculated the "exact" steady state of system (15) by solving it with 200 cumulants taken into account. Then we found the steady solutions of approximations (9), (12), (16,17), (16,18): R 2 1 = 1 − K cr K ,(20)R 2 1 = 1 − K cr K ,(21)R 2 1 = 1 2 − 3K cr 4K + (2K − K cr ) 2 + 16D(K − K cr ) 4K ,(22)R 2 1 = 2 − 3K cr 2K − 4(K − K cr )(K − 2D) + K 2 cr 2K ,(23) respectively; the approximation (16,19) yields a cubic equation for R 2 1 with a cumbersome analytical solution. The deviations from the exact state are shown in Fig. 2. One can see, that the Gaussian approximation yields better accuracy than the OA ansatz only for strong noise (D = 0.99, we remind that according to the normalization adopted 0 ≤ D ≤ 1) and strong coupling. The 2C approximation with the closure κ 3 = 0 works as a plain first-order correction to the OA solution. The 2C approximation with the closure κ 3 = (3/2)κ 2 2 /Z 1 (18) is the best one in all situations. The 2C approximation with the closure κ 3 = (3/2)κ 2 2 Z * 1 (19) is approaching the one with (18) for high synchrony, but yields the same accuracy as the closure κ 3 = 0 for small R 1 ; for a strong noise and moderate synchrony, it is only slightly less accurate than the Gaussian approximation. Close to the synchronization threshold K cr , the inaccuracy of the Gaussian approximation reaches 0.2 and exceeds the value of order parameter R 1 , while the inaccuracy of R 1 with the OA ansatz is always reasonably small. As the 2C approximations are based on the correction to the OA one, the former are always superior to the latter. One can also see, that the Gaussian approximation is accurate where the synchrony is high, which is also suggested by the von Mises distribution with |h| ≫ D. Noteworthy, for high synchrony, the 2C approximations with closures (18) and (19) contain the Gaussian distribution as an admissible particular case. Moreover, these 2C truncations employ the Gaussian scaling only in the expression for the third cumulant κ 3 , while the second cumulant κ 2 is allowed to deviate from the value dictated by the first cumulant Z 1 for the Gaussian distribution. Hence, these truncations also encompass a first-order correction for the case of Gaussian approximation under high synchrony. The closure (18) decently approximates the wrapped Gaussian distribution also for non-high synchrony. Being not less accurate than the first-order corrections to both the OA and Gaussian reductions, the 2C D = 0.01 reduction with closure (18) becomes superior to them for the Kuramoto model with intrinsic noise. B. Active rotators model A population of active rotators with the Kuramototype coupling (3) can exhibit diverse regimes of collective dynamics, depending on parameter values 9 . Following Ref. 9 , we consider identical elements (γ = 0) and focus on the case which is impossible for the Kuramoto ensemble: an excitable state of individual elements, a > ω 0 . Noteworthy, in this case the synchrony imperfectness is owned solely by intrinsic Gaussian noise. For all the cases presented in Figs. 3 and 4, the accurate solution is calculated from system (15) with 200 cumulants. In Fig. 3, we evaluate accuracy of approximations (9), (12), (16,17), (16,18), and (16,19). For high synchrony (which is observed for a weak noise), the Gaussian approximation is more accurate than the OA one. Where the OA approximation fails, the plain 2C approximation with closure κ 3 = 0 is not more accurate than the OA solution: in Fig. 3 for D = 0.01 and 0.1, the inaccuracy of the OA solution is of the same order of magnitude as the deviation of R 1 from 1 for the exact solution. The 2C approximations with Gaussian closures always provide much better accuracy than both the OA and the Gaussian ones. C. Testing scaling laws In Fig. 4 we test how well the scaling laws (8) and (10), which lie at the basis of the OA and the Gaussian approximations, are valid. To check the OA ansatz (8), we plot the values of the cumulants: the cumulants κ n with n ≥ 2 should vanish if the OA ansatz is exact. To check the Gaussian approximation, we compare the n-dependence of R n = |Z n | with a parabola. Panel Fig. 4(a) shows the scaling for the Kuramoto model. One can see that although high-order cumulants do not vanish, there is a gap between the first and the second cumulants. This means that the OA ansatz is relatively good, but can be definitely improved by taking into account the second cumulant. The Gaussian approximation is valid for small n 7 only. Panels Fig. 4(b,c) show the cumulants and the moments for the active rotator model. In panel (b) we illustrate the situation where the Gaussian approximation is superior to the OA one. One can see that the system practically perfectly obeys the n 2 -scaling law for R n . On the other hand, the gap between the first and the second cumulants is not large, which means that the OA ansatz is poor (see Fig. 3 for D = 0.01, K = 1); the inaccuracy of the OA solution is compatible to the deviation of R 1 from 1. The case in panel (b) is the case of high synchrony. The plots in panel (c) are similar to those in panel (a); here only a few first values R n follow the n 2 -scaling law (10). On the other hand, the gap between the first and the second cumulants is present, and the OA ansatz becomes acceptably accurate (see Fig. 3 for D = 1, K = 1); here the inaccuracy of the OA solution is one order of magnitude smaller than the deviation of R 1 from 1. Remarkably, in all the cases one observes that higher cumulants decay exponentially κ n ∝ exp[−const n]. This law has been derived in Ref. 12 for small D only; here we see that it is valid for moderate and strong noise as Gaussian approximation (12) The Gaussian approximation is superior to the OA one for high synchrony, if the distortion of the perfect synchrony is not dominantly due to a non-Gaussian disorder (e.g. Lorentzian distribution of natural frequencies). Two-cumulant reduction with κ3 = 0 (16,17) This plain first-order correction to the OA solution is frequently superior to the Gaussian approximation, but may have the same (low) accuracy as the OA ansatz, where the latter completely fails. well. For a strong noise, there is a small parameter (1/D) which can serve for a hierarchy in the system, κ n+1 ∼ (1/D)κ n . For a moderate noise strength, there is no small parameter, but nevertheless, a hierarchy is present. VI. CONCLUSION We have compared five low-dimensional approximations describing the dynamics of large populations of noisy phase oscillators (or active rotators) with global sine-coupling: Eqs. (9), (12), (16,17), (16,18), and (16,19); the latter two cases are novel two-cumulant truncations within the framework of circular cumulant formalism. As prototypic examples, we have chosen the standard Kuramoto model and the active rotator model in the excitable state regime. Tabel I summarizes applicability of different low-dimensional reductions. The truncation with the closure according to κ 3 = (3/2)κ 2 2 /κ 1 , which most accurately corresponds to the Gaussian reduction under high synchrony, deserves special attention. By construction, this truncation is simultaneously a first-order correction to the Ott-Antonsen ansatz, and comprises the wrapped Gaussian distribution of phases, where the latter can be formed. In all the cases considered, this two-cumulant approximation is significantly superior to all other approximations. Re-markably, even for the cases, where R n nearly perfectly follows the n 2 -scaling law, this two-cumulant approximation enhances the accuracy of the Gaussian one, by a few orders of magnitude. Generally, a high synchrony is not a sufficient condition for applicability of the Gaussian ansatz. In this paper, our analysis has been restricted to the situations of synchronization by coupling. However, for synchronization by a common noise [13][14][15] , in nonideal situations (i.e., with intrinsic noise and/or nonidentity of elements), it is known that the phase distribution possesses heavy power-law tails even in the limit of high synchrony 16,17 . For such systems, the Gaussian ansatz is never natural. k 2 2FIG. 1 . 21) exp[kZ 1 ], and obtain for the moments Z m = The normalized perturbation of the phase distribution wC/|κ2| (solid lines, right axis) is compared to the OA distribution wOA for |Z1| = 0.4 (black dashed line, left axis). below). The OA equation(9) is exact for noiseless populations D = 0. The Gaussian ansatz (12) is exact for noisy oscillators without coupling h = 0. The two-cumulant (2C) approximations(16) with closures (17), (18), or (19) reduce to the OA ansatz for D = 0 if one sets κ 2 = 0, and in fact are the first-order corrections in the noise intensity D; we will use them, however, for large values of D as well. FIG. 2 . 2The accuracy of solutions for the noisy Kuramoto ensemble vs coupling strength K is plotted with blue solid squires for the Ott-Antonsen ansatz (20), red solid diamonds for the Gaussian approximation(21), black open squares for the 2C truncation with closure κ3 = 0 (22), black open circles for closure κ3 = 1.5κ 2 2 Z * 1 , black solid circles for closure κ3 = 1.5κ 2 2 /Z1 (23) (the cusp at a large noise strength is due to the change of sign of the error). Bold solid lines: the exact solution for the order parameter R1. Parameters are rescaled so that γ + D = 1: noise intensity D is specified in plots, Kcr = 2(γ + D) = 2. The case of vanishing intrinsic noise corresponds to D = 0, the case of identical natural frequencies (or extremely strong noise) corresponds to D = 1. FIG. 3 . 3The difference between the exact steady-state solution for the population of noisy active rotators with Kuramoto-type coupling (3) and different approximations vs the coupling strength K. Blue solid squares: the OA ansatz(9); red solid diamonds: the Gaussian ansatz(12); black open squares: the two-cumulant reduction(16) with closure(17); black open circles: reduction (16) with (19); black solid circles: reduction (16) with (18). Bold solid lines: the order parameter R1 for the accurate solution. Parameters: ω0 = 1, a = 3, noise intensity D is specified in the panels. FIG. 4 . 4The scaling law for the Kuramoto-Diado order parameters Rn (circles) and the hierarchy of smallness for the circular cumulants κn (squares) are plotted for the noisy Kuramoto system with D = 0.75, γ = 0.25, K = 3Kcr (a) and the population of active rotators with ω = 1, a = 3, K = 1, γ = 0, D = 0.01 (b) and D = 1 (c). The solutions of system(15) are calculated with 200 cumulants. Dotted lines show the trends Rn ∼ R n 2 1 and κn ∝ s 2(n−1) . TABLE I . ILow-dimensional model reductionsReduction Eqs. Comments OA ansatz (9) /Z1. For low synchrony, it is as accurate as the plain twocumulant truncation (κ3 = 0).Two-cumulant reduction with κ3 = 3 2 κ 2 2 /Z1 (16,18) It works as a first-order correction to the best of OA and Gaussian ap- proximations. Not to be used for problems where Z1 can approach 0 without κ 2 2 /Z1 remaining finite. Two-cumulant reduction with κ3 = 3 2 κ 2 2 Z * 1 (16,19) For high synchrony, its accuracy approaches the accuracy of closure κ3 = 3 2 κ 2 2 ACKNOWLEDGMENTSThe authors thank M. Zaks for fruitful discussions and Z. Levnajic for bringing paper 8 to our attention. Work of A.P. on Secs. II, III, V.C was supported by Russian Science Foundation (Grant Nr. 17-12-01534). Work of D.S.G. and L.S.K. on Secs. IV, V.A-B was supported by Russian Science Foundation (Grant Nr. 14-21-00090). A Pikovsky, M Rosenblum, J Kurths, Synchronization: A Universal Concept in Nonlinear Sciences. CambridgeCambridge University PressA. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001, 2003). Sync (Hyperion. S Strogatz, S. Strogatz, Sync (Hyperion, 2003). Analysis of a power grid using a Kuramoto-like model. G Filatrella, A H Nielsen, N F Pedersen, Eur. Phys. J. B. 61G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B 61, 485-491 (2008). A T Winfree, The Geometry of Biological Time. SpringerA. T. Winfree, The Geometry of Biological Time (Springer, 2001). . J A Acebrón, L L Bonilla, C J P Vicente, F Ritort, R , J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. The Kuramoto model: A simple paradigm for synchronization phenomena. Spigler, Rev. Mod. Phys. 77Spigler, The Kuramoto model: A simple paradigm for synchro- nization phenomena, Rev. Mod. Phys. 77, 137-185 (2005). A Pikovsky, M Rosenblum, Dynamics of globally coupled oscillators: progress and perspectives. 2597616A. Pikovsky and M. Rosenblum, Dynamics of globally coupled oscillators: progress and perspectives, Chaos 25, 097616 (2015). Low dimensional behavior of large systems of globally coupled oscillators. E Ott, T M Antonsen, Chaos. 1837113E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos 18, 037113 (2008). Macroscopic models for networks of coupled biological oscillators. K M Hannay, D B Forger, V Booth, Sci. Adv. 41701047K. M. Hannay, D. B. Forger, and V. Booth, Macroscopic mod- els for networks of coupled biological oscillators, Sci. Adv. 4, e1701047 (2018). Noise-controlled oscillations and their bifurcations in coupled phase oscillators. M A Zaks, A B Neiman, S Feistel, L Schimansky-Geier, Phys. Rev. E. 6866206M. A. Zaks, A. B. Neiman, S. Feistel, and L. Schimansky-Geier, Noise-controlled oscillations and their bifurcations in coupled phase oscillators, Phys. Rev. E 68, 066206 (2003). Approximate solution to the stochastic Kuramoto model. B Sonnenschein, L Schimansky-Geier, Phys. Rev. E. 8852111B. Sonnenschein and L. Schimansky-Geier, Approximate solu- tion to the stochastic Kuramoto model, Phys. Rev. E 88, 052111 (2013). Collective dynamics in two populations of noisy oscillators with asymmetric interactions. B Sonnenschein, Th K Dm, F A Peron, J Rodrigues, L Kurths, Schimansky-Geier, Phys. Rev. E. 9162910B. Sonnenschein, Th. K. DM. Peron, F. A. Rodrigues, J. Kurths, and L. Schimansky-Geier, Collective dynamics in two popula- tions of noisy oscillators with asymmetric interactions, Phys. Rev. E 91, 062910 (2015). Dynamics of Noisy Oscillator Populations beyond the Ott-Antonsen Ansatz. I V Tyulkina, D S Goldobin, L S Klimenko, A Pikovsky, Phys. Rev. Lett. 120264101I. V. Tyulkina, D. S. Goldobin, L. S. Klimenko, and A. Pikovsky, Dynamics of Noisy Oscillator Populations beyond the Ott- Antonsen Ansatz, Phys. Rev. Lett. 120, 264101 (2018). Synchronization and stochastization of array of self-excited oscillators by external noise. A S Pikovskii, Radiophys. Quantum Electron. 27390A. S. Pikovskii, Synchronization and stochastization of array of self-excited oscillators by external noise, Radiophys. Quantum Electron. 27, 390 (1984). Robustness of the Noise-Induced Phase Synchronization in a General Class of Limit Cycle Oscillators. J N Teramae, D Tanaka, Phys. Rev. Lett. 93204103J. N. Teramae and D. Tanaka, Robustness of the Noise-Induced Phase Synchronization in a General Class of Limit Cycle Oscil- lators, Phys. Rev. Lett. 93, 204103 (2004). Synchronization of periodic self-oscillations by common noise, Radiophys. Quantum Electron. D S Goldobin, A S Pikovsky, 47910D. S. Goldobin and A. S. Pikovsky, Synchronization of periodic self-oscillations by common noise, Radiophys. Quantum Elec- tron. 47, 910 (2004). Synchronization and desynchronization of self-sustained oscillators by common noise. D S Goldobin, A Pikovsky, Phys. Rev. E. 7145201D. S. Goldobin and A. Pikovsky, Synchronization and desyn- chronization of self-sustained oscillators by common noise, Phys. Rev. E 71, 045201(R) (2005). Interplay of coupling and common noise at the transition to synchrony in oscillator populations. A V Pimenova, D S Goldobin, M Rosenblum, A Pikovsky, Sci. Rep. 638518A. V. Pimenova, D. S. Goldobin, M. Rosenblum, and A. Pikovsky, Interplay of coupling and common noise at the tran- sition to synchrony in oscillator populations, Sci. Rep. 6, 38518 (2016).
[]
[ "Spin and quadrupole contributions to the motion of astrophysical binaries", "Spin and quadrupole contributions to the motion of astrophysical binaries" ]
[ "Jan Steinhoff \nCentro Multidisciplinar de Astrofísica (CENTRA)\nInstituto Superior Técnico (IST)\nAvenida Rovisco Pais 11049-001LisboaPortugal\n" ]
[ "Centro Multidisciplinar de Astrofísica (CENTRA)\nInstituto Superior Técnico (IST)\nAvenida Rovisco Pais 11049-001LisboaPortugal" ]
[]
Compact objects in general relativity approximately move along geodesics of spacetime. It is shown that the corrections to geodesic motion due to spin (dipole), quadrupole, and higher multipoles can be modeled by an extension of the point mass action. The quadrupole
10.1007/978-3-319-18335-0_19
[ "https://arxiv.org/pdf/1412.3251v3.pdf" ]
37,252,982
1412.3251
fe0e43553c6b0ef5cf9ca05d386b9df525302ca8
Spin and quadrupole contributions to the motion of astrophysical binaries March 19th, 2014 Jan Steinhoff Centro Multidisciplinar de Astrofísica (CENTRA) Instituto Superior Técnico (IST) Avenida Rovisco Pais 11049-001LisboaPortugal Spin and quadrupole contributions to the motion of astrophysical binaries March 19th, 2014 Compact objects in general relativity approximately move along geodesics of spacetime. It is shown that the corrections to geodesic motion due to spin (dipole), quadrupole, and higher multipoles can be modeled by an extension of the point mass action. The quadrupole Introduction The problem of motion is among of the most fundamental ones in general relativity. As a part of the present proceedings on "Equations of Motion in Relativistic Gravity" this does probably not require any explanations. The problem is addressed using multipolar approximation schemes, the most prominent are due to Mathisson [1,2] and Dixon [3], and another one is due to Papapetrou [4]. These particular methods have in common that equations of motion for extended bodies are derived from the conservation of energy-momentum. In the present contribution, the focus lies on theoretical models for compact stars and black holes based on point-particle actions. There equations of motions follow from a variational principle instead of conservation of energy-momentum. These point-particle actions were probably first discussed in general relativity by Westpfahl [5] for the case of a pole-dipole particle and later generalized by Bailey and Israel [6] to generic multipoles. However, without further justification, it is not obvious how a pointparticle action relates to an actual extended body. Most important is the effacing principle [7], which indicates that a nonrotating star can be represented by a point mass up to a high order within the post-Newtonian approximation. (More details on the use of point-masses for self-gravitating bodies within this approximation can be found in other contributions to these proceedings, see, e.g., the contribution by G. Schäfer.) This suggests that extensions of the point-mass action can serve as models for extended bodies, even in the self-gravitating case. A similar conclusion arises from the framework of effective field theory applied to gravitating compact bodies [8] (which is also covered by a different contribution to these proceedings). Indeed, the effective action belonging to a compact body naturally takes on the form of a point-particle action, which puts previous works on similar actions [5,6] into a different light. This provides enough motivation for us to further elaborate the action approach of [6] in Sec. 2, where it is combined with useful aspects of more recent literature [9][10][11][12][13]. An application to the post-Newtonian approximation of self-gravitating extended bodies is omitted, because various formalisms exist for it and the aim is to highlight aspects that are independent of (and hopefully useful for) all of them. It is worth mentioning that the effective field theory framework offers a machinery which can be used, at least in principle, to compute the effective point-particle action from a complete microphysical description of the extended body. In practice, however, this procedure is not viable for realistic astrophysical objects and one must be satisfied with a more phenomenological construction of the effective action. This is in fact analogous to other situations in physics. For instance, it is usually admitted that thermodynamic potentials can be derived from a microscopic description. Yet an explicit calculation is often too complicated, or the microscopic description is even unclear. But a phenomenological construction of thermodynamic potentials or equations of state is usually possible. This analogy is further elaborated in Sec. 5. There an adiabatic quadrupole deformation due to spin [14] is discussed. An application to a binary system in the extreme mass ratio case is given. Quadrupole deformation due to an external gravitational fields is discussed in Secs. 6-4, both in an adiabatic [15][16][17] and a dynamical situation [18,19]. A main critique against point-particles arises from the fact that Dirac delta distributions are ill-defined sources for the nonlinear Einstein equations. But the situation changes once one softens the Dirac delta using regularization techniques. It is then possible to solve the field equations iteratively within some approximation, like the post Newtonian one. If one regards the chosen regularization prescription as a part of the phenomenological model, then point-particles must be accepted as viable sources in general relativity (at least for applications within approximation schemes). This point is further stressed in Sec. 6.4. It is important that a weak field approximation for the point-particle mimics the field of the actual self-gravitating extended body away from the source. (This is precisely the criterion for the phenomenological construction of the effective point-particle source.) Hence, though one applies the effective source to weak field approximations, e.g., to compute predictions for a binary, strong-field effects from the interior of the bodies are taken into account. The signature of spacetime is taken to be +2. Units are such that the speed of light c is equal to one. The gravitational constant is denoted by G. We are going to utilize three different frames, denoted by different indices. Greek indices refer to the coordinate frame, lower case Latin indices from the beginning of the alphabet belong to a local Lorentz frame, and upper case Latin indices from the beginning of the alphabet denote the so called body-fixed Lorentz frame. Round and square brackets are used for index symmetrization and antisymmetrization, respectively, e.g., A (µν) ≡ 1 2 (A µν + A νµ ). The convention for the Riemann tensor is R µ ρναβ = Γ µ νβ,α − Γ µ να,β + Γ ρ νβ Γ µ ρα − Γ ρ να Γ µ ρβ . (1) Point-particle actions Action principles for spinning point particles have a long tradition, see, e.g., [5,6,[9][10][11][12][13][20][21][22][23][24]. In this section, the advantages from several of these references are brought together. Our approach is most similar to [6]. Compared to the presentation in [11], a simpler (manifestly covariant) variation technique is applied and the transition to tetrad gravity is discussed at a later stage. This makes the derivation more transparent. Manifestly covariant variation Before we start to formulate the action principle, let us introduce a useful notation due to B. S. DeWitt [25], see also [12, appendix A]. One can define a linear operator G ν µ such that the covariant derivative ∇ α and the Lie derivative L ξ read ∇ α := ∂ α + Γ µ να G ν µ , L ξ := ξ µ ∂ µ − (∂ ν ξ µ )G ν µ .(2) For instance, G ν µ operates on a tensor T α β as G ν µ T α β := −δ ν α T µ β + δ β µ T α ν . That is, G ν µ is a linear operator that acts on the spacetime indices of a tensor. Notice that G ν µ does not act on indices of the body-fixed frame. Further, G ν µ obeys a product rule like a differential operator. Similarly, we can construct a covariant differential D and a covariant variation ∆ of quantities defined along a worldline z α by D := d + Γ µ να (dz α )G ν µ , ∆ := δ + Γ µ να (δz α )G ν µ .(3) For scalars the contributions from the connection vanish. Notice that a variation of the worldline δz α is not manifestly covariant if the component values of tensors defined on the worldline are held fixed. The variation ∆ instead parallel transports to the varied worldline. When it is applied to a tensor field taken at the worldline, e.g., T α β (z), then the variation δ splits into a part due to the shift of the worldline δz ρ and a part coming from the variation of the field itself. Let us denote the latter part by δ z T α β := (δT α β )(z), so we have δ ≡ δ z + (δz ρ )∂ ρ , ∆ ≡ δ z + (δz α )∇ α , (for fields).(4) For instance, the metric compatibility of ∇ α then leads to ∆g µν = δ z g µν . Action principle We envisage an action principle localized on a worldline z ρ (λ). Here λ is an arbitrary parameter, not necessarily identical to the proper time τ . (Let us require that the action is invariant under reparametrizations of the worldline). We further assume that the action is varied with respect to a "body-fixed" frame defined by Lorentz-orthonormal basis vectors Λ A µ (λ) labeled by A, Λ A µ Λ B ν g µν ≡ η AB , Λ A µ Λ B ν η AB ≡ g µν .(6) Now, stars are in general differentially rotating and it is difficult to interpret a body-fixed frame. Such a frame is thus rather an abstract element of our theoretical model, inspired by the Newtonian theory of rigid bodies (see, e.g., [11,Sec. 3.1.1]). The constraint (6) implies that Λ A µ and g µν are in general not independent and one should take special care when both are varied at the same time. In order to address this problem, we split the variation ∆Λ A ν as Λ Aµ ∆Λ A ν = Λ A[µ ∆Λ A ν] + 1 2 ∆(Λ Aµ Λ A ν ) = ∆Θ µν − 1 2 g µα g νβ δ z g αβ . (7) where we used δg µν = −g µα g νβ δg αβ and (5). In the last step, we also introduced the abbreviation ∆Θ µν := Λ A[µ ∆Λ A ν] ,(8) which is similar to the antisymmetric variation symbol used in [22], see also [13,Eq. (2.7)]. The independence of ∆Θ µν from the metric variation δ z g αβ will be made more manifest in Sec. 2.4. For now let us just appeal to the fact that the 6 degrees of freedom of the antisymmetric symbol ∆Θ µν exactly matches the degrees of freedom of a Lorentz frame (3 boosts and 3 rotations). Thus ∆Θ µν corresponds to the independent variation of the body-fixed Lorentz frame. Let us consider an action that is as generic as possible, W = W F + W M , W F [g µν , . . . ] = 1 16πG d 4 x √ −gR + . . . , (9) W M [g µν , z ρ , Λ A µ , . . . ] = dλ L M (g µν , u µ , Λ A µ , Ω µν , φ I ),(10) Here φ I collectively denotes other dependencies of the Lagrangian L M and the dots denote other fields, like the electromagnetic one. (In this section I is a multi-index that may comprise any sort of spacetime, Lorentz, or label indices.) Notice that fields like g µν are taken at the worldline position z ρ in W M . The 4-velocity u µ and the angular velocity Ω µν are defined by u µ := dz µ dλ , Ω µν := Λ Aµ DΛ A ν dλ ,(11) Notice that Ω µν is antisymmetric due to (6) and Dg µν /dλ = 0. Variation For the sake of deriving equations of motion, we may assume δλ = 0. Then the variation can be commuted with ordinary or partial λ-derivatives. Furthermore, the Lagrangian L M is a scalar and we can make use of δL M ≡ ∆L M to write its variation in a manifestly covariant manner, δL M = p µ ∆u µ + 1 2 S µν ∆Ω µν + ∂L M ∂Λ A µ ∆Λ A µ + ∂L M ∂g µν ∆g µν + ∂L M ∂φ I ∆φ I ,(12) where we have defined the linear momentum p µ and spin S µν = −S νµ as generalized momenta belonging to the velocities u µ and Ω µν , p µ := ∂L M ∂u µ , S µν := 2 ∂L M ∂Ω µν .(13) It should be noted that (12) can be checked using a usual variation δ together with the identity (21), but here it is a simple consequence of the chain rule for ∆. Obviously this method nicely organizes the Christoffel symbols. The 5 individual terms in (12) are transformed as follows: • The 1st term of (12) is evaluated with the help of ∆u µ ≡ δu µ + Γ µ αβ u α δz β = Dδz µ dλ .(14) • The 2nd term of (12) requires the most work. In order to evaluate ∆Ω µν , we need to commute ∆ with the covariant differential D contained in Ω µν , Eq. (11). The definitions in (3) lead to [∆, D] = [(δ z Γ µ να ) − (δz β )R µ ναβ ](dz α )G ν µ .(15) Notice the analogy to the commutator of covariant derivatives, which also gives rise to curvature. It is useful to derive intermediate commutators first, for instance [G ν µ , G β α ] = δ β µ G ν α − δ ν α G β µ .(16) Next, we express δ z Γ µ να in (15) with the help of δΓ ν βα = 1 2 g νρ [∇ β δg αρ + ∇ α δg βρ − ∇ ρ δg αβ ] .(17) Now it is straightforward to evaluate ∆Ω µν . In the result, we replace ∆Λ A µ using (7), make use of Dδ z g ρσ dλ = u α (∇ α δg ρσ )(z),(18) and finally arrive at ∆Ω µν = D(∆Θ µν ) dλ + 2Ω α [µ ∆Θ ν]α + R µν αβ u α δz β + Ω α[µ g ν]β δ z g αβ + g β[µ g ν]ρ u α (∇ β δg ρα )(z).(19) • Before proceeding to the 3rd term of (12), let us recall the transformation property of a tensor under an infinitesimal coordinate transformation x µ = x µ − ξ µ , φ I − φ I = −(∂ ν ξ µ )G ν µ φ I , e.g., u µ − u µ = −u ν ∂ ν ξ µ .(20) The Lagrangian is a scalar and thus invariant, but it depends on tensors which transform. As ∂ ν ξ µ is quite arbitrary, the invariance of the Lagrangian L M leads to the identity p µ u ν + S µα Ω να + ∂L M ∂Λ A µ Λ A ν − 2 ∂L M ∂g να g µα + ∂L M ∂φ I G ν µ φ I ≡ 0. (21) We eliminate the partial derivative of L M with respect to Λ A µ using this relation and we replace ∆Λ A µ using (7) to arrive at ∂L M ∂Λ Aµ ∆Λ Aµ = 1 2 p µ u ν − S α µ Ω να + (G µν φ I ) ∂L M ∂φ I − 2 ∂L M ∂g µν δ z g µν + p µ u ν − S αµ Ω ν α − (G µν φ I ) ∂L M ∂φ I ∆Θ µν .(22) • In the 4th term of (12) we use (5). • The 5th term of (12) is not touched for now, as this requires a specialization of φ I . This is discussed in the Sec. 3.1. All these transformations are now applied to (12). Furthermore, we insert a unity in the form of 1 ≡ d 4 x δ (4) , δ (4) := δ(x µ − z µ ),(23) into the terms containing field variations of type δ z . This allows one to rewrite these variations at the spacetime point x µ and perform partial integrations. Notice that δ (4) has compact support for finite λ-intervals, so these partial integrations do not require assumptions on field variations at the spatial boundary. Finally, (12) turns into δL M = d 4 x p µ u ν δ (4) + (G µν φ I ) ∂L M ∂φ I δ (4) − ∇ α (S αµ u ν δ (4) ) δg µν (x) 2 + ∂L M ∂φ I ∆φ I + p µ u ν − (G µν φ I ) ∂L M ∂φ I − 1 2 DS µν dλ ∆Θ µν(24)+ 1 2 S αβ R αβ ρµ u ρ − Dp µ dλ δz µ + d dλ p µ δz µ + 1 2 S µν ∆Θ µν . Notice that the covariant and ordinary derivatives with respect to λ are identical for the last term. Metric versus tetrad gravity The separation of metric and body-fixed-frame variations by means of (7) is an elegant trick to derive equations of motion. However, if further calculations at the level of the action are performed, one often needs an explicit split between gravitational and body-fixed-frame degrees of freedom. This can be achieved by introducing a tetrad gravitational field e a µ (x), that is, a field of Lorentz-orthonormal basis vectors labeled by a and defined at every spacetime point x ρ , e a µ e b ν g µν ≡ η ab , e a µ e b ν η ab ≡ g µν .(25) Tetrad gravity replaces the metric by virtue of the latter relation and regards e a µ as the fundamental gravitational field in the variation principle. This allows us to split Λ A µ as Λ A µ = Λ A a e a µ (z),(26) where Λ A a is now just a usual (flat-spacetime) Lorentz matrix Λ A a Λ B b η ab ≡ η AB , Λ A a Λ B b η AB ≡ η ab .(27) Thus Λ A a is independent of the gravitational field e a µ and the announced manifest split is indeed given by (26). Based on this split, we can understand the meaning of ∆Θ µν in more detail. As Λ A a is a usual Lorentz matrix, we can follow Ref. [22] and describe its independent variations by an antisymmetric symbol δθ ab := Λ Aa δΛ A b . Then ∆Θ µν reads explicitly ∆Θ µν = e a µ e b µ δθ ab + (Γ [νµ] α + e a[µ ∂ α e a ν] )δz α + e a[µ δ z e a ν] .(28) One can be even more explicit and write δθ ab as a linear combination of six independent variations of angle variables parameterizing Λ Aa , see [22,Sec. 3.A]. Anyway, ∆Θ µν is in fact a linear combination of the independent frame variations δθ ab with other variations. Now, it is legitimate to regard ∆Θ µν , δz α , and δe a µ as independent variations instead of δθ ab , δz α , and δe a µ . This just corresponds to a linear recombining of the equations of motion following from the variation. Equation (28) shows that this recombination manifestly removes noncovariant terms related to the δz α -variation and an antisymmetric part of the energy-momentum tensor due to e a[µ δ z e a ν] (the symmetric part arises from e a(µ δe a ν) = 1 2 δg µν as usual). All of this is important for the next section, where equations of motion are deduced from (24). In the next step it is possible to return to metric gravity by a partial gauge fixing of the tetrad. For instance, a possible gauge condition is to require that the matrix (e aµ ) is symmetric (in spite of the different nature of its indices). Then e a µ is given by the matrix square-root of the metric. This gauge choice leads to the same conclusions as in [9, Sec. IV.B], where a more direct construction was followed. In the end, the partially gaugefixed tetrad is a function of the metric, so we have obtained a metric gravity theory. It might look like the introduction of a tetrad field accompanied by an enlarged gauge group of gravity is just extra baggage. However, more gauge freedom is important for applications, as gauges can and should be adopted to the problem at hand. For instance, for an ADM-like canonical formulation of spinning particles, it is a wise choice to adopt the Schwinger time-gauge for the tetrad [10]. Further, some subtle aspects of the consistency of the theory can be analyzed more easily within tetrad gravity (e.g., the algebra of gravitational constraints, because after reduction to metric gravity the gravitational field momentum receives complicated corrections [10]). Spinning particles should always be coupled to tetrad gravity in the first place. Equations of motion In order to draw conclusions from (24), one must further specialize the so far arbitrary φ I . The assumptions we are going to introduce in the following are not the least restrictive, but already allow important insights on the structure of the equations of motion. Further assumptions Let us assume from now on that the φ I can be split into two groups. We denote by φ field I the part that contains spacetime fields (functions of x), so its variation ∆φ field I can be evaluated using (4). The second group φ wl I contains variables defined on the worldline only (functions of λ) and its first order derivativesφ wl I , where˙:= D/dλ. Most importantly, we assume that the δφ wl I correspond to independent variational degrees of freedom, like Lagrange multipliers or the dynamical multipoles introduced in Sec. 6. Without loss of generality, one can then assume that the φ wl I carry indices of the body-fixed frame instead of spacetime indices, so that G ν µ φ wl I = 0. Notice that our assumptions do not allow time (i.e., λ) derivatives of u µ and Ω µν as part of the φ wl I . If such accelerations would appear in subleading contributions of the Lagrangian (within some approximation scheme), then one can often remove them by a redefinition of variables [26]. Further, acceleration-dependent Lagrangians are often problematic due to Ostrogradsky instability. For these reasons, we also assumed that at most first-order time derivatives of the φ wl I appear in L M . However, our assumptions here are not entirely exhaustive. For instance, a concrete situation for which our assumptions should be relaxed in the future is discussed at the end of Sec. 5.1. Equations of motion for linear momentum and spin With these assumptions, we have ∂L M ∂φ I ∆φ I = ∂L M ∂φ field I [δ z φ field I + (δz α )∇ α φ field I ] + ∂L M ∂φ wl I − dψ I wl dλ δφ wl I + d dλ ψ I wl δφ wl I ,(29) where we used that the worldline variables do not carry spacetime indices and we introduced their canonical generalized momenta, ψ I wl := ∂L M ∂φ wl I .(30) The second line leads to the usual Euler-Lagrange equtions for the worldline degrees of freedom φ wl I , which are discussed in Sec. 6.4. Let us focus on the other terms for now. Using the arbitrariness and independence of δz µ and ∆Θ µν , we can read off the equations of motion for the linear momentum and the spin from (24) and (29), Dp µ dλ = 1 2 S αβ R αβ ρµ u ρ + (∇ µ φ field I ) ∂L M ∂φ field I ,(31)DS µν dλ = 2p [µ u ν] − 2(G [µν] φ field I ) ∂L M ∂φ field I .(32) The total λ-derivative in the last line of (24) was ignored here. (Here we assume that the variation vanishes at the end points of the worldline). The energy-momentum tensor density √ −gT ab is simply given by the coefficient in front of δg µν /2 in (24). However, an explicit determination requires yet another specialization of φ field I , because the fields can dependent on the metric. In the absence of φ field I , one immediately recovers the result of Tulczyjew [27]. Quadrupole Let us now explore the case that φ field I = {R µναβ }. It is useful to introduce an abbreviation for the corresponding partial derivative of the Lagrangian, J µναβ := −6 ∂L M ∂R µναβ .(33) The conventional factor of −6 is motivated by comparing (31), now reading Dp µ dλ = 1 2 S αβ R αβ ρµ u ρ − 1 6 ∇ µ R νρβα J νρβα ,(34) with the corresponding result of Dixon at the quadrupolar approximation level. An identification of J µναβ with Dixon's reduced quadrupole moment is tempting, as this makes (34) formally identical to Dixon's result. It is important that J µναβ inherits the symmetries of the Riemann tensor. From these symmetries and the properties of the operator G µ ν , we obtain J σραβ G µ ν R σραβ = −4J σραβ δ µ σ R νραβ = −4J µραβ R νραβ .(35) This simplifies (32) to DS µν dλ = 2p [µ u ν] + 4 3 R αβρ[µ J ν] ρβα ,(36) and formally agrees with Dixon's spin equation of motion, too. Finally, the energy-momentum tensor agrees with the explicit result in [28] (in the present conventions, see (5.3) in [11]). This derives from δR µ ναβ = ∇ α δΓ µ νβ − ∇ β δΓ µ να ,(37) which must be further expanded using (17) and then leads to ∂L M ∂R µναβ δ z R µναβ = d 4 x − 1 3 J µραβ R ν ραβ δ (4) − 2 3 ∇ β ∇ α (J µαβν δ (4) ) δg µν 2 . (38) This is the contribution coming from the first term in (29). Another contribution arises from the second term in the first row of (24), which is evaluated using (35). Collecting all terms in front of δg µν in (24), we can read off the energy momentum tensor density as √ −gT µν = dλ u (µ p ν) δ (4) − ∇ α (S α(µ u ν) δ (4) ) + 1 3 R βαρ (µ J ν)ραβ δ (4) − 2 3 ∇ β ∇ α (J µ(αβ)ν δ (4) ) .(39) Other multipoles Dixon's moments are essentially defined as integrals over the energy-momentum tensor of the extended body. Though these definitions can be applied to selfgravitating bodies, the derivation of the equations of motions based on these definitions only succeeds for test-bodies [3]. It was shown in [29] (see also the corresponding contribution by A. Harte in these proceedings) using methods for self-force calculations that for self-gravitating objects the equations of motions are still of the same form, but the multipole moments must be renormalized. The multipoles arising from the effective action should therefore be related to these renormalized moments. For self-gravitating bodies, one can not in general calculate the moments in the equations of motion using Dixon's integral formulas any more. In the language of effective field theory, the multipoles are calculated through a "matching" procedure instead, which will be explained in Sec. 5. Other gravitational multipoles can be incorporated by including symmetrized covariant derivatives of the curvature in φ field I . Similarly, electromagnetic multipoles arise from an analogous construction based on the Faraday tensor F µν . A quite exhaustive case is therefore φ field I = {R µναβ , ∇ ρ R µναβ , ∇ (σ ∇ ρ) R µναβ , . . . , F µν , ∇ (ρ F µ)ν , ∇ (σ ∇ ρ F µ)ν , . . . }. (40) Notice that the commutation of covariant derivatives results in curvature terms and that, e.g., 3∇ α F µν = 2∇ (α F µ)ν − 2∇ (α F ν)µ , which can be checked using ∇ [α F µν] = 0. Again the partial derivatives of L M with respect to the φ field I can be called multipole moments. However, these multipoles and also p µ are probably not unique, because L M in not unique. For instance, contractions of covariant derivatives with u µ can be written as λ-derivatives and one can partially integrate them. Notice that Dixon's multipole moments have the same symmetries as ours, but satisfy additional orthogonality relations to a timelike vector defined on the worldline. It is also possible to include a term proportional to A µ u µ in the Lagrangian, as this combination transforms into a total λ-derivative under a gauge transformation of the electromagnetic potential A µ . It just leads to the well-known Lorentz force. However, in the present approach a part of the Lorentz force is hidden in the definition of p µ , making the equations of motion not manifestly gauge-invariant. Symmetries, transformations, and conditions In this section we discuss symmetries, conservations laws, various transformations of the action, and conditions it must fulfill. Symmetries and conserved quantities Action principles have the advantage that one can easily derive conserved quantities from the Noether theorem [30]. Here we are going to consider only symmetry transformations where the fields are not transformed. Further, we assume δλ = 0, so the variational formula (24) together with (29) is still valid. On the one hand, we require that the Lagrangian transforms under such a symmetry into a total derivative δL M = dK dλ ,(41) without making use of the equations of motion. On the other hand, if we assume that the equations of motion hold, then only the total time derivatives from (24) with (29) inserted contribute to δL M . These total derivatives are located in the last lines of (24) and (29). (The first lines of (24) and (29) vanishes because fields are not transformed here.) We therefore have the conservation law d dλ p µ δz µ + 1 2 S µν ∆Θ µν + ψ I wl δφ wl I − K = 0.(42) A simple example is given by the global symmetry under a change of the body-fixed frame. In order to make things even more simple, we assume that L M does not depend on Λ A µ and on theφ wl I , so that ψ I wl = 0. But L M still implicitly depends on Λ A µ through Ω µν . A constant infinitesimal Lorentz transformation of the body-fixed frame then reads δz a = 0, δΛ Aµ = ω AB Λ B µ ,(43) where ω AB is a constant infinitesimal antisymmetric matrix. Obviously Ω µν is invariant under this transformation, so (41) is fulfilled with K = 0. Further, we have ∆Θ ab = Λ A a Λ B b ω AB and (42) reads 1 2 ω AB d dλ S µν Λ A µ Λ B ν = 0.(44) As ω AB is arbitrary, we see that the components of the spin in the body- fixed frame S AB ≡ S µν Λ A µ Λ B ν are constant. A corollary of this fact is that the spin length S is constant, where 2S 2 = S AB S AB = S µν S µν . The next important example is a symmetry of the spacetime described by a Killing vector field ξ µ , L ξ g µν = 0. (Notice that also L ξ R abcd = 0 etc.) Other fields entering L M are assumed to be invariant under this symmetry, too, e.g., L ξ F µν = 0. We consider an infinitesimal shift of the worldline coordinate δz µ = ξ µ , ∆Λ A ν = − L ξ Λ A ν = Λ A µ ∇ µ ξ ν , δφ wl I = 0,(45) where is an infinitesimal constant and we assume parallel transport of Λ A µ along ξ ν , i.e., ξ ν ∇ ν Λ A µ = 0. Notice that the fields are not transformed, but their symmetry along ξ µ is important. Recall that − L ξ generates an infinitesimal coordinate transformation. Therefore, L M is invariant under this transformation if all the variables it depends on, including the fields, would be transformed by − L ξ . But the shift (45) only applies to z µ , Λ A µ , and φ wl I , so the result of (45) on L M is exactly opposite to the case when all variables except z µ , Λ A µ , and φ wl I are transformed. These variables are all the fields, so the δL M produced by (45) can be obtained by transforming all the fields using + L ξ . But the fields were assumed to be invariant under this transformation. Hence we have argued that (45) is a symmetry of the action, δL M = 0, and K = 0. Combining (42) and (45), we find the conserved quantity E ξ := p µ ξ µ + 1 2 S µν ∇ µ ξ ν = const,(46) where ∆Θ µν = ∇ [µ ξ ν] was used. It is interesting that this covers to all multipole orders. This was also shown in [31, p. 210] based on the equations of motion. A special kind of conserved quantities that is not covered by the Noether theorem here are mass-like quantities. We will see later on that masses enter the action as parameters and are therefore constant by assumption. Legendre transformations Before proceeding, it is worth to point out that of course not every conceivable Lagrangian L M is acceptable. Some choices are mathematically inconsistent or physically unacceptable for other reasons. Some Lagrangians L M are technically more difficult to handle and it makes sense to assume simplifying conditions for L M for a first study. One such assumption we make here is that the relation between spin and angular velocity is a bijection. Notice that this relation is fixed by L M through our definition of S µν in (13). A violation of our assumption can have the interesting implication that the spin supplementary condition follows from (13), see [22], but we will not considering this scenario here. The supplementary conditions are discussed in the next section. With this assumption on the relation between spin and angular velocity, we can solve for Ω µν in terms of S µν (and probably other variables). This allows a Legendre transformation in Ω µν , i.e., W M [e a µ , z ρ , Λ A µ , S µν , . . . ] = dλ 1 2 S µν Ω µν + R M (g µν , u µ , Λ A µ , S µν , φ I ) . (47) It is important that the spin is varied independently now. Notice that this notion of Legendre transformation is unusual in mechanics, as Ω µν is not a time derivative, but a combination of time derivatives. Still Legendre transformations are applicable in much more generic situations, which is heavily used, i.e., in thermodynamics. The function R M establishes the connection to Routhian approaches [32][33][34]. The Routhian is a mixture of a Hamiltonian and a Lagrangian. Here it is essentially the sum of R M and the connection part in 1 2 S µν Ω µν . Notice that therefore the Routhian is not manifestly covariant and covariance only becomes apparent at the level of the equations of motion. In contrast, in our construction R M is manifestly covariant. A consequence of reparametrization invariance is that L M must be a homogeneous function of degree one in all (first-order) λ-derivatives. For our assumptions in Sec. 3.1 this applies only to u µ , Ω µν , andφ wl I , so Eulers theorem on homogeneous functions reads L M = ∂L M ∂u µ u µ + ∂L M ∂Ω µν Ω µν + ∂L M ∂φ wl Iφ wl I = p µ u µ + 1 2 S µν Ω µν + ψ I wlφ wl I . (48) This is a consequence of reparametrization-gauge invariance, so in this sense it is analogous to (21), which follows from coordinate-gauge invariance. Let us proceed with a Legendre transformation in u µ . This is more subtle, as the relation between u µ and p µ can not be a bijection. To see this, first notice that (48) can be interpreted as a constraint on the component p µ u µ of p µ . This can be formulated as the famous mass-shell constraint p µ p µ + M 2 = 0,(49) where M is called dynamical mass and usually depends on the dynamical variables. Thus the momentum only has three independent components. On the other hand, u µ has four independent components: three physical and one gauge degree of freedom due to reparametrization invariance in λ. (If we would choose λ to be the proper time, then just 3 components are independent. But the constraint u µ u µ = −1 makes the variational principle more subtle.) That is, the constraint (49) produces a mismatch in degrees of freedom between u µ and p µ , so they can not be connected by a bijection. However, the Legendre transformation can in fact be generalized to the case where constraints appear. One can perform the Legendre transformation "as usual" if all constraints are added to the action using Lagrange multipliers [35,36]. Here we need one Lagrange multiplier α for (49), which together with the three independent components of p µ provides a total of four independent variables. This exactly matches the four independent degrees of freedom of u µ . The Lagrange multiplier α isolates the reparametrization-gauge degree of freedom, while p µ represents the physical degrees of freedom. Again we require L M to be such that no pathologies for this "constraint" Legendre transformation arise. Finally, the result of the transformation is W M [e a µ , z ρ , p µ , α, S µν , Λ A µ , φ wl I , ψ I wl , . . . ] = dλ p µ u µ + 1 2 S µν Ω µν + ψ I wlφ wl I − α 2 (p µ p µ + M 2 ) ,(50) where M = M(g µν , p µ , Λ A µ , S µν , φ wl I , ψ I wl , φ field I ).(51) We assume that we can also Legendre transform in theφ wl I without giving rise to further constraints or pathologies. From the variations of p µ , S µν , and ψ I wl , we obtain u µ = αp µ + α 2 ∂M 2 ∂p µ , Ω µν = α ∂M 2 ∂S µν ,φ wl I = α 2 ∂M 2 ∂ψ I wl .(52) These are just the inverses to variable transformations used in the Legendre transformations. Because we did not touch the variables g µν , φ wl I , Λ A µ and φ field I , it is clear that ∂L M ∂g µν ≡ − α 2 ∂M 2 ∂g µν , ∂L M ∂Λ A µ ≡ − α 2 ∂M 2 ∂Λ A µ ,(53)∂L M ∂φ wl I ≡ − α 2 ∂M 2 ∂φ wl I , ∂L M ∂φ field I ≡ − α 2 ∂M 2 ∂φ field I .(54) The Lagrange multiplier α is determined by choosing a normalization for u µ , which corresponds to a gauge choice for λ. For a given dynamical mass function M, one can then evaluate the equations of motion (31) and (32). Coming back to the plan outlined in the introduction, we have the option to construct either L M , R M , or M in a phenomenological manner. Let us explore the last option here, e.g., because it promotes both p µ and S µν to dynamical variables, which are probably easier to identify in realistic situations compared to u µ and Ω µν . Further, it is suggestive that the mass M of the object as a function of the dynamical variables completely determines the macroscopic dynamics of the body. This situation is analogous to a thermodynamic potential (like the internal energy) describing the largescale behavior of a thermodynamic system. This is the first indication that thinking in terms of thermodynamic analogies is very useful here. Supplementary conditions The model for spinning bodies developed up to now comprises too many degrees of freedom. We expect three rotational degrees of freedom instead of six provided by the Lorentz frame Λ A µ . Similarly, the spin should only have three independent components, too. It is suggestive to impose that the time direction of the body-fixed frame is aligned to a (to be defined) rest frame described by a unit time-like vector r µ , and that the spin only has spatial components in this rest frame, Λ 0 µ = r µ , S µν r ν = 0.(55) One can also envision different time-like vectors in each of these conditions. However, using the same vector seems to fit well to the interpretation of r µ as a rest frame. The condition on the spin is usually called spin supplementary condition. Two specific options are r µ = u µ / √ −u ν u ν or r µ = p µ / √ −p ν p ν . The latter condition is usually considered as the best choice, as it uniquely fixes the representative worldline of the extended object [37][38][39] (if Dixon's definitions for the multipoles are used). A more detailed discussion of supplementary conditions is given in the contributions by D. Giulini, L.F. Costa and J. Natário. But notice that in flat space the choice of this condition can be related to the choice of the representative worldline for the extended body. In curved spacetime this relation could not be established yet. From a careful perspective one should therefore reckon that different spin supplementary conditions may lead to inequivalent models. As long as the relation to the choice of center is not clarified for curved spacetimes, one must regard this condition as a constitutive relation of the model. For this reason, one should also avoid conditions which are not manifestly covariant. The most straightforward way to implement (55) into a given action is to add these conditions using Lagrange multipliers. In general, this will modify the dynamics by constraint forces. As in classical mechanics, one requires that (55) is preserved in time, which should fix the Lagrange multipliers. This can lead to inconsistencies, in which case one should revise the action or the choice for r µ . It can also lead to further constraints, which we regard as unphysical here as they further reduces the number of independent variables (we want exactly three rotational degrees of freedom). Similarly, if some of the Lagrange multipliers remain undetermined, then the degrees of freedom are increased, which we also regard as unphysical. The last possibility is that the Lagrange multipliers are uniquely fixed by requiring that (55) is preserved. In the end, we can insert this solution for the Lagrange multipliers into the action. In this way we obtain an action without Lagrange multipliers which preserves (55). Conditions on the dynamical mass Having this said, we can try to directly construct an action which preserves (55). This approach is in fact very natural here. For instance, one can make an ansatz for M 2 and use this requirement to fix some of the coefficients. The first condition in (55) can be written as η 0A = Λ Aµ r µ and is preserved in time if 0 = Dr µ dλ + Ω µν r ν , where Ω µν = α ∂M 2 ∂S µν .(56) Using (21) and (53), we can write the spin equation of motion (32) in the form DS µν dλ = 2S α[µ Ω ν] α + α ∂M 2 ∂Λ A α δ [µ α Λ A ν] .(57) With the help of this equation, we see that the spin supplementary condition is preserved in time if it holds (56) and additionally 0 = ∂M 2 ∂Λ A α δ [µ α Λ A ν] r ν .(58) This condition is often trivially fulfilled, namely when M 2 does not explicitly depend on Λ A µ . We are going to construct a simple example now, in order to show that functions M exist which are consistent with all of our requirements. A simple construction of the dynamical mass Instead of constructing an action which fulfills (56) and (58) for a specific choice of r µ , one can look at a specific action and construct a r µ such that the requirements (56) and (58) are fulfilled. Let us consider a simple example where M 2 is a nonconstant analytic function f depending only on S 2 := S µν S µν /2, i.e., M 2 = f (S 2 ). It is clear that (58) is fulfilled, because there is no explicit dependence on the body-fixed frame. We still need to satisfy (56), which reads explicitly 0 = Dr µ dλ + αf S µν r ν = Dr µ dλ .(59) That is, the vector r µ must be parallel transported along the worldline. This does indeed characterize a suitable spin supplementary condition, which was first discussed in [40,Sec. 3.4]. Although, with this condition, r µ lacks an immediate interpretation as a rest frame, the numerical results in [40] show that it leads to similar predictions as the choice r µ = p µ / √ −p ν p ν . Further discussions on this supplementary condition are given in other contributions to these proceedings. For the case of a black hole, the laws of black hole dynamics [41,Box 33.4] suggests that M 2 BH = f (S 2 ) = m 2 0 + S 2 (2Gm 0 ) 2 .(60) where m 0 is the constant irreducible mass related to the horizon area. We can now have a look at the angular velocity with respect to asymptotic time, so we have α = M −1 (for a body at rest). Evaluating (52), we find agreement with what is usually identified as the angular velocity of the horizon. This is a nice check for the consistency of the interpretation of our variables. Notice that the laws of black hole dynamics owe their name to their similarity to the laws of thermodynamics. Further, an action principle similar to the one presented here can be used to derive the so called first law of black hole binary dynamics [13]. Again we encounter the thermodynamic character of the approach. For objects other than black holes, we can derive M from the moment of inertia. One usually defines the moment of inertia I(S 2 ) as the proportionality factor between spin and angular velocity, which can be read off from (52). Again we have α = M −1 = f −1/2 , so (52) leads to the differential equation I −1 = f −1/2 f . Its solution reads M 2 = f (S 2 ) = m 0 + S 2 0 dx 2I(x) 2 ,(61) where the irreducible mass m 0 enters as an integration constant. For neutron stars, the function I(S 2 ) can be obtained numerically, e.g., using the RNS code [42,43]. Alternatively, one can numerically compute the gravitating mass M directly as a function of S for a fixed number of baryons in the star. It would be interesting to see if both methods lead to compatible results. It should be noted that both black holes and neutron stars posses a quadrupole (and other multipoles) when they are spinning, which was neglected here. It will be included in the next section. Interestingly, it is implied by [44] that for the pole-dipole case one can construct a M 2 such that (56) and (58) are fulfilled for r µ = p µ / √ −p ν p ν without the need for approximations or truncations of M 2 . Then, however, M 2 is not solely dependent on S. The details on this are left for a future work. Spin-induced quadrupole In this section, we are going to develop a simple phenomenological model for M 2 describing the spin-induced quadrupole of a star. This is the quadrupole of a star arising from a deformation away from spherical symmetry due to rotation. We start with a reasonable ansatz for M 2 . The main idea for this ansatz is to include all possible covariant (general coordinate invariant) terms up to a certain power in spin and curvature. The unknown coefficients in this ansatz are then fixed by comparing to the Kerr metric and to numerical solutions for the gravitational field of a rotating neutron star. One should emphasize that a truncation of M 2 requires negligibly small interaction energies, not small multipoles. Construction of the action We are going to include in our ansatz the quadratic order in spin and second order derivatives of the metric. This means that we include terms linear in the curvature and covariant derivatives of the curvature are not allowed. This implies that we exclude λ-derivatives of the curvature for now, but this restriction will be loosened below. Symbolically we have φ field I = {R µναβ }, which according to Sec. 3.4 implies that we neglect interaction terms involving octupole and higher multipoles. Finally, let us assume the absence of further worldline degrees of freedom in this section, or φ wl I = ∅, so we have no need for a dependence of M 2 on Λ A µ . [Then (58) is already fulfilled.] The main task is to collect all possible interaction terms. One must take care of including only independent terms, which can by tricky due to the symmetries of the Riemann tensor. A procedure for this was applied to the construction of effective Lagrangians or Routhians in [8,34,45]. Instead, we are going to construct M 2 directly, but the arguments are essentially the same. We will follow a different approach to implement the spin supplementary condition, too, by making an ansatz for r µ around the case r µ = p µ √ −p ν p ν + O(R µναβ ).(62) As a first simplification, one can replace R µναβ by its tracefree version, the Weyl tensor C µναβ . The traces are given by R µν := g αβ R µανβ and R := g µν R µν , which are related to the energy momentum tensor T µν through Einstein's gravitational field equations R µν = 8πG T µν − 1 2 T α α g µν .(63) The energy momentum tensor can contain contributions from fields penetrating the compact object, like electromagnetic or dark matter fields. We assume that these can be neglected, i.e., the bodies are mainly interacting via the gravitational field. But in the case of self-gravitating bodies, the energy momentum tensor also includes a singular contribution from the point-particle (39) itself. Let us assume that these singular self-interactions can be dropped. Then we can effectively make use of the vacuum field equations R µν = 0 at the particle location, so we have R µναβ = C µναβ . However, in general one is not allowed to use field equations at the level of the action. But in the current context this is essentially a valid procedure, as it is equivalent to a field redefinition in the action, see [8], [26], or [46,Appendix A]. Without loss of generality, we can therefore restrict to φ field I = {C µναβ } in the quadrupole case. The most important and most obvious requirement on the allowed interaction terms in M 2 is general coordinate invariance. Further restrictions on the terms and transformations identifying equivalent terms (equivalent within our truncation) are: 1. In four spacetime dimensions, the Weyl tensor can be split into an electric E µν and a magnetic part B µν with respect to a time-like unit vector. Choosing this vector to be r µ , it holds E µν = C µανβ r α r β , B µν = 1 2 η µαρσ C νβ ρσ r α r β ,(64) where η µναβ is the volume form. These tensors have the properties E µν = E νµ , E µν g µν = 0, E µν r ν = 0,(65)B µν = B νµ , B µν g µν = 0, B µν r ν = 0.(66) These properties make E µν and B µν much easier to handle compared to C µναβ . 2. We include only terms invariant under parity transformations. In this respect it is important to notice that B µν is of odd parity. We conclude that any terms with an odd number of magnetic Weyl tensors must also include an odd number of volume forms η µναβ . Due to the antisymmetry of η µναβ , it will always be contracted with both indices of the spin at the current level of truncation. Then we can rewrite all terms involving η µναβ in terms of the dual of the spin tensor * S αβ := 1 2 S µν η µναβ . S µα * S αν = − 1 4 δ ν µ S αβ * S αβ .(68) As a consequence, it holds B µν S µ α * S αν = 0. It is customary to define a spin vector S µ := r ν * S νµ . 3. It should be noticed that one is in general not allowed to neglect terms involving the combination S µν r ν , though these numerically vanish due to the spin supplementary condition (55): A variation of these terms can lead to nonvanishing contributions to the equations of motion. Instead, terms in the action which are at least quadratic in S µν r ν can be neglected, as their contributions to the equations of motion are at least linear in the spin supplementary condition and thus always vanish. 4. The quadrupole interaction terms can be simplified using the leading order truncation of the mass-shell constraint p µ p µ + M 2 = 0. (For the ansatz in (70) given below, this implies that we can set p µ p µ ≈ −µ 2 in the higher order terms of M 2 .) This transformation does in fact just correspond to a redefinition of the Lagrange multiplicator α, and the idea is therefore similar to the field redefinitions mentioned above [26]. 5. Time derivatives of p µ and S µν can also be removed by redefinitions of variables, which follows from the ideas in [26] and is again analogous to the mentioned field redefinitions. Besides that, the absence of higher order time derivatives was already assumed in Sec. 3.1. The last point also shows that our ansatz will automatically cover time derivatives of E µν and B µν of arbitrary order. As we work at linear level in the curvature, the time derivatives can always be removed from the curvature through partial integration. After this transformation, all time derivatives finally apply to p µ and S µν only, which can be removed by virtue of the argument 5 above. This suggests that these terms belong to the quadrupole level, too, although time derivatives of the fields are in fact covariant derivatives D/dλ = u µ ∇ µ . This is of course related to the ambiguity of the multipoles pointed out in Sec. 3.4. At linear level in the curvature, one can assume that the covariant derivatives are projected orthogonal to r µ , because r µ ∇ µ ≈ u µ ∇ µ = D/dλ up to higher order terms, which can be partially integrated. The first point mentioned above suggests to include just E µν and B µν in φ field I . However, this is currently not possible, because we assumed in Sec. 3.1 that the φ field I contain just fields, but r µ in (64) is only defined on the worldline. For instance, one would have to clarify the meaning of ∇ α r µ arising from ∇ α E µν in (31). For simplicity, let us stick to φ field I = {C µναβ } here, but have in mind that M 2 depends on C µναβ only through the combinations E µν and B µν . The equations of motion are initially expressed in terms the quadrupole moment related to C µναβ , J µναβ := −6 ∂L M ∂C µναβ .(69) see (33), but these are at once related to the moments belonging to E µν and B µν through the chain rule. The interpretation of the latter moments as quadrupoles is much more obvious than for (69), as E µν and B µν are symmetric tracefree spatial tensors in the rest frame defined by r µ . These moments can be called electric and magnetic quadrupoles, respectively. They match the quadrupole degrees of freedom of the gravitational field outside the body [47], in contrast to (33), which in general is not tracefree. This approach to define electric and magnetic quadrupoles was briefly discussed in [11]. An explicit split into electric and magnetic quadrupoles at the level of the equations of motion was performed in [48]. Ansatz The most general ansatz for M 2 now reads M 2 = µ 2 + C BS 2 p B µν S µ S ναp α + C ES 2 E µν S µα S ν α + O(E 2 , B 2 , S 3 ),(70) where we introduced the abbreviationp µ = p µ /µ. We assume that µ, C BS 2 p , and C ES 2 are constants. Remember that within the curvature terms we can set r µ ≈p µ , which is due to (62) and point 4 of the last section. Notice that µ must be a function of the constant spin length, µ 2 = f (S 2 ), cf. (61). Otherwise the Legendre transformation would be problematic. Consistent with our truncation, we may write µ 2 = m 2 0 + m 0 I 0 S 2 + O(S 4 ),(71) where I 0 ≡ I(0) is the moment of inertia in a slow rotation limit S → 0. Furthermore, the constants C BS 2 p and C ES 2 will in general depend on µ and S. This is further discussed below. Next, we want to check if (56) is fulfilled. Notice that (56) is required to hold at linear order in spin only. For this purpose, let us make an ansatz for r µ to linear order in S, r µ = C rpp µ + GC rBS B µν S ν + O(E 2 , B 2 , S 2 ).(72) The normalization r µ r µ = −1 leads to C rp = 1. Inspecting (56), we see that most of the contributions from the C rBS -term are shifted to higher orders, namely quadratic level in spin: This is due to Ω = O(S) andṠ ν = O(S 2 ). The only C rBS -term linear in spin containsḂ µν . Though this is a derivative of the curvature, it is not of higher order, because we realized that our ansatz effectively also covers λ-derivatives of the curvature. We conclude that C rBS = 0, or r µ ≈p µ . For calculatingṙ µ in (56), it is useful to rewrite (31) as Dp µ dλ = α(E α[µ S ν] α − B α[µ r ν] S α ) 2p ν + ∂M 2 ∂p ν − α 2 (∇ µ φ field I ) ∂M 2 ∂φ field I .(73) Finally, the condition (56) is fulfilled to linear order in spin if C BS 2 p = 2 in our ansatz (70). The condition (58) is of course also fulfilled. In summary, Figure 1: The coefficient C ES 2 as a function of the dimensionless spin a = S/Gµ 2 , where µ is identified with the gravitating mass and is given by µ = 1.4M here. The data points were generated using the RNS code [42,43], where a multipole extraction according to [49] was used. The labels SLy, APR, FPS, and AU refer to equations of state considered in [50]. we must have C rp = 1, C rBS = 0, C BS 2 p = 2,(74) while C ES 2 is not determined by basic principles, but depends on the specific object. Instead of fixing r µ algebraically like in (72), it would be interesting to view (56) as an evolution equation for r µ in the future, analogous to (59) in the pole-dipole case. In Figure (1) the numerical value of C ES 2 is shown as a function of the spin length for fixed mass µ but different neutron star models. It is apparent from the plot that C ES 2 is approximately independent of the spin length. However, one should be careful and check this assumption for the specific case of interest. This determination of C ES 2 is actually a simple example of a matching procedure. The quadrupole moment J of the effective point particle is parametrized through the ansatz (70) as J ∼ C ES 2 S 2 . This is compared (or matched) to the quadrupole moment of a numeric neutron star spacetime computed with the RNS code [42,43]. Here the quadrupole moment is identified through the exterior spacetime. This means that the effective point particle mimics the exterior spacetime of a numerically constructed neutron star model, which depends crucially on strong field effects in the interior. This makes C ES 2 an interesting indicator for both the neutron star equation of state and strong-field modifications of gravity. For black holes, a comparison with the Kerr metric leads to C ES 2 = 1. Finally, we come back to the thermodynamic analogy to our approach. The quadrupole relation J ∼ C ES 2 S 2 can be viewed as a simple (idealized) "equation of state" relating the macroscopic variables J and S. As in the case of the ideal gas, this model can be improved to meet the required accuracy. This can be done systematically here by extending the ansatz (70) to higher orders. Application As an application for the spin-induced quadrupole constructed in the last section, we consider the case of a test particle moving in a Kerr spacetime. This test particle can be characterized as a pole-dipole-quadrupole particle. We aim at an estimate for the relevance of the spin-squared contributions, so we may consider a specific orbital configuration that simplifies the discussion. This is obviously a circular orbit in the equatorial plane of the Kerr geometry. Let us further assume that the spin of the test body is aligned with the rotation axis of the background spacetime. In the absence of a quadrupole, these orbits can be constructed in a simple manner, which was first used in [51]. This method is in fact still applicable for the considered quadrupole model [52]. It requires that conserved quantities, spin supplementary condition, and constraints on the orbital configuration are enough to uniquely fix the 10 dynamic variables contained in p µ and S µν . This is just an algebraic calculation, in contrast to solving the differential equations of motion. A numeric study for Schwarzschild spacetime is given in [53]. The spin supplementary condition (S µν p ν = 0) contains three independent equations. The constraint on the orbit provides three further independent conditions: one due to equatorial orbits (p θ = 0) and two due to spin alignment (S µθ = 0). So we need to identify 10−3−3 = 4 conserved quantities in order to solve for p µ and S µν algebraically. Three conserved quantities were already identified in Sec. 4.1. These are the spin-length S := 1 2 S ab S ab and the quantities derived from the two Killing vectors of Kerr spacetime (∂ t and ∂ θ ) through (46). Well call the latter two the energy E := E ∂t and total angular momentum J φ := E −∂ φ of the particle. The last remaining conserved quantity is just the mass-like parameter µ, which in the action approach is constant by assumption. However, one should remember that (70) is truncated and thus only approximately valid. One can equivalently say that µ is only conserved approximately, corresponding to the truncation of (70). This point of view was taken in [52]. Now we are in a position to solve for p a and S ab . Most important is the equation for p r . After some algebra [52], one finds that (p r ) 2 is given by a polynomial of second order in E. We denote the roots of this polynomial by U + and U − , i.e., (p r ) 2 ∝ (E − U + )(E − U − ).(75) For p r to be a real number, we need to have both E ≤ U + and E ≤ U − , or both E ≥ U + and E ≥ U − . It turns out that the important relation is just E ≥ U + for the most relevant part of the parameter space. This justifies to call U + effective potential: The test body can only move in the region where E ≥ U + and its turning points are given by E = U + , because then p r = 0 (which implies u r = 0, see [52]). Therefore the minimum of U + as a function of r defines circular orbits. This completes our construction. The various contributions to the dimensionless binding energy e := E/µ − 1 are plotted in Fig. 2 for the case of a very rapidly rotating (small) black hole in a Schwarzschild background. A comparison with recent results for the conservative part of the self-force [54] is also included. In a Kerr background, the last stable circular orbit can be very close to the horizon, so that the discussed effects can be some orders of magnitude stronger. The reader is referred to [52] for a more complete discussion. Dynamical quadrupole and tidal forces For the model developed in the last section, the quadrupole adiabatically follows the spin evolution. Thus, the quadrupole is not an independent dynamical variable. In this section, we are going to investigate dynamical quadrupoles, but restrict to the nonspinning case for simplicity. Basic idea We have already discovered that the dynamical mass M plays a role similar to a thermodynamic potential. From this perspective, one can compare the variables it depends on, like p µ and S µν , to thermodynamic state variables. Noticing that p µ and S µν are the monopole and dipole moment, a natural extension is to introduce dynamical "state" variables for other multipoles, too. A possible motivation arises from the realization that stars have oscillation modes and that these modes can be excited by tidal forces from an external time-dependent gravitational field. This phenomenon is well understood in Newtonian gravity [55], see also [56][57][58] and references therein. If one wants to capture it by our approach, one obviously must introduce dynamical worldline variables corresponding to these oscillation modes. Suitable point-particle actions were already discussed in [45,59], though with applications to absorption or binary systems in mind. The key to find a model for dynamical multipoles is to understand the reaction of the multipoles to external fields. We focus here on the response of the quadrupole to external tidal fields. In fact, we will encode the quadrupole dynamics in terms of a response function. This function can equivalently be called the propagator of the quadrupole [45], which better highlights the fact that it is a necessary ingredient for deriving predictions using perturbative calculations, e.g., in the post-Newtonian approximation. A third possible naming is correlation function between quadrupole and external field. This better accentuates the parallels to statistical mechanics or thermodynamics. The idea is that if one would be able to model the correlations of the most important multipoles among each other and with external fields, then one can in principle predict the motion of extended objects (with complicated internal structure) to any desired precision. It is important to notice that the multipole moments of a compact object can be defined through their exterior field. The response functions of the multipoles to externally applied tidal fields can therefore be obtained by analyzing the gravitational field outside of the body. The final goal is to extract these functions from numerical simulations of a single compact object. However, for a first simpler investigation one can restrict to linear perturbations of nonrotating compact objects. The unperturbed metric in the exterior is then just the Schwarzschild one. Because this metric is static and spherically symmetric, its linear perturbations can then be decomposed into Fourier basis in the time direction and spherical harmonic basis Y lm (θ, φ) in angular directions. Then their radial dependence is described by the famous Zerilli [60] or Regge-Wheeler [61] equations for electric-or magnetic-paritytype perturbations, respectively. The Zerilli equation can be transformed into the simpler Regge-Wheeler form [62], so we can focus just on the latter one. It reads d 2 X dr 2 * + 1 − R S r l(l + 1) − 3R S r r 2 + ω 2 X = 0,(76) where ω is the frequency of the perturbation, l is the angular momentum quantum number, r is radial coordinate in the Regge-Wheeler gauge, R S is the Schwarzschild radius (representing the mass of the body), r * = r + R S log(r/R S − 1) is the tortoise radial coordinate, and X denotes the Regge-Wheeler master function. Given some boundary values for X at the surface of the body (which result from a solution to the more complicated interior perturbation equations), it is straightforward to integrate this equation numerically. The question is how one can decompose X into external (applied) tidal field and multipolar field generated by the body in response to the external field. This is a complicated problem in the general relativistic case. Let us therefore start with the Newtonian theory in order to get a better understanding of the problem [19]. Newtonian case The Newtonian case can be obtained as a weak field and slow motion approximation of general relativity. That is, we have to set R S = 0 (weak field) and ω = 0 (slow motion) in (76). The perturbation of the Newtonian potential Φ pert can be reconstructed as Φ pert = − 1 2π dω lm e iωt Y lm 1 2 d dr + l(l + 1) 2r X lmω ,(77) where the X lmω are solutions to the Newtonian limit of (76) for all values of the parameters l, m, and ω. The generic solution to the Newtonian limit of (76) reads X = C 1 r l+1 + C 2 r −l ,(78) where C 1 and C 2 are integration constants. The r l+1 part diverges asymptotically, which means that its source is located at infinity. Therefore, C 1 is the strength of the external field. Similarly, the r −l part is singular at the origin and emanates from the compact body, so C 2 describes the l-polar field of the body. The frequency-domain responseF l of the multipoles to external fields is then proportional to the ratio of C 2 and C 1 . In the conventions used in [18,19], it holdsF l (ω) = l(l − 1) G(l + 1)(l + 2)(2l − 1)!! C 2 C 1(79) This response must in general be computed numerically. The first step is to numerically solve the interior problem of a perturbed body, including the interior gravitational field perturbation. Then the gravitational field is matched to (78) at the surface, which leads to numeric values for the integration constants and thus for the response (79). This response can in general acquire a complicated frequency dependence through the internal dynamics. Usually one defines normal oscillation modes by requiring that the body keeps up a multipolar field without external excitation, i.e., for C 1 = 0. Therefore the response (79) has a pole at normal mode frequencies. In the case of linear perturbations of a nonrotating barotropic star, the response turns out to be quite simple. For the quadrupolar case l = 2, the outcome is shown in Fig. 3. In fact, the form of the response can even be computed analytically and reads [19] F l = n I 2 nl ω 2 nl − ω 2 .(80) This is just the sum of response functions of harmonic oscillators with resonance frequencies (poles) at ω nl . Here n labels the type and overtone number of the oscillation modes. The constants I nl are the so called overlap integrals, which here simply take the role of coupling constants between the oscillators and the external driving forces. As a consequence, the internal dynamics can be captured by an effective action through just a set of harmonic oscillators, which are coupled to the tidal force of the gravitational field [19] (with coupling constants I nl ). By fitting the numeric result forF l to (80), one can extract the constants ω nl and I nl . It is worth to point out that the presented Newtonian setup is simple enough to perform explicitly the effective field theory procedure of integrating out small scales, see [19]. This turns a compact fluid configuration into a point particle on macroscopic scales. Relativistic case at zero frequency Let us now return to the relativistic case, but restrict to even parity and the adiabatic case ω = 0. The connection between the relativistic tidal constants defined in [15][16][17]63] and the response function is given by a Taylor-expansion,F l (ω) l! = µ l + iλ l ω + µ l ω 2 + O(ω 3 ),(81) see [19]. Here the constants µ l are named after the astronomer A. E. H. Love, who introduced them for tidal effects in the Earth-Moon system. A dimensionless version of the Love numbers µ l is often defined as k l = (2l − 1)!! 2R 2l+1 Gµ l ,(82) where R is the radius of the star. The λ l -term in (81) is related to absorption [45] and µ 2 was introduced in [63]. It remains to define how the response should be computed in the adiabatic relativistic case. First, we again solve (76), this time for ω = 0, and find an analytic result in terms of the Gauss hypergeometric function 2 F 1 , X = C 1 r l+1 2 F 1 (−l − 2, 2 − l, −2l; R S /r) + C 2 r −l 2 F 1 (l − 1, l + 3, 2(l + 1); R S /r),(83) see, e.g., [64]. Again we can obtain numeric values for the integration constants by solving the perturbation equations inside the body and then match the gravitational field to (83) at the surface. In the limit of 1/r → 0 the hypergeometric functions are equal to 1, so (83) turns into (78). This implies that the interpretation of the integration constants as magnitudes of external field and response is still valid. The even-parity response in the adiabatic case ω = 0 then follows from (79) as before. A plot of the outcome in terms of the dimensionless Love number k 2 is given in Fig. 4. An extension of the application from Sec. 5.3 to adiabatic tidal deformations can be found in [52]. For integer values of l, the hypergeometric functions in (83) turn into polynomials (which possibly contain logarithms). Then one might worry that the exponents on r from the two independent solutions in (83) can overlap and spoil an unique identification of external field and response. However, this is avoided by examining X for generic values of l, in the sense of an analytic continuation. This is in spirit similar to working in generic dimension, as done in [64]. Relativistic case for generic frequency We now turn our attention to the case of generic frequency in the even parity sector [18]. One can still solve (76) analytically [65], this time in terms of a series involving hypergeometric functions. We write the generic solution schematically as X = A 1 X l MST + A 2 X −l−1 MST ,(84) where we denote the solution from [65] by a subscript MST. Note here that X l MST ∼ r l and X −l−1 MST ∼ r −l , which means that (79) essentially still works. Of course, one has to take into account the normalization of the X MST in order to rewrite the C i in (79) in terms of the A i . This introduces complicated ω-dependent corrections into (79). These are computed through a matching of the asymptotic field of the extended body to the field of the point-particle model. The details of this procedure can be found in [18]. The basic steps are as follows: • The field of the effective theory is obtained from an inhomogeneous version of (76) with a point particle source. It is understood that the post-Minkowskian approximation is applied, as this removes the singular point of (76) at the Schwarzschild radius. The explicit form of the source term derives from (39). • The solution to the inhomogeneous equation is constructed from the homogeneous solution (84) using the method of variation of parameters. This method involves integrals over products of singular source and the X MST . The integration constants just represent a generic solution to the homogeneous solution that can always be added. • Here the integration constants must be restricted further. Due to the singular behavior of the differential equation at r = 0, the homogeneous solution might actually not be homogeneous at r = 0. But the externally applied field is homogeneous everywhere, including r = 0. The restriction of the integration constants is therefore equivalent to the identification of the external part of the field and the part generated by the particle. • Notice that an l-pole source involves l partial derivatives of a delta distribution. This suggests to identify the self-field by X −l−1 MST ∼ r −l and the external field by X l MST ∼ r l for dimensional reasons. Here the idea of analytic continuation in l is again crucial. • The integrals arising in the variation of parameters are actually singular. This is not surprising, as the self-field of point-particles always leads to this kind of problem. A regularization method must be introduced. These steps lead to a refined (frequency dependent) version of (79) expressing the response function in terms of A 1 and A 2 . The final step is again to obtain numeric values for A 1 and A 2 for an actual (extended) neutron star. The result for the general relativistic response function is shown in Fig. 3. It can still be fitted by (80) very well. This implies that the internal dynamics can be approximated by a set of harmonic oscillators. Restricting to the quadrupolar level l = 2 for simplicity, this translates to a dynamical mass of the form M 2 ≈ µ 2 + µ n (ψ n AB ψ nAB + ω 2 n φ nAB φ n AB + 2I n E AB φ nAB ),(85) where the internal dynamical variables φ nAB and ψ n AB only have spatial components in the body-fixed frame (φ n0B = 0 = ψ n 0B ) and are symmetric tracefree in the indices A and B. The dynamical equations for the quadrupolar worldline variables can be extracted from (52), (29), and (54), φ n AB = α 2 ∂M 2 ∂ψ nAB ,ψ nAB = − α 2 ∂M 2 ∂φ n AB .(86) In the linear perturbation regime, the contributions of the internal dynamical variables are small compared to µ 2 . The index n still labels the type of the oscillation mode. The mass quadrupole is the coefficient in front of E AB , i.e., Q AB := n I n φ nAB . Now the frame enters through E AB = λ A µ λ B ν E µν , so we need to check if (58) is fulfilled. Using Λ 0 µ = r µ and E µν r ν = 0 it is easy to see that this is the case. In fact, (58) is always fulfilled if the time direction of the body-fixed frame Λ 0 µ drops out of the action. Some final remarks on the problem of regularization of point particles are in order. It was shown already in [59] that the quadrupole diverges at order ω 2 in dimensional regularization. It is therefore not surprising that poles appear in the generalization of (79) at order ω 2 , which must be subtracted within some renormalization scheme. At the same time, the poles give rise to an explicit appearance of a renormalization scale parameter, which in a sense parametrizes the ambiguity in the choice of the renormalization scheme. An important point is that this scale parameter is in fact fixed by the requirement that the response function has an asymptotic behavior for ω → ∞ compatible with (80). Different regularization and renormalization schemes will in general lead to slightly different numeric values for this scale parameter. However, within a given scheme its value can be uniquely matched and is therefore not ambiguous. In this sense, the regularization and renormalization scheme is a part of the phenomenological model. Conclusions We considered point-particle models for extended bodies in gravity, in particular for black holes and neutron stars. The multipoles of the point particles are adjusted such that their field predicted from a weak field approximation matches an exact/numerical solution for the extended object in question. This incorporates strong field effects from the interior of the extended object in the model. This is of particular importance when binary systems are considered using weak field approximations, e.g., for gravitational wave source modeling or pulsar timing. Therefore, point-particle actions are far more powerful than what was probably envisioned when they were first investigated [5,6]. The resulting equations of motion are similar to Dixon's results. Here we developed astrophysical realistic models for the multipoles in these equations. The latest development is the inclusion of oscillation modes in relativistic tidal interaction of neutron stars. An interesting topic not discussed here are universal relations for various neutron star properties. Here "universal" refers to an approximate independence among various proposed realistic equations of state. In [66,67] universal relations between the dimensionless moment of inertia I/G 2 µ 3 , the quadrupolar Love number µ 2 /G 4 µ 5 , and the quadrupole constant C ES 2 were found and coined I-Love-Q relations. Further investigations, also including higher multipoles, followed shortly afterwards [68][69][70][71][72][73]. This indicates that coefficients in (70) arising at higher orders are actually not independent, but are (approximately) fixed by universality. (For black holes, in fact all coefficients are fixed, which is guaranteed by the no hair theorem.) This makes the expansion (70) a meaningful tool to study the impact of the equation of state on observations, as predictions of the effective model are then parametrized by only a small set of constants. The most interesting development for the future is probably the description of oscillation modes for rotating bodies, which can be tried in a slow rotation approximation. It is also interesting to investigate if universal properties hold for the ingredients of the response function, e.g., for the overlap integrals. Note added in arXiv version: In Ref. [74] the equations of motion for the center of mass were obtained in a manifestly covariant manner using a family of wordlines and explicit expressions for the equations of motion including all gravitational multipoles are given in the appendix. Figure 2 : 2Various corrections to the binding energy e for a maximally spinning (small) black hole in a Schwarzschild background. Here l c := J φ − S is the orbital angular momentum. The mass ratio is formally taken to be q = 1 in the plot, though the result are only valid for q 1. The curves can be scaled to the case of interest (q 10 −2 ): self-force and linear spin effects scale as ∝ q, the others as ∝ q 2 . Figure 3 : 3Response function of the quadrupole, l = 2, for a one solar mass star. The equation of state is a polytrope with index 1 and such that the radius R is 17.7 km in the Newtonian case or 15.7 km in the relativistic case. Figure 4 : 4Dimensionless quadrupolar Love number k 2 as a function of the compactness c = R S /2R for two different equations of state (SLy, FPS). Neue Mechanik materieller Systeme. M Mathisson, Acta Phys. Pol. 6M. Mathisson, "Neue Mechanik materieller Systeme," Acta Phys. Pol. 6 (1937) 163-200. Republication of: New mechanics of material systems. M Mathisson, 10.1007/s10714-010-0939-yGen. Relativ. Gravit. 42M. Mathisson, "Republication of: New mechanics of material systems," Gen. Relativ. Gravit. 42 (2010) 1011-1048. Extended bodies in general relativity: Their description and motion. W G Dixon, Proceedings of the International School of Physics Enrico Fermi LXVII. J. Ehlersthe International School of Physics Enrico Fermi LXVIIAmsterdamNorth HollandW. G. Dixon, "Extended bodies in general relativity: Their description and motion," in Proceedings of the International School of Physics Enrico Fermi LXVII, J. Ehlers, ed., pp. 156-219. North Holland, Amsterdam, 1979. Spinning test-particles in general relativity. I. A Papapetrou, 10.1098/rspa.1951.0200Proc. R. Soc. A. 209A. Papapetrou, "Spinning test-particles in general relativity. I," Proc. R. Soc. A 209 (1951) 248-258. Relativistische Bewegungsprobleme. VI. Rotator-Spinteilchen und allgemeine Relativitätstheorie. K , 10.1002/andp.19694770706Ann. Phys. (Berlin). K. Westpfahl, "Relativistische Bewegungsprobleme. VI. Rotator-Spinteilchen und allgemeine Relativitätstheorie," Ann. Phys. (Berlin) 477 (1969) 361-371. Lagrangian dynamics of spinning particles and polarized media in general relativity. I Bailey, W Israel, 10.1007/BF01609434Commun. math. Phys. 42I. Bailey and W. Israel, "Lagrangian dynamics of spinning particles and polarized media in general relativity," Commun. math. Phys. 42 (1975) 65-82. Gravitational radiation and the motion of compact bodies. T Damour, Gravitational Radiation, N. Deruelle and T. PiranNorth HollandAmsterdamT. Damour, "Gravitational radiation and the motion of compact bodies," in Gravitational Radiation, N. Deruelle and T. Piran, eds., pp. 59-144. North Holland, Amsterdam, 1983. An effective field theory of gravity for extended objects. W D Goldberger, I Z Rothstein, 10.1103/PhysRevD.73.104029arXiv:hep-th/0409156Phys. Rev. D. 73104029W. D. Goldberger and I. Z. Rothstein, "An effective field theory of gravity for extended objects," Phys. Rev. D 73 (2006) 104029, arXiv:hep-th/0409156. Post-Newtonian corrections to the motion of spinning bodies in nonrelativistic general relativity. R A Porto, 10.1103/PhysRevD.73.104031arXiv:gr-qc/0511061Phys. Rev. D. 73104031R. A. Porto, "Post-Newtonian corrections to the motion of spinning bodies in nonrelativistic general relativity," Phys. Rev. D 73 (2006) 104031, arXiv:gr-qc/0511061. Canonical formulation of self-gravitating spinning-object systems. J Steinhoff, G Schäfer, 10.1209/0295-5075/87/50004arXiv:0907.1967Europhys. Lett. 8750004gr-qcJ. Steinhoff and G. Schäfer, "Canonical formulation of self-gravitating spinning-object systems," Europhys. Lett. 87 (2009) 50004, arXiv:0907.1967 [gr-qc]. Canonical formulation of spin in general relativity. J Steinhoff, 10.1002/andp.201000178arXiv:1106.4203Ann. Phys. (Berlin). 523gr-qcJ. Steinhoff, "Canonical formulation of spin in general relativity," Ann. Phys. (Berlin) 523 (2011) 296-353, arXiv:1106.4203 [gr-qc]. . B S Dewitt, 10.1007/978-3-540-36911-0Bryce DeWitt's Lectures on Gravitation. 826SpringerLecture Notes in Physics. 1st ed.B. S. DeWitt, Bryce DeWitt's Lectures on Gravitation, vol. 826 of Lecture Notes in Physics. Springer, Berlin, 1st ed., 2011. First law of mechanics for black hole binaries with spins. L Blanchet, A Buonanno, A Le Tiec, 10.1103/PhysRevD.87.024030arXiv:1211.1060Phys. Rev. D. 8724030gr-qcL. Blanchet, A. Buonanno, and A. Le Tiec, "First law of mechanics for black hole binaries with spins," Phys. Rev. D 87 (2013) 024030, arXiv:1211.1060 [gr-qc]. Quadrupole moments of rotating neutron stars. W G Laarakkers, E Poisson, 10.1086/306732arXiv:gr-qc/9709033Astrophys. J. 512W. G. Laarakkers and E. Poisson, "Quadrupole moments of rotating neutron stars," Astrophys. J. 512 (1999) 282-287, arXiv:gr-qc/9709033. Relativistic tidal properties of neutron stars. T Damour, A Nagar, 10.1103/PhysRevD.80.084035arXiv:0906.0096Phys. Rev. D. 8084035gr-qcT. Damour and A. Nagar, "Relativistic tidal properties of neutron stars," Phys. Rev. D 80 (2009) 084035, arXiv:0906.0096 [gr-qc]. Relativistic theory of tidal Love numbers. T Binnington, E Poisson, 10.1103/PhysRevD.80.084018arXiv:0906.1366Phys. Rev. D. 8084018gr-qcT. Binnington and E. Poisson, "Relativistic theory of tidal Love numbers," Phys. Rev. D 80 (2009) 084018, arXiv:0906.1366 [gr-qc]. Tidal Love numbers of neutron stars. T Hinderer, 10.1086/533487arXiv:0711.2420Astrophys. J. 677astro-phT. Hinderer, "Tidal Love numbers of neutron stars," Astrophys. J. 677 (2008) 1216-1220, arXiv:0711.2420 [astro-ph]. New perspectives on neutron star and black hole spectroscopy and dynamic tides. S Chakrabarti, T Delsate, J Steinhoff, arXiv:1304.2228gr-qcS. Chakrabarti, T. Delsate, and J. Steinhoff, "New perspectives on neutron star and black hole spectroscopy and dynamic tides," arXiv:1304.2228 [gr-qc]. Effective action and linear response of compact objects in Newtonian gravity. S Chakrabarti, T Delsate, J Steinhoff, 10.1103/PhysRevD.88.084038arXiv:1306.5820Phys. Rev. D. 8884038gr-qcS. Chakrabarti, T. Delsate, and J. Steinhoff, "Effective action and linear response of compact objects in Newtonian gravity," Phys. Rev. D 88 (2013) 084038, arXiv:1306.5820 [gr-qc]. Relativistische Bewegungsprobleme. II. Der starre Rotator. H Goenner, K Westpfahl, 10.1002/andp.19674750505Ann. Phys. (Berlin). H. Goenner and K. Westpfahl, "Relativistische Bewegungsprobleme. II. Der starre Rotator," Ann. Phys. (Berlin) 475 (1967) 230-240. Relativistische Bewegungsprobleme. IV. Rotator-Spinteilchen in schwachen Gravitationsfeldern. H Römer, K Westpfahl, 10.1002/andp.19694770506Ann. Phys. (Berlin). H. Römer and K. Westpfahl, "Relativistische Bewegungsprobleme. IV. Rotator-Spinteilchen in schwachen Gravitationsfeldern," Ann. Phys. (Berlin) 477 (1969) 264-276. The relativistic spherical top. A J Hanson, T Regge, 10.1016/0003-4916(74)90046-3Ann. Phys. (N.Y.). 87A. J. Hanson and T. Regge, "The relativistic spherical top," Ann. Phys. (N.Y.) 87 (1974) 498-566. Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation. M Leclerc, 10.1088/0264-9381/22/16/006arXiv:gr-qc/0505021Class. Quant. Grav. 22gr-qcM. Leclerc, "Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation," Class. Quant. Grav. 22 (2005) 3203-3222, arXiv:gr-qc/0505021 [gr-qc]. Tangent Euler top in general relativity. J Natario, 10.1007/s00220-008-0475-8arXiv:gr-qc/0703081Commun.Math.Phys. 281gr-qcJ. Natario, "Tangent Euler top in general relativity," Commun.Math.Phys. 281 (2008) 387-400, arXiv:gr-qc/0703081 [gr-qc]. Dynamical theory of groups and fields. B S Dewitt, Relativity, Groups, and Topology. Les Houches; New YorkGordon and BreachB. S. DeWitt, "Dynamical theory of groups and fields," in Relativity, Groups, and Topology, Les Houches 1963. Gordon and Breach, New York, 1964. Redefinition of position variables and the reduction of higher order Lagrangians. T Damour, G Schäfer, 10.1063/1.529135J. Math. Phys. 32T. Damour and G. Schäfer, "Redefinition of position variables and the reduction of higher order Lagrangians," J. Math. Phys. 32 (1991) 127-134. Motion of multipole particles in general relativity theory. W M Tulczyjew, Acta Phys. Pol. 18W. M. Tulczyjew, "Motion of multipole particles in general relativity theory," Acta Phys. Pol. 18 (1959) 393-409. Multipolar equations of motion for extended test bodies in general relativity. J Steinhoff, D Puetzfeld, 10.1103/PhysRevD.81.044019arXiv:0909.3756Phys. Rev. D. 8144019gr-qcJ. Steinhoff and D. Puetzfeld, "Multipolar equations of motion for extended test bodies in general relativity," Phys. Rev. D 81 (2010) 044019, arXiv:0909.3756 [gr-qc]. Mechanics of extended masses in general relativity. A I Harte, 10.1088/0264-9381/29/5/055012arXiv:1103.0543Class. Quant. Grav. 2955012gr-qcA. I. Harte, "Mechanics of extended masses in general relativity," Class. Quant. Grav. 29 (2012) 055012, arXiv:1103.0543 [gr-qc]. Invariante Variationsprobleme. E Noether, arXiv:physics/0503066Nachr. Akad. Wiss. Gött. E. Noether, "Invariante Variationsprobleme," Nachr. Akad. Wiss. Gött. (1918) 235-257, arXiv:physics/0503066. http://resolver.sub.uni-goettingen.de/purl?GDZPPN00250510X. Dynamics of extended bodies in general relativity -center-of-mass description and quasirigidity. J Ehlers, E Rudolph, 10.1007/BF00763547Gen. Relativ. Gravit. 8J. Ehlers and E. Rudolph, "Dynamics of extended bodies in general relativity -center-of-mass description and quasirigidity," Gen. Relativ. Gravit. 8 (1977) 197-217. Equations of motion for spinning particles in external electromagnetic and gravitational fields. K Yee, M Bander, 10.1103/PhysRevD.48.2797arXiv:hep-th/9302117Phys. Rev. D. 48K. Yee and M. Bander, "Equations of motion for spinning particles in external electromagnetic and gravitational fields," Phys. Rev. D 48 (1993) 2797-2799, arXiv:hep-th/9302117. Spin(1)spin(2) effects in the motion of inspiralling compact binaries at third order in the post-Newtonian expansion. R A Porto, I Z Rothstein, 10.1103/PhysRevD.78.044012arXiv:0802.0720Phys. Rev. D. 7844012gr-qcR. A. Porto and I. Z. Rothstein, "Spin(1)spin(2) effects in the motion of inspiralling compact binaries at third order in the post-Newtonian expansion," Phys. Rev. D 78 (2008) 044012, arXiv:0802.0720 [gr-qc]. Next to leading order spin(1)spin(1) effects in the motion of inspiralling compact binaries. R A Porto, I Z Rothstein, 10.1103/PhysRevD.78.044013arXiv:0804.0260Phys. Rev. D. 7844013gr-qcR. A. Porto and I. Z. Rothstein, "Next to leading order spin(1)spin(1) effects in the motion of inspiralling compact binaries," Phys. Rev. D 78 (2008) 044013, arXiv:0804.0260 [gr-qc]. Zur Quantelung der Wellenfelder. L Rosenfeld, 10.1002/andp.19303970107Ann. Phys. (Berlin). 397L. Rosenfeld, "Zur Quantelung der Wellenfelder," Ann. Phys. (Berlin) 397 (1930) 113-152. Generalized Hamiltonian dynamics. P A M Dirac, 10.4153/CJM-1950-012-1Canad. J. Math. 2P. A. M. Dirac, "Generalized Hamiltonian dynamics," Canad. J. Math. 2 (1950) 129-148. The center-of-mass in Einsteins theory of gravitation. W Beiglböck, 10.1007/BF01646841Commun. math. Phys. 5W. Beiglböck, "The center-of-mass in Einsteins theory of gravitation," Commun. math. Phys. 5 (1967) 106-130. The center of mass in general relativity. R Schattner, 10.1007/BF00760221Gen. Relativ. Gravit. 10R. Schattner, "The center of mass in general relativity," Gen. Relativ. Gravit. 10 (1979) 377-393. The uniqueness of the center of mass in general relativity. R Schattner, 10.1007/BF00760222Gen. Relativ. Gravit. 10R. Schattner, "The uniqueness of the center of mass in general relativity," Gen. Relativ. Gravit. 10 (1979) 395-399. Spinning test particles in a Kerr field -II. K Kyrian, O Semerák, 10.1111/j.1365-2966.2007.12502.xMon. Not. R. Astron. Soc. 382K. Kyrian and O. Semerák, "Spinning test particles in a Kerr field - II," Mon. Not. R. Astron. Soc. 382 (2007) 1922-1932. C W Misner, K S Thorne, J A Wheeler, Gravitation. W. H. Freeman and Company. New YorkC. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. W. H. Freeman and Company, New York, 1973. . N Stergioulas, S Morsink, N. Stergioulas and S. Morsink, RNS code. http://www.gravity.phys.uwm.edu/rns/. Comparing models of rapidly rotating relativistic stars constructed by two numerical methods. N Stergioulas, J L Friedman, 10.1086/175605arXiv:astro-ph/9411032Astrophys. J. 444astro-phN. Stergioulas and J. L. Friedman, "Comparing models of rapidly rotating relativistic stars constructed by two numerical methods," Astrophys. J. 444 (1995) 306-311, arXiv:astro-ph/9411032 [astro-ph]. Canonical dynamics of spinning particles in gravitational and electromagnetic fields. H P Künzle, 10.1063/1.1666045J. Math. Phys. 13H. P. Künzle, "Canonical dynamics of spinning particles in gravitational and electromagnetic fields," J. Math. Phys. 13 (1972) 739-744. Dissipative effects in the worldline approach to black hole dynamics. W D Goldberger, I Z Rothstein, 10.1103/PhysRevD.73.104030arXiv:hep-th/0511133Phys. Rev. D. 73104030W. D. Goldberger and I. Z. Rothstein, "Dissipative effects in the worldline approach to black hole dynamics," Phys. Rev. D 73 (2006) 104030, arXiv:hep-th/0511133. Gravitational-wave versus binary-pulsar tests of strong-field gravity. T Damour, G Esposito-Farèse, 10.1103/PhysRevD.58.042001arXiv:gr-qc/9803031Phys. Rev. D. 5842001T. Damour and G. Esposito-Farèse, "Gravitational-wave versus binary-pulsar tests of strong-field gravity," Phys. Rev. D 58 (1998) 042001, arXiv:gr-qc/9803031. Multipole expansions of gravitational radiation. K S Thorne, 10.1103/RevModPhys.52.299Rev. Mod. Phys. 52K. S. Thorne, "Multipole expansions of gravitational radiation," Rev. Mod. Phys. 52 (1980) 299-339. Deviation of quadrupolar bodies from geodesic motion in a Kerr spacetime. D Bini, A Geralico, 10.1103/PhysRevD.89.044013arXiv:1311.7512Phys. Rev. D. 8944013gr-qcD. Bini and A. Geralico, "Deviation of quadrupolar bodies from geodesic motion in a Kerr spacetime," Phys. Rev. D 89 (2014) 044013, arXiv:1311.7512 [gr-qc]. Revising the multipole moments of numerical spacetimes, and its consequences. G Pappas, T A Apostolatos, 10.1103/PhysRevLett.108.231104arXiv:1201.6067Phys. Rev. Lett. 108231104gr-qcG. Pappas and T. A. Apostolatos, "Revising the multipole moments of numerical spacetimes, and its consequences," Phys. Rev. Lett. 108 (2012) 231104, arXiv:1201.6067 [gr-qc]. S Chakrabarti, T Delsate, N Gürlebeck, J Steinhoff, arXiv:1311.6509The I-Q relation for rapidly rotating neutron stars. gr-qcS. Chakrabarti, T. Delsate, N. Gürlebeck, and J. Steinhoff, "The I-Q relation for rapidly rotating neutron stars," arXiv:1311.6509 [gr-qc]. Black holes and spinning test bodies. S N Rasband, 10.1103/PhysRevLett.30.111Phys. Rev. Lett. 30S. N. Rasband, "Black holes and spinning test bodies," Phys. Rev. Lett. 30 (1973) 111-114. Influence of internal structure on the motion of test bodies in extreme mass ratio situations. J Steinhoff, D Puetzfeld, 10.1103/PhysRevD.86.044033arXiv:1205.3926Phys. Rev. D. 8644033gr-qcJ. Steinhoff and D. Puetzfeld, "Influence of internal structure on the motion of test bodies in extreme mass ratio situations," Phys. Rev. D 86 (2012) 044033, arXiv:1205.3926 [gr-qc]. Dynamics of quadrupolar bodies in a Schwarzschild spacetime. D Bini, A Geralico, 10.1103/PhysRevD.87.024028Phys. Rev. D. 8724028D. Bini and A. Geralico, "Dynamics of quadrupolar bodies in a Schwarzschild spacetime," Phys. Rev. D 87 (2013) 024028. Gravitational self-force correction to the binding energy of compact binary systems. A Le Tiec, E Barausse, A Buonanno, 10.1103/PhysRevLett.108.131103arXiv:1111.5609Phys. Rev. Lett. 108131103gr-qcA. Le Tiec, E. Barausse, and A. Buonanno, "Gravitational self-force correction to the binding energy of compact binary systems," Phys. Rev. Lett. 108 (2012) 131103, arXiv:1111.5609 [gr-qc]. On formation of close binaries by two-body tidal capture. W H Press, S A Teukolsky, 10.1086/155143Astrophys. J. 213W. H. Press and S. A. Teukolsky, "On formation of close binaries by two-body tidal capture," Astrophys. J. 213 (1977) 183-192. Tidal resonances in binary star systems. M E Alexander, Mon. Not. R. Astron. Soc. 227M. E. Alexander, "Tidal resonances in binary star systems," Mon. Not. R. Astron. Soc. 227 (1987) 843-861. A variational formalism for tidal excitation: Non-rotating, homentropic stars. Y Rathore, A E Broderick, R Blandford, 10.1046/j.1365-8711.2003.06140.xarXiv:astro-ph/0209003Mon. Not. Roy. Astron. Soc. 339astro-phY. Rathore, A. E. Broderick, and R. Blandford, "A variational formalism for tidal excitation: Non-rotating, homentropic stars," Mon. Not. Roy. Astron. Soc. 339 (2003) 25-32, arXiv:astro-ph/0209003 [astro-ph]. Gravitomagnetic resonant excitation of Rossby modes in coalescing neutron star binaries. É E Flanagan, Racine, 10.1103/PhysRevD.75.044001arXiv:gr-qc/0601029Phys. Rev. D. 7544001É. E. Flanagan andÉ. Racine, "Gravitomagnetic resonant excitation of Rossby modes in coalescing neutron star binaries," Phys. Rev. D 75 (2007) 044001, arXiv:gr-qc/0601029. Gravitational radiative corrections from effective field theory. W D Goldberger, A Ross, 10.1103/PhysRevD.81.124015arXiv:0912.4254Phys. Rev. D. 81124015gr-qcW. D. Goldberger and A. Ross, "Gravitational radiative corrections from effective field theory," Phys. Rev. D 81 (2010) 124015, arXiv:0912.4254 [gr-qc]. Effective potential for even parity Regge-Wheeler gravitational perturbation equations. F J Zerilli, 10.1103/PhysRevLett.24.737Phys. Rev. Lett. 24F. J. Zerilli, "Effective potential for even parity Regge-Wheeler gravitational perturbation equations," Phys. Rev. Lett. 24 (1970) 737-738. Stability of a Schwarzschild singularity. T Regge, J A Wheeler, 10.1103/PhysRev.108.1063Phys. Rev. 108T. Regge and J. A. Wheeler, "Stability of a Schwarzschild singularity," Phys. Rev. 108 (1957) 1063-1069. On the equations governing the perturbations of the Schwarzschild black hole. S Chandrasekhar, 10.1098/rspa.1975.0066Proc. R. Soc. A. 343S. Chandrasekhar, "On the equations governing the perturbations of the Schwarzschild black hole," Proc. R. Soc. A 343 (1975) 289-298. Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description. D Bini, T Damour, G Faye, 10.1103/PhysRevD.85.124034arXiv:1202.3565Phys. Rev. D. 85124034gr-qcD. Bini, T. Damour, and G. Faye, "Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description," Phys. Rev. D 85 (2012) 124034, arXiv:1202.3565 [gr-qc]. Black hole stereotyping: Induced gravito-static polarization. B Kol, M Smolkin, 10.1007/JHEP02(2012)010arXiv:1110.3764JHEP. 120210hep-thB. Kol and M. Smolkin, "Black hole stereotyping: Induced gravito-static polarization," JHEP 1202 (2012) 010, arXiv:1110.3764 [hep-th]. Analytic solutions of the Teukolsky equation and their properties. S Mano, E Takasugi, 10.1143/PTP.97.213arXiv:gr-qc/9611014Prog. Theor. Phys. 97gr-qcS. Mano and E. Takasugi, "Analytic solutions of the Teukolsky equation and their properties," Prog. Theor. Phys. 97 (1997) 213-232, arXiv:gr-qc/9611014 [gr-qc]. I-Love-Q: Unexpected universal relations for neutron stars and quark stars. K Yagi, N Yunes, 10.1126/science.1236462arXiv:1302.4499Science. 341gr-qcK. Yagi and N. Yunes, "I-Love-Q: Unexpected universal relations for neutron stars and quark stars," Science 341 (2013) 365-368, arXiv:1302.4499 [gr-qc]. I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves and fundamental physics. K Yagi, N Yunes, 10.1103/PhysRevD.88.023009arXiv:1303.1528Phys. Rev. D. 8823009gr-qcK. Yagi and N. Yunes, "I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves and fundamental physics," Phys. Rev. D 88 (2013) 023009, arXiv:1303.1528 [gr-qc]. Equation-of-state-independent relations in neutron stars. A Maselli, V Cardoso, V Ferrari, L Gualtieri, P Pani, 10.1103/PhysRevD.88.023007arXiv:1304.2052Phys. Rev. D. 8823007gr-qcA. Maselli, V. Cardoso, V. Ferrari, L. Gualtieri, and P. Pani, "Equation-of-state-independent relations in neutron stars," Phys. Rev. D 88 (2013) 023007, arXiv:1304.2052 [gr-qc]. Relations between neutron-star parameters in the Hartle-Thorne approximation. M Bauböck, E Berti, D Psaltis, F Özel, 10.1088/0004-637X/777/1/68arXiv:1306.0569Astrophys. J. 77768astro-ph.HEM. Bauböck, E. Berti, D. Psaltis, and F.Özel, "Relations between neutron-star parameters in the Hartle-Thorne approximation," Astrophys. J. 777 (2013) 68, arXiv:1306.0569 [astro-ph.HE]. On the universality of I-Love-Q relations in magnetized neutron stars. B Haskell, R Ciolfi, F Pannarale, L Rezzolla, 10.1093/mnrasl/slt161arXiv:1309.3885Mon. Not. R. Astron. Soc. Lett. 438astro-ph.SRB. Haskell, R. Ciolfi, F. Pannarale, and L. Rezzolla, "On the universality of I-Love-Q relations in magnetized neutron stars," Mon. Not. R. Astron. Soc. Lett. 438 (2014) L71-L75, arXiv:1309.3885 [astro-ph.SR]. Breakdown of I-Love-Q universality in rapidly rotating relativistic stars. D D Doneva, S S Yazadjiev, N Stergioulas, K D Kokkotas, 10.1088/2041-8205/781/1/L6arXiv:1310.7436Astrophys. J. Lett. 7816gr-qcD. D. Doneva, S. S. Yazadjiev, N. Stergioulas, and K. D. Kokkotas, "Breakdown of I-Love-Q universality in rapidly rotating relativistic stars," Astrophys. J. Lett. 781 (2014) L6, arXiv:1310.7436 [gr-qc]. Universal behavior of rotating neutron stars in GR: Even simpler than their Newtonian counterparts. G Pappas, T A Apostolatos, arXiv:1311.5508gr-qcG. Pappas and T. A. Apostolatos, "Universal behavior of rotating neutron stars in GR: Even simpler than their Newtonian counterparts," arXiv:1311.5508 [gr-qc]. Multipole Love relations. K Yagi, arXiv:1311.0872gr-qcK. Yagi, "Multipole Love relations," arXiv:1311.0872 [gr-qc]. Cubic order spin effects in the dynamics and gravitational wave energy flux of compact object binaries. S Marsat, arXiv:1411.4118gr-qcS. Marsat, "Cubic order spin effects in the dynamics and gravitational wave energy flux of compact object binaries," arXiv:1411.4118 [gr-qc].
[]
[ "Do we live in an anthropic universe?", "Do we live in an anthropic universe?" ]
[ "Domingos Soares \nPhysics Department\nFederal University of Minas Gerais Belo Horizonte\nBrazil\n" ]
[ "Physics Department\nFederal University of Minas Gerais Belo Horizonte\nBrazil" ]
[]
I cast doubt upon the desired consistency between the anthropic principle and modern cosmology.
null
[ "https://arxiv.org/pdf/physics/0209094v3.pdf" ]
118,967,244
physics/0209094
a9a1dd63c7b3c512e1cab6e5f5932eaaa979add5
Do we live in an anthropic universe? 23 Nov 2021 September 26, 2002 Domingos Soares Physics Department Federal University of Minas Gerais Belo Horizonte Brazil Do we live in an anthropic universe? 23 Nov 2021 September 26, 2002arXiv:physics/0209094v3 [physics.gen-ph] I cast doubt upon the desired consistency between the anthropic principle and modern cosmology. Amongst all possible universes we live in one that deserves us. This is what could be called the naive version of the anthropic principle. At what extent this view is consistent with modern scientific results obtained from theoretical and observational work in cosmology? The anthropic principle was originally put forward by the cosmologist Brandon Carter (1974) with the statement that 'our location in the universe is necessarily privileged to the extent of being compatible with our existence as observers'. (The italic is mine). The definite status as a consensual principle of nature has been crowned with the thorough account of its implications in seemingly unpaired areas of human knowledge such as philosophy, quantum mechanics, cosmology, biochemistry, the search for extraterrestrial life and ultimately the future of the universe, by John D. Barrow and Frank J. Tipler, in the now classical book entitled The Anthropic Cosmological Principle (Barrow & Tipler 1986 but see Soares 2001). Since then, two major achievements in cosmology lie on our pathway, two brilliant milestones. On the theoretical side, Alan Guth invented the inflationary theory (Guth 1997) in the early 80's, and on the observational side, the first results from the Cosmic Background Explorer satellite were published in the early 90's 1 . The expanding universe paradigm gained strength with renewed blood from these sources. Two recent reviews by Michael S. Turner (2002) and Max Tegmark (2002) give a clear picture of the present situation. The evidence for a flat global topology comes from both inflation and measurements of the anisotropy of the cosmic microwave background on angular scales of about 1 degree. The measurements were triggered by COBE's spectacular results, from a plethora of satellite and balloon experiments (see Tegmark 2002). Current cosmological models should be at least reassuring of an anthropic universe. But, what does the cosmic budget tell us? Following Turner one has: • Bright stars: 0.5% • Baryonic dark matter: 3.5% • Nonbaryonic dark matter: 30% • Dark energy: 66% Flatness requires that everything adds up to 100% of the closure density. Except for a half per cent of visible, ordinary, observable matter, we are left with dark, exquisite, unobservable stuff. Now, back right to the beginning: aren't we in the wrong universe? See COBE's homepage at http://space.gsfc.nasa.gov/astro/cobe/ J D Barrow, F J Tipler, The Anthropic Cosmological Principle. OxfordOxford University PressBarrow, J.D. & Tipler, F.J. 1986, The Anthropic Cosmological Principle (Ox- ford University Press, Oxford) B Carter, Confrontation of Cosmological Theories with Observation. M.S. Longair (ReidelDordrecht291Carter, B. 1974, in Confrontation of Cosmological Theories with Observation, ed. M.S. Longair (Reidel, Dordrecht), p. 291 A H Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. ReadingAddison-Wesley Publishing CompanyGuth, A.H. 1997, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (Addison-Wesley Publishing Company, Reading) . D S L Soares, arXiv:astro-ph/0108180Soares, D.S.L. 2001, arXiv:astro-ph/0108180 . M Tegmark, arXiv:astro-ph/0207199Tegmark, M. 2002, arXiv:astro-ph/0207199 . M S Turner, arXiv:astro-ph/0207297Turner, M.S. 2002, arXiv:astro-ph/0207297
[]
[ "Dopin Controlle g d Superconductor-Insulator Transition in Bi 2 Sr 2-x La x CaCu 2 O 8+δ", "Dopin Controlle g d Superconductor-Insulator Transition in Bi 2 Sr 2-x La x CaCu 2 O 8+δ" ]
[ "* Eckstein \nDepartment of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA\n", "Seongshik Oh \nDepartment of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA\n", "Trevis A Crane \nDepartment of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA\n", "D J Van Harlingen \nDepartment of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA\n", "J N \nDepartment of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA\n" ]
[ "Department of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA", "Department of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA", "Department of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA", "Department of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA", "Department of Physics\nUniversity of Illinois\n61801UrbanaIllinoisUSA" ]
[]
SIT) in a high mentally l doping, the sheet resistance seems to diverge in the zero temperature limit. Above the critical doping, the transport is universally scaled by a two-component conductance model. Below, it continuously evolves from weakly to strongly insulating behavior. The two-component conductance model suggests that a collective electronic phase separation mechanism may be responsible for this unconventional SIT behavior.We show that the doping-controlled superconductor-insulator transition critical temperature cuprate system (Bi 2 Sr 2-x La x CaCu 2 O 8+δ ) exhibits a funda different behavior than is expected from conventional SIT. At the critica * [email protected], Present address: National Institute of Standards and Technology, Boulder CO 80305.
10.1103/physrevlett.96.107003
[ "https://arxiv.org/pdf/cond-mat/0509521v4.pdf" ]
13,591,162
cond-mat/0509521
8372f85640545c14b07db194becf134677948263
Dopin Controlle g d Superconductor-Insulator Transition in Bi 2 Sr 2-x La x CaCu 2 O 8+δ * Eckstein Department of Physics University of Illinois 61801UrbanaIllinoisUSA Seongshik Oh Department of Physics University of Illinois 61801UrbanaIllinoisUSA Trevis A Crane Department of Physics University of Illinois 61801UrbanaIllinoisUSA D J Van Harlingen Department of Physics University of Illinois 61801UrbanaIllinoisUSA J N Department of Physics University of Illinois 61801UrbanaIllinoisUSA Dopin Controlle g d Superconductor-Insulator Transition in Bi 2 Sr 2-x La x CaCu 2 O 8+δ numbers: 7130+h7425Dw7478Bz SIT) in a high mentally l doping, the sheet resistance seems to diverge in the zero temperature limit. Above the critical doping, the transport is universally scaled by a two-component conductance model. Below, it continuously evolves from weakly to strongly insulating behavior. The two-component conductance model suggests that a collective electronic phase separation mechanism may be responsible for this unconventional SIT behavior.We show that the doping-controlled superconductor-insulator transition critical temperature cuprate system (Bi 2 Sr 2-x La x CaCu 2 O 8+δ ) exhibits a funda different behavior than is expected from conventional SIT. At the critica * [email protected], Present address: National Institute of Standards and Technology, Boulder CO 80305. Since the late 1980s, the superconductor-insulator transition has been intensively studied For doping levels much higher than the critical doping, the resistance monotonically decreases as a function of decreasing temperature and drops to zero below a doping, the peratures, crease as the temperature is further reduced. However, for a small range of doping levels just above the critical doping, the temperature dependence of the resistance is composed of three regions [10][11][12][13]. As a function of decreasing temperature, the resistance first decreases at high temperatures, reaches a minimum at an intermediate temperature, starts to increase at lower temperatures until it reaches a local maximum, and then it drops to zero resistance below the superconducting transition temperature. Although such a re-entrant behavior is both experimentally [1][2][3][4] and theoretically [5][6][7][8][9] in homogeneously disor dimensional (2D) thin films. These studies suggest that there exists a defect-sc dominated metallic (that is, resistance being temperature-independen zero-temperature critical sheet resistance at the transition from super perconducting as T 0, otherwise they are insulating. High-T c cuprates (HTCs) are a quasi-2D system, and magnetic field [1 doping control [12][13][14] can continuously transform superconducting cuprate insulators as in the conventional SIT. In the case of doping-controlled superconducting transition temperature. For doping levels below the critical resistance first decreases as a function of decreasing temperature at high tem reaches a minimum at an intermediate temperature and then it starts to in 2 universally observed in almost all underdoped cuprates [10][11][12][13], its origin or significance has not attracted much attention so far. SIT study with n hidden in this phenomenon. First of all, the sheet resistance tends to diverge as T 0 at the critical doping, contrary to the common notion of finite critical sheet resistance not only in conventional disordered metals [1][2][3][4] but also in cuprates [12,13]. Above the critical ling function ng phase. This separation mechanism driven by carrier-carrier correlations [15][16][17][18][19]. We attribute the observed divergence of the critical sheet resistance to this phase separation phenomenon. [20]. Each curve in Fig. 2 is obtained after a few minutes of (~10 -8 Torr) or ozone (~10 -6 Torr) annealing at ~400 ºC. Fig. 2 shows sheet . Above T min , all the curves exhibit metallic behavior (dR/dT > 0), and dop provides a temperature-ind resistance either switches to a metallic behavior (dR/dT > 0) when critical value or remains insulating. This is more easily seen in Fig. 2(b) whe resistance versus logT. We interpret the change to metallic behavior as t superconducting fluctuations and expect these samples will eventually 4 In conventional superconductors, the temperature dependence of resistance just A key finding of our work is that the bottom 8 curves (curves 6 throu Fig. 2(b), which show resistance peaks above 4.2 K, can be rescaled by divi temperature by T p and the resistance by R p . When this is done, the low tem of the curves overlap nicely as shown in Fig. 3(a). This suggests that th conductance model, where the total conductance, g(t) = r(t) -1 , is due to tw c t t − 0.065 found as the parameters leading to the best fit of the data. Here T c re superconducting critical temperature, below which the resistance In AL theory, the fluctuation contribution, however, is not referenced to prope is term to the overall conductance is scaled by 1/ p which is strongly d doping. The two-component scaling function determined from curves 6-13 in plotted in Fig. 3 This result is interesting because it implies that, for the doping contro critical temperature dependence is insulating (dR/dT < 0), and a zero-tempe may doubt if the analysis obtained from data above 4.2 K can be applied to the zero temperature. The best way to resolve this issue would be to exten investigation down to much lower temperatures. However, we believe th y deviation from the two component scaling form in the temperature ran studied. Making use of the normal state conductance, G, which is monoton the doping level as shown in Fig. 1, it is possible to estimate the doping lev curve, since within a reasonable approximation G must be proportional t 9 scaling predicts T c ∝ |d-d c | νz [6], a power law relationship is expected between T p and doping. From Fig. 3(d), the estimated critical exponent is νz = 1.8 ± 0.3. curves 1 and 2 of the sheet resistance for a barely insulating sample between 2 and 0.2 K; this is labelled curve a. For curves a and b, the resistance is well described by 2D variable-range hopping (VRH), where [31,32]. Here Transport on the insulating side of the SIT is shown in Fig. 4, where from Fig. 2 are reproduced. We also measured the temperature dependence 1/ 3 0 0 ( ) ( ) exp / R T R T T = ( ) 2 10 / T k N E ξ = 0 B F density for all cuprates in the temperature dependence of the in-plane resistance. with A = 0.881, q = 0.145, B = 1.72 and t c (≡ T c /T p ) = presents the le turns zero.Since by the definition of T p , among the four parameters only two (for example, q and t c ) are independent, and the other two can be obtained from A = (1 + q(1 -t c )) -1 and B = Aq(1 -t c ) 2 . The two independent model parameters, q and t c . The first term represents the conductance of 2D weakly-insulating normal carriers from the hole-poor background, while the second is due to the conductance of fluctuating superconducting pairs from the hole-rich islands, similar in kin (AL)[29].any material rty and is scaled only by 4e 2 /h. In our case, since 1/R = g(t)/R p , the a range of R p and T p pairs in which T p can take any values between 0 and ~0.5K. In order to show this, we have selected three representative values, 0.1 K, 0.01 K and 0.001 K for T p and presented the corresponding fitting curves inFig. 3(b). All three curves are equally well fit to the scaling function even if their T p values are orders of magnitude different with one another. This means that the doping level, d, in curve 3 is Both terms of this two-component conductance formula correspond identifiable conductance channels and this is consistent with the above-ment form to the 2D superconducting fluctuations treated by Aslamasov and Lar to 0.01, we believe the obtained value is effectively zero. If α were not zero, then as T c 0, R would tend either toward zero or infinity for all temperatures depending on the sign of α, either of which is unphysical. Since we find α ~ 0, we believe our scaling c The temperature dependence corresponding to the critical doping, d c , obtained by taking the limit of the scaling function as T ∝T 0the best fit for the 8 data points (curves 6-13) of R p v of R and T for the other two (curves 4 and 5) with T < 4.value of α is significant because α = 0 implies that there ex asymptotic critical-doping temperature dependence to which R(T) tends as th approaches d from the superconducting side. Since the error in determ8 function and empirical relationship between R p and T p provide a well-behaved limiting form as T c goes to zero. lled SIT, the rature critical sheet resistance does not exist. This is in direct contrast to the conventional picture of the SIT, according to which a temperature-independent and finite sheet resistance exists at a critical value of the control parameter[1][2][3][4]12,13]. With respect to this interpretation, remain unchanged, because there is no indication in the transport data ofan ge we have ically related to el, d, of each o d. Assuming that d of curve 3 is the critical value, d c and taking 0.05 hole/Cu, the widely-accepted value for the critical doping [30], as the critical value, we have converted G at 150 K into d inFig. 3(d) and presented T p as a function of the doping level. Since T c (∝T p ) is the energy scale that governs transport on the superconducting side and renormalization of states at the Fermi level, and ξ is the characteristic size of the local between which hopping occurs. In our data T 0 tends to zero as the doping le approaches the critical doping from the insulating side. Curve 2 is better fit VRH form at very low temperatures. This observation suggests the existen crossover temperature, T i , at which a weakly insulating (power law dependent) measured temperature range. Curve 1 tends to deviate from a high-temperature power-law toward a low-temperature 2D-VRH dependence. Even curve 2 may take a 2Dce of a igh havior transforms into a strongly (quasi-exponential such as 2D-VRH) in e and ξ very hich is the case for All the above observations can be summarized into a (T, d) phase diagram shown inFig. 5. It is composed of metallic, insulating, superconductive fluctuating (i.e. local superconducting), and global superconducting regions divided by three crossover sulating low temperature one. In the other extreme, when T 0 gets very larg small, the transport starts to diverge faster than is expected by VRH, w curves c, d and e. 10 temperatures, T , T and T . The insulating region can again be divided into starts to segregate into superconductive fluctuating islands and the sea of insulating background at temperatures below T . The resistance is first dominated by , a global superconducting state emerges. Along the doping axis, as the doping level is reduced from ~0.058 (hole/Cu), the relative portion of the superconductive own to lower ping, 0.050 the way down to the zero temperature and this results in absence of the zero temperature critical sheet resistance, r temperatures, considering that our measurement was obtained only above 4 K. Figure Captions wide doping range. The solid arrow on the vertical axis indicates the pair quantum resistance, h/4e 2 . ery small range of doping levels. (a) and (b) are the same data plotted on different horizontal scales. Note that each curve is indexed by a number 1 through 13 in the order of increasing doping. Fig. 3 (Fig 3color online). Scaling of superconducting curves. Each index represents the right as examples. (c) R p vs T p for the 10 superconducting curves (4-13). The solid line represents the best fitting for the 8 measured data points (curves 6-13), which corresponds to R p = 1.72T p -0.149 . (d) T p vs doping. Conductance at 150 K is also shown on the top axis. The solid line, T p = 4.02⋅10 5 ⋅(d -d c ) 1.8 , is the best fitting for d > d c . Fig. 1 ( 14 Fig 114color online). Doping controlled SIT over a Fig. 2 (color online). Detailed view of doping-controlled SIT over a v corresponding curve from Fig. 2(b). (a) Rescaled plot of curves 6-13 from The two-component scaling function plotted together with curves 3 -13. Curv used to obtain the scaling function. Curves 4-5 are fit to the obtained scalin T p and R p as the fitting parameters. As for curve 3, any choice of T p b reasonably good fittin three curves with T = 0.comparison. Inset shows how curve a compares with other curves from Fig. 2.Fig. 5 (color online). (T, d) phase diagram near the critical doping. T and T in are ing to the scaling analysis. Fitting curves for each of these crossover temperatures are plotted as well. Unlike other crossover temperatures, T i is only marginally defined in our experiment. p m determined by the resistance data, and T c is evaluated as 0.065⋅T p accord tunnelling energy and the Josephson coupling energy overcome the Coulomb energy at reduced temperatures, a superconducting transition can occur from a phase. We propose that the insulating normal-state temperature dependen here even in superconducting samples, is of a similar origin. Although H fect[24][25][26][27], and scanning-tunnelling experiments have verified that su electronic states prevail in underdoped HTCs[18,19].According to this phase separation scenario, the re-entrant behavior o background until superconducting islands start to couple with each other at lower where r ≡ R/R p and t ≡ T/T p , which describes the transition from insulating to superconducting temperature dependence for all of the superconducting curves. One can find a well-fitting scaling function, r(t), using a simple two-component o contributions and is given by temperatures, ultimately giving rise to a global superconducting transitio re-entrant behavior is universally observed in almost all underdoped HTC s 13,28], its origin has not been well-known. Below, using a two-compon model, we present a quantitative ana echanism behind this re-entrant behavior.perature- cting transition or, has been ds are embedded in an insulating background [21-23]. In such a case, even when the high temperature dependence shows insulating behavior due to Coulomb charging energy, if the intergrain n insulating normal ce, which we see TCs are structurally homogeneous, theories have predicted that hole carriers may segregate into hole-rich and hole-poor islands at low doping due to strong carrier-carrier correlation ef ch granular bserved for doping levels above the critical value can be qualitatively understood. At temperatures above T min , strong thermal fluctuations wash out any phase separation mechanism, leaving only metallic behavior. Below T min , however, phase separation occurs such that hole-rich (presumably superconducting) islands are formed inside a hole-poor (presumably insulating) background. In this case, the resistance is dominated by the insulating above the superconducting transition temperature is generally flat due to tem independent defect scattering [1,4]. However, emergence of a supercondu from an insulating temperature dependence, which we call re-entrant behavi observed in granular superconductors, where superconducting islan 5 n. Although such ystems [10- ent conductance lysis supporting the phase-separation as the main m gh 13) from ding the perature parts ere exists a single scaling function r = r(t), (b) together with curves 3-13. Curves 4 and 5 can be nicely scaling function with T p values less than 4.2 K and corresponding R p values 2 cannot be fit to the scaling function for any choice of T and R value very near the T = 0 transition value, but it is impossible to know whether it is exactly zero or just very nearly zero.7 can be c P h corresponds to R = 1.14R p (T p /T) . Once the relationship between R p and T p is known, temperature dependence at critical doping can be determined. Our two-component conductance model does not give any a priori information about how T and R are related, and they should be R T s T p . Values The authors thank Philip W. Phillips, Anthony J. Leggett, and Michael B. Weissman for useful discussions. We also acknowledge the US Office of Naval Research for strongly insulating regions by T i .Fig. 5presents a detailed view right aro doping, which has not been well treated in the conventional HTC phase dia According to the above analysis, this phase diagram is interpreted as f doping level gets lower than ~0.058 (hole/Cu), the high temperature homo min weakly-insulating background. However, as the temperature is further increasing conductivity of the superconductive fluctuating islands overturns temperature dependence into metallic (dR/dT >0) behavior below T p . At eve temperatures (below T ), when Josephson coupling is formed among the sup islands gets smaller and the insulating temperature dependence survives d temperatures. Eventually when the doping level reaches the critical do (hole/Cu), the weakly-insulating behavior survives all which, however, needs to be verified by further studies at much lowe11 supporting under grant N00014-00-1-0840 and the use of the Frederick Seitz MRL-CMM at UIUC through U.S. DOE, Div. of Mat. Sci. award #DEFG02-91ER45439.min p c . hys. Rev. Lett. 622180hys. Rev. Lett. 62, 2180 (1989). . ys. Rev. Lett. 743037ys. Rev. Lett. 74, 3037 (1995). . T Pang, Phys. Rev. Lett. 622176T. Pang, Phys. Rev. Lett. 62, 2176 (1989). . M P A Fisher, G Grinstein, S M Girvin, Phys. Rev. Lett. 6414440M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys. Rev. Lett. 64, 587 (1990). 14440 (1998). . S Ono, Phys. Rev. Lett. 85638S. Ono et al., Phys. Rev. Lett. 85, 638 (2000). . K Semba, A Matsuda, Phys. Rev. Lett. 86163K. Semba and A. Matsuda, Phys. Rev. Lett. 86, 496 (2001). 351, 163 (2001). . Te Superconductors, Springer-Verlangte Superconductors. (Springer-Verlang, . E Dagotto, Rev. Mod. Phys. 66763E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). . S A Kivelson, Rev. Mod. Phys. 751201S. A. Kivelson et al., Rev. Mod. Phys. 75, 1201 (2003). . K M Lang, Nature. 415412K. M. Lang et al., Nature 415, 412 (2002). . S H Pan, Nature. 413282S. H. Pan et al., Nature 413, 282 (2001). . S Oh, T Di Luccio, J N Eckstein, Phys. Rev. B. 7152504S. Oh, T. Di Luccio, and J. N. Eckstein, Phys. Rev. B 71, 52504 (2005). . D B Haviland, Y Liu, A M Goldman, D. B. Haviland, Y. Liu, and A. M. Goldman, P . N Markovic, Phys. Stat. Sol. (B). 218N. Markovic et al., Phys. Stat. Sol. (B) 218, 221 (200 . A Yazdani, A Kapitulnik, PhA. Yazdani and A. Kapitulnik, Ph . A M Goldman, Physica E. 182A. M. Goldman, Physica E 18, 1 (2 . M C Cha, S M Girvin, Phys. Rev. B. 499794M. C. Cha and S. M. Girvin, Phys. Rev. B 49, 9794 (1994). . D Das, S Doniach, Phys. Rev. B. 57D. Das and S. Doniach, Phys. Rev. B 57, . P Phillips, D Dalidovich, Science. P. Phillips and D. Dalidovich, Science . Y Ando, Phys. Rev. Lett. 754662Y. Ando et al., Phys. Rev. Lett. 75, 4662 (1995). . Z Konstantinovic, Z Z Li, H Raffy, Physica CZ. Konstantinovic, Z. Z. Li, and H. Raffy, Physica C . T Tamegai, Jpn. J. Appl. Phys. T. Tamegai et al., Jpn. J. Appl. Phys Phase Separation in Cupra Berlin. E Sigmund, K A , E. Sigmund, K. A. Phase Separation in Cupra Berlin, 1994). Y Shapira, G Deutscher, Granular and Quantum Effects ). 274463Y. Shapira and G. Deutscher, Phys. Rev. B 27, 4463 (1983). perconductors: Granular and Quantum Effects ). . V I Yukalov, E P Yukalova, Phys. Rev. B. 70224516V. I. Yukalov and E. P. Yukalova, Phys. Rev. B 70, 224516 (2004). . V J Emery, S A Kivelson, H Q Lin, Phys. Rev. Lett. 64475V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990). . S A Emery, Kivelson, Physica C. 209597Emery and S. A. Kivelson, Physica C 209, 597 (1993). . hys. Rev. Lett. 94207004hys. Rev. Lett. 94, 207004 . M R Presland, Physica C. 17695M. R. Presland et al., Physica C 176, 95 (1991). N F Mott, Conduction in Non-Crystalline Materials. OxfordClarendon PressN. F. Mott, Conduction in Non-Crystalline Materials (Clarendon Press, Oxford, 1993). B I Shklovskii, A L Efros, Electronic Properties of Doped Semiconductors. Berlin, HeidelbergSpringer-VerlagB. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin, Heidelberg, 1984). . M Ma, P A Lee, Phys. Rev. B. 325658M. Ma and P. A. Lee, Phys. Rev. B 32, 5658 (1985). . E Simánek, Inhomogeneous Su, Oxford Univ. PressNew YorkE. Simánek, Inhomogeneous Su (Oxford Univ. Press, New York, 1994). . L P Gor&apos;kov, A V Sokol, JETP Lett. 46420L. P. Gor'kov and A. V. Sokol, JETP Lett. 46, 420 (1987 . V , V. J . S Komiya, H D Chen, S C Zhang, Y Ando, P. S. Komiya, H. D. Chen, S. C. Zhang, and Y. Ando, P (2005). . L G Aslamasov, A I Larkin, Phys. Lett. 26238L. G. Aslamasov and A. I. Larkin, Phys. Lett. 26A, 238 (1968).
[]
[ "Ultralong distance coupling between asymmetric resonant microcavities", "Ultralong distance coupling between asymmetric resonant microcavities" ]
[ "Fang-Jie Shu \nDepartment of Physics\nShangqiu Normal University\n476000ShangqiuP. R. China\n", "Chang-Ling Zou \nKey Lab of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiP. R. China\n", "Wen-Cong Chen \nDepartment of Physics\nShangqiu Normal University\n476000ShangqiuP. R. China\n", "Fang-Wen Sun \nKey Lab of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiP. R. China\n" ]
[ "Department of Physics\nShangqiu Normal University\n476000ShangqiuP. R. China", "Key Lab of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiP. R. China", "Department of Physics\nShangqiu Normal University\n476000ShangqiuP. R. China", "Key Lab of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiP. R. China" ]
[]
The ultralong distance coupling between two Asymmetric Resonant Microcavities (ARCs) is studied. Different from traditional short distance tunneling coupling between microcavities, the high efficient free space directional emission and excitation allow ultralong distance energy transfer between ARCs. In this paper, a novel unidirectional emission ARC, which shows directionality I40 = 0.54, is designed for materials of refractive index n = 2.0. Compared with regular whispering gallery microresonators, the coupled unidirectional emission ARCs show modulations of resonance frequency and linewidth even when the distance between cavities is much longer than wavelength. The performances and properties of the ultralong distance interaction between ARCs are analyzed and studied by coupling mode theory in details. The ultralong distance interaction between ARCs provides a new way to free-space based optical interconnects between components in integrated photonic circuits.
10.1364/josab.31.000478
[ "https://arxiv.org/pdf/1310.1992v1.pdf" ]
118,387,550
1310.1992
3c411adffffb56c461ef2df9f936646757c26586
Ultralong distance coupling between asymmetric resonant microcavities 8 Oct 2013 (Dated: October 9, 2013) Fang-Jie Shu Department of Physics Shangqiu Normal University 476000ShangqiuP. R. China Chang-Ling Zou Key Lab of Quantum Information University of Science and Technology of China 230026HefeiP. R. China Wen-Cong Chen Department of Physics Shangqiu Normal University 476000ShangqiuP. R. China Fang-Wen Sun Key Lab of Quantum Information University of Science and Technology of China 230026HefeiP. R. China Ultralong distance coupling between asymmetric resonant microcavities 8 Oct 2013 (Dated: October 9, 2013) The ultralong distance coupling between two Asymmetric Resonant Microcavities (ARCs) is studied. Different from traditional short distance tunneling coupling between microcavities, the high efficient free space directional emission and excitation allow ultralong distance energy transfer between ARCs. In this paper, a novel unidirectional emission ARC, which shows directionality I40 = 0.54, is designed for materials of refractive index n = 2.0. Compared with regular whispering gallery microresonators, the coupled unidirectional emission ARCs show modulations of resonance frequency and linewidth even when the distance between cavities is much longer than wavelength. The performances and properties of the ultralong distance interaction between ARCs are analyzed and studied by coupling mode theory in details. The ultralong distance interaction between ARCs provides a new way to free-space based optical interconnects between components in integrated photonic circuits. I. INTRODUCTION Atoms are bonded together to form molecules, which greatly enrich our world. The same story is true for microcavities [1]. Coupling of two microdisks [2][3][4][5] provide appealing characters, such as high-Q unidirectional emission, low-threshold and single-frequency laser. These bonded microcavities, known as the photonic molecule [6] [7], are formed relying on the bond interaction which comes from the short distance evanescent coupling. Nevertheless, the typical length of evanescent tail of optical modes reaching out of the cavities is smaller than the wavelength, thus it is required a subtle control on subwavelength interval between cavities for forming of a designed photonic molecule. Therefore, the experiment difficulties are increased and the applications are limited. For example, Stock et al. [8] presented recently an on-chip integrated quantum optical system made up with two micropillars: one micropillar is an electrically pumped microlaser which serves as the integrated light source, the other micropillar is a microcavity containing quantum dots which serves as quantum optical device. In this configuration, the energy transfered from the light source to the device is crucial. However, due to the fact that each pillar cavity is covered by an electrode or a micro-aperture which much larger than themselves, the gap between two cavities is naturally much longer than the wavelength, thus the short distance interaction is very inefficient. As suggested by the authors, the asymmetric resonant cavities (ARCs), instead of regular circular shaped cavities, can be applied for efficient energy transfer. In this paper, we numerically demonstrated the ultralong distance interaction between ARCs through radiation coupling. The key issue of the long distance interaction is the high directionality emission of ARC, which determines the light emission and collection efficiency [10][11][12]. First of all, we proposed a novel ARC for unidirectional emission. Then, we carried out a strong far-field coupling system composed by two ARCs placed as emission direction opposite to each other. Next, numerical and theoretical studies are performed to investigate the unidirectional emission character of individual cavity, coupling induced resonance variation, and energy transferring between two cavities. The ultralong coupling scheme provides a new energy saving way to exchange energy on-chip. II. ARC WITH UNIDIRECTIONAL EMISSIONS Directional emission is essential for free space radiation and excitation of microcavity, so we should carefully choose a proper cavity. Here, we focus on the coupling between two cavities, thus the unidirectional emission is necessary. Different from the most design aiming on a high (n = 3.3) [13,15] or a low (n = 1.5) [14,15] refractive index, we focus on the cavity with middle refractive index around n = 2.0 in which condition the previous cavity design is not applicable, for the relative refractive index of the semiconductor cavity embedded in polymer or silica in Ref. [8] and also refractive index of some materials, such as SiN and diamond, are near 2.0. Based on the analysis of cavity's symmetry, the cavity shape for high quality factor and unidirectional emission can be written as [15] B(φ) = R(1 − i a i cos i φ), −π/2 < φ ≤ π/2 R(1 − i b i cos i φ), π/2 < φ ≤ 3π/2 (1) where φ is the polar angle respected to right direction of horizontal axis, and R is the radius . In the frame of ray optics, the boundary B(φ) of a cavity with unidirectional emission is optimized by using a hill-climbing algorithm [16]. We obtain a set of parameters for unidirectional emission. These are a 2 = 0.03693, a 3 = 0.09501, b 2 = 0.09791, b 3 = −0.02404, and zero for other a i , b i . Now we turn to wave optics. In the designed ARC, a transverse magnetic (TM) mode, which has only one nozero electric field component in direction perpendicular to the cavity plane, characterized as TM 37,1 , which is found through boundary elements method [17,18]. In the distribution of normalized intensity of this mode in logarithmic scale [ Fig. II(a)], two main emission regions lie in the top and the bottom sites. Through these sites, the radiation from the mode TM 37,1 mainly propagates toward the left direction, and forms a narrow and high emission peak near the angle of 180 • in the far-field angular distribution ( Fig. II(b)). From the far-field distribution, it is found that 54% of total emission energy is enclosed in the angle of 40 • around the main peak or say technically I 40 = 0.54 [16]. The dimensionless eigenfrequency of TM 37,1 is kR = 22.32621 − 0.00011i, where i stands for the imaginary unit. The imaginary part Im(kR) characterizes the dissipation of the cavity mode, satisfy the relation Im(kR) = −Re(kR) 2Q (2) where Q is quality factor of a mode. Note that since the numerical error raised in our calculation (total boundary element number is 1600) is much larger than Im(kR), it is hard to solve the tiny imaginary part Im(kR) directly [6,17]. In this paper, we solve the accurate Q with the help of energy flow method [19], then obtain the Im(kR) through Eq. (2). III. NUMERICAL RESULTS ABOUT THE COUPLING Place two aforementioned ARCs together with mirror symmetry about the vertical axis, and let their main emission directions pointing to each other to form a double-cavities system. Since the system is linear, we can deduce the dynamics of ARCs through the eigenmodes of the system. Therefore, we resort eigenvalue equation to investigate this system rather than research the complicated dynamic process of coupling directly. When the gap L (Fig. IV) between the cavities is 3R, an intensity distribution of the eigenmode, whose resonant frequency kR EE = 22.32617 − 0.00011i corresponding to TM 37,1 in single ARC, of the system is calculated by solving the eigenfunction of kR ( Fig. III(a)) [18]. As we can see in the figure, intensity along the vertical and horizontal axis of symmetry is maximum, so corresponding complex amplitude distribution of optical field has reflective symmetry about the two axes. This mode is classified as even-even (EE) mode. Near the resonant frequency of EE mode, there always exist other three classes of mode (odd-even, even-odd, and odd-odd) in the weak coupling system with four-fold reflective symmetry. In our To compare with coupling of circular cavities without directional emission, two EE modes in coupled circular cavities are calculated with parameters: gap L = 3R, radius R = 1, refractive index n = 2.0. The TM 40,1 and the TM 30,3 modes are found around kR = 22.3 with kR c1 = 22.70596 − 5.7 × 10 −13 i and kR c3 = 22.02345 − 0.00004i, respectively. Because the Q of TM 40,1 mode is as high as 2.0 × 10 13 , in other words, little energy can leak out from the cavities. Intensity outside the cavities is failed to be depicted even in logarithmic coordinates ( Fig. III(b)). In this case, the coupling between the modes of two cavities can be ignored, and the field distribution and the resonant frequency consisted with that of the isolated cavity. For the TM 30,3 mode, its Q ∼ 2.5 × 10 5 is much less than that of TM 40,1 mode, while it is in the same order of magnitude with the Q ∼ 1.0 × 10 5 of TM 37,1 mode of coupled ARCs. At this point, a regular intensity lattice between the two cavities is formed by interference. Moreover, the distribution of the regular pattern is consistent with the interference field between the emissions of two isolated cavity modes. Further more, by changing the gap L between the two cavities, we investigate how the resonant frequency of the three EE modes varies with L. In this paper, we only concern the long distance radiation coupling instead of near distance tunneling coupling, so the case of L < λ, which has been well investigated in many studies of optical molecular, is omitted. Figures III(d) and (e) show the real and imaginary part of the kR relative to the isolated one of three modes varying with L ∈ [λ, 10λ], respectively. The Re(∆kR) and Im(∆kR) of TM 37,1 mode of coupled ARCs vary periodically with L, and the variation period is λ. It suggests that an efficient coupling channel is established between the ARCs. In addition, the maximum value of the Re(∆kR) curve shift left about λ/4 to the Im(∆kR) curve. Their variation amplitudes are the same in magnitude of 10 −4 at L = λ, and decrease slowly as increasing of L. Besides, the variation is around resonant frequency kR = 22.32621 − 0.00011i of isolated cavity, and the limited frequency at L = ∞ tend to the frequency of isolated cavity also. All above is in line with physical expectations. Now, we turn to the variation of resonant frequency of coupled circular cavities. The Re(∆kR) of TM 40,1 mode does not show any oscillation in the order of 10 −6 , when the L is larger than 1.5λ. As to its Im(∆kR), the no oscillation order is down to 10 −11 . The oscillation of resonant frequency of TM 30,3 mode is irregular. The amplitude of oscillation is about 10 −5 , an order of magnitude less than that of ARC case. The oscillation amplitude positively associated with the coupling strength between the two cavities, even count the ratio of Qs ofTM 30,3 to TM 37,1 , the coupling strength is enhanced four times in coupled ARCs. We can draw a conclusion safely that the coupling strength and stability of ARCs are both better than that of circular cavities because of the character of the unidirectional emission. IV. ANALYSIS BY THE COUPLED-MODE THEORY Based on the image of the long distance coupling between ARCs in Fig. 2(a), we can establish a simple model of radiation coupled cavities [see Fig. IV]. The eigenmode of separated ARC is described by the wave function Ψ a(b) (r), with the frequency and intrinsic loss of ω a(b) = Re(k a(b) ) × c and κ a(b) = −Im(k a(b) ) × c, where c is the speed of light in vacuum. For isolated ARCs, the dynamics of cavity field can be expressed as Φ a (r, t) = a(t)Ψ a (r) = a(0)e (−iωa−κa)t Ψ a (r) and Φ b (r, t) = b(t)Ψ b (r) = b(0)e (−iω b −κ b )t Ψ b (r) , with dimensionless coefficients of a(t) and b(t) . With weak coupling perturbation, the wave function of a composite system can be written as Φ(r, t) = a(t)Ψ a (r) + b(t)Ψ b (r) [20]. Considering the radiation coupling between two ARCs, and according to the coupled-mode theory, the dynamics of the system satisfy the following differential equations d dt a(t) = (−iω a − κ a )a(t) + ge iθ b(t) d dt b(t) = (−iω b − κ b )b(t) + ge iθ a(t)(3) where ge iθ is the term of radiation coupling , in which g ≥ 0 is the coupling strength and θ is the phase shift that the coupling wave undergoes from a cavity to another. According to the coupled mode equations, we can solve the new normal modes of the composite system as Ψ ± (r) = c a Ψ a (r) + c b Ψ b (r), which frequency ω ± and coefficients c a c b are just the eigenvalues and eigenvectors of the matrix M = ω 1 γ γ ω 2 = −iω a − κ a ge iθ ge iθ −iω b − κ b .(4) It is easy to solve the eigenfrequency ω ± = 1 2 (ω 1 + ω 2 ± (ω 1 − ω 2 ) 2 + 4γ 2 ),(5) and corresponding coefficients For example, in the case of two identical ARCs, i.e. ω a = ω b and κ a = κ b , we have ω ± = −iω a − κ a ± ge iθ and Ψ ± = 1 √ 2 (Ψ a ± Ψ b ). To the radiation coupling, the phase θ is proportional to the propagation distance L as θ = L λ × 2π + θ 0 , where θ 0 is constant phase according to coupling phase when L = 0. Therefore, when the θ increases with the increasing of L, the points of ω ± draw a circle centered at ω 1 and with radius g in the complex plan of ω. c a c b = 1 |ω ± − ω 1 | 2 + |γ| 2 γ ω ± − ω 1 .(6) Noting that a pair of ω + and ω − always locate at the ends of a diameter of the circle. Additionally, g is not a constant but a function of L. Since g stands for the magnitude of coupling, i.e. a portion of emitting energy from one cavity coupled into the other cavity, we can assume the function as g(L) = g(0)/ √ L + 2R.(7) The magnitude of g must less than that of Im(ω 1 ), which refers to the emission amount of one cavity. Consequently, the data range of g is [0, 1] × Im(ω 1 ). Let us give an example. Assuming θ 0 = 0, λ = 0.3, g(0) = 0.9, and L ∈ [1, 10] × λ, we get the double spiral curves of ω ± in ω plane ( Fig. IV(a)), which starts from the outer points and gradually approaches the center cross ω 1 with increasing of L. The ω + is associated with k, which is calculated in the last section, with proportional relationship ω + = −ikc. Transporting the curves of ARCs in Fig. III to a spiral curve in kR complex plane, we get the kR track of varying with L ( Fig. IV(b) blue solid line). The point on the spiral curve closes to center cross with increasing L, which indicates that the coupling strength is not a constant but a decreasing value too. In addition, the density of points in upper part of the curve is higher than that in the under part. It reveals that there is no strictly linear relationship between the θ and the L. Furthermore, a kR track of odd-even mode corresponding to ω − is calculated (Fig. IV(b) red dash line). Again, it is a spiral curve like the track of EE mode, and intertwined with the spiral line of EE mode around the center cross. Though the pictures of analytical and numerical results are qualitatively consistent with each other, the last one is more complicated. The reason is the real coupling term ge iθ as a function of L is more complex than we assumed. For example, not like the waveguide coupling case, in free space coupling case, coupling phase θ is related with the shape and location of the beam emitted by isolated cavity. So, the linear relationship with L is a simplified model. Before a further discussion, we evaluate the g from the data shown in Fig. IV(b), for 2g = |ω + − ω − |. As is shown in Fig. 5(a) (solid blue line), when the L increases the envelope of g decreases slowly. Obviously, a relatively high g (> 0.4) last in a wide range to L = 10λ. Besides, a fitted line (Fig. 5(a) dash red line) according to Eq. (7) agrees with simulation data in certain degree, if we ignore the oscillation. Here, the oscillation stems from multi-modes combination induced by multi-channel coupling in free space [12], which is not included in our simplified model of single mode coupling. V. DYNAMICS OF COUPLED ARCS A. On-resonance Since the eigenfunction Φ ± of coupled mode is known, the time evolution of any complex function Φ in the subspace can be determined. Here, as an example, we set the initial state of coupled system to Φ a , and discuss the energy transportation between the two coupled cavities. When t = 0, we have Φ(r, 0) = Φ a (r, 0) = [Φ + (r, 0) + Φ − (r, 0)]/ √ 2. Later the state function at time t reads Φ(r, t) = 1 √ 2 [Φ + (r, 0)e ω+t + Φ − (r, 0)e ω−t ](8) To find the energy in the two cavities, last formula can be rewritten as: Φ(r, t) = 1 2 [(e ω+t + e ω−t )Ψ a + (e ω+t − e ω−t )Ψ b ](9) The normalized energy in cavity A or B reads E a,b = 1 2 (e ω+t ± e ω−t ) 2 = 1 2 e −2κat [cosh(2g cos θ · t) ± cos(2g sin θ · t)](10) where positive and negative sign correspond to a and b, respectively. Taking some examples, the mode TM 37,1 in isolated ARC has a Q ∼ 10 5 , i.e. κ a = 5 × 10 −6 ω a . We pick a middle g = 0.68κ a and observe the time evolution of energy in cavity A and B at some coupling angles of θ (Figs. 5(b) and (d)). When θ = 0, the curves show there is already considerable energy transported from cavity A into cavity B at the time ω a t ∼ 2.5 × 10 5 , though the initial energy is wholly in cavity A. Then after a period of time the energy is balanced in both cavity, and attenuates exponentially with slope of −2(κ a − g)/ω a . Enlarge the θ gradually, the feature of transportation of energy is similar with case of θ = 0 except the final attenuation slope which changes to −2(κ a − g cos θ)/ω a . When the θ close to π/2, such as 0.45π, the augment 2g cos θ · t of attenuation (first) term bracketed in Eq. (10) is small, while the oscillating (second) term plays a major role when t is not too large. At this time, energy oscillates between the two cavities several times, then the amplitude of the oscillation reduces to zero, and energy becomes equal in two cavities. When the θ equals π/2, only the oscillating term left in square bracket of Eq. (10). Thus, energy attenuates with oscillation ( Fig. 5 solid line), that is to say the energy in the two cavities empty in turn. The attenuation slope reach the maxima value −2κ a /ω a . The case of θ ∈ [π/2, π] repeats the result in interval [0, π/2] with the reversal manner π − θ. Furthermore, π is the period of the dynamic rule of the evolution of modes varying with θ. Then we change the g to 0.5κ a and 0.9κ a for comparison (Fig. 5(c)). To the smaller g = 0.5κ a case, the speed of energy exchange slows down, and the time requirement for energy balance in coupled cavities increases. Moreover, the proportion of the energy transported from one cavity to another decreases, i.e. more energy emits out of the coupling system. Hence, the slope of energy attenuation line decreases except the constant slope in θ = π/2 case. If θ = π/2, the period of energy emptying in cavities increases with decreasing of g. On the contrary, if we change the mode used in coupling with different Q, then the time to achieve energy balance is proportional to the Q. B. Off-resonance In practice, two coupled cavities may not be identical but have slight different geometry, so the resonant frequencies of cavities have a small mismatch, i.e. ω 1 = ω 2 . Keeping κ a and κ b equal, we study the detuning δ = ω a − ω b influence on the dynamics of the coupling. Then, like in the on-resonance case, after derivation from Eq. (6) we get the evolution equation of energy and depict it in Fig. 6 with g = 0.68κ a and Q = 10 5 again. If the detuning induced by size difference is slight, say δ = 0.1κ a , the resonances in two cavities overlap significantly with each other. Then, the resonant frequency of coupled mode still varies around the average value of isolated frequencies. Yet the reduction of the coupling strength shrinks the splitting of two coupled modes. Besides, energy of modes distribute unequally in two cavities after a transient state. When θ increases from 0 to 0.5π, the unbalance of the energy distribution becomes worse and the slope of the decay curve becomes steeper (Fig. 6 first and second columns). For the Fig. 6(c), though the cavity A keep more energy than cavity B in the oscillation, the oscillation period is nearly as same as that in on-resonance case (Fig. 5(c) thin line). The imbalance of energy distribution becomes more serious when the detuning δ gets larger. In the middle detuning case δ = 1κ a (Fig. 6 second column), we found whether the over coupling or under coupling occurs depends on the coupling phase θ. For instance, when θ = 0.25π the energy flow from cavity A to cavity B can not reflow back to cavity A completely, but when θ = 0.75π the energy in cavity A can not transport to B entirely (Fig. 6 middle panel and its insert). In addition, the evolution of energy about the θ is also in the period of π but is different in the cases of θ with of π − θ . In fact, when the detuning is much larger than the linewidth, say δ ≫ κ a , the resonated mode in one cavity is merely influenced by another cavity (Fig. 6 last column). In other words, energy cannot deliver from cavity A to cavity B efficiently, so mode a decays exponentially without oscillation. At this time, a very few energy in cavity B oscillates intensely, as there arises a beat oscillation, whose period is inversely proportional to δ. VI. CONCLUSION In summary, an efficient coupling in long distance is established by employing unidirectional ARCs. We designed a unidirectional cavity with refractive index 2.0 for coupling. Comparing with the free-space far-field coupling of two circular cavities, the coupling of ARCs is strong and stable, which makes ARC being the first choice for far-field coupling. The coupled mode theory works well on solving the coupled problem based on the isolated mode. Results show that the coupled mode is the superposition of two single modes, and the corresponding resonant frequency is the sum (difference) of single mode frequency and a coupling term. Moreover, based one the result of the coupled mode theory, the time revolution of the light field, which is described by an arbitrary complex function, can be calculated by decomposing it with bases of normal mode. Then under on-and off-resonance coupling conditions, the transmission of energy in the coupling system is depicted in time domain. It need to note that coupling mode or time domain characters both extremely rely on the coupling phase θ. So far we have solved the low coupling efficiency problem encountered in Ref. [8].This paper only discusses two coupled passive ARCs. Next, coupling system composed of a laser cavity and a passive cavity [9] in ARC form need to be studied, and the coupling of multi-cavities [21] is interesting too. FIG. 1 : 1The near field (a) and far-field (b) distributions of TM37,1. Red-green-blue false colors indicate intensity from high to low in logarithmic scale. FIG. 2 : 2Far-field coupling patterns of TM37,1 (a) of the ARC, TM40,1 (b), and TM30,3 (c) of circular cavity. The real (d) and imaginary (e) parts of ∆kRs vary with gap L between two cavities. classification, the first and last symmetry indicators (E or O) refer to symmetry state about vertical and horizontal axis, respectively. InFig. III(a), obvious interference fringes with regular periodic oscillation are shown outside the cavities. Furthermore, the interference fringes inside the cavities are different to single cavity pattern either (Fig. II(a)). FIG. 3 : 3Schematics of coupled cavities. FIG. 5 : 5(a) The values of g/κa get from simulation results (solid blue line) and Eq. (7) with g(0) = 1.4 (dash red line). Time evolution of normalized energy of two coupled cavities. Energy in cavity A (b) and B (d) vary with time when g = 0.68κa and θ = 0, 0.1π, 0.2π, 0.3π, 0.4π, 0.45π, 0.5π. (c) Evolution of energy at different combination of parameters. FIG. 6 : 6Time evolution of normalized energy of two mismatched cavities. Three columns, from left to right, week, middle, and strong represent detuning, respectively. Each row cope with a typical value of θ. Insert in (c): middle detuning and with θ = 0.75π. Optical microcavties. K J Vahala, Nature. 424K. J. Vahala, "Optical microcavties," Nature 424, 839-846 (2003). Directional interacting whispering-gallery modes in coupled dielectric microdisks. J.-W Ryu, S.-Y Lee, C.-M Kim, Y.-J Park, Phys. Rev. A. 7413804J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, "Directional interacting whispering-gallery modes in coupled dielectric microdisks," Phys. Rev. A 74, 013804 (2006). Mode coupling between first-and second-order whispering-gallery modes in coupled microdisks. J J Li, J X Wang, Y Z Huang, Opt. Lett. 32J. J. Li, J. X. Wang, and Y. Z. Huang, "Mode coupling between first-and second-order whispering-gallery modes in coupled mi- crodisks," Opt. Lett. 32, 1563-1565 (2007). Designing coupled microcavity lasers for high-Q modes with unidirectional light emission. J W Ryu, M Hentschel, Opt. Lett. 36J. W. Ryu, and M. Hentschel, "Designing coupled microcavity lasers for high-Q modes with unidirectional light emission," Opt. Lett. 36, 1116-1118 (2011). High-Q unidirectional emission properties of the. X Tu, X Wu, L Liu, L Xu, J. Opt. Soc. Am. B. 27X. Tu, X. Wu, L. Liu, and L. Xu, "High-Q unidirectional emission properties of the," J. Opt. Soc. Am. B 27, 300-304 (2010). Photonic atoms and molecules. Y P Rakovich, J F Donegan, Laser Photonics Rev. 4Y. P. Rakovich, and J. F. Donegan, "Photonic atoms and molecules," Laser Photonics Rev. 4, 179-191 (2009). Polymeric photonic molecule super-mode lasers on silicon. T Grossmann, T Wienhold, U Bog, T Beck, C Friedmann, H Kalt, T Mappes, Light: Sci. & Appl. 282T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, "Polymeric photonic molecule super-mode lasers on silicon," Light: Sci. & Appl. 2, e82 (2013). On-chip quantum optics with quantum dot microcavities. E Stock, F Albert, C Hopfmann, M Lermer, C Schneider, S Hfling, A Forchel, M Kamp, S Reitzenstein, Adv. Mater. 25E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, "On-chip quantum optics with quantum dot microcavities," Adv. Mater. 25, 707-710 (2013). Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes. E I Smotrova, A I Nosich, Opt. Lett. 38E. I. Smotrova, and A. I. Nosich, "Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes," Opt. Lett. 38, 2059-2061 (2013). Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers. S.-B Lee, J Yang, S Moon, J.-H Lee, K An, Appl. Phys. Lett. 9041106S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, "Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers," Appl. Phys. Lett. 90, 041106 (2007). Dynamic process of free space excitation of asymmetric resonant microcavity. F.-J Shu, C.-L Zou, . F.-W Sun, J. Lightwave Technol. 31F.-J. Shu, C.-L. Zou, and a. F.-W. Sun, "Dynamic process of free space excitation of asymmetric resonant microcavity," J. Lightwave Technol. 31, 1884-1889 (2013). Theory of free space coupling to high-Q whispering gallery modes. C.-L Zou, F.-J Shu, F.-W Sun, Z.-J Gong, Z.-F Han, G.-C Guo, Opt. Express. 21C.-L. Zou, F.-J. Shu, F.-W. Sun, Z.-J. Gong, Z.-F. Han, and G.-C. Guo, "Theory of free space coupling to high-Q whispering gallery modes," Opt. Express 21, 9982-9995 (2013). Combining directional light output and ultralow loss in deformed microdisks. J Wiersig, M Hentschel, Phys. Rev. Lett. 10033901J. Wiersig, and M. Hentschel, "Combining directional light output and ultralow loss in deformed microdisks," Phys. Rev. Lett. 100, 033901 (2008). Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities. X F Jiang, Y F Xiao, C L Zou, L He, C H Dong, B B Li, Y Li, F W Sun, L Yang, Q Gong, Adv. Mater. 24X. F. Jiang, Y. F. Xiao, C. L. Zou, L. He, C. H. Dong, B. B. Li, Y. Li, F. W. Sun, L. Yang, and Q. Gong, "Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities," Adv. Mater. 24, OP260-OP264 (2012). High Q and unidirectional emission whispering gallery modes: principles and design. C.-L Zou, F.-W Sun, C.-H Dong, F.-J Shu, X.-W Wu, J.-M Cui, Y Yang, Z.-F Han, G.-C Guo, IEEE J. Sel. Top. Quantum Electron. 199000406C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, "High Q and unidirectional emission whispering gallery modes: principles and design," IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013). An optimization method of asymmetric resonant cavities for unidirectional emission. F.-J Shu, C.-L Zou, F.-W Sun, J. Lightwave Technol. 31F.-J. Shu, C.-L. Zou, and F.-W. Sun, "An optimization method of asymmetric resonant cavities for unidirectional emission," J. Lightwave Technol. 31, 2994-2998 (2013). Boundary element method for resonances in dielectric microcavities. J Wiersig, J. Opt. A: Pure. Appl. Opt. 5J. Wiersig, "Boundary element method for resonances in dielectric microcavities," J. Opt. A: Pure. Appl. Opt. 5, 53-60 (2003). Quick root searching method for resonances of dielectric optical microcavities with the boundary element method. C L Zou, H G L Schwefel, F W Sun, Z F Han, G C Guo, Opt. Express. 19C. L. Zou, H. G. L. Schwefel, F. W. Sun, Z. F. Han, and G. C. Guo, "Quick root searching method for resonances of dielectric optical microcavities with the boundary element method," Opt. Express 19, 15669-15678 (2011). Accurately calculating high quality factor of whisperinggallery modes with boundary element method. C.-L Zou, Y Yang, C.-H Dong, Y.-F Xiao, Z.-F Han, G.-C Guo, J. Opt. Soc. Am. B. 26C.-L. Zou, Y. Yang, C.-H. Dong, Y.-F. Xiao, Z.-F. Han, and G.-C. Guo, "Accurately calculating high quality factor of whispering- gallery modes with boundary element method," J. Opt. Soc. Am. B 26, 2050-2053 (2009). Coupled-mode theory. H A Haus, W Huang, Proc. IEEE 79. IEEE 79H. A. Haus, and W. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991). Symmetry Considerations for Closed Loop Photonic Crystal Coupled Resonators. M D Weed, C G Williams, P J Delfyett, W V Schoenfeld, J. Lightwave Technol. 31M. D. Weed, C. G. Williams, P. J. Delfyett, and W. V. Schoenfeld, "Symmetry Considerations for Closed Loop Photonic Crystal Coupled Resonators," J. Lightwave Technol. 31, 1426-1432 (2013).
[]
[ "Toward a General Theory of Societal Collapse. A Biophysical Examination of Tainter's Model of the Diminishing Returns of Complexity", "Toward a General Theory of Societal Collapse. A Biophysical Examination of Tainter's Model of the Diminishing Returns of Complexity" ]
[ "Ugo Bardi \nDipartimento di Chimica\nUniversità di Firenze\nItaly\n", "Sara Falsini \nConsorzio Interuniversitario per la Scienza e Tecnologia dei Materiali (INSTM)\nScientifico di Sesto Fiorentino\nvia della Lastruccia 3Sesto F. 50019 (Fi)Italy\n", "Ilaria Perissi \nConsorzio Interuniversitario per la Scienza e Tecnologia dei Materiali (INSTM)\nScientifico di Sesto Fiorentino\nvia della Lastruccia 3Sesto F. 50019 (Fi)Italy\n" ]
[ "Dipartimento di Chimica\nUniversità di Firenze\nItaly", "Consorzio Interuniversitario per la Scienza e Tecnologia dei Materiali (INSTM)\nScientifico di Sesto Fiorentino\nvia della Lastruccia 3Sesto F. 50019 (Fi)Italy", "Consorzio Interuniversitario per la Scienza e Tecnologia dei Materiali (INSTM)\nScientifico di Sesto Fiorentino\nvia della Lastruccia 3Sesto F. 50019 (Fi)Italy" ]
[]
The collapse of large social systems, often referred to as "civilizations" or "empires," is a well-known historical phenomenon, but its origins are the object of an unresolved debate. In this paper, we present a simple biophysical model which we link to the concept that societies collapse because of the "diminishing returns of complexity" proposed by Joseph Tainter 1 . Our model is based on the description of a socio-economic system as a trophic chain of energy stocks which dissipate the energy potential of the available resources. The model produces various trajectories of decline, in some cases rapid enough that they can be defined as "collapses." At the same time, we observe that the exploitation of the resource stock ("production") has a strongly nonlinear relationship with the complexity of the system, assumed to be proportional to the size of the stock termed "bureaucracy." These results provide support for Tainter's hypothesis. 1
10.1007/s41247-018-0049-0
[ "https://arxiv.org/pdf/1810.07056v1.pdf" ]
119,471,147
1810.07056
03197711be9403cec6aa1b4f93377bab3d404077
Toward a General Theory of Societal Collapse. A Biophysical Examination of Tainter's Model of the Diminishing Returns of Complexity Ugo Bardi Dipartimento di Chimica Università di Firenze Italy Sara Falsini Consorzio Interuniversitario per la Scienza e Tecnologia dei Materiali (INSTM) Scientifico di Sesto Fiorentino via della Lastruccia 3Sesto F. 50019 (Fi)Italy Ilaria Perissi Consorzio Interuniversitario per la Scienza e Tecnologia dei Materiali (INSTM) Scientifico di Sesto Fiorentino via della Lastruccia 3Sesto F. 50019 (Fi)Italy Toward a General Theory of Societal Collapse. A Biophysical Examination of Tainter's Model of the Diminishing Returns of Complexity The collapse of large social systems, often referred to as "civilizations" or "empires," is a well-known historical phenomenon, but its origins are the object of an unresolved debate. In this paper, we present a simple biophysical model which we link to the concept that societies collapse because of the "diminishing returns of complexity" proposed by Joseph Tainter 1 . Our model is based on the description of a socio-economic system as a trophic chain of energy stocks which dissipate the energy potential of the available resources. The model produces various trajectories of decline, in some cases rapid enough that they can be defined as "collapses." At the same time, we observe that the exploitation of the resource stock ("production") has a strongly nonlinear relationship with the complexity of the system, assumed to be proportional to the size of the stock termed "bureaucracy." These results provide support for Tainter's hypothesis. 1 Introduction The collapse of large social systems, also called "civilizations" or "empires," is a well-known and highly studied subject. In many cases, the historical record does not provide quantitative data on these events, but in some cases it is possible to quantify the collapse phenomenon in terms, for instance, of the extent of the areas controlled by the central government as reported by Tageepera 2 or of the output of the economic system as reported by Sverdrup 3 and McCollen et al. 4 In these studies, we can observe how collapses are often rapid in comparison to the build-up of the social and economic structures of a civilization. This behaviour is consistent with Diamond's definition of collapse as, "a drastic decrease in human population size and or political/economic/social complexity, over a considerable area, for an extended time." 5 Nevertheless, despite the number of studies in this area, there is little agreement on the causes of societal collapses and, in particular, on the possibility of a common mechanism causing them. Edward Gibbon was probably the first to attempt an interpretation of the fall of a large empire, the Roman one, attributing it mainly to the decline of the traditional values 6 . Later authors explained the fall of Rome in widely different ways and Demandt 7 (1984) lists about 210 different theories on this subject, probably an incomplete list. The same variety of interpretations affects the studies of the collapse of other societies in history, as described, for instance, by Tainter 8 in 2008. No consensus appears to exist in this field but, overall, we can divide the interpretations of collapses into two main subsets: theories based on several independent causes (concauses) and theories based on a single cause that generates a cascade of different effects. An example of the first approachseveral independent concauses -is the study by E. H. Cline on the collapse of the Late Bronze Age Mediterranean Civilization 9 . According to Cline, multiple negative effects occurred at the same time, including climate change, earthquakes, foreign invasions, and more. An extreme example of the multiplication of causes is the study Bury published in 1923 10 who argued that the collapse of the Roman Empire resulted from several contingent events all occurring at about the same time. Tainter comments 11 stating that Bury considers that "The collapse was just bad luck". There are several examples of the second approach, single cause followed by a cascade of related events. One is Douglas Reynolds' interpretation of the fall of the Soviet Union in 1991 12 . Reynolds attributes it to mineral depletion and, specifically, to the cascade of negative effects generated by the growing costs of oil production which affected the whole Soviet economic system. Another singlefactor model of civilization collapse factor has been proposed by Joseph Tainter in his study "The Collapse of Complex Societies" 1 and in later papers 8,13 , 14 . Tainter identifies "diminishing returns," a well-known concept in economics, as the general factor in the decline and fall of civilizations. The idea is that, as societies become larger, more complex control structures are needed to maintain the cohesion of society and solve the problems that appear along their path. These structures can be described in terms of governments, the nobility, armies, bureaucracy, and the like. According to Tainter, as these structures become larger, they become less efficient, to the point that the economic returns they provide are smaller than their cost. At this point, society becomes unable to cope with the challenges it faces and must decline, or even collapse. The contrast between single/multiple causes in the interpretation of the fall of societies highlights a general methodological problem. Not only data are often scarce on these historical phenomena, but their interpretation is often based on the author's personal judgment of the relative importance of the events he studies. It goes without saying that the collapse of civilizations is not amenable to experimental studies but, even taking this point into account, one may ask how proposing a specific interpretation of the fall of -say -the Roman Empire can be justified. Here, we have several problems, including the fact that the very concept of "causation" is hard to approach in a quantitative manner 15 . Nevertheless, we can choose to rely on the basic scientific concept that the preferable interpretation of an event is not only one that's compatible with the available data, but also which is of general validity -that is, can explain more than a single event of the same class. In this sense, Tainter's interpretation of "diminishing returns of complexity" provides a general framework to interpret a large number of cases and it is, therefore, an interesting idea in view of understanding the general phenomenon of societal collapse. In the present study, we looked at Tainter's ideas using the modern concept of "System Science." 16 By using the modelling method known as "system dynamics" 17 we developed a simple biophysical model describing the evolution of a society. The model includes the effects of overshoot 18 and of diminishing returns in the exploitation of natural resources. It is not supposed to describe specific social systems but to provide a "mind-sized" 19 model the main factors that cause collapse. We find that the complexity of the system assumed to be proportional to the size of a stock such as "bureaucracy" follows a trajectory that makes the model compatible with the one proposed by Tainter. That is, the system shows a hysteresis that makes its trajectory non-reversible: reducing the costs of bureaucracy doesn't return society to the previous conditions of prosperity. Tainter's model: Tainter describes his model in his 1988 book "The Collapse of Complex Societies." 1 Here is an excerpt from the book. More complex societies are more costly to maintain than simpler ones, requiring greater support levels per capita. <..> It is the thesis of this chapter that return on investment in complexity varies, and that this variation follows a characteristic curve. More specifically, it is proposed that, in many crucial spheres, continued investment in sociopolitical complexity reaches a point where the benefits for such investment begin to decline, at first gradually, then with accelerated force. Thus, not only must a population allocate greater and greater amounts of resources to maintaining an evolving society, but after a certain point, higher amounts of this investment will yield smaller increments of return. Diminishing returns, it will be shown, are a recurrent aspect of sociopolitical evolution, and of investment in complexity. The graphic representation of Tainter's mechanism is shown in figure 1 (redrawn from Tainter's book). Tainter's thesis is not directly based on quantitative data or models, but historical data are used to support it. For example, the decline of the content of silver in the Roman denarius for a period that goes from the 1 st to the 3 rd century AD is interpreted by Tainter as an indication that the Roman government was experiencing increasing financial difficulties. Tainter attributes this phenomenon to the increasing cost of the Roman bureaucracy, including the imperial court, resulting from the expansion of the Roman civilization during the 1 st century BC and the 1 st century AD. Surprisingly, Tainter never mentions in his 1988 book the depletion of the Roman silver and gold mines in Spain, another possible cause of diminishing returns. The subject of mineral depletion in the ancient world is difficult to study for the lack of specific data on mineral production, but it is known that depletion was a problem for the Roman mines and that mining required progressively more efforts, for instance in terms of deeper mines 20 . Recent data show that the Roman mining of various metals and the Roman industrial activity collapsed together with the third century crisis. 4 These data don't necessarily imply that depletion was the problem, but do indicate that the lack of precious metals was due to a real decline in availability, not just to the expansion of the system. In a more recent discussion, 13 Tainter himself identifies the diminishing returns of mineral exploitation as related to the concept of "EROI" (energy returned for energy invested) developed by Hall and others 21 . The concept of EROI cannot be directly applied to the extraction of metals and other nonenergy producing minerals, but the basic mechanism that leads to diminishing EROI for fossil fuels is valid for all mineral resources. The less costly resources are exploited first, and this leads to a gradual and irreversible increase in the energetic and monetary costs of extraction. This is one of the causes of diminishing returns in a complex society 22 . Collapse: the systems science approach By developing the concept of "diminishing returns," Tainter links historical collapses to the modern concept of "systems science" 16 a field dedicated to the study of complex systems, all different, but all tending to show a similar behaviour in similar circumstances. Complex systems are often described in the framework of system dynamics 17 as described in terms of "stocks" and "flows," while the dynamical evolution of the system is determined by the concepts defined as "forcing" and "feedback." A forcing is an external perturbation to the system which generates a series of enhancing and/or damping feedbacks. Enhancing feedbacks tend to amplify the forcing, damping feedbacks have the opposite effect. The result of enhancing feedbacks may well be to perturb the system to such a point that it crosses a "tipping point" 23 and moves to a different state. This transition can be described as a collapse if the new state corresponds to a condition of lower complexity. In particular, the concept of enhancing feedback may explain the wide variety of attribution of societal collapses to different causes: the initial perturbation which unbalanced the system goes unnoticed when it is masked by the large feedbacks it generates. In the present study, we use systems dynamics to analyze a well-known biophysical concept that we apply to the concept of civilization: the trophic chain. It assumes that a socio-economic system can be represented as an ecosystem where different trophic levels participate in the degradation, or the dissipation, of the thermodynamic potential associated with the highest potential stock (or lowest trophic level). In the case of civilizations, the lowest trophic level is the one defined as "natural resources," e. g., fossil fuels in the case of the modern civilization. Fossil fuels have a high thermodynamic potential which can be dissipated in the form of the heat produced when fuels combine with atmospheric oxygen. Then, the resulting cascade of trophic levels corresponds to different elements of the economic system: the extractive industry, the manufacturing industry, the bureaucracy, and -lastly -pollution (or waste): the end result of the economic process. This approach is part of the concept of "world modelling" developed first by Jay Forrester 24 and then applied to studies such as the well-known 1972 study titled "The Limits to Growth" 25,26 . A modern example of a world model is the "MEDEAS" model (www.medeas.eu), which aims at describing the whole world system in detail with several hundred parameters. Other models based on the same approach describe the behavior of a complex system by a small number of parameters and may be termed "mind sized" 19 . One of these relatively simple models is the "HANDY" model created by Motesharrei et al. 27 The model developed here is a simple trophic chain of stocks, in its simplest possible form is a twostock model, already described in a previous paper 19 . The model is shown in figure 2. Note that we represent the flows in the model as going "down" from higher thermodynamic potentials toward lower thermodynamic potentials. It is a convention described in ref. 19 The model is represented in a form that emphasizes the unidirectional flows from higher potential stocks to lower potential stocks, going "down" in the arrangement. We can say that the stock termed "The Economy" dissipates the energy potentials contained in the "Resources" stock, turning them into waste heat and non-recyclable materials. For instance, if the resource is crude oil, then the economy transforms it into all the products that can be made using crude oil, fuels, plastics, chemicals, etc. Note also that the model as shown involves no entropic loss for the transformation, that is it assumes a 100% efficient of transformation -such a loss can be easily introduced in the model, but that doesn't change the qualitative results of the model. Note also that the model does not consider the effects of pollution, while other factors, such as human population, social structure, bureaucracy, etc., are all aggregated in the "Economy" stock. In the model, the flow between stocks is regulated by feedbacks. We assume that each flow is linearly proportional to the size of the stocks connected by it, multiplied by a proportionality constant. It would be also possible to assume that there exist exponents to the stock sizes analogous to the ones called "elasticities" in some macroeconomic models. Here, however, we will use a simple assumption of linear proportionality. The size of the stocks of the model can be measured in terms of energy: natural resources such as crude oil can be measured in terms of the chemical energy they contain, while other mineral resources can be measured in terms of the "embedded energy" 28 necessary for their extraction, refining, and production. The stocks aggregated in the "economy" stock can also be described in terms of embedded energy -e.g. a piece of machinery can be measured in terms of the energy necessary to create it, including the cost of producing the metals and the other materials needed. The stock sizes can also be measured in monetary units, assumed to be "proxies" of energy units. Here, no quantitative assessment of real-world stocks is attempted, but the fact that they can be all measured in the same units ensures that the transformations described in the model are physically possible. If the natural resources are supposed to be non-renewable (i.e. the flow into the resource stock is set to zero) this simple two-stock model generates a symmetric, bell-shaped curve for the flow of energy that goes from the upper stock to the lower stock, a flow that we may call "production". This curve corresponds to the "Hubbert curve" which approximately describes the cycle of extraction of mineral resources 29 . If, instead, the natural resource stock is assumed to reform at a rate proportional to the stock size, as it happens for a biological stock, the model is equivalent to the well-known Lotka-Volterra one 30 , 31 and it generates continuous oscillations in the size of both stocks. Here, we assume that the Resources stock is slowly renewable, so we consider only one cycle of growth and decline. The simple two-stock model is the basis of more complex models where we consider further trophic levels. The next step involves adding a third stock, labelled as "pollution" -this model has been described in a previous paper 18 where it was termed the "Seneca Model" because it produces asymmetric bell-shaped curves where decline is faster than growth, corresponding to an observation put forward long ago by the Roman philosopher Lucius Annaeus Seneca 32 . A further step consists in adding one more level which we term here as "Bureaucracy" supposed to be a stock that aggregates all the non-productive societal structures: the army, the nobility, the court, the priests, etc. The trophic chain, at this point, is the following. Resources 2. The Economy 3. Bureaucracy 4. Pollution The model is shown in figure 4. The flow is unidirectional, and it goes in this way: Natural Resources → The Economy → Bureaucracy → Pollution. As in the simpler two-stock model, resources are transformed into economic capital. This capital is partly turned into "bureaucracy" whose damping effect may be taken into account as an indirect effect caused by the drawdown of resources from the economy stock, which is not available to exploit natural resources. Note that this stock is connected to the "production" flow. We may assume that bureaucracy may have an enhancing effect on production. In the real world, this effect could play out, as an example, by providing the extractive industry with a legal framework that allows them to exploit the resource they control without the need of defending them from competitors. This effect can be neglected in a simplified form of the model. All stocks generate pollution. the necessary result of all the ongoing operation of society. Pollution may take the form of gases emitted by the combustion of fossil fuels, heavy metals dispersed in the environment, as well as the destruction of the fertile soil and the general disturbance of the ecosystem. The pollution stock may be assumed to abate slowly as the polluting substances are reabsorbed by the environment. All stocks in the model are affected by feedbacks which make flows proportional to the size of the stocks connected by it. Note that Bureaucracy affects production in terms of a multiplying factor assumed to be (1+Bureaucracy). This assumption accounts for the fact that Bureaucracy is a less important factor than economic capital in facilitating production. If the resources are assumed to be non-renewable, all the stocks of the system go to zero one after the other. However, the 4-stock model produces a more abrupt collapse than the 2-stock model. This behaviour often takes the form of the "Seneca Collapse" 32 . It is the result of a stock being subjected to a feedback-dominated drawdown by another stock, while at the same time being unable to maintain a replenishing flow from a depleted stock (this can be termed the "candle burning at both ends" effect). The initial values are 1 unit for "Resources", 0.1 units for "The Economy," 0.01 units for "Bureaucracy," and 0.001 units for "Pollution". The constants are 0.38 for the "Production" flow, 0.15 for the "Depreciation" flow, 0.3 for the "Bureaucracy Creation Flow" and 0.5 for the "Bureaucracy Decay" flow. All the other flows constants are assumed to be zero -in other words, the resources are assumed to be non-renewable and pollution to be persistent. A robust feature of the model is the how the decline in the stock of natural resources is related to a cycle of growth and decline of the other stocks. In most cases, the decline turns out to be faster than growth, a characteristic defined here as "collapse." Note also that the behavior of the model can vary depending on the initial assumptions. Depending on the stock parameter, the bureaucratic stock may go to zero before the capital stock does. In this case, we may see the model as describing a civilization that loses some of its centralized control structure (e.g. an imperial court) and moves toward simpler delocalized structures. It could describe how the Middle Ages feudal structure developed after the collapse of the Western Roman Empire. A different behavior may also occur, that is the bureaucracy stock surviving the collapse of the capital stock. In this case, society maintains for a certain time an overgrown central control structure while the productive structures have disappeared. This might be the case of the Eastern Roman Empire which saw the capital city of Constantinople surviving even though most of the territory of the empire had been lost. The question is now how these models can be related to the qualitative descriptions given by Tainter. We can note first that Tainter describes complexity as 1 : Complexity is generally understood to refer to such things as the size of a society, the number and distinctiveness of its parts, the variety of specialized social roles that it incorporates, the number of distinct social personalities present, and the variety of mechanisms for organizing these into a coherent, functioning whole. From this definition, it seems that we can identify a proxy for the concept of complexity in terms of "The Economy" for the simple 2-stock model and "Bureaucracy" for the case of the four-stock model. In the latter case, the stock is surely composed of those people whom Tainter describes "specialists not directly involved in resource production." Then, what is that we should see as "benefits of complexity" in the model? The stocks of the system can be replenished only as long as the producing capital can extract energy and materials from the resource stock. Therefore, we can quantify the benefits of complexity using the "production" parameter as a proxy. We can therefore use the model to reproduce Tainter's model of the diminishing returns to complexity by plotting production as a function of the size of the bureaucracy stock. We do this first for the simplified 2-stock model ( figure 6) and then for the more complex 4-stock model ( figure 7). In both cases, the parameters are those described in the captions for the figures 4 and 5. The results are qualitatively similar. figure 5. This diagram has a different shape in comparison with the 2-stock model, but the qualitative dependency of the two stocks, bureaucracy and production, remains the same: growth and decline, followed by an inversion in the trend. The curve for the simpler 2-stock model is qualitatively similar to the one proposed by Tainter, (fig 7) although the two-stock model shows the whole loop ( fig. 6) whereas Tainter's illustration shows only the growing phase of complexity. Note the hysteresis of the curve: the bureaucracy stock declines after having reached a maximum value, but its relation to the productivity of the system doesn't go back to the earlier values. In other words, halving the costs of bureaucracy during the decline phase doesn't lead back to the same conditions when the system was growing. The behaviour is qualitatively similar to the 4-stock model although, in this case, Bureaucracy initially favours growth but later on becomes a burden. This phenomenon appears to correspond to the current conditions of modern society. Governments everywhere are cutting their bureaucratic expenses, but the system is not returning to the efficiency of the earlier times. We may also use a more detailed model to check the behaviour that we observe for our "mind sized" model. A well-known such model is the one called "World3," which was used for the 1972 study "The Limits to Growth." We used the most recent available version of World3 to calculate a Tainter plot for the "business as usual" or "base case" scenario. The world3 model has an "industrial output" that we may consider a proxy for Tainter's "benefits of complexity." It doesn't have a "bureaucracy" or "complexity" stock but, after examining the model, we believe that the parameter defined as "Relative Services Output" can be considered proportional to the size of the bureaucracy in the model and hence as a proxy for Tainter's "complexity" concept. The results are shown in figure 8 and are very similar to those obtained using the simpler models described here: we can see the typical hysteresis of these phenomena. Historical examples. In this section, we apply to historical cases the considerations developed in the previous sections. A first example is the Roman empire where the data have been taken from Sverdrup et al 33 who studied the connection between resources depletion and the stability of Roman civilization. In particular, the depletion of silver content in coins that started at the beginning of the 1 st century AD continued well into the 3 rd century AD and then, decreased. From that moment on, the social structures of empire, such as the army, became so expensive that it was impossible to maintain them at the previous level of organization and size. This phenomenon reflects exactly the concept of diminishing return of complexity: the state becomes larger and more and larger non-producing social structures are needed to maintain it. This is true until the expenses to maintain the bureaucracy and the army does not become too high. In this case, we took the "cash flow" as a measure of the productivity of the Roman Empire and the army size as proportional to the size of the non-productive sector of the Roman economy. Again (figure 10) Figure 10: Tainter plot for the Roman empire where the relative cash flow in coinage (state income) is reported as a function of the relative Roman army size. The cash flow is taken as a proxy for Tainter's "benefits of complexity" whereas the army size is taken as a proxy for the complexity of the state system. (data from Sverdrup et al. 34 ) A similar behaviour can be observed in the dynamic of exploitation, or better said, in the dynamic of overexploitation of fish. Fish has been, and still is, one of the most precious natural resources that feed the humankind since prehistorical age, placing fishing, and subsequently the fishery sector, as a historical driver in the development of the economy of several countries. However, fisheries are also a good example of the tendency of overexploiting resources, in this case, overfishing 35 , 36 , 37 , 38 , 39 . The phenomenon of overfishing has been recently studied by system dynamics in the recent work of Perissi and al. 40 revealing how overshoot is the crucial factor that leads to the destruction of the fisheries. This situation has important consequences in the fisheries economy that can be described exactly by the Tainter's concept of diminishing returns of complexities. The fishing companies tend to invest more and more (fleets, machinery, crews) to stay competitive, a strategy which, incidentally, generates a considerable waste and inefficiency with the fish reaching the market is only about 50% 41 of the total captured. The result is often a dramatic reduction of the population of some species -a phenomenon already observed during the 19 th century when the US whale fishery collapse because of overexploitation 42 . More recently, the entire fish population of marine areas has been destroyed or is being destroyed, as it is happening to Japan and Iceland 40 . In the following figures, we report data on fisheries based on the concept of "diminishing returns" proposed by Tainter, taking fish production as a measure of the efficiency of the system and the capital investment as a proxy for the size of the industry (data source Perissi et. Al. 43 ). Again, we observe a qualitatively similar behaviour to the others reported before -including the typical hysteresis of the curves. Here, obviously, the fishing industry is not a social system in the same sense the Roman Empire was, but all these systems are evidently similar. Sperm Oil (production) is reported as a function of the Tonnage capacity of fishing boats (complexities) as the effort necessary to maintain the whales' oil production. Left Japanese Fishery Industry (years from 1962 to 2000). Total Catch (production) and the Disbursement of Fishery (complexity) the effort necessary to maintain a competitive landing (data source 43 ) Conclusion The models studied here are not supposed to describe specific cases of the collapse of human societies, rather, they are thought of as a simplified playground to examine the effect of some parameters on the trajectory of complex systems intended as dissipative structures based on finite or slowly renewable resources. The models are based on a simple concept: that of the trophic chain. If we assume that the natural resources available are non-renewable, as they are in the case of mineral resources (e.g. gold and silver for the Roman Empire and fossil fuels for the modern global empire), the disappearance of the trophic structures exploiting the resources is unavoidable -unless new resources can be found. The same is true for those resources which are slow to renew in comparison to the rate of exploitation. The models tell us how the dissipation of the natural resources goes by the progressive filling and emptying of the stocks at lower thermodynamic potential -every step implying a loss of exploitable potential energy which disappears in the form of pollution, e.g. low temperature heat. This phenomenon generates "bell-shaped" curves for the filling/emptying of the stocks. These curves can also take the "Seneca shape" 32 when the decline is faster than the growth. As this phenomenon goes on, the stocks interact with each other. The time delay in the filling/emptying of the stocks generates a trajectory where stocks move in the phase space along a hysteresis curve. The system continuously evolves in an irreversible manner and it can never return to an earlier condition, unless the resources are assumed to be renewable and, in this case, the system circles around an attractor in phase-space. In other words, simply reducing the size of the "Bureaucracy" stock will not return the system to a condition in which it was during the growing cycle, an observation which seems to correspond to the current situation. Overall, as long as a society exploits resources in a condition of unbridled feedback, as it happens when it tries to maximize yields, then the overexploitation collapse is unavoidable even though the resources are theoretically renewable. Only an intelligent control able to plan for the future can avoid this destiny. Such a control was not modelled in the present study, but the records of history tell us that it is rarely -if ever -utilized in human societies in history. Figure 1 : 1The diminishing returns to complexity, graphically described by Tainter 1 . (redrawn from Tainter's book). Figure 2 : 2A simple 2-stock System Dynamics model describing the flow and the dissipation of natural resources in a complex society. All the flows in the model are determined by constant multipliers, not shown in the diagram. Figure 3 : 3Typical behavior of the stocks in the "2-stock" model for nonrenewable natural resources. The initial values for the stocks are assumed to be 1 for the Resources stock and 0.01 for the Economy stock. The constants determining the flows are 0.1 for the "production" flow and 0.5 for the depreciation flow. Figure 4 : 4The four-stock SD model used in this paper. All the flows are regulated by constants, not shown in the diagram. Figure 5 : 5Typical behaviour of the 4-stock model. In this figure, all the four stocks are shown. Figure 6 : 6Production vs "The Economy" for the two-stock model. The values of the constant are the same as those described infigure 3. Figure 7 , 7a comparison of the results of the two-stock model reported here and Tainter's model as reported in his 1988 book 1 . The two curves are not identical, but the similarity is evident. Figure 8 : 8The curve of Production vs. Bureaucracy for the 4-stock model. The values of the constants in the model are the same as those in Figure 9 : 9Tainter plot for World 3 where the relative industrial output is reported as a function of the relative services output. Figure 11 : 11Right. Data for the American Whaling industry during (1820-1880) (data source44 ). The collapse of complex societies. Joseph Tainter, A , Cambridge University PressTainter, Joseph, A. The collapse of complex societies. (Cambridge University Press, 1988). Size and Duration of Empires: Growth-Decline Curves, 600 B.C. to 600 A.D. R Taagepera, Soc. Sci. Hist. 3115Taagepera, R. Size and Duration of Empires: Growth-Decline Curves, 600 B.C. to 600 A.D. Soc. Sci. Hist. 3, 115 (1979). Peak Metals, Minerals, Energy, Wealth, Food and Population: Urgent Policy Considerations for a Sustainable Society. H U Sverdrup, D Koca, K V Ragnarsdóttir, J. Environ. Sci. Eng. 2Sverdrup, H. U., Koca, D. & Ragnarsdóttir, K. V. Peak Metals, Minerals, Energy, Wealth, Food and Population: Urgent Policy Considerations for a Sustainable Society. J. Environ. Sci. Eng. 2, 189-222 (2013). Lead pollution recorded in Greenland ice indicates European emissions tracked plagues, wars, and imperial expansion during antiquity. J R Mcconnell, Proc. Natl. Acad. Sci. U. S. A. 115McConnell, J. R. et al. Lead pollution recorded in Greenland ice indicates European emissions tracked plagues, wars, and imperial expansion during antiquity. Proc. Natl. Acad. Sci. U. S. A. 115, 5726-5731 (2018). Collapse: How Societies Choose to Fail or Succeed. J Diamond, Viking PressDiamond, J. Collapse: How Societies Choose to Fail or Succeed. (Viking Press, 2005). History of the Decline and Fall of the Roman Empire. E Gibbon, Strahan & CadellGibbon, E. History of the Decline and Fall of the Roman Empire. (Strahan & Cadell). Der Fall Roms. (Beck'sche. A Demandt, Demandt, A. Der Fall Roms. (Beck'sche, 1984). Collapse, Sustainability, and the Environment: How Authors Choose to Fail or Succeed. J A Tainter, Rev. Anthropol. 37TAINTER, J. A. Collapse, Sustainability, and the Environment: How Authors Choose to Fail or Succeed. Rev. Anthropol. 37, 342-371 (2008). The Year Civilization Collapsed. E H Cline, B C , Princeton University PressCline, E. H. 1177 B.C.: The Year Civilization Collapsed. (Princeton University Press, 2014). History of the Later Roman Empire. J B Bury, MacMillanBury, J. B. History of the Later Roman Empire. (MacMillan, 1923). Collapse, Sustainability, and the Environment: How Authors Choose to Fail or Succeed. J A Tainter, Rev. Anthropol. 37TAINTER, J. A. Collapse, Sustainability, and the Environment: How Authors Choose to Fail or Succeed. Rev. Anthropol. 37, 342-371 (2008). Cold War Energy: The Rise and Fall of the Soviet Union. D B Reynolds, Alaska Chena LLCReynolds, D. B. Cold War Energy: The Rise and Fall of the Soviet Union. (Alaska Chena LLC, 2016). Social complexity and sustainability. J A Tainter, Ecological Complexity. 3Tainter, J. A. Social complexity and sustainability. Ecological Complexity 3, 91-103 (2006). Social complexity and sustainability. J A Tainter, Ecological Complexity. 3Tainter, J. A. Social complexity and sustainability. Ecological Complexity 3, 91-103 (2006). The book of why : the new science of cause and effect. J Pearl, D Mackenzie, Basic BooksPearl, J. & Mackenzie, D. The book of why : the new science of cause and effect. (Basic Books, 2018). . G E Mobus, M C Kalton, 10.1007/978-I-4939-1920-8Principles of System Science. SpringerMobus, G. E. & Kalton, M. C. Principles of System Science. (Springer, 2015). doi:10.1007/978-I-4939-1920-8 System dynamics. G Richardson, 10.1007/978-1-4419-1153-7_1030Encyclopedia of operations research and management … 1519-1522. New YorkSpringerRichardson, G. System dynamics. in Encyclopedia of operations research and management … 1519-1522 (Springer New York, 2013). doi:10.1007/978-1-4419-1153- 7_1030 Overshoot, the ecological basis of revolutionary change. W Catton, Illinin Books EditionCatton, W. Overshoot, the ecological basis of revolutionary change. (Illinin Books Edition, 1982). . U Bardi, Mind Sized World Models. Sustainability. 5Bardi, U. Mind Sized World Models. Sustainability 5, 896-911 (2013). Mining in the Later Roman Empire and Beyond: Continuity or Disruption?. J C Edmondson, J. Rom. Stud. 79Edmondson, J. C. Mining in the Later Roman Empire and Beyond: Continuity or Disruption? J. Rom. Stud. 79, 84-102 (1989). . A K Gupta, C A S C A S Hall, A K Gupta, C A S C A S Hall, A Review of the Past and Current State of EROI Data. Sustainability. 3Gupta, A. K., Hall, C. A. S. C. A. S., Gupta, A. K. & Hall, C. A. S. C. A. S. A Review of the Past and Current State of EROI Data. Sustainability 3, 1796-1809 (2011). Complexity, Problem Solving, and Sustainable Societies. Joseph Tainter, A , Getting Down to Earth -Practical Applications of Ecological Economics. Island PressTainter, Joseph, A. Complexity, Problem Solving, and Sustainable Societies. in Getting Down to Earth -Practical Applications of Ecological Economics (Island Press, 1996). The Tipping Point: How Little Things Can Make a Big Difference: Malcolm. M Gladwell, Back Bay BooksGladwell, M. The Tipping Point: How Little Things Can Make a Big Difference: Malcolm. (Back Bay Books, 2002). . Forrester, J. W. World dynamics. Wright-Allen PressForrester, J. W. World dynamics. (Wright-Allen Press, 1971). The Limits to Growth. D H Meadows, 10.1111/j.1752-1688.1972.tb05230.xClub Rome. 211Meadows, D. H. M. The Limits to Growth. Club Rome 211 (1972). doi:10.1111/j.1752- 1688.1972.tb05230.x The Limits to Growth Revisited. U Bardi, SpringerBardi, U. The Limits to Growth Revisited. (Springer, 2011). Human and nature dynamics (HANDY): Modeling inequality and use of resources in the collapse or sustainability of societies. S Motesharrei, J Rivas, E Kalnay, Ecol. Econ. 101Motesharrei, S., Rivas, J. & Kalnay, E. Human and nature dynamics (HANDY): Modeling inequality and use of resources in the collapse or sustainability of societies. Ecol. Econ. 101, 90-102 (2014). . H T Odum, Energy, Ecology, and Economics. Ambio. 2Odum, H. T. Energy, Ecology, and Economics. Ambio 2, 220-227 (1973). A simple interpretation of Hubbert's model of resource exploitation. U Bardi, A Lavacchi, 2Bardi, U. & Lavacchi, A. A simple interpretation of Hubbert's model of resource exploitation. Energies 2, 646-661 (2009). Elements of Physical Biology. Alfred J Lotka, 10.2105/AJPH.15.9.812-bWilliams and Wilkins CompanyAlfred J. Lotka. Elements of Physical Biology. Williams and Wilkins Company (Williams and Wilkins Company, 1925). doi:10.2105/AJPH.15.9.812-b Fluctuations in the abundance of a species considered mathematically. V Volterra, Nature. 118Volterra, V. Fluctuations in the abundance of a species considered mathematically. Nature 118, 558-560 (1926). The Seneca Effect. Why Growth Is Slow but Collapse Is Rapid. U Bardi, Springer VerlagBardi, U. The Seneca Effect. Why Growth Is Slow but Collapse Is Rapid. (Springer Verlag, 2017). Food and Population: Urgent Policy Considerations for a Sustainable Society. H U Sverdrup, D Koca, K V Ragnarsdóttir, Peak Metals, Minerals, Energy, Wealth, J. Environ. Sci. Eng. 2Sverdrup, H. U., Koca, D. & Ragnarsdóttir, K. V. Peak Metals, Minerals, Energy, Wealth, Food and Population: Urgent Policy Considerations for a Sustainable Society. J. Environ. Sci. Eng. B2 189-222 (2013). Peak Metals, Minerals, Energy, Wealth, Food and Population; Urgent Policy Considerations for A Sustainable Society. H U Sverdrup, D Koca, K V Rganarsdottir, J. Environ. Sci. Eng. B. 1Sverdrup, H. U., Koca, D. & Rganarsdottir, K. V. Peak Metals, Minerals, Energy, Wealth, Food and Population; Urgent Policy Considerations for A Sustainable Society. J. Environ. Sci. Eng. B 1, 499-533 (2012). Catch reconstructions reveal that global marine fisheries catches are higher than reported and declining. D Pauly, D Zeller, Nat. Commun. 7Pauly, D. & Zeller, D. Catch reconstructions reveal that global marine fisheries catches are higher than reported and declining. Nat. Commun. 7, 1-9 (2016). Global marine yield halved as fishing intensity redoubles. R A Watson, Fish Fish. 14Watson, R. A. et al. Global marine yield halved as fishing intensity redoubles. Fish Fish. 14, 493-503 (2013). Short-term projection of global fish demand and supply gaps. J Cai, P Leung, Fao. 607Cai, J. & Leung, P. Short-term projection of global fish demand and supply gaps. Fao 607, (2017). Landing the blame: The influence of EU Member States on quota setting. G Carpenter, R Kleinjans, S Villasante, B C O&apos;leary, Mar. Policy. 64Carpenter, G., Kleinjans, R., Villasante, S. & O'Leary, B. C. Landing the blame: The influence of EU Member States on quota setting. Mar. Policy 64, 9-15 (2016). Global marine fisheries discards: A synthesis of reconstructed data. D Zeller, T Cashion, M Palomares, D Pauly, Fish Fish. 19Zeller, D., Cashion, T., Palomares, M. & Pauly, D. Global marine fisheries discards: A synthesis of reconstructed data. Fish Fish. 19, 30-39 (2018). Dynamic patterns of overexploitation in fisheries. I Perissi, U Bardi, T Asmar, A El &amp; Lavacchi, 23Perissi, I., Bardi, U., Asmar, T. El & Lavacchi, A. Dynamic patterns of overexploitation in fisheries. 23 (2016). Global Food Losses and Food Waste -extent , causes and prevention " -FAO. J Gustavsson, C Cederberg, U Sonesson, A Emanuelsson, SIK report. The methodology of the FAO studyGustavsson, J., Cederberg, C., Sonesson, U. & Emanuelsson, A. The methodology of the FAO study : " Global Food Losses and Food Waste -extent , causes and prevention " -FAO , 2011. SIK report, (2013). History of the American Whale Fishery. A Starbuck, CastleStarbuck, A. History of the American Whale Fishery. (Castle, 1989). Dynamic patterns of overexploitation in fisheries. I Perissi, U Bardi, U Bardi, T El Asmar, Ecol. Modell. 359Perissi, I., Bardi, U., Bardi, U. & El Asmar, T. Dynamic patterns of overexploitation in fisheries. Ecol. Modell. 359, (2017). History of the American Whale Fishery. A Starbuck, CastleStarbuck, A. History of the American Whale Fishery. (Castle, 1989).
[]
[ "Variable density preserving topology grids and the digital models for the plane", "Variable density preserving topology grids and the digital models for the plane" ]
[ "Alexander V Evako [email protected]. \nVolokolamskoe Sh\n1, kv. 157125080MoscowRussia\n" ]
[ "Volokolamskoe Sh\n1, kv. 157125080MoscowRussia" ]
[]
We define LCL decompositions of the plane and investigate the advantages of using such decompositions in the context of digital topology. We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object. We prove that for any LCL tiling of the plane, the digital model is necessarily a digital 2-manifold. We show that elements of an LCL tiling can be of an arbitrary shape and size. This feature generates a variable density grid with a required resolution in any region of interest, which is extremely important in medicine. Finally, we describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions.
null
[ "https://arxiv.org/pdf/1302.3809v1.pdf" ]
16,163,683
1302.3809
0f60b15393d4bcdea981e3403254977c1d393e90
Variable density preserving topology grids and the digital models for the plane Alexander V Evako [email protected]. Volokolamskoe Sh 1, kv. 157125080MoscowRussia Variable density preserving topology grids and the digital models for the plane 1 We define LCL decompositions of the plane and investigate the advantages of using such decompositions in the context of digital topology. We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object. We prove that for any LCL tiling of the plane, the digital model is necessarily a digital 2-manifold. We show that elements of an LCL tiling can be of an arbitrary shape and size. This feature generates a variable density grid with a required resolution in any region of interest, which is extremely important in medicine. Finally, we describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions. introduction Integrating topological features into discretization and segmentation procedures in order to generate topologically correct digital models of anatomical structures is critical for many clinical and research applications [1,3,14]. Sometimes, particular regions of the object require a dense grid while a relatively coarse grid can be used over the rest of the object of interest. In such cases, it is suitable to use variable density grids according to external requirements and geometrical and topological features of the object. A considerable amount of works has been devoted to building two-, three-and n-dimensional grids, e.g., [2,[11][12][13]. In the present paper, we use an approach, which was introduced and studied in [5][6][7] and was based on LCL discretization of n-dimensional objects. This type of discretization has several obvious advantages. In the discrete model (the grid), topology equivalent elements (n-tiles) are used and at the same time, the shape and the size of an individual n-tile can be arbitrary (an n-tile is not necessarily a convex set) within the framework of an LCL tiling. This allows obtaining more detailed geometrical and topological information about the regions of interest. Another feature is that the intersection graph (digital model) of the grid is a digital ndimensional manifold preserving the topology of the object. The material to be presented below begins with basic definitions and results related to digital objects in section 2. We study in section 3 discretization of the plane by LCL tilings. We formulate conditions for a tiling for the plane to be the LCL cover. We show that one can choose an LCL grid with a required density in any region of interest which is extremely important in medicine. We prove that for any LCL tiling for the plane, the intersection graph is necessarily a digital 2-manifold. A trivial result of this consideration is that the quantity of non-isomorphic digital models (and LCL grids) of the plane is not restricted by a number. We provide a simple algorithm, which constructs digital models of areas of interest with any required resolution. Preliminaries A digital object G is a simple undirected graph G=(V,W), where V={v 1 ,v 2 ,...v n ,…} is a finite or countable set of points, and W={(v р v q ),....}VV is a set of edges provided that (v р v q )=(v q v p ) and (v р v p )W [7]. Such notions as the connectedness, the adjacency, the dimensionality and the distance on a graph G are completely defined by sets V and W. Further on, if we consider a graph together with the natural topology on it, we will use the phrase 'digital space". We use the notations v p G and (v р v q )G if v p V and (v р v q )W respectively if no confusion can result. Since in this paper we use only subgraphs induced by a set of points, we use the word subgraph for an induced subgraph. Points v р and v q are called adjacent if (v р v q )W. The subgraph O(v)G containing all points adjacent to v (without v) is called the rim or the neighborhood of point v in G, the subgraph v O(v) is called the ball of v. Graphs (digital spaces) can be transformed from one into another in a variety of ways. Contractible transformations of graphs [10] seem to play the same role in this approach as a homotopy in algebraic topology. If a graph G is obtained from a graph H by a sequence of contractible transformations, then we say that G is homotopic (or homotopy equivalent) to H. A graph is called contractible if it is homotopy equivalent to a point. Contractible transformations retain the Euler characteristic and homology groups of a graph [10]. Let us remind some necessary definitions. A digital 0-dimensional sphere is a disconnected graph S 0 (a,b) with just two points a and b. [4,6]. A connected space M is called a digital n-sphere, n>0, if for any point vM, the rim O(v) is a digital (n-1)-sphere and the space M-v is contractible. For any terminology used but not defined here, see Harary [9]. A connected space M is called a digital n-dimensional manifold, n>0, if the rim O(v) of any point v is a digital (n-1)-dimensional sphere LCL tilings and digital models of the plane In this section, we use intrinsic topology of an object, without reference to an embedding space if no confusion will result. A set D is called an n-tile, if it is homeomorphic to a closed unit n-dimensional cube on R n . A set S is called an n-sphere, if it is homeomorphic to a unit n-dimensional sphere on R n+1 . We denote the interior and the boundary of an n-tile D by IntD and D respectively, D=IntDD. Note that the boundary D of an n-tile D is an (n-1)-sphere. The 0-tile D is a single point for which D=. If S is a circle and D is a 1-tile contained in S, then B=S-IntD is a 1-tile and D∩B=S 0 is a pair of points at the ends of D and B. Definition 3.1. Let W={D 1 ,D 2 ,… } be a collection of n-tiles, n=1, 2.  W is called a locally centered collection (LC collection) if from condition D i(k) D i(m) , mk, m, k=1,2,...s, it follows that D i(1) D i(2) …D i(s) .  W is called a locally lump collection (LL collection) if any nonempty intersection of s distinct n-tiles is an (n-s+1)-tile: D i(1) D i(2) …D i(s) =D i(1) D i(2) …D i(s) = D n-s+1 .  W is called a locally centered lump collection (LCL collection) if W is a locally centered collection and a locally lump collection at the same time. As it follows from definition 3.1, if W={D 1 ,D 2 ,… }is an LCL collection of 1-tiles and D 1 D 2 , then D 1 D 2 =D 1 D 2 = D 0 is a point. The intersection of three or more distinct 1tiles is empty ( fig.1). If W={D 1 ,D 2 ,… } is an LCL collection of 2-tiles and D 1 D 2 , then D 1 D 2 =D 1 D 2 = D 1 is a 1-tile. If D 1 D 2 D 3 , then D 1 D 2 D 3 =D 1 D 2 D 3 = D 0 is a point. The intersection of four or more distinct 2-tiles is empty ( fig. 2). Evidently, an individual n-tile can be of an arbitrary shape and size within the framework of an LCL collection. In paper [6], a locally centered collection is called continuous and it is shown that for a given object, the intersection graphs of all continuous, regular and contractible covers are homotopic to each other. In paper [13], a normal set W of convex nongenerate polygons (intersection of any two of them is an edge, a vertex, or empty) is called strongly normal if for all P, P 1 ,…P n (n>0)W, if each P i intersects P and I=P 1 …P n is nonempty, then I intersects P (fig. 3). Several papers, e.g. [1,11,12] extended basic results about strong normality to collections of polyhedra in R n . There are obvious differences between SN collections of polygons and LCL collections. For example, elements of an SN collection are convex sets. On the contrary, any 2-tile in an LCL collection can be of an arbitrary form and size ( fig. 2, 3) and the local topology is determined (d) (b) (a) (c) (e) by the neighborhood of the tile. The following proposition is a direct consequence of definition 3.1. Proposition 3.1 (1) Let W={D 0 ,D 1 ,…} be an LCL collection of n-tiles, n=1, 2. Then any subcollection of W is an LCL collection of n-tiles. Regard now the set of n-tiles, n=1, 2, adjacent to a given n-tile in an LCL collection. This set specifies local topological properties of the collection. Proposition 3.2 Let W={D 0 ,D 1 ,…} be an LCL collection of n-tiles, n=1,2, and C i =D 0 D i , for i=1,…s. Then the collection V={C 1 ,C 2 ,…C s } of (n-1)-tiles is an LCL collection and collections U={D 1 ,D 2 ,…D s } and V={C 1 ,C 2 ,…C s } are isomorphic ( fig.4). Proof. It is obvious for n=1 ( fig.1). Let n=2 . Suppose that C m C r ≠, m,r=1,…t, m≠r. Then C 1 C 2 =D 0 D 1 D 2 =x is a point. By construction, x is an endpoint of 1-tiles C 1 and C 2 , i.e., C 1 C 2 =C 1 C 2 =x. Assume now that C m C r ≠, m,r=1,2,3, Then C 1 C 2 C 3 =D 0  D 1 D 2 D 3 =. Therefore, t=2 and V is an LCL collection of 1-tiles. The isomorphism of U and V is evident.  Fig. 4(a) shows an LCL collection W={D 0 ,…} of 2-tiles, the collection U of 2-tiles adjacent (a) (D 0 (b) (c) to D 0 is depicted in fig. 4(b), the collection V of 1-tiles C i =D 0 D i is shown in fig. 4(c). Collections U and V are isomorphic. Definition 3.2. Let W={D 0 ,D 1 ,…} be an LCL collection of n-tiles, n=1,2. Then W is called a tiling of M=D 1 D 2 …. The intersection graph G(W) of W is called the digital model of M=D 0 D 1 … in regard to W. Proposition 3.3.  Let an LCL collection W={D 0 ,D 1 ,D 2 …} of 1-tiles be a tiling of a circle. Then the intersection graph G(W) of W is a digital 1-sphere ( fig. 3.4).  Let an LCL collection W={D 0 ,D 1 ,D 2 …} of 1-tiles be a tiling of the line R 1 . Then the intersection graph G(W) of W is a digital 1-manifold ( fig. 3.4). The proof follows from figure 5. Proposition 3.4. Suppose that an LCL collection W={D 0 ,D 1 ,…} of 2-tiles is a tiling of the plane ( fig. 6). Then the intersection graph G(W) of W is a digital 2-manifold. Proof. Let W={D 0 ,D 1 ,…} be an LCL tiling of the plane and consider the collection U={D 1 ,D 2 …D s } of all 2-tiles, which intersect D 0 , D 0 D i , i=1,…s. Then the collection V={C 1 ,C 2 ,…C s }, where C i =D 0 D i , i=1,,...s, is an LCL collection of 1-tiles according to proposition 3.2. By construction, V is a tiling of the circle S=D 0 and by proposition 3.3, the intersection graph G(V) of V is a digital 1-sphere S 1 0 . Therefore, the intersection graph G(U) of U={D 1 ,…D s } is a digital 1-sphere S 1 0 . Since this is applicable to any D i W, then G(U i )=S 1 i is a digital 1- (a) (b) (c) (d) (e) sphere. Therefore, G(W) is a digital 2-manifold.  As it follows from propositions 3.3. and 3.4, the digital model retains the same topology regardless of what LCL tilings are introduced. LCL tilings of the plane and their digital models Z 2 are depicted in fig. 3(e) and fig. 6. Consider the LCL tilings of the plane depicted in fig. 6. In tiling (c), 2-tiles of different shape and size are used. Digital models (b) and (d) of these tilings are digital 2-manifolds Z 2 . The tiling (e) is a grid with variable density. In the middle of the picture the density is the highest one. Note that all these tilings are not SN. Now we are able to describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions. On the first step, construct an LCL grid W with the variable density in accordance with conditions and restrictions imposed by requirements defined by the accuracy and correctness of the representation. On the second step, build the digital model (intersection graph) of W. Using an LCL cover guaranties that G(W) is a digital 2-manifold preserving the topology of the object. Figure 1 .Figure 2 . 12Collections of 1-tile: (a) is an LC, non-LL collection. (b) is an LL, non-LC collection. (c), (d), (e) are LCL collections. Collections of 2-tiles: (a) is an LL, non-LC collection. (b) is an LC, non-LL collection. (c), (d), (e) are LCL collections. Figure 3 . 3(a)-(c) are SN collections. The unions of tiles in these collections are topologically different, but the intersection graphs of the unions are identical. The collection (d) is SN but not LCL one. The collection (e) is LCL and SN one. Figure 4 . 4(a) An LCL collection W of 2-tiles. (b) The LCL collection U of 2-tiles adjacent to D 0 . (c) The LCL collection V of 1-tiles C i =D i ∩D 0 . Collections U and V are isomorphic. Figure. 5 .Figure. 6 . 56(a) LCL tilings of a circle their digital models. The intersection graphs of tilings are digital 1-spheres. (b) An LCL tiling of the line. The intersection graph of the tiling is a digital 1-manifold. LCL tilings of the plane and their digital models. (b) The digital model Z 2 of the tiling (a) is a digital 2-manifolds of (6,6) type. (d) The digital model Z 2 of the tiling (c) is a digital 2-manifold of (4,8) type. (e ) The LCL tiling is a grid with variable density. In the middle of the picture the density is the highest one. Summary of resultsIn this paper we introduce an LCL tiling for the plane and investigate its properties. We show that the intersection graph of any LCL tiling of the plane is a digital 2-manifold preserving the local topological structure of the plane. We present a simple algorithm for building digital counterparts of the plane with the required resolution in specific reagons of interest. Y Bai, X Han, J Prince, Digital Topology on Adaptive Octree Grids. 34Y. Bai, X. Han, J. Prince: Digital Topology on Adaptive Octree Grids. Journal of Mathematical Imaging and Vision 34(2), (2009), pp. 165-184. On strongly normal tesselations. P Brass, Pattern Recognition Letters. 209P. Brass, On strongly normal tesselations, Pattern Recognition Letters, 20(9), (1999), pp.957-960. Cortical surface-based analysis i: Segmentation and surface reconstruction. A Dale, B Fischl, S M , NeuroImage. 9A. Dale, B. Fischl, and S. M.I. Cortical surface-based analysis i: Segmentation and surface reconstruction. NeuroImage, 9:179{194, 1999. Dimensional properties of graphs and digital spaces. A V Evako, R Kopperman, Y V Mukhin, Journal of Mathematical Imaging and Vision. 6A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, Journal of Mathematical Imaging and Vision 6 (1996), pp. 109-119. One method of constructing digital models of closed continuous surfaces by using covers. A V Evako, Computer Vision and Image Understanding. 102Topological properties of closed digital spacesA.V. Evako, Topological properties of closed digital spaces. One method of constructing digital models of closed continuous surfaces by using covers, Computer Vision and Image Understanding 102 (2006), pp. 134-144. Topological properties of the intersection graph of covers of ndimensional surfaces. A V Evako, Discrete Mathematics. 147A.V. Evako, Topological properties of the intersection graph of covers of n- dimensional surfaces, Discrete Mathematics 147 (1995), pp. 107-120. The consistency principle for a digitization procedure. An algorithm for building normal digital spaces of continuous n-dimensional objects. A V Evako, 16A.V. Evako, The consistency principle for a digitization procedure. An algorithm for building normal digital spaces of continuous n-dimensional objects, http://www.arxiv.org/abs/math., CV/0511064, 16 Nov 2005. Dimension on Discrete Spaces. A V Evako, International Journal of theoretical. 7A. V. Evako, Dimension on Discrete Spaces, International Journal of theoretical 7 . Physics. 33Physics 33 (1994), pp. 1553-1568. F Harary, Graph Theory. Addison-Wesley, Reading, MAF. Harary, Graph Theory, Addison-Wesley, Reading, MA (1969). Representation of smooth surfaces by graphs. Transformations of graphs which do not change the Euler characteristic of graphs. A V Ivashchenko, Discrete Mathematics. 122A.V. Ivashchenko, Representation of smooth surfaces by graphs. Transformations of graphs which do not change the Euler characteristic of graphs, Discrete Mathematics 122 (1993), pp. 219-233. Strongly normal sets of contractible tiles in N dimensions. T Kong, P Saha, A Rosenfeld, Pattern Recognition. 402T. Kong, P. Saha, A. Rosenfeld, Strongly normal sets of contractible tiles in N dimensions, Pattern Recognition 40(2) ( 2007), pp. 530-543 Strongly Normal Sets of Tiles in N Dimensions. P K Saha, T Y Kong, A Rosenfeld, Electronic Notes in Theoretical Computer Science. 46P. K. Saha, T. Y. Kong, A. Rosenfeld, Strongly Normal Sets of Tiles in N Dimensions, Electronic Notes in Theoretical Computer Science 46 (2001), pp. 1- 12. Strongly normal sets of convex polygons or polyhedra. P Saha, A Rosenfeld, Pattern Recognition Letters. 1912P. Saha , A. Rosenfeld, Strongly normal sets of convex polygons or polyhedra, Pattern Recognition Letters, 19 (12), (1998), p.1119-1124. Integration of Topological Constraints in Medical Image Segmentation. F Segonne, B Fischl, Biomedical Image Analysis: Methodologies and Applications. F. Segonne, B. Fischl, Integration of Topological Constraints in Medical Image Segmentation. Biomedical Image Analysis: Methodologies and Applications, Dec 2007
[]
[ "Pseudoscalar decays to gauge bosons at the LHC and at a future 100 TeV collider", "Pseudoscalar decays to gauge bosons at the LHC and at a future 100 TeV collider" ]
[ "Abdesslam Arhrib [email protected] \nDépartement de Mathématiques\nFaculté des Sciences et Techniques\nUniversité Abdelmalek Essaadi\nB. 416TangierMorocco\n", "Rachid Benbrik \nLPHEA\nMSISM Team Départment of Physics\nFaculté Polydisciplinaire de Safi\nBP 4162Sidi Bouzid, SafiMorocco\n", "† ", "Jaouad El Falaki \nDépartement de Mathématiques\nFaculté des Sciences et Techniques\nUniversité Abdelmalek Essaadi\nB. 416TangierMorocco\n\nCenter for Fundamental Physics\nZewail City of Science and Technology\nSheikh Zayed12588Giza, Egypt\n", "Marco Sampaio §e-mail:[email protected]¶e-mail:[email protected] \nDepartamento de Física da Universidade de Aveiro\nCampus de Santiago3810-183AveiroPortugal\n\nCIDMA -Center for Research & Development in Mathematics and Applications\nCampus de Santiago3810-183AveiroPortugal\n", "Rui Santos \nISEL -Instituto Superior de Engenharia de Lisboa\nInstituto Politécnico de Lisboa\n1959-007LisboaPortugal\n\nCentro de Física Teórica e Computacional\nFaculdade de Ciências\nUniversidade de Lisboa\nEdifício C81749-016Campo Grande, LisboaPortugal\n" ]
[ "Département de Mathématiques\nFaculté des Sciences et Techniques\nUniversité Abdelmalek Essaadi\nB. 416TangierMorocco", "LPHEA\nMSISM Team Départment of Physics\nFaculté Polydisciplinaire de Safi\nBP 4162Sidi Bouzid, SafiMorocco", "Département de Mathématiques\nFaculté des Sciences et Techniques\nUniversité Abdelmalek Essaadi\nB. 416TangierMorocco", "Center for Fundamental Physics\nZewail City of Science and Technology\nSheikh Zayed12588Giza, Egypt", "Departamento de Física da Universidade de Aveiro\nCampus de Santiago3810-183AveiroPortugal", "CIDMA -Center for Research & Development in Mathematics and Applications\nCampus de Santiago3810-183AveiroPortugal", "ISEL -Instituto Superior de Engenharia de Lisboa\nInstituto Politécnico de Lisboa\n1959-007LisboaPortugal", "Centro de Física Teórica e Computacional\nFaculdade de Ciências\nUniversidade de Lisboa\nEdifício C81749-016Campo Grande, LisboaPortugal" ]
[]
We discuss the search for a CP-odd scalar decaying into gauge bosons in the framework of a CP-conserving two-Higgs doublet model (2HDM) and of a 2HDM extended with a vector-like quark (VLQ) at the Large Hadron Collider and at a future 100 TeV collider. The rate of decay of a pseudoscalar to Z-bosons could be important to ascertain the CP-nature of the scalars in the model. In the 2HDM A → ZZ will be extremely hard to detect even at a future 100 TeV pp collider while in the 2HDM+VLQ this decay can be probed even during the present LHC run. We further discuss all decays of the pseudoscalar into gauge bosons at the LHC and at a future 100 TeV collider in the alignment limit where the lightest scalar is the 125 GeV Higgs with SM-like couplings to the fermions and gauge bosons. *
10.1103/physrevd.99.035043
[ "https://arxiv.org/pdf/1809.04805v2.pdf" ]
118,918,597
1809.04805
07a1714010647f055f1dec97c5601925e0bdb183
Pseudoscalar decays to gauge bosons at the LHC and at a future 100 TeV collider 13 Sep 2018 Abdesslam Arhrib [email protected] Département de Mathématiques Faculté des Sciences et Techniques Université Abdelmalek Essaadi B. 416TangierMorocco Rachid Benbrik LPHEA MSISM Team Départment of Physics Faculté Polydisciplinaire de Safi BP 4162Sidi Bouzid, SafiMorocco † Jaouad El Falaki Département de Mathématiques Faculté des Sciences et Techniques Université Abdelmalek Essaadi B. 416TangierMorocco Center for Fundamental Physics Zewail City of Science and Technology Sheikh Zayed12588Giza, Egypt Marco Sampaio §e-mail:[email protected]¶e-mail:[email protected] Departamento de Física da Universidade de Aveiro Campus de Santiago3810-183AveiroPortugal CIDMA -Center for Research & Development in Mathematics and Applications Campus de Santiago3810-183AveiroPortugal Rui Santos ISEL -Instituto Superior de Engenharia de Lisboa Instituto Politécnico de Lisboa 1959-007LisboaPortugal Centro de Física Teórica e Computacional Faculdade de Ciências Universidade de Lisboa Edifício C81749-016Campo Grande, LisboaPortugal Pseudoscalar decays to gauge bosons at the LHC and at a future 100 TeV collider 13 Sep 2018 We discuss the search for a CP-odd scalar decaying into gauge bosons in the framework of a CP-conserving two-Higgs doublet model (2HDM) and of a 2HDM extended with a vector-like quark (VLQ) at the Large Hadron Collider and at a future 100 TeV collider. The rate of decay of a pseudoscalar to Z-bosons could be important to ascertain the CP-nature of the scalars in the model. In the 2HDM A → ZZ will be extremely hard to detect even at a future 100 TeV pp collider while in the 2HDM+VLQ this decay can be probed even during the present LHC run. We further discuss all decays of the pseudoscalar into gauge bosons at the LHC and at a future 100 TeV collider in the alignment limit where the lightest scalar is the 125 GeV Higgs with SM-like couplings to the fermions and gauge bosons. * Introduction After the discovery of the Higgs boson by the ATLAS [1] and CMS [2] collaborations at the Large Hadron Collider (LHC) the high energy physics community focused on the search for signs of extended scalar sectors [3]. Such extended sectors, with extra Higgs singlets, doublets or triplets, is a common feature of several Beyond the Standard Model (BSM) models. Finding a new scalar would be a clear signal of BSM physics with extended Higgs sectors. Two-Higgs doublet models (2HDM) [4], both in their CP-conserving and CP-violating versions, have been used as benchmark models to search for new scalars at the LHC. The 2HDM has four extra degrees of freedom with two extra neutral scalars and two charged scalars. In the CP-conserving version of the model the two neutral states are h and H (CP-even) and A (CP-odd) while in the CP-violating version the three neutral states are a mixture of CP-even and CP-odd states and are referred to as h 1 , h 2 and h 3 . In this work we focus on the production of the pseudoscalar via gluon fusion plus bb initiated process with the subsequent decay to gauge bosons, with focus on A → γγ and A → ZZ. Although A → γγ is a loop induced process, it is nevertheless competitive with other final states like τ + τ − in large portions of the parameter space of the model. Hence, a pseudoscalar could first be detected in the two photon final state. However, even if A is discovered in some other final states, the remaining possible decays have either to be confirmed or excluded. This study will therefore give us further information on the model. The decay of scalars to ZZ can be used as a means to search for signals of CP-violation at the LHC [5]. Several classes of processes may hint a signal of CP-violation. One such example is the combined observation of the three decays h 2 → h 1 Z, h 2 → ZZ and h 1 → ZZ, where h 1 is the 125 GeV Higgs. Except for the already measured h 1 → ZZ, the other two processes that occur at tree-level in a CP-violating model, can be mistaken by the loop processes A → hZ and A → ZZ from the corresponding CP-conserving model. In the alignment limit, where all h 1 /h couplings to the SM particles mimic the SM ones, A → hZ is zero at tree-level. Also, A → ZZ is a loop induced process and therefore very small. Hence, if a new scalar is found, its CP-numbers could be hard if not at all impossible to determine if the rates are too small. However, there are simple extensions of the 2HDM with the addition of vector like quarks (VLQ) [6][7][8], which lead to very significant enhancement of the pseudoscalar production cross section with the introduction of an extra loop contribution. Also the loop decays to gauge bosons are modified by extra contributions from the vector like quarks. Therefore, the rate pp → A → ZZ can differ by several orders of magnitude in the 2HDM and in the 2HDM+VLQ. In this work we examine the prospects of detecting a pseudoscalar decaying into two vector bosons at the 14 TeV LHC and also at a future 100 TeV pp collider with a planned luminosity of the order of 10 ab −1 . Previous works have discussed the pseudoscalar decays into gauge bosons in a variety of models such as the 2HDM [9][10][11], the 2HDM with a sequential fourth generation of quarks [12] and in Supersymetric Models [13]. In this work, our main focus either diverges or completes the previous studies. Regarding A → γγ we present the latest experimental results from the LHC and we include all other experimental constraints in the theoretical calculations. We discuss what is expected by the end of the LHC and also at a future 100 TeV pp collider. The other focus is on A → ZZ because these are decays that can give information on the CP nature of the scalars. So it is important to understand if either in the next LHC run or at a future 100 TeV collider a scalar decays into two Z bosons if the model can distinguish between a scalar and a pseudoscalar. The paper is organized as follows. In the next section we discuss the 2HDM setup and list the theoretical and experimental constraints we will be using. In section 3 we present the 2HDM extended by an up-type vector-like quark (2HDM+T). In section 4 we present our results for the 2HDM while in section 5 results for the 2HDM+T are presented for the LHC Run 2. A discussion on the prospects for a future 100 TeV collider are examined in section 6. Finally, in section 7 we address the problem of the contribution of these searches to understand the CP nature of a new scalar. Our conclusions are presented in the last section. The two-Higgs doublet model The 2HDM was proposed by T.D. Lee [4] in an attempt to explain the matter anti-matter asymmetry of the Universe through the addition of an extra source of CP-violation. In this work we discuss the CP-conserving version of the model that contains two CP-even states denoted by h (the lightest) and H, one CP-odd state denoted by A and two charged states, H ± . As tree-level flavour changing neutral currents (FCNC) are very constrained by experiments, we impose a Z 2 symmetry Φ 1 → Φ 1 and Φ 2 → −Φ 2 on the scalar fields. The resulting Higgs potential (softly broken by the dimension two term m 2 12 ) can be written as V = m 2 11 Φ † 1 Φ 1 + m 2 22 Φ † 2 Φ 2 − m 2 12 Φ † 1 Φ 2 + h.c. + 1 2 λ 1 Φ † 1 Φ 1 2 + 1 2 λ 2 Φ † 2 Φ 2 2 +λ 3 Φ † 1 Φ 1 Φ † 2 Φ 2 + λ 4 Φ † 1 Φ 2 Φ † 2 Φ 1 + 1 2 λ 5 Φ † 1 Φ 2 2 + h.c. .(1) Choosing real vacuum expectation values (VEVs), v 1 and v 2 and demanding m 2 12 and λ 5 to be real as well, the potential is CP-conserving. One should note that the CP-conserving minimum of any 2HDM is stable at tree-level, that is, any other stationary point, if it exists, is a saddle point [14,15]. Still two CP-conserving minima can coexist but the existence of a global minimum can be easily enforced by a simple condition [16,17]. The free independent parameters are the four masses, m h , m H , m A and m H ± , the soft breaking parameter m 2 12 , the angle tan β = v 2 /v 1 and the rotation angle α that diagonalizes the CP-even mass matrix. When we impose that no tree-level FCNCs are present in the theory by extending the Z 2 symmetry [18,19] to the Yukawa sector, we end up with four independent versions of the model. These are: Type I -only Φ 2 couples to all fermions; Type II -Φ 2 couples to up-type quarks and Φ 1 couples to charged leptons and down-type quarks; Flipped or Type Y -Φ 2 couples to charged leptons and up-type quarks and Φ 1 couples to down-type quarks; Lepton Specific or Type X -Φ 2 couples to quarks and Φ 1 couples to charged leptons. The scan in the 2HDM parameter space was performed fixing m h = 125 GeV , sin(β − α) = 1, since a small misalignment has no phenomenological consequences on the CP-odd decay, and m H ± = m H = 600 GeV and varying m A , tan β and m 2 12 in the allowed parameter space. This is the exact alignment limit and it is in agreement with the most relevant experimental and theoretical constraints: • The potential is bounded from below at tree-level [20,21]; • Perturbative unitarity is enforced [22][23][24] to the quartic couplings of the Higgs potential; • The parameter space complies with electroweak precision observables [25] via S and T parameters [26][27][28][29][30] because m H ± = m H and cos(β − α) = 0 [29]; • Collider bounds from LEP, Tevatron and from LHC Run 1 are taken into account. Since we work in the alignment limit, sin(β − α) = 1, automatic agreement with the constraints on the Higgs couplings to the other SM particles is attained, because all Higgs couplings become SM-like. Regarding the searches, the tree-level decays to gauge bosons of both H and A are forbidden. The decays to fermions are considered and of particular relevance are the bounds arising from the search pp → A → τ + τ − [31][32][33]. These imply that in the Type II 2HDM, the values of tan β cannot be too large especially for low m A bounds. Therefore we choose to take tan β < 10 in the entire mass range for Type II. However, one should note that the largest cross sections are the gluon fusion ones, with the maximum value for tan β = 1. Also the BR(A → γγ) decreases with tan β. Hence, overall the largest rates are obtained for tan β ≈ 1. We will check directly in our study how the searches for a pseudoscalar decaying into gauge bosons affects the parameter space. • We consider the most relevant indirect constraints on the parameter space in the plane (m H ± , tan β). These are mainly loop processes where cancellations could occur in the loops if other sources of new physics are considered. The bounds arise mainly from B-physics observables [34][35][36][37][38] and R b = Γ(Z → bb)/Γ(Z → hadrons) [39][40][41]. These constraints result in a rough bound of tan β ≥ 1 for all Yukawa types. Regarding the charged Higgs mass the most relevant bound comes from b → sγ (Type II and Y only) and is at present m H ± ≥ 570 GeV [42,43]. The same constraints forces tan β ≥ 2 in Type I and X. LHC run 1 has contributed with direct bounds in the (m H ± , tan β) plane with the process pp → tt(H + W − bb) [44,45]. Finally, LEP provided the only direct bound on the charged Higgs mass of roughly m H ± ≥ 90GeV for all Yukawa types with e + e − → H + H − assuming only BR(H ± → cs) + BR(H ± → τ ν) + BR(H ± → AW ) = 1 [46]. The 2HDM extended by an up-type vector-like quark Vector-like quarks (VLQs) appear naturally in various extension of the SM, such as some supersymetric models [47], models with extra-dimensions [48], little Higgs models [49] and composite Higgs models [50]. VLQs are also well motivated by the fact that they can solve the Higgs boson mass instability resulting from large radiative corrections at high scales. In fact, a vector-like top quark partner (T) could play the same role as the superpartner of the top quark in supersymetric models. A particular feature of VLQs is that their left and right-handed components transform in the same way under the SM gauge group. Consequently, their mass terms are allowed in the Lagrangian without violating gauge invariance transformations. There have been many studies on the phenomenology of the SM extended with VLQs [6,7,51]. Moreover, both ATLAS and CMS have performed several experimental searches for such new quarks. Direct searches by the ATLAS and CMS Collaborations have set lower limits on the mass of the single vector-like T top partner in the range of 550 − 900 GeV at 8 TeV [52][53][54] through one of its main decay channels: T → ht, W b, Zt since the new top is expected to couple predominantly to the third generation quarks. The above limit was improved in the 13 TeV run. A lower limit on the mass of the T-quark was derived and found to be in the range 1170 − 1295 GeV by ATLAS and CMS at 13 TeV [55][56][57][58]. This lower limit can be lowered if the new T-quark has a non-negligible mixing with the first and second generation quarks [59]. In the present study we consider an extension of the 2HDM by adding a vector-like top quark (T) with charge + 2 3 . This extention was already studied in detail in [8,60,61]. Similarly to the SM, we introduce left and right components of the new top: T 0 L , T 0 R . The 2HDM-VLQ Lagrangian with the new top-quark T is given by: −L Y ⊃ y T Q 0 L H 2 T 0 R + λ T Q 0 L H 1 T 0 R + M T T 0 L T 0 R + h.c. = y T (t 0 L , b 0 L ) ϕ 0 2 −iA √ 2 −H − T 0 R + λ T (t 0 L , b 0 L ) v+ϕ 0 1 −iG 0 √ 2 −G − T 0 R + M T T 0 L T 0 R + h.c.,(2) whereH i ≡ iτ 2 H * i and Q 0 L is the left handed third generation quark doublets. Note that this Lagrangian is valid for all 2HDM-VLQ types because the couplings to the top are the same in all models. After spontaneous symmetry breaking, the top quark mixes with T and the mass matrix of the mixing between (t 0 L , T 0 L ) and (t 0 R , T 0 R ) is given by (where we can rotate away one off-diagonal element of the mass mixing matrix). M = ytv √ 2 λ T v √ 2 0 M T ,(3) where y t and λ T are the Yukawa couplings for the top quark and T. This matrix can be diagonalized by rotating the weak eigenstates (t 0 L , T 0 L ) into the mass eigenstates (t L , T L ) using a bi-unitarity transformations. t L,R T L,R = U L,R t 0 L,R T 0 L,R ,(4) where the unitarity matrices are given by U L,R = c L,R −s L,R s L,R c L,R ,(5) with c L,R = cos(θ L,R ) and s L,R = sin(θ L,R ). Thus the mass mixing matrix M is diagonalized as follows: U L MU † R = M diag = m t 0 0 m T ,(6) or similarly U R,L M † MU † R,L = M 2 diag .(7) From the fact that the off-diagonal elements of (7) vanish, one obtains the following relations for the mixing angles, tan(2θ L ) = 4m t m T 2m 2 T − 2m 2 t − λ 2 T v 2 , tan(2θ R ) = 2 √ 2λ T m t v 2m 2 T + 2m 2 t − λ 2 T v 2 .(8) We stress that the above mixing angles are not independent, by using eq (6) one can derive the following relations tan θ R = m t m T tan θ L , λ T y t = sin θ L cos θ L m 2 T − m 2 t m t m T .(9) It is important to mention that both the interaction of the top-quark and of the T vector quark with the electroweak gauge bosons depend on the mixing angle θ L and are given by L N C = g cos θ W Z µ cos θ 2 L 2 − 2 3 sin 2 θ W t L γ µ t L + sin θ 2 L 2 − 2 3 sin 2 θ W T L γ µ T L + sin θ L cos θ L 2 T L γ µ t L + h.c .(10)L CC = g √ 2 cos θ L t L + sin θ L T L bγ µ W + µ + h.c. .(11) After EWSB one can derive the following Yukawa couplings of the CP-odd Higgs A to both t and T Att = −i c L c R tan β − c L s R y T y t , ATT = −i s L s R tan β + s L c R y T y t , At L T R = −i c L s R tan β + c L c R y T y t , At RTL = −i s L c R tan β − s L s R y T y t .(12) After this brief review of the couplings of the heavy top to gauge bosons and Higgs bosons, we list hereafter the most important theoretical and phenomenological constraints on the parameters of the model. From the theoretical side, the scalar sector of 2HDM+T is subject to the same unitarity and vacuum stability constraints as the usual 2HDM [16,[22][23][24]. On the other hand, y T is also constrained from unitarity to be less than 4π while λ T is a derived quantity (see equation 9). • The above interactions of t R and T R with the Z and W bosons in the 2HDM+T are the same as those in the SM. Since the new top will contribute to gauge bosons self energies, the mixing angle θ L can be constrained from electroweak observables such as S and T parameters [8]. It has been shown in [8] that sin θ L < 0.2 (resp 0.12) for m T = 400 GeV (resp m T = 1 TeV.) • The interaction of the heavy top with charged Higgs and bottom quark can affect the rate of BR(b → sγ) [8]. This can be translated into constraint on the charged Higgs mass and/or mixing angle between heavy top and top quark. In the case where we assume that sin θ L ≈ V T b , it has been shown in [8] that in 2HDM-II, V T b must be smaller than 0.03 for m T > 600 GeV. This limit can be weakened for light m T . • Regarding compatibility with the couplings of the h 125 Higgs to the remaining SM particles, since we are working in the alignment limit all couplings are SM-like except the couplings to top quarks for which we choose a small mixing angle to force compatibility with the measured Higgs couplings. Regarding the searches for heavy scalars, the situation is exactly the same as for the 2HDM. So again we use the limits derived for the scalar decay to fermions and study the effect of the decay to gauge bosons in the parameter space of the model. Moreover, since the new heavy top couples to the all Higgs bosons, the decay patterns T → bW, ht, Zt will be modified. It is well known that in the SM extended with a heavy top the values of Br(T → bW ), Br(T → ht) and Br(T → Zt) are respectively 50%, 25% and 25%. In the 2HDM with the presence of new Higgs bosons A, H and H ± , the above pattens are modified [8,61] and the limits obtained by ATLAS and CMS have to be reinterpreted in the framework of the 2HDM+T. Results for the 2HDM In this section we present the results for the pp → A → V V production rates, where V = γ, Z, W , evaluated in the narrow width approximation. The pseudoscalar production cross section was calculated using SUSHI [62] at NNLO and it includes gluon and bb fusion. The branching ratios were calculated using the HDECAY [63,64] version for the 2HDM [65]. The widths for the pseudoscalar decays into vector bosons are loop induced and were calculated with the packages FeynArts [66], FormCalc [67] and LoopTools [67,68] for loop integrals evaluation. The loop calculation is performed in the 'tHooft-Feynman gauge using dimensional regularization. At the one loop level, only fermionic loops contribute to A → V V . The reason is that in the bosonic sector, the electroweak theory conserves CP while after adding fermions CP is no longer conserved. Therefore, there is no contribution to A → V V from the SM bosons [13], or from the spin zero scalars but only from the fermions. Analytical and numerical check of UV finiteness have been performed. As the measurements of the Higgs couplings to the other SM particles become increasingly precise, the 2HDM approaches more and more the alignment limit where sin(β − α) ≈ 1 if h, the lightest CP-even scalar, is the SM-like Higgs boson. Considering the lower limit on the charged Higgs mass (about 100 GeV for Type I and X and 600 GeV for Type II and Y) and that m h = 125 GeV, the decay A → hZ is kinematically allowed for a pseudoscalar mass of more than about 200 GeV for all types while A → H ± W ∓ would be allowed for a pseudoscalar mass of about 200 (700) GeV for Type I and X (Type II and Y). In this study we will choose the heavy Higgs masses, m H ± and m H , to be equal and such that the decay A → H ± W ∓ is kinematically disallowed, which is already the case for models Type II and Y due to the combined experimental and theoretical limits. The decay A → hZ is exactly zero in the alignment limit. By extending this condition to all models we are being conservative in the chances for the detection of a pseudoscalar in V V final states. In fact, as more decays of the pseudoscalar are kinematically available, the chances of detecting a pseudoscalar in V V final states become smaller. The range of variation of the pseudoscalar mass is chosen to be in the range 50 GeV < m A < 600 GeV for A → γγ and 2 m V < m A < 600 GeV for A → V V , V = W, Z. In this mass range the main decay channels of the pseudoscalar are A → bb, A → τ + τ − and A → γγ (also A → ZZ(W + W − ) are possible but at much lower rate as will be discussed later). Clearly, the A → γγ rate is at least two orders of magnitude below the tree-level decays so one could ask how important this decay really is. Taking into account the analysis performed for the 125 GeV SM Higgs where the ratio BR(h → γγ)/BR(h → τ + τ − ) ≈ 0.0362 (13) holds and still the two photons channel was the first one to be measured, we can expect that for a factor of about 100 the A → γγ channel will still be competitive due to sharp resolution in the di-photon invariant mass achievable by the CMS and ATLAS detectors. The above number will be used a rough guide to what can be expected from future analyses. In figure 1 we show BR(A → γγ)/BR(A → τ τ ) as a function of m A for tan β = 1 and tan β = 10 in Type I and Type II. Also shown is the SM line for the same ratio. In Type I the two lines for tan β = 1 and tan β = 10 overlap because A → γγ is mediated only through fermionic loops and the Aff ∝ 1/ tan β coupling factorizes out and cancels with the same factor coming from A → τ τ . In Type I the SM line is crossed for m A ≈ 250 GeV independently of tan β. However, since both the production cross section and the luminosity will be much higher during Run 2 it is to be expected that at very high luminosity all values of the pseudoscalar mass will be probed by the end of the LHC 14 TeV run in the low tan β region. The same behaviour is seen in Type II for low tan β. As tan β increases it will become increasingly harder to detect a pseudoscalar in the two-photons final state. It should be mentioned that the width of A → V V , whatever the final state is, is controlled by the top-quark loop. This loop is always proportional to 1/ tan 2 β and therefore the width is the same for all Yukawa versions of the model except for very large values of tan β in Type II and Y. The behaviour we see in the right panel of figure 1 is due to the width of A → τ τ that increases with tan β in Type II. A → γγ We now move to the detailed study of the production rates of a pseudoscalar decaying into two photons at the LHC at 8 TeV and 14 TeV. In figure 2 we plot the pseudoscalar production cross section multiplied by the branching ratio Br(A → γγ) for all four Yukawa types as a function of m A . The remaining masses are fixed at m h = 125 GeV and m H + = m H = 600 GeV. Regarding the angles we take the exact alignment limit sin(β − α) = 1 and we scan over tan β from 1 to 40, except for Type II, where the scans stops at 10. As previously discussed, the largest rates are for small tan β and therefore the upper bound is not relevant for the discussion. In the same plots we show the limits obtained by ATLAS [69] and CMS [70,71] after Run 1. The situation is similar for all models: only a small region of the parameter space where tan β is small is excluded with this search. If we move to the large tan β and/or to large pseudoscalar mass the number of events becomes negligible. The increase in cross section in Type II and Y due to bb production (the bbA coupling is proportional to tan β) is not enough to compensate the decrease in branching ratio. Therefore the number of events will be small even for large tan β except for the region of very small pseudoscalar mass. In figure 3 we again show a scan in the Type I and Type II models for 14 TeV and with the exact same conditions of the previous figure 2. The pseudoscalar production cross section increases from a factor of about 2 for m A = 50 GeV to a factor of about 4 for m A = 600 GeV. An extra factor coming from the bb initiated process will further increase the production cross section for Type II and Y especially for pseudoscalar masses above the tt threshold. Still this factor is quite small: taking for instance m A = 500 GeV the cross section increases by about 20 % for tan β = 40. Regarding the luminosity, which is expected to be about 300f b −1 at the end of run 2, when compared to the 8 TeV run where the total luminosity collected was about 30f b −1 , there is an approximate 10 fold increase. Overall a factor below 20 can be foreseen for the low mass region. Therefore, it is clear that it will be hard to probe this channel above about tan β = 10 (and probably less) by the end of Run 2. A → ZZ(W + W − ) The decay rate of a pseudoscalar to massive gauge bosons in the 2HDM is extremely small. The reason is clear: the decays of pseudoscalar bosons to massive gauge bosons can only occur at the loop level and massive gauge bosons are heavy. In fact, Γ(A → ZZ) is always smaller than 10 −5 GeV below the tt threshold and smaller than 10 −4 GeV above the same threshold, independently of the Yukawa version of the model. It is therefore clear that these are not competitive channels when compared to the ones with two fermions or even two photons final states. Since we are considering pseudoscalar production via fermion loops or bb initiated processes, a pseudoscalar decaying to two massive gauge bosons is expected to be observed well after being detected in some fermion final state (τ + τ − , bb or tt) or in γγ. Similarly to the two-photon final state, also here we can use exclusion bounds from searches for a scalar decaying to either two Z bosons or to W + W − performed by the ATLAS [72,73] and CMS [74] collaborations. In figure 4 we present the pseudoscalar production cross section multiplied by the branching ratio A → ZZ (top) and A → W + W − (bottom) for Type I (left) and Type II (right). We also present the best experimental upper exclusion bound for these channels [72,73]. It is clear that the experimental bounds are still about one order of magnitude away from the points with the largest rates in the scan. Moreover, also for these final states only points below the tt threshold and in the low tan β region have some chances of being probed at the next LHC run. As previously discussed for the two photon final state there is an overall factor of about 20 for the low mass region when considering both the increase in cross section and in luminosity. However, figure 4 clearly shows that even if the results are better by two orders of magnitude we will barely start to probe a few scenarios in the low tan β region. In Before we present our results for 2HDM with a vector like top, we first show in figure 6 the allowed range for y T , tan β, the mixing angles α and sin θ L for a fixed m T = 1 TeV. From the left plot of figure 6 one can see that |y T | ≤ 15 is allowed and that the mixing angle sin θ L should be less than about 0.2 for any value of 1 ≤ tan β ≤ 40. In the right panel we show tan β as a function of sin α where we can see a very similar behaviour to the 2HDM for the same variables, indicating that we are again very close to alignment except for a few points in the region where sin α > 0 where the coupling to b-quarks change sign, known as the wrong sign limit [75,76]. Extending the 2HDM to include a vector-like top quark, results in an enhancement both in the pseudoscalar production cross section and in its decay width into γγ, ZZ and W + W − . Hence, even with all constraints taken into account, in the case of large mixing between the top quark and new top T, the production rates of loop-induced processes can be several orders of magnitude above the 2HDM ones. In order to see the effects of the new top in the production rates of the gauge bosons we plot in figure 7 the rate [σ(gg → A) + σ(bbA)] Br(A → γγ, ZZ, W W ) at √ s = 8 TeV as a function of m A (and tan β), where we chose m T = 1 TeV, y T = 10 and sin(θ L ) = 0.12. One can see from this plot that the effect of the new top on the production and the decay of the CP-odd Higgs is a significant enhancement in both the production cross section by gluon fusion and in the decays A → γγ, A → ZZ and A → W + W − when compared to the the ones in the 2HDM (note that the Higgs production in bottom-quark annihilation is not altered by the new top). As it is clear from figure 7, the largest enhancement in the rates is again obtained for low tan β, as in the 2HDM. Like in the 2HDM, this behaviour can be understood by looking at the couplings given in equation (12), valid both for 2HDM+T Types I and II, which show that all couplings contain a term proportional to 1/ tan β. Another source of enhancement relative to the 2HDM is the choice of large Yukawa y T , which is still well below the perturbativity limit of 4π, as well as large mixing sin θ L . Also shown in the plots are the exclusion lines from the experiments at CERN. In the upper plots we can see the diphoton exclusion line from ATLAS. This plot is presented to show a very interesting result: in a Type I 2HDM+T a pseudoscalar with m A < 2m t ≈ 350 GeV and large mixing sin θ L ≈ 0.12−0.06 is excluded for any value of tan β. This means that the two-photon final state search is important to further constrain models with vector like quarks. One should note however that away from tt threshold the exclusion is valid only for rather small tan β. In the Type II 2HDM+T, one can see that for m A ≤ 350 GeV small tan β is excluded from diphoton events. We note that there is no exclusion for small sin θ L ≤ 0.01 both for Type I and Type II. In the middle and lower panels of figure 7 we present the rates for σ(pp → A) × Br(A → V V ) for V=Z or W together with the exclusion line from ATLAS. In both cases, the total rates are still about one order of magnitude smaller than the exclusion line. We note that there is a slight enhancement in the rate σ(pp → A) × Br(A → ZZ) with respect to the 2HDM case, due to the extra loop contribution. Still depending on the parameters chosen, the rate σ(pp → A) × Br(A → V V ) for V = Z, W may also be suppressed compared to the 2HDM. At a 100 TeV pp collider In the quest for new physics, there is a consensus among the community of particle physicists in favour of the construction a high energy machine with 100 TeV center of mass energy. One question that has been raised about a future 100 TeV pp collider is what is the luminosity needed to address the physics that is not within the reach of the LHC, even at high luminosity. In [77] several physics scenarios were analysed and a luminosity of about 10-20 ab −1 was shown to be a good compromise in extending the discovery reach for new phenomena relative to the high luminosity LHC. The production cross sections pp → A at a 100 TeV collider is increased relative to the 8 TeV LHC from a factor of about 20 for m A = 50 GeV to about 220 for m A = 600 GeV. This behaviour is shown in the left panel of figure 8 where the ratio of the cross sections for 100 TeV and for 8 TeV σ(pp → A) 100 T eV /σ(pp → A) 8 T eV is shown as a function of the pseudoscalar mass for Type I and two values of tan β, 1 and 10. The plots for all other Yukawa types show exactly the same behaviour as the one for Type I for tan β = 1 and for large tan β the contribution of the bb initiated process slightly changes this ratio with no meaningful changes in the conclusions. As previously discussed, so far analyses were only performed for 8 TeV with a total luminosity of about 30f b −1 . Therefore, in the low mass region the cross section is increased by a factor of 20 while the luminosity is incremented by about 1000. Overall, an improvement of at least four orders of magnitude is expected. The increase is more significant for higher masses but the branching ratios are smaller. Furthermore, the ratio of the cross sections is almost independent of tan β and of the Yukawa type. Considering figure 2 it is clear that most of the parameter space will be probed in the case of pp → A → γγ for the 2HDM, and for any Yukawa type. However, when examining figure 4 for the case of the decays into massive gauge bosons, we see that only a small portion of the parameter space will be probed, mainly for low tan β and for pseudoscalar mass below the tt threshold. The same is true for the W + W − final state. The left plots are for Type I and the right plots are for Type II. In the case of A → γγ part of the parameter space, for the low mass region is already excluded for the parameters shown. For the same set of parameters almost all values of m A and tan β are within the reach of a 100 TeV colliders. However, as y T decreases, the model will resemble the 2HDM and therefore as previously discussed only the low tan β region will have some chances to be probed. Mimicking a CP-violating 2HDM As discussed in the introduction, it is possible to search for signals of CP-violation at the LHC with a combination of three decays. In reference [5] it was shown that three simultaneous decays of Higgs such as h 2 → h 1 Z, h 1 → ZZ and h 2 → ZZ are a sign of CP-violation in any model. Also h 1 → ZZ, h 2 → ZZ and h 3 → ZZ, where h 1 , h 2 , h 3 are generic Higgs bosons, is a clear signal of CP-violation in the 2HDM except if there is a CP-odd state that decays to ZZ with a significant rate. That is, the combination of these three last decays can distinguish a CP-conserving 2HDM from a CP-violating one. Also the simultaneous processes pp → Zh 1 , Zh 2 , Zh 3 was shown to be a sign of CP-violation in [78,79]. Note however that there are many models with 3 CP-even scalars that can decay to ZZ, like the singlet extension or the 2HDM extended with a singlet. However, when the rate of h i → ZZ becomes too small we no longer know if this is just a very suppressed tree-level process or one that only appears at one-loop, as happens for A → ZZ. In this work we are considering a scenario where sin(β − α) = 1 (the alignment limit), the lightest scalar is the SM-like Higgs with a mass m h = 125 GeV and m H + = m H = 600 GeV. In this scenario the pseudoscalar will decay mainly to fermions. In the right panel of figure 8 we present the partial width Γ(A → ZZ) as a function of m A for the Type I model for values of tan β between 1 and 40. We see that below the tt threshold, where σ(pp → A) BR(A → ZZ) is largest, the width is always below 10 −5 GeV. In the alignment limit Γ(H → ZZ) is zero at tree-level. However, using the prescription in [80] it can be shown that when the tree-level coupling g HZZ is zero the one loop Γ(H → ZZ) is of the order 10 −5 to 10 −4 . This means that the BR(A → ZZ) and the BR(H → ZZ) will be of the same order of magnitude. Even if BR(H → ZZ) can be slightly larger, also the production cross section of a pseudoscalar is larger than that of a scalar in gluon fusion. Hence, if it is true that a 100 TeV collider will be able to probe very small branching ratios of scalars to ZZ it will most certainly be unable to tell the CP number of the new scalar particles. Furthermore, in the alignment limit the branching ratio for the decay A → hZ is exactly zero. However, when moving away from this limit BR(A → hZ) will again be small but non-zero. In conclusion, if new scalars are found at these very low rates, great precision is needed both from the experimental side and from the theoretical side with the calculation of higher order corrections. Any conclusion will be therefore extremely model dependent. Conclusions In this work we have analysed the detection of a pseudoscalar produced in gluon fusion plus bb initiated process and decaying to a pair of gauge bosons. We worked in the alignment limit, where sin(β − α) = 1, driven by the precision measurements of the Higgs couplings that have shown that one of the scalars has to resemble the SM Higgs. The experimental search for a generic scalar particle at the LHC Run 1 has been already performed by ATLAS and CMS. A small portion of parameter space of the 2HDM has already been probed in the search with two-photons in the final state. For the case of the final states with two massive gauge bosons we are still at least one order of magnitude away from highest possible rates in the model. We have also analysed the 2HDM model with an extra vector like quark, 2HDM+T. Due to the extra loop contribution from the top partner we can have an enhancement of both the production cross section and of the decay widths. In fact we have shown that the results for pp → A → γγ already exclude a substancial region of the parameter space below the tt threshold. As for the decays to massive vector bosons the enhancement is not enough to reach the exclusion limit obtained during Run 1. We have shown that in a future 100 TeV collider with a luminosity of 10-20 ab −1 almost all parameter space of four 2HDM Yukawa types will be probed in the case of the decay A → γγ. However, for the pseudoscalar decays into massive gauge bosons, possibly only a small portion of the parameter space will be at experimental reach. Also it is important to note that when all rates in the ZZ final state become very small it will be extremely hard to use them to search for CP-violation. In the case of the 2HDM+T as the rates are much higher it is expected that a larger region of the parameter space will be probed. By the end of Run 2, and if no new physics is found, the model will be closer to align, and the limits on heavy scalars will be stronger. In that case, the 100 TeV collider will start operation with severe constraints both on the Higgs couplings and on the masses/couplings of extra scalars. Note that in the limit where the new top decouples, the results are similar to the 2HDM ones. One of the main ideas that triggered this work was the search for CP-violation. In the CPviolating 2HDM, the decays of any scalar to ZZ are allowed. However, a CP-odd particle cannot decay to ZZ. We have shown that for the 2HDM the process A → ZZ is of the same order of magnitude as H → ZZ if the tree level HZZ coupling is zero. If such a final state is detected at Run 2 or at a future 100 TeV collider with a very low rate, it will be very hard to conclude anything about CP-violation. If a new scalar is detected in this final state with a higher rate it can then be a scalar form a 2HDM but also a pseudoscalar from an extended version of the 2HDM, the 2HDM+T. Hence, and fortunately, a lot of work is expected to pinpoint the underlying model. Finally we note that many other models with an extended Higgs sector will behave exactly like the 2HDM. In fact, if we extend the 2HDM with a singlet we end up with a model that in the alignment limit has a pseudoscalar that couples to the remaining SM particles exactly like the 2HDM.Therefore, our conclusions are valid for all extensions of the SM where alignment leads to a pseudoscalar with 2HDM-like couplings. Figure 1 : 1BR(A → γγ)/BR(A → τ τ ) as a function of m A for tan β = 1 and tan β = 10 in Type I and Type II. Figure 2 : 2[σ(gg → A)+σ(bbA)]BR(A → γγ) at √ s = 8 TeV as a function of m A with m H + = m H = 600 GeV . The values of tan β are color coded as indicated on the right of the plots. Also shown is the exclusion line from ATLAS (see text). Figure 3 : 3[σ(gg → A) + σ(bbA)]BR(A → γγ) at √ s = 14 TeV as a function of m A with m H + = m H = 600 GeV . The values of tan β are color coded as indicated on the right of the plots. figure 5 we present [σ(gg → A) + σ(bbA)]BR(A → ZZ) at √ s = 14 TeV as a function of m A with m H + = m H = 600 GeV . As discussed, both for Type I and Type II, there is an increase in the maximum values of the rates but still well below the experimental result line and an increase in Figure 4 : 4[σ(gg → A) + σ(bbA)]BR(A → ZZ) (top) and [σ(gg → A) + σ(bbA)]BR(A → W W ) (bottom) at √ s = 8 TeV as a function of m A with m H + = m H = 600 GeV . The values of tan β are color coded as indicated on the right of the plots.more than order of magnitude is needed to start probing the largest values of the rates. Figure 5 : 5[σ(gg → A) + σ(bbA)]BR(A → ZZ) at √ s = 14 TeV as a function of m A with m H + = m H = 600 GeV . The values of tan β are color coded as indicated on the right of the plots. Figure 6 : 6Results of the scan described in the text in the (sin θ L , y L ) plane (left) and in the (sin α, tan β) plane (right), after imposing the most relevant theoretical and experimental constraints as previously described in detail (m T = 1 TeV). Figure 7 : 7Scatter plots of [σ(gg → A) + σ(bbA)] Br(A → γγ, ZZ, W W ) at √ s = 8 TeV in the 2HDM+T as a function of m A where m H + = m H 0 = 600 GeV, sin(θ L )=0.12, m T = 1TeV and y T = 10,the values of tan β are color coded as indicated on the right of the plots Figure 8 : 8Left panel: ratio of cross section σ(pp → A) 100 T eV /σ(pp → A) 8 T eV as a function of m A for tan β = 1 and tan β = 10; Right panel: partial width Γ(A → ZZ) as a function of m A for Type I. The values of tan β are color coded as indicated on the right of the plot. Figure 9 : 9Scatter plots for [σ(gg → A) + σ(bbA)] Br(A → γγ, ZZ, W W ) for √ s = 100 TeV in the 2HDM+T as a function of m A where m H + = m H 0 = 600 GeV, sin(θ L )=0.12, m T = 1TeV and y T = 10,the values of tan β are color coded as indicated on the right of the plots. Finally in figure 9 we present scatter plots for [σ(gg → A) + σ(bbA)] Br(A → γγ, ZZ, W W ) for √ s = 100 TeV in the 2HDM+T as a function of m A where m H + = m H 0 = 600 GeV, sin(θ L )=0.12, m T = 1TeV and y T = 10. AcknowledgmentsThe authors are supported by the grant H2020-MSCA-RISE-2014 no. 645722 (NonMinimalHiggs). This work is also supported by the Moroccan Ministry of Higher Education and Scientific Research MESRSFC and CNRST: Projet PPR/2015/6. J.E would like to thank Shaaban Khalil for the hospitality extended to him during his stay in the Center for Fundamental Physics (CFP) at Zewail City of Science and Technology where part of this work has been done. He also acknowledges the receipt of the grant from the Abdus Salam International Center for Theoretical Physics, Trieste, Italy. R.S. is also supported in part by the National Science Centre, Poland, the HARMONIA project under contract UMO-2015/18/M/ST2/00518. . G Atlas, Aad, 1207.7214Phys.Lett. 7161ATLAS, G. Aad et al., Phys.Lett. B716, 1 (2012), 1207.7214. . S Cms, Chatrchyan, 1207.7235Phys. Lett. 71630CMS, S. Chatrchyan et al., Phys. Lett. B716, 30 (2012), 1207.7235. . D De Florian, 1610.07922LHC Higgs Cross Section Working GroupLHC Higgs Cross Section Working Group, D. de Florian et al., (2016), 1610.07922. . T D Lee, Phys. Rev. 81226T. D. Lee, Phys. Rev. D8, 1226 (1973). . D Fontes, J C Romão, R Santos, J A P Silva, 1506.06755Phys. Rev. 9255014D. Fontes, J. C. Romão, R. Santos, and J. a. P. Silva, Phys. Rev. D92, 055014 (2015), 1506.06755. . J A Aguilar-Saavedra, R Benbrik, S Heinemeyer, M Pérez-Victoria, 1306.0572Phys. Rev. 8894010J. A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer, and M. Pérez-Victoria, Phys. Rev. D88, 094010 (2013), 1306.0572. . G Cacciapaglia, 1502.00370JHEP. 0912G. Cacciapaglia et al., JHEP 09, 012 (2015), 1502.00370. . A Arhrib, 1607.08517Phys. Rev. 9795015A. Arhrib et al., Phys. Rev. D97, 095015 (2018), 1607.08517. . A Arhrib, R Benbrik, hep-ph/0610184A. Arhrib and R. Benbrik, (2006), hep-ph/0610184. . J L Diaz-Cruz, C G Honorato, J A Orduz-Ducuara, M A Perez, 1403.7541Phys. Rev. 9095019J. L. Diaz-Cruz, C. G. Honorato, J. A. Orduz-Ducuara, and M. A. Perez, Phys. Rev. D90, 095019 (2014), 1403.7541. . D Chowdhury, O Eberhardt, 1711.02095JHEP. 05161D. Chowdhury and O. Eberhardt, JHEP 05, 161 (2018), 1711.02095. . W Bernreuther, P Gonzalez, M Wiebusch, 1003.5585Eur. Phys. J. 6931W. Bernreuther, P. Gonzalez, and M. Wiebusch, Eur. Phys. J. C69, 31 (2010), 1003.5585. . J F Gunion, H E Haber, C Kao, Phys. Rev. 462907J. F. Gunion, H. E. Haber, and C. Kao, Phys. Rev. D46, 2907 (1992). . P M Ferreira, R Santos, A Barroso, hep-ph/0406231Erratum: Phys. Lett. 603219Phys. Lett.P. M. Ferreira, R. Santos, and A. Barroso, Phys. Lett. B603, 219 (2004), hep-ph/0406231, [Erratum: Phys. Lett.B629,114(2005)]. . A Barroso, P M Ferreira, R Santos, hep-ph/0507224Phys. Lett. 632684A. Barroso, P. M. Ferreira, and R. Santos, Phys. Lett. B632, 684 (2006), hep-ph/0507224. . A Barroso, P M Ferreira, I P Ivanov, R Santos, 1303.5098JHEP. 0645A. Barroso, P. M. Ferreira, I. P. Ivanov, and R. Santos, JHEP 06, 045 (2013), 1303.5098. . I P Ivanov, Phys. Rev. 79I. P. Ivanov, Phys. Rev. E79, 021116 (2009), 0802.2107. . S L Glashow, S Weinberg, Phys. Rev. 151958S. L. Glashow and S. Weinberg, Phys. Rev. D15, 1958 (1977). . E A Paschos, Phys. Rev. 151966E. A. Paschos, Phys. Rev. D15, 1966 (1977). . N G Deshpande, E Ma, Phys. Rev. 182574N. G. Deshpande and E. Ma, Phys. Rev. D18, 2574 (1978). . I P Ivanov, hep-ph/0609018Phys. Rev. 7535001Erratum: Phys. Rev.D76,039902(2007)I. P. Ivanov, Phys. Rev. D75, 035001 (2007), hep-ph/0609018, [Erratum: Phys. Rev.D76,039902(2007)]. . G C Branco, 1106.0034Phys. Rept. 5161G. C. Branco et al., Phys. Rept. 516, 1 (2012), 1106.0034. . S Kanemura, T Kubota, E Takasugi, hep-ph/9303263Phys. Lett. 313155S. Kanemura, T. Kubota, and E. Takasugi, Phys. Lett. B313, 155 (1993), hep-ph/9303263. . A G Akeroyd, A Arhrib, E.-M Naimi, hep-ph/0006035Phys. Lett. 490119A. G. Akeroyd, A. Arhrib, and E.-M. Naimi, Phys. Lett. B490, 119 (2000), hep-ph/0006035. . Delphi Cdf, Sld Electroweak, Aleph Heavy Flavour Groups, Electroweak Working, Group, Sld, L E W L3, Group, Tevatron Electroweak Working GroupTevatron Electroweak Working Group, CDF, DELPHI, SLD Electroweak and Heavy Flavour Groups, ALEPH, LEP Electroweak Working Group, SLD, OPAL, D0, L3, L. E. W. Group, (2010), 1012.2367. . M E Peskin, T Takeuchi, Phys. Rev. 46381M. E. Peskin and T. Takeuchi, Phys. Rev. D46, 381 (1992). . C D Froggatt, R G Moorhouse, I G Knowles, Phys. Rev. 452471C. D. Froggatt, R. G. Moorhouse, and I. G. Knowles, Phys. Rev. D45, 2471 (1992). . W Grimus, L Lavoura, O M Ogreid, P Osland, 0802.4353Nucl. Phys. 801W. Grimus, L. Lavoura, O. M. Ogreid, and P. Osland, Nucl. Phys. B801, 81 (2008), 0802.4353. . H E Haber, D O&apos;neil, 1011.6188Phys. Rev. 8355017H. E. Haber and D. O'Neil, Phys. Rev. D83, 055017 (2011), 1011.6188. . M Baak, 1107.0975Eur. Phys. J. 72M. Baak et al., Eur. Phys. J. C72, 2003 (2012), 1107.0975. Cms Atlas, CMS CollaborationsP Bruckman De Renstrom, CMS CollaborationsStatus of Beyond Standard Model Higgs searches at the LHC On behalf of the ATLAS and. St. Petersburg, Russia; Gatchina; Kurchatov Institute, Kurchatov InstituteProceedings, 3rd Large Hadron Collider Physics Conference (LHCPATLAS, CMS, P. Bruckman De Renstrom, Status of Beyond Standard Model Higgs searches at the LHC On behalf of the ATLAS and CMS Collaborations, in Proceedings, 3rd Large Hadron Collider Physics Conference (LHCP 2015): St. Petersburg, Russia, August 31-September 5, 2015, pp. 192-206, Gatchina, 2016, Kurchatov Institute, Kurchatov Institute. . A Arhrib, K Cheung, J S Lee, C.-T Lu, 1509.00978JHEP. 0593A. Arhrib, K. Cheung, J. S. Lee, and C.-T. Lu, JHEP 05, 093 (2016), 1509.00978. . A M Cms, Sirunyan, 1803.06553CMS, A. M. Sirunyan et al., (2018), 1803.06553. . A Wahab El Kaffas, P Osland, O M Ogreid, Phys. Rev. 76A. Wahab El Kaffas, P. Osland, and O. M. Ogreid, Phys. Rev. D76, 095001 (2007), 0706.2997. . M Aoki, S Kanemura, K Tsumura, K Yagyu, 0902.4665Phys. Rev. 8015017M. Aoki, S. Kanemura, K. Tsumura, and K. Yagyu, Phys. Rev. D80, 015017 (2009), 0902.4665. . S Su, B Thomas, 0903.0667Phys. Rev. 7995014S. Su and B. Thomas, Phys. Rev. D79, 095014 (2009), 0903.0667. . F Mahmoudi, O Stal, Phys. Rev. 81F. Mahmoudi and O. Stal, Phys. Rev. D81, 035016 (2010), 0907.1791. . O Deschamps, 0907.5135Phys. Rev. 8273012O. Deschamps et al., Phys. Rev. D82, 073012 (2010), 0907.5135. . A Denner, R J Guth, W Hollik, J H Kuhn, Z. Phys. 51695A. Denner, R. J. Guth, W. Hollik, and J. H. Kuhn, Z. Phys. C51, 695 (1991). . H E Haber, H E Logan, hep-ph/9909335Phys. Rev. 6215011H. E. Haber and H. E. Logan, Phys. Rev. D62, 015011 (2000), hep-ph/9909335. . A Freitas, Y.-C Huang, 1205.0299JHEP. 0850Erratum: JHEP10,044(2013)A. Freitas and Y.-C. Huang, JHEP 08, 050 (2012), 1205.0299, [Erratum: JHEP10,044(2013)]. . M Misiak, 1503.01789Phys. Rev. Lett. 114221801M. Misiak et al., Phys. Rev. Lett. 114, 221801 (2015), 1503.01789. . M Misiak, M Steinhauser, 1702.04571M. Misiak and M. Steinhauser, (2017), 1702.04571. . S Cms, Chatrchyan, 1205.5736JHEP. 07143CMS, S. Chatrchyan et al., JHEP 07, 143 (2012), 1205.5736. . G Atlas, Aad, 1204.2760JHEP. 0639ATLAS, G. Aad et al., JHEP 06, 039 (2012), 1204.2760. . Delphi Lep, Opal, Aleph, G L3, Abbiendi, 1301.6065Eur. Phys. J. 732463LEP, DELPHI, OPAL, ALEPH, L3, G. Abbiendi et al., Eur. Phys. J. C73, 2463 (2013), 1301.6065. . S P Martin, 0910.2732Phys. Rev. 8135004S. P. Martin, Phys. Rev. D81, 035004 (2010), 0910.2732. . K Kong, S C Park, T G Rizzo, 1004.4635JHEP. 0759K. Kong, S. C. Park, and T. G. Rizzo, JHEP 07, 059 (2010), 1004.4635. . N Arkani-Hamed, A G Cohen, E Katz, A E Nelson, hep- ph/0206021JHEP. 0734N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson, JHEP 07, 034 (2002), hep- ph/0206021. . D B Kaplan, H Georgi, S Dimopoulos, Phys. Lett. 136187D. B. Kaplan, H. Georgi, and S. Dimopoulos, Phys. Lett. 136B, 187 (1984). . A Angelescu, A Djouadi, G Moreau, 1510.07527Eur. Phys. J. 7699A. Angelescu, A. Djouadi, and G. Moreau, Eur. Phys. J. C76, 99 (2016), 1510.07527. . G Atlas, Aad, 1504.04605JHEP. 10150ATLAS, G. Aad et al., JHEP 10, 150 (2015), 1504.04605. . G Atlas, Aad, 1505.04306JHEP. 08105ATLAS, G. Aad et al., JHEP 08, 105 (2015), 1505.04306. . V Cms, Khachatryan, 1509.04177Phys. Rev. 9312003CMS, V. Khachatryan et al., Phys. Rev. D93, 012003 (2016), 1509.04177. . M Atlas, Aaboud, 1606.03903Phys. Rev. 9452009ATLAS, M. Aaboud et al., Phys. Rev. D94, 052009 (2016), 1606.03903. . M Atlas, Aaboud, 1705.10751JHEP. 0852ATLAS, M. Aaboud et al., JHEP 08, 052 (2017), 1705.10751. . A M Cms, Sirunyan, 1710.01539CMS, A. M. Sirunyan et al., (2017), 1710.01539. . M Atlas, Aaboud, 1707.03347JHEP. 10141ATLAS, M. Aaboud et al., JHEP 10, 141 (2017), 1707.03347. . G Atlas, Aad, 1202.3389Phys. Rev. 8612007ATLAS, G. Aad et al., Phys. Rev. D86, 012007 (2012), 1202.3389. . A Angelescu, A Djouadi, G Moreau, 1512.04921Phys. Lett. 756126A. Angelescu, A. Djouadi, and G. Moreau, Phys. Lett. B756, 126 (2016), 1512.04921. . J A Aguilar-Saavedra, D E López-Fogliani, C Muñoz, 1705.02526JHEP. 0695J. A. Aguilar-Saavedra, D. E. López-Fogliani, and C. Muñoz, JHEP 06, 095 (2017), 1705.02526. . R V Harlander, S Liebler, H Mantler, 1212.3249Comput. Phys. Commun. 1841605R. V. Harlander, S. Liebler, and H. Mantler, Comput. Phys. Commun. 184, 1605 (2013), 1212.3249. . A Djouadi, J Kalinowski, M Spira, hep- ph/9704448Comput.Phys.Commun. 108A. Djouadi, J. Kalinowski, and M. Spira, Comput.Phys.Commun. 108, 56 (1998), hep- ph/9704448, http://people.web.psi.ch/spira/hdecay/. . A Djouadi, M M Muhlleitner, M Spira, hep- ph/0609292Acta Phys. Polon. 38A. Djouadi, M. M. Muhlleitner, and M. Spira, Acta Phys. Polon. B38, 635 (2007), hep- ph/0609292. . R Harlander, M Mühlleitner, J Rathsman, M Spira, O Stål, 1312.5571R. Harlander, M. Mühlleitner, J. Rathsman, M. Spira, and O. Stål, (2013), 1312.5571. . T Hahn, hep-ph/0012260Comput. Phys. Commun. 140T. Hahn, Comput. Phys. Commun. 140, 418 (2001), hep-ph/0012260. . T Hahn, M Perez-Victoria, hep-ph/9807565Comput. Phys. Commun. 118T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999), hep-ph/9807565. . G J Van Oldenborgh, J A M Vermaseren, Z. Phys. 46425G. J. van Oldenborgh and J. A. M. Vermaseren, Z. Phys. C46, 425 (1990). . G Atlas, Aad, 1407.6583Phys. Rev. Lett. 113171801ATLAS, G. Aad et al., Phys. Rev. Lett. 113, 171801 (2014), 1407.6583. . V Cms, Khachatryan, 1506.02301Phys. Lett. 750494CMS, V. Khachatryan et al., Phys. Lett. B750, 494 (2015), 1506.02301. . G Atlas, Aad, 1509.00389JHEP. 0132ATLAS, G. Aad et al., JHEP 01, 032 (2016), 1509.00389. . G Atlas, Aad, 1507.05930Eur. Phys. J. 76ATLAS, G. Aad et al., Eur. Phys. J. C76, 45 (2016), 1507.05930. . V Cms, Khachatryan, 1504.00936JHEP. 10144CMS, V. Khachatryan et al., JHEP 10, 144 (2015), 1504.00936. . P M Ferreira, J F Gunion, H E Haber, R Santos, 1403.4736Phys. Rev. 89115003P. M. Ferreira, J. F. Gunion, H. E. Haber, and R. Santos, Phys. Rev. D89, 115003 (2014), 1403.4736. . P M Ferreira, R Guedes, M O P Sampaio, R Santos, 1409.6723JHEP. 1267P. M. Ferreira, R. Guedes, M. O. P. Sampaio, and R. Santos, JHEP 12, 067 (2014), 1409.6723. . I Hinchliffe, A Kotwal, M L Mangano, C Quigg, L.-T Wang, 1504.06108Int. J. Mod. Phys. 301544002I. Hinchliffe, A. Kotwal, M. L. Mangano, C. Quigg, and L.-T. Wang, Int. J. Mod. Phys. A30, 1544002 (2015), 1504.06108. . A G Akeroyd, A Arhrib, hep-ph/0107040Phys. Rev. 6495018A. G. Akeroyd and A. Arhrib, Phys. Rev. D64, 095018 (2001), hep-ph/0107040. . A Arhrib, D K Ghosh, O C W Kong, hep-ph/0112039Phys. Lett. 537217A. Arhrib, D. K. Ghosh, and O. C. W. Kong, Phys. Lett. B537, 217 (2002), hep-ph/0112039. . M Krause, R Lorenz, M Muhlleitner, R Santos, H Ziesche, 1605.04853JHEP. 09143M. Krause, R. Lorenz, M. Muhlleitner, R. Santos, and H. Ziesche, JHEP 09, 143 (2016), 1605.04853.
[]
[ "Catastrophic ice lake collapse in Aram Chaos, Mars", "Catastrophic ice lake collapse in Aram Chaos, Mars" ]
[ "Manuel Roda ", "Maarten G Kleinhans ", "Tanja E Zegers ", "Jelmer H P Oosthoek " ]
[]
[]
Hesperian chaotic terrains have been recognized as the source of outflow channels formed by catastrophic outflows. Four main scenarios have been proposed for the formation of chaotic terrains that involve different amounts of water and single or multiple outflow events. Here, we test these scenarios with morphological and structural analyses of imagery and elevation data for Aram Chaos in conjunction with numerical modeling of the morphological evolution of the catastrophic carving of the outflow valley. The morphological and geological analyses of Aram Chaos suggest large-scale collapse and subsidence (1500 m) of the entire area, which is consistent with a massive expulsion of liquid water from the subsurface in one single event. The combined observations suggest a complex process starting with the outflow of water from two small channels, followed by continuous groundwater sapping and headward erosion and ending with a catastrophic lake rim collapse and carving of the Aram Valley, which is synchronous with the 2.5 Ga stage of the Ares Vallis formation. The water volume and formative time scale required to carve the Aram channels indicate that a single, rapid (maximum tens of days) and catastrophic (flood volume of 9.3·10 4 km 3 ) event carved the outflow channel. We conclude that a sub-ice lake collapse model can best explain the features of the Aram Chaos Valley system as well as the time scale required for its formation.
10.1016/j.icarus.2014.03.023
[ "https://arxiv.org/pdf/1404.3858v1.pdf" ]
118,443,620
1404.3858
752e158184dc8f6df1594e9830424bda6fedca86
Catastrophic ice lake collapse in Aram Chaos, Mars Manuel Roda Maarten G Kleinhans Tanja E Zegers Jelmer H P Oosthoek Catastrophic ice lake collapse in Aram Chaos, Mars MarssurfaceGeological ProcessesIces Hesperian chaotic terrains have been recognized as the source of outflow channels formed by catastrophic outflows. Four main scenarios have been proposed for the formation of chaotic terrains that involve different amounts of water and single or multiple outflow events. Here, we test these scenarios with morphological and structural analyses of imagery and elevation data for Aram Chaos in conjunction with numerical modeling of the morphological evolution of the catastrophic carving of the outflow valley. The morphological and geological analyses of Aram Chaos suggest large-scale collapse and subsidence (1500 m) of the entire area, which is consistent with a massive expulsion of liquid water from the subsurface in one single event. The combined observations suggest a complex process starting with the outflow of water from two small channels, followed by continuous groundwater sapping and headward erosion and ending with a catastrophic lake rim collapse and carving of the Aram Valley, which is synchronous with the 2.5 Ga stage of the Ares Vallis formation. The water volume and formative time scale required to carve the Aram channels indicate that a single, rapid (maximum tens of days) and catastrophic (flood volume of 9.3·10 4 km 3 ) event carved the outflow channel. We conclude that a sub-ice lake collapse model can best explain the features of the Aram Chaos Valley system as well as the time scale required for its formation. Introduction Martian chaotic terrains are deeply collapsed areas (> 1 km deep) that stretch for up to hundreds of kilometers and show a bumpy floor characterized by an irregular pattern of fractures and tilted blocks of different sizes (from meter to kilometer scale). Chaotic terrains predominantly occur along the dichotomy boundary between the southern highlands and northern lowlands (Sharp, 1973;Chapman and Tanaka, 2002;Rodriguez et al., 2005;Glotch and Christensen, 2005;Meresse et al., 2008;Warner et al., 2011). Outflow channels represent the largest systems carved by liquid water on Mars. They are thousands of kilometers long, more than a kilometer deep (Baker, 2001)and show attributes such as grooves, terraces, teardrop islands, streamlined terraces and high width-to-depth ratios that are consistent with the erosive origin of the channels (e.g., Nelson and Greeley, 1999;Baker, 2001;Coleman, 2005;Pacifici et al., 2009;Warner et al., 2010b). In many cases, chaotic terrains represent the source area of Hesperian (approximately 3.7-3.3 Ga) outflow channels (Nelson and Greeley, 1999;Tanaka et al., 2003) and several authors (e.g., Carr, 1979Carr, , 1996Baker, 2001) argue that those chaotic terrains were formed by a rapid discharge of water from the subsurface, resulting in collapsed and fractured areas and massive flows carving the large outflow channels. However, the actual evolutionary process leading to chaotic terrain formation and collection and the discharge of catastrophic volumes of water (≥10 5 km 3 ) has remained controversial. Several evolutionary processes have been proposed that can be grouped in four different scenarios (Fig. 1). In the first hypothesis (Fig. 1a), fracturing in the bedrock led the water expulsion to the surface. The water was generated by partial melting of the cryosphere after magmatic intrusions that increased the subsurface temperature (e.g., Sharp, 1973;Chapman and Tanaka, 2002;Ogawa et al., 2003;Rodriguez et al., 2005;Leask et al., 2006;Meresse et al., 2008). In a second group of mechanisms (Fig. 1b), the release of water from the cryosphere was the result of an increase in pressure of a global pressurized sub-cryospheric aquifer (e.g., Carr, 1979;Clifford, 1993;Harrison and Grimm, 2009). The discharge from the pore space relied on the flow of water through a permeable subsurface layer, where water that was discharged was replaced by recharge. In the majority of martian aquifer models, water is generally assumed to have recharged from a great distance (≥2000 km, Clifford, 1993;Harrison and Grimm, 2008). Numerical flow models Harrison and Grimm, 2008) indicate that the large total volume of liquid water (10 5 -10 6 km 3 ) and high discharge rates (10 6 -10 9 m 3 /s) required to form the morphology of the outflow channels were not achievable by a single discharge from a porous medium. Because of the flow volume discharge from aquifers, a scenario was proposed wherein a large number of small flooding events were Meresse et al., 2008); b) Aquifer model (redrawn after Harrison and Grimm, 2009); c) Gas-hydrate model (redrawn after Kargel et al., 2007); d) sub-ice lake model (redrawn after Zegers et al., 2010). followed by a sudden release of previously ponded water (≥600, Harrison and Grimm, 2008). Another hypothesis (Fig. 1c) suggests that dewatering of fluids from gas and/or salt hydrate buried deposits (Max and Clifford, 2001; Montgomery and Gillespie, 2005) or the hydrologic processes triggered by clathrates (Kargel et al., 2007) could be responsible for the water outflow. Finally, Zegers et al. (2010) propose that chaotic terrains developed by the catastrophic collapse of sediments induced as a consequence of the melting of buried ice sheets (Fig. 1d). Thermal modeling results show that even under very low crustal heat flux conditions, ice sheets will melt if buried under thick sed-iments (up to 2 km) because of the difference in thermal conductivity between the basin fill and surrounding crust. When the buried sub-ice lake reaches a critical thickness, the overburden collapses and subsides, resulting in a massive expulsion of water to the surface. To distinguish between those four scenarios and test the validity of the evolution models for chaotic terrains and their outflow channels, the amount and timing of the water release, the amount of subsidence, and the fracture distributions are fundamental variables. To constrain those variables for a single chaotic terrain and its outflow channel, we analyzed the morphological and geological features characterizing Aram Chaos and its valley. Furthermore we estimated the flow volume and formative time scale required to carve the Aram channel. Finally we discuss the four scenarios proposed for chaotic terrain formation in view of these results, propose the most likely scenario for the evolution of Aram Chaos and Aram channel and discuss the significance of our findings for other chaotic terrains. Aram Chaos morphology and fracturing In this section, we present the geological and morphological observations related to Aram Chaos in conjunction with a structural analysis performed on the fractures to determine the sequence of depositional and fracturing events during its formation. Geology and pre-faulting morphology Aram Chaos is situated in a circular basin with a diameter of 280 km centered at 2.5 • N and 338.5 • E (Fig. 2) suggesting that it is developed in a large impact crater (Schultz et al., 1982;Glotch and Christensen, 2005). It is connected to Ares Vallis by Aram Valley, a 15 km wide and 2.5 km deep outflow channel (Glotch and Christensen, 2005;Masse et al., 2008;Oosthoek et al., 2007). The formation time of the original impact crater is not constrained; however, given the size of the crater, the impact likely occurred in the Noachian (≥ 3.7 Ga, Zegers et al., 2010).It has been suggested that the outflow event of Aram Valley was synchronous with the final erosive event of Ares Vallis (approximately 2.5 Ga, Warner et al., 2009). Glotch and Christensen (2005) divide the Aram chaotic terrain into three possibly lateral main units. The Fractured Plains unit is the largest and almost completely surrounds the crater rim. It consists of up to tens of kilometer-sized slumped blocks forming a curvilinear fracture pattern. The Knobby Terrain unit is the second unit and occurs in the central part of Aram Chaos and at certain locations surrounding the crater rim. It consists of km-scale irregular blocks (knobs). The High Thermal Inertia Chaotic Terrain unit occurs in the central part of Aram Chaos and underlies outcrops of layered material. Those two (Masse et al., 2008) or three (Lichtenberg et al., 2010) layered subunits (500-m thick) are composed by mono-and polyhydrated minerals such as sulfates and ferric-oxides. Observations in 3D, particularly from HRSC stereo data (12 m/pixel), provide evidence that the current Aram Chaos terrain was originally a large impact basin that was almost entirely filled by sediments before fracturing and subsidence occurred and resulted in the present-day morphology. Figure 3 shows a part of the rim where evidence for this sequence of events is clearly visible from a view based on HRSC data. In the foreground of figure 3a, a location is visible where the original boundary between the crater rim highlands and crater fill is not disrupted by fractures. The transition from crater rim to crater fill in this location is similar to many filled craters in the highlands of Mars (Warner et al., 2010a), which include a crater rim terrain showing irregular erosional features with an onlap transition to smooth crater fill material. There is a gradual slope from the crater rim into the crater fill of almost 500 m (Fig. 3b, white arrow), which suggests that the crater was likely filled to a level of 500 m below the rim in this part. In the background of figure 3b, the transition from unfractured highland to chaotic terrain is visible, and the transition is marked by an abrupt escarpment of 1500 m. Some impact craters along the northern rim ( Fig. 4a and b) indicate that there was a continuous surface between the surrounding highland terrain and Aram crater fill material prior to fracturing. We conclude that the pre-faulting geometry of the basin was similar to many other large impact craters on Mars: the crater was almost entirely filled, but the outline of the crater is still visible (Fig. 4). In the case of Aram Chaos, the crater was likely filled to a level 500 m below the rim in the south part and almost completely filled in the northern part, as indicated by the different characters of the transition between highland and crater fill (Figs 4b and 5b). This suggests that the difference in elevation between the northern and southern part was almost 500 m before the collapse, supposing a homogeneous rim degradation. Aram Chaos is in a region thought to have experienced massive erosion and deposition of a sedimentary overburden during the Late Noachian-early Hesperian (Hynek and Phillips, 2001). Based on crater counting, Warner et al. (2010b) suggest that the intense degradation and infill occurred during a short 200 Myr interval in the Late Noachian, from 3.8 Ga to 3.6 Ga. Although sedimentation rates were likely higher in the Noachian and spatially variable (Grotzinger and Milliken, 2012), even very low sedimentation rates of 0.01 mm/yr would have been sufficient to fill a deep (2 km) depression in a 200 Myr time span. The original depth of the Aram Chaos crater was estimated using an empirical relation between crater diameter and pristine depth (van Kan Parker et al., 2010): H(m) = 350 · D 0.44 (km)(1) For Aram Chaos, with an estimated diameter (D) of 280 km, this resulted in an original depth (H) of 4.2 km. This suggests that prior to fracturing and subsidence, the original Aram Chaos crater of approximately 4 km depth was filled by sediments to at least 500 m below the rim (south part) or until the rim (north part). Fracturing and depositional events Aram Chaos shows a fractured and collapsed morphology that resulted in a geometry of faulted blocks that is characteristic of chaotic terrains. The rim faults have experienced most of the displacement related to subsidence and are up to 1500 m on the western rim. Fault displacements between individual fault blocks in the basin are minor compared to the rim fault. The fracture pattern was analyzed in terms of fracture density and fracture orientation (Fig. 5a) to infer the fracture mechanism. The fracture spacing is highly variable and ranges from tens of kilometers down to locally small values below the resolution limit where no single block is detectable. No preferred orientation is obvious in the fracture patterns of the fractured There is a gradual slope from the crater rim into the crater fill. Lower image (b) shows the elevation contour lines. In the background the elevation drop between unfractured and fractured terrain is at least 1000 m between. Image based on HRSC mosaic (orbits H0401 0001 and H1000 0000) draped over HRSC-derived Digital Elevation Model. units ( Fig. 5b). Only a weak alignment of fractures with the rim orientation exists in the outer zone. The fracture pattern includes interfering patches of radial fractures, (Fig. 5a, arrows), which partly originated from loci of very high fracture density. The fracture density increases in the area around the outflow channel (north and south), and a more abrupt escarpment characterizes the southern rim of the Aram crater with respect to the northern part, which presents a more gradual slope of the rim (Fig. 6). On the basis of fracture pattern and morphology, Oosthoek et al. (2007) differentiated between two different fractured units: Highland and Chaos Formation and Lower Aram Chaos Formation (Fig. 7). The latter is further divided into three lateral subunits: fractured, broken and smooth. The fractured subunit is cross-cut by relatively small-scale fractures (1-2 km scale) compared to the fractures of the Highland and Chaos Formation. Some fractures have raised rims and others show small thrusts at the base of the rim (Fig. 8). The broken subunit is highly fractured with approximately 1-km sized irregular mesas. This subunit always occurs at the boundary of the fractured Highland and Chaos Formation. The fractured and broken In the center of the image the continuity between the crater filling and the highlands is clearly visible suggesting that the crater was almost entirely filled before fracturing. Image based on HRSC mosaic (orbits H0967 0000) draped over HRSC-derived Digital Elevation Model. b) Detail of the Aram Chaos northern rim: along the boundary two small impact craters illustrate the ancient (crater 1) and present-day (crater 2) continuity between Aram Chaos sedimentary fill and the highland terrains. The rim is clearly identifiable on the floor of crater 1 but it is deleted in crater 2. CTX image P03 002272 1829 XI 02N020W. units are contained by the approximately 150 km diameter inner ring (Figs. 5 and 7) of the Aram Chaos crater. The smooth subunit is non-fractured and may in fact be a relatively thin unit covering the fractured subunits (see Fig. 7). These fracture-geological unit relationships may either be interpreted as two distinct fracture events or as one continuous fracture event, during which the fractured Aram unit was deposited. The first event is represented by the fracturing and collapse of the Highland and Chaos Formation. The fractured and broken subunit of the Lower Aram Chaos Formation was deposited during (or just after) the collapse of the underlying Highland and Chaos Formation and has subsequently been fractured and broken up. The smooth subunit of the Lower Aram Chaos Formation represents the first depositional event after the collapse covering the fractured subunits. Intermediate and Upper Aram Chaos Formations are not fractured and are deposited on top of the fractured units, closing the depositional sequence. Geological, morphological and structural analyses on Aram Chaos suggest a large-scale collapse and subsidence of the entire area (1000-1500 m). Outflow channels morphology A large valley (Aram Valley) cuts through the entire eastern rim of Aram Chaos. On the northeastern rim, two minor channels are visible. In this section, we describe the morphological and morphometric features of the outflow channels to determine the chronological sequence of the outflow events. Minor channels Along the northeast rim of Aram Chaos, two small channels are visible (Fig. 9). They have a similar and constant slope (0.019 for the northern and 0.020 for the southern, measured from HRSC dataset) and a general U-shaped profile (Fig. 10). The northern channel is 5 km wide with a width/depth ratio of 50; the southern channel is 4 km wide with a width/depth ratio of 40. Their inlets are located at approximately -1800 m elevation (Fig. 10b). The outlets are truncated by the erosional features of Ares Vallis ( Fig. 9) at an elevation between -3500 and -4000 m. These features represent the remnants of the late erosive events in Ares Vallis (2.8-2.5 Ga, Warner et al., 2009). This indicates that the two outflow channels were active before or during the late carving of Ares Vallis. Furthermore, the high channel slopes that are quite similar to the slopes of the pristine Aram Chaos rim, suggest that they could be representatives of the early outflow stages recorded in the Aram Chaos. They may also have resulted from groundwater sapping toward Ares Vallis that occurred along Aram crater walls saturated with fluid. Whether these small channels formed as water sloshed over the rim after collapse or as sapping channels in the Aram crater wall saturated with fluid water, water must have been present within Aram Chaos. Assuming an empty crater and watersaturated rims, water flow from the rims toward the crater would be expected because of a higher subsurface gradient along its direction, with a consequent opposite carving direction of minor channels. A rainfall origin for those channels (unlikely during the Late Hesperian, i.e., after the collapse) is inconsistent with the absence of other small channels around the Aram Chaos rim. Aram Valley The Aram Valley is a deep (2.5 km) V-shaped valley ( Fig. 11a and b) with a low width/depth ratio (6-8, measured from HRSC dataset) that represents an outflow channel from Aram Chaos to Ares Vallis (Pacifici et al., 2009). The inlet of the valley along the Aram Chaos boundary is characterized by a high number of relatively small and deep channels and radial and cross-cutting grooved terrains overlying the fractured and knobby units (Figs 11b and 12). The distal part of the inlet stands at a higher elevation with respect to part of the Aram Valley floor (Fig. 11c). This complex structure can be interpreted as an erosive remnant generated by flow converging into the channel and resulting in cross-cutting converging ridges and channels extending into the lake just upstream of the outflow 4 N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N point. The flow converges in the middle of the erosive remnant to create a deeply incised scour hole (Figs 11c and 12). At this location, the valley is narrower than further downstream. The different elevations of the radial grooves suggest progressive erosion of the valley inlet. Initially, the valley floor was at a higher elevation than the chaos floor. With the progressive erosion of the inlet and valley floor, the difference in elevation decreases, whereas certain older ridges and channels are abandoned as the channel incises further until the valley reaches the present-day elevation. These observations are surprisingly consistent with analogue experiments of catastrophic, single-event crater outflows (Marra et al., 2014), including the formation of a converging erosive remnant composed by multiple crosscutting ridges and channels. The valley slope, obtained by removing from the profile two landslides that occur along the northern rim of the Aram Valley, is quite constant with a gentle gradient (0.004, measured from HRSC dataset) toward Ares Vallis (Fig. 11c). The present-day valley slope is lower than the initial slope prior to incision suggested by the profile along the north and south rims (0.047 for the rim N and 0.028 for rim S, measured from HRSC dataset) of the outflow valley (Fig. 11c). In the valley cross-sections 1 and 2 ( Fig. 11b), abandoned flow terraces are visible on the northeastern part of the Aram channel, and their depth below the surrounding plateau (from 230 to 520 m) is a reasonable estimate of the channel water depth. The largest portion of the Aram Chaos terrain stands at a higher elevation compared to the upper level of the Aram Valley floor (-4000 m, Figs 11 and 13) with the exception of certain basins that can reach a depth below the lowest part of the valley (≤ -4300 m, Fig. 13), particularly at the southern edge of Aram Chaos. The differences in elevation between the chaotic area and the Aram Valley floor, alley slope and flow convergence zone support the hypothesis of outflow and erosion processes from Aram Chaos to Ares Vallis (Fig. 11c). To understand why the Aram Valley is located where it is found today, we analyzed the elevation pattern of the Aram Chaos rim. At Aram valley, the crater rim before the incision of the valley was likely at approximately -1500 m (Fig. 11b, cross-section 3). The elevation map of Aram Chaos (Fig. 13) reveals the occurrence of lower elevation sections of the crater rim, especially in the northern and north-western side, which could represent a favorable outflow path way. However, no outflow remnants were found on the rim, with the exclusion of two small channels along the north-eastern side (see section 3.1 above). The occurrence of a pre-existing channel carved from the Ares Vallis during the early stages of its formation (3.6 Ga, Warner et al., 2009) and reused for the outflow from the Aram Chaos seem unlikely; the higher elevation of the western rim of Aram Chaos with respect to the surrounding terrain and present-day direction of the Aram Valley are incompatible with this interpretation (Fig. 13). As an alternative to surface run-off, outbursts of groundwater from a pressurized aquifer before the collapse of the sediment cap may explain the breaching of the eastern rim. Pre-existing buried fractures concentrated along the eastern rim may have been used by the water as upwelling conduits. The pressurized water would generate overpressure in the sediment cap and cause fracturing and outbursts along the eastern rim. However, the fractures generated by impact should be developed symmetrically along the entire crater rim, especially for a circular-shape crater such as Aram crater. Several outbursts would then be expected along the entire crater boundary. However, the eastern rim at the current location of the Ares Figure 6: a) Location of three N-S cross-sections across Aram Chaos on DEM map. b) Cross-sections across Aram Chaos: a more abrupt escapement characterizes the southern rim of the Aram crater with respect to the northern part, which presents a more gradual slope of the rim (black arrows). HRSC datasets: orbit H0401 0001 and H0926 0000. 1 N−S 2 NNW−SSE 3 NW−SE North Rim South Rim 19° W 19° W 20° W 20° W 21° W 21° W 22° W 22° W 23° W 23° W 6° N 6° N 5° N 5° N 4° N 4° N 3° N 3° N 2° N 2° N 1° N 1° N 0° 0° 0 50 100 m K 5 2 Legend -1246 m -4950 m 1 2 3 a 0 Vallis is characterized by a shorter distance between the inner wall of the pristine crater and topographic low lying terrain (Fig. 13). The W-E profile along Aram Chaos (Fig. 14) indicates that the Western Valley (Pacifici et al., 2009), that surrounds the western part of Aram Chaos and the tributary of Hydapsis Chaos has an elevation comparable to that shown by Ares Vallis before the Aram Valley outflow event, such as -3500 m. This elevation, however, was likely obtained by the last flood sourced from Hydapsis Chaos and dated at 2.6 Ga (Warner et al., 2010b). The first outflow event carving the Hydapsis tributary was coeval with the first event of Ares Vallis (Warner et al., 2010b), and the erosive surface was most likely located at -2600 m (Fig. 14a). The steepest subsurface gradient was likely localized between Aram Chaos and Ares Vallis (0.018-0.023 compared to 0.012-0.016 of the western part), which most likely made the groundwater flow towards Ares Vallis with the outlet located between -3000 and -3500 m (Fig. 14). It is therefore likely that the outflow started as groundwater seepage and evolved into an outflow valley by groundwatersapping and headcutting from Ares Vallis through the crater rim (Howard and McLane, 1988;Brocard et al., 2011). After the breaching of the crater rim, the process would have developed much faster to evolve as a massive outflow, and the channel would have deepened rapidly to complete the carving of Aram Valley. The observations and analogue modeling support the hypothesis of a single and continuous outflow carving the valley, although subsequent small outflow events may have occurred because of groundwater upwelling. As observed by Warner et al. (2009), the Aram Valley is graded to the final erosive surface of Ares Vallis with no evidence of intersecting or truncating flood grooves or a knickpoint occurrence at the confluence. Furthermore, the topography of Ares Vallis (Fig. 15) shows an abrupt increase in slope at the confluence with Aram Valley as well as an increase of width. These observations clearly support the interpretation of Warner et al. (2009), who suggested that the water outflow from the Aram Valley was synchronous with the final erosive event of Ares Vallis (approximately 2.5 Ga). It is now possible to infer a relative chronology of the Aram Chaos outflows, which started with two relatively small channels along the north-eastern part of the rim and continued in the larger Aram Valley as massive outflow once the groundwater headcutting was completed. This process was active for a relatively short period and was coeval with the late stages of Ares Vallis formation. 5°00'N 4°00'N 3°00'N 2°00'N 1°00'N 5°00'N 4°00'N 3°00'N 2°00'N 1°00'N 23°00'W 22°00'W 21°00'W 20°00'W 19°00'W 23°00'W 22°00'W 21°00'W 20°00'W 19°00'W Lower Outflow time scale and water volume estimates In this section, we describe the method for deriving flow volume and formative time scales from the morphology and morphometric characteristics of outflow channels. A reconstructed flow rate will be combined with the reconstructed volume of water to derive an estimate of event duration. The rate of sediment removal calculated on the basis of the reconstructed flow rate will be combined with the volume of the valley to derive a second, independent estimate of event duration, which we will then compare to the first estimate to arrive at a best estimate of the event duration. Sources of uncertainty are discussed in combination with a likely course of events in the excavation of the channel. Time scale and water volume determination The principle of the calculation of formative time scale is that a flow requires a certain time to remove or deposit a known volume of sediment (Kleinhans, 2005). The volume of sediment eroded from the valleys is estimated from cross-sections and the length of the valleys. The estimates of cross-section surface area, averaged over several of HRSC profiles, have been multiplied by the length of the valleys (Figs 10 and 11). This yields an average volume of 460 km 3 for the Aram Valley and 20 and 12 km 3 for channels 1 and 2, respectively ( Table 1). The flow flux from the Aram Chaos crater was likely (nearly) clear water that resulted from ponding after the collapse so that the sediment transport capacity of the flow was entirely available for erosion of the channels. This clear water scour was basically the inverse of the deposition of crater lake deltas from a sediment-laden flow that enters a crater lake (Kleinhans, 2005;Kraal et al., 2008). The sudden transition from sediment transport to zero transport in the delta case and vice versa in the 5 km (Kleinhans, 2005). The sediment transport rate is calculated from the flow flux through the channel. Flow flux is calculated by the following steps. First, the width, flow depth and gradient of the channel are estimated, and the width, cross-sectional valley shape and valley gradients are estimated from HRSC topography (Figs 10 and 11). Maximum flow depth is estimated from terraces heights, and hydraulic roughness is then calculated, f = 8 (2.2( h D 50 ) −0.055 S −0.275 ) 2(2) from which the flow velocity u = 8ghS f(3) and flow discharge Q w = uhW(4) follow. h is the channel depth, D 50 is the median grain size, S is the channel slope, g is the martian gravity and W the channel width. The water depth inferred from terraces (h) is within the expected range based on the resulting width-depth ratio of the flow (20 for narrow terrestrial gravel bed rivers) and results in reasonable Froude numbers and sediment mobilities (expressed as non-dimensional Shields number, Table 1) (Kleinhans, 2005). To estimate the total water volume V w that must have come out of the Aram Chaos crater to form the observed channels, the formative time scale T s for channel excavation can be multiplied by the flow flux Q w so that V w = T s Q w (Kleinhans, 2005). This yields a water volume estimate of 9.3·10 4 km 3 for the Aram Valley and 2.3·10 3 and 2.1·10 3 km 3 for the two small channels. Sediment flux is calculated by two methods: one assuming a bed load-dominated transport (with mostly rolling and saltating particles and limited energy) and one assuming a suspended load-dominated event (non-cohesive granular material). The sediment mobility, which depends on flow and sediment properties, is used to determine which of the two transport modes is valid. A recent granular debris flow in the middle of the present channel (Fig. 16) indicates that failure of the side walls would immediately collapse the material into non-cohesive granular sediment. This debris flow formed as a dry granular flow because only this rheology allows for a debris flow that goes slope-upward on the opposite valley wall. This is evident in that the material is neither cohesive nor strongly lithified in addition to the extensive analysis in Grotzinger and Milliken (2012),which allows the application of transport capacity predictors. Furthermore, these predictors are reasonably accurate in cases of weakly lithified material (see discussion in Kleinhans, 2005). Classical sediment transport capacity predictors are employed corrected for gravity, and sediment properties such as estimated in Kleinhans (2005) are used. The ratio of suspended and bed load transport is much larger than unity, so that the system is suspension dominated and the appropriate predictor is used (Kleinhans, 2005). The volumetric transport rate [m 3 /s] is Q s = 1 1 − n Φ s RgD 3/2 50 W (5) where Φ s = 0.4 f θ 2.5(6) is the non-dimensional suspended load transport predictor, θ is the shear stress, n is the porosity, R is the relative submerged density. The maximum time scale for channel formation (T s = V/Q s ) then becomes of the order of tens of days (Table 1 -Formative time scale) with an order of magnitude in the uncertainty range (Kleinhans, 2005). Analysis of uncertainties For their morphometry and position, the two small channels are treated as representative scenarios for the early outflow stages. For the Aram Valley, instead, various scenarios for the event are presented in Table 1 and cover the likely range of conditions, formative time scales and volumes involved. Five different scenarios are presented and referred to as follows: (i) a best guess scenario, where the best guesses were used for channel geometry and gradient and for the outflow volume from the crater and eroded sediment volume from the channel; (ii) a slower scenario, where a larger estimate for water depth and smaller estimate of gradient were combined to produce the (iii) a faster (larger gradient) and smaller (lower water depth) scenario representative of the initial stages of aerial water flow; (iv-vii) the best guess scenario combined with the larger and smaller water volume estimate or the larger and smaller sediment volume, also bracketing the likely time scales. The comparison indicates a maximum formative time scale lower than 30 days and a water volume greater than 1·10 4 km 3 . Furthermore, a sensitivity analysis was performed for the Aram Valley for the parameters affected by significant uncertainties and was performed for water depth in particular, but also for valley shape and valley gradient (Fig. 17). The resulting time scales for the water flow and sediment removal are plotted against nine water depths for 3 valley gradients and 3 sediment volumes that reflect three different valley shapes. Because it is the water flow that removes the sediment, the two time scales must be equal (T w = T s ) and the measured amount of water must be exactly sufficient to erode the measured amount of sediment based on predicted flow and sediment transport rates. The result is remarkably consistent (Fig. 17): the formative time scales for release of water and removal of sediment are equal at tens of days for a water depth range of approximately 250-400 m, which takes into account all the conservative estimates of uncertainties. This water depth is in accurate agreement with observed terrace heights in the valley (Fig. 11b), which are indicative of a water depth that was found in the experiments of Marra et al. (2014). The order of magnitude of the uncertainty in time scale for sediment removal T s results from the uncertainty of water surface gradient (0.004), which was estimated from the final bed surface gradient (ignor-ing later debris flow deposits) and maximum likely slope from the crater rim surface just outside the channel (0.02) after the breaching. Neither is likely to be representative for the entire period, so we selected a gradient magnitude in between for the best guess scenario (0.0045). The estimated uncertainty of an order of magnitude also includes the contributions of the chosen friction relation and the total load sediment transport predictor. Comparison with evolutionary scenarios From the analysis performed on Aram Chaos and Aram Valley it is clear that four main elements are important for the validation of the proposed evolutionary scenarios of Aram Chaos and chaotic terrains in general: fracturing, the amount of subsidence of the chaotic terrains, total water volume and water flux (i.e., time scale) of the outflow channels (Table 2). For Aram Chaos, at least 1.5 km of collapse is estimated by the structural and morphological analysis. The Aram Valley outflow channel was likely carved with a large amount of water in less than 30 days and in a maximum 1 or 2 outflow events. The first group of proposed mechanisms (Fig. 1) accounts for the interaction of volcanic activity with the cryosphere (e.g., Ogawa et al., 2003;Meresse et al., 2008). Although these models could explain the occurrence of hydrothermal minerals in terms of magma/ice interactions (Ogawa et al., 2003), the general lack of evidence of magmatic activity within the chaotic terrains raises questions about the applicability of the model. However, the relative proximity of the Tharsis region with the Xanthe Terra chaotic region could provide evidence of the occurrence of active volcanism in that chaotic terrain. In those models, the subsidence is partially related to the volume loss melt and discharge to achieve a subsidence of 1500 m, as observed for the Aram Chaos. Using a much more reliable depthporosity relation (Clifford et al., 2010) and taking into account a very high surface porosity of 30%, the maximum amount of subsidence achievable for 20 km of thick cryosphere is less than 1 km. Although the water volume and water flux generated by the volcanic-cryosphere model is not estimated, the amount of water released from the cryosphere, which depends primarily on porosity and permeability, is low (10 0 -10 2 km 3 , 10 3 km 3 only for extreme values of permeability, Harrison and Grimm, 2008) in comparison with the volume of water required to carve the outflow channels (9.3·10 4 km 3 for Aram Valley). Only with a large number of outflow events is it possible to archive a comparably large water volume. This implies a continuous recharging and freezing of cryosphere and repeated intrusions. Furthermore, considering the low mean discharge suggested by the numerical modeling of the martian aquifer (10 3 -10 4 m 3 /s, Andrews-Hanna and Phillips, 2007), the duration of the outflow events (10 0 -10 2 years, Andrews-Hanna and Phillips, 2007) is not compatible with the catastrophic event (tens of days) suggested by the hydrologic analysis performed for the Aram Valley. For the second mechanism (Fig. 1), the aquifer model (e.g., Carr, 1979;Clifford, 1993;Harrison and Grimm, 2009)and numerical and analogue modeling suggests that a very high number of outflow events are required to achieve the amount of water sufficient to carve the outflow channels. The catastrophic characteristics of the outflow mechanism and its formative time scale suggested by the Aram Valley analysis are in conflict with this interpretation. The water flood from a subsurface aquifer initiates when superlithostatic pore pressures within a confined aquifer lead to the propagation of hydrofractures through the confining cryosphere to the surface (Andrews- . This could explain the fractures occurring in the chaotic terrains. However, the mechanism of hydrostatic fracturing has not been directly implemented in the hydrologic models. Furthermore, the collapse and subsidence over at least 1500 m that was simultaneous with fracturing affecting the chaotic terrains region is not compatible with hydrofracturing because only minor subsidence and collapse is acceptable if the flow properties of the aquifer are to be maintained to achieve multiple outflow events. In this model, the high subsidence observed in the chaotic terrains area is treated as a pre-existing topographic low along which the hydraulic head can reach the lithostatic pressure and lead to fracturing. Morphological analysis of Aram Chaos, instead, suggests that the subsidence, which is mainly taken up by the rim fault (1500 m displacement), is coeval with overall fracturing of the chaotic terrain. With respect to the aquifer model, the gas (or salt) hydrated mechanism (Max and Clifford, 2001;Montgomery and Gillespie, 2005) implies the increase in pore-pressure (and the consequent cryosphere hydrofracturing) because of the gas and/or water released by salt hydrate deposits or clathrates. However, for the same reasons as the required amount of water, time scale of floods and amount of subsidence discussed for the aquifer model, the gas (or salt) hydrated model can not explain the sequence of events in Aram Chaos. Furthermore, the large number of floods and relative time scale required by the permeability of the aquifer contrasts with the rapid degassing (or dewatering) proposed for this mechanism. Although this model could explain the occurrence of mono and poly-hydrated minerals, a model to explain gas/salt hydrated occurrence on Mars is not yet available. In the buried sub-ice lake scenario proposed by Zegers et al. (2010), melting of a buried ice layer followed by mechanical destabilization of the sediment cap and massive water outflow is able to explain the strong subsidence and fracturing of the chaotic terrains as well as the large amount of water required to carve the outflow channels in a catastrophic way. Analogue modeling of the process (Manker and Johnson, 1982) appears to confirm the resulting chaotic morphology, and analogue modeling of the outflow process (Marra et al., 2014) confirms all diagnostic features in the resulting valley morphology. Furthermore, numerical modeling suggests that a heat flow of 25 mW/m 2 together with the thermal insulation of overburden is sufficient to induce relative slow melting of the ice, without the need of a magmatic event Schumacher and Zegers, 2011). Zegers et al. (2010) estimated the volume of liquid water that was produced in a single chaotization event by taking the subsidence as a measure of the water escaped. Given a crater diameter of 280 km and a subsidence of 1-2 km and assuming a simple cylindrical shape of the crater, a volume of 0.6-1.2·10 5 km 3 results. The uncertainty in the calculation of the water volume results from an uncertainty in the magnitude of subsidence (between 1 and 2 km, with local variations). As a best estimate of the water volume released assuming a single event, we therefore use 9.2·10 4 km 3 (Table 1 -measured flood volume). The total predicted volume of flood obtained for the Aram Valley and two small channels by the hydrological analysis (9.3·10 4 km 3 , Table 1 -predicted volume of flood) is in the same order of magnitude of the independent estimate of the volume from crater geology that assume a single event. In conclusion we propose that a buried sub-ice lake scenario can best explain the observations and the water volume and outflow time scale estimates made for the Aram Chaos-Valley system. Scenario for the evolution of the Aram Chaos-Valley system In this section, we use the morphological and chronological information together with quantitative estimates of the outflows (time scale and water volume) presented in the previous sections to obtain a description of the events involved during the Aram Chaos and Aram Valley outflow channel formation. The morphological and hydrological analysis of the Aram Chaos -Valley system suggests fracturing, subsidence and collapse of terrain within the crater and a catastrophic water outflow along the eastern rim. Applying the scenario presented by Zegers et al. (2010), Aram Chaos would be the result of destabilization of a buried sub-ice lake with a consequent collapse of the sediment cap and massive expulsion of water toward the surface, followed by temporary ponding. Two small channels were carved by surface run-off, and at the same time, groundwater flow started from the lake to Ares Vallis, generating seepage erosion. After the breaching of the Aram Chaos rim, a massive outflow of water carved Aram Valley in a very short time scale. The formative time scale and the volume of water involved in carving the channel were calculated as if the fluxes were constant during the event. It is likely that the fluxes were not constant but rather a result of the likely process of channel formation, which are described below and can be approached with constant fluxes for our purpose. In the following sections, we discuss in detail the evolutionary events of the Aram Chaos-Valley system. Table 1: Calculation of formative time scale for channel excavation and the necessary water volume involved (to be compared to the geologically estimated water volume) for different scenarios. Slower = deeper and small gradient; Faster = shallower and large gradient. See the italic values for the variables changed within each scenario. The friction factor is obtained using the equation 13 of Kleinhans (2005 Aram Chaos collapse Geological, morphological and structural analyses of Aram Chaos suggest a large scale collapse and subsidence of the entire area of 1000-1500 m, which is compatible with a massive expulsion of liquid water from the subsurface in one (or possibly two) event(s). It is not possible to determine the timing of fracturing and subsequent collapse; however, the processes took place between the crater formation (≥ 3.7 Ga) and late erosive events in Ares Vallis (2.5 Ga). The order of magnitude of the total time span is therefore hundreds of million years within which sedimentation, melting of buried ice, fracturing and collapse occurred. The amount of subsidence in Aram Chaos is directly related to the thickness of the buried sub-ice lake before mechanical destabilization. The fracturing was a result of fast collapse of the sediment cap that occurred during the release of water toward the surface (Fig. 13a). The formative time scale and water volume determination indicates the occurrence of very rapid (tens of days) and catastrophic (volume of flood of 9.3·10 4 km 3 ) events capable of carving the Aram Valley and two other small channels. This means that a large amount of water would have to have been available at the source of the outflows. In a context such as that observed in Aram Chaos, a temporary lake generated by a massive expulsion of liquid water from the subsurface of the chaos and confined within the crater rims could be a valid solution to explain the morphology of the area (subsidence and fracturing), the time scale and amount of water required to carve Aram Valley. In the present-day atmospheric pressure conditions of Mars, the ponding of water for a long time is not feasible. Furthermore, the thermo-chronology of martian meteorites suggests that for most of the past 4 Ga, ambient near-surface temperatures on Mars were unlikely to have been much higher than the present cold state (Shuster and Weiss, 2005). For these reasons, a rapid outflow of water followed directly by rapid valley formation is more consistent with the Hesperian condition. Slow and multi-stage outflow over millions of years require higher atmospheric pressure (and possible higher surface temperatures) for which no direct evidence exists. Under present-day conditions, lakes on Mars can remain largely liquid for thousands of years if an ice cap hinders the evaporation of water (Newsom et al., 1996;Kreslavsky and Head, 2002), which would make the storage of large amounts of water possible for a relative short period and allow for the initial process of seepage erosion through the crater rim. In fact, given a typical groundwater flow velocity, seepage from the lake to Ares Vallis would have taken thousands of years. Early outflow event: two small channels After the collapse and rise of water, the first flow over the rim between Aram Chaos and Ares Vallis would have occurred locally and briefly at the topographically lowest point along the rim, which was localized in the NE sector at an elevation between -1800 and -2000 m (Figs 10 and 14a). Figure 13 shows that the -2000 m surface, excluding successive erosion, was likely mostly continuous around the crater, whereas the -1500 surface was localized in certain isolated areas. For this reason, an elevation between -1800 and -2000 m is a good approximation of the initial outflow channel inlet. Most initial erosion would have taken place at the downstream end where the flow descended steeply into Ares Vallis. The backward steps might have had certain antidunes in the initial stages, such as in the experiments of Kraal et al. (2008) and Marra et al. (2014) where the flow initially debouched onto a dry floor. The channel would have been narrower where it cut deeper into the plateau. On the upstream rim there would not have been much erosion because the flow essentially went uphill initially. With little or no sediment supply at the upstream, the channels slope quickly decreased from the initial values (0.028 -0.047, Figs 11c and green line of Fig. 14a) to the present-day slope (0.020, Fig. 10b). The lake water level decreased down to -2000 m and the outflow channels were abandoned (Fig. 14b). Groundwater sapping as a precursor of the Aram Valley Temporary ponding of water would likely result in groundwater seepage and sapping along the eastern rim, where the subsurface gradient was steeper ( Fig. 14a -blue lines). Therefore, together with the first outflow event, groundwater flow would have been present through the crater rim because of the relatively high porosity of the ejecta blanket. Where the groundwater emerged on the surface, certain erosional mechanisms became active because of the seepage (Howard and McLane, 1988). Three main processes of sediment mobilization, related to the seepage erosion, are described by Schorghofer et al. (2004) and Marra et al. (2014): channelization, slumping, and fluidization. Seepage erosion starts with a quick incision of an initial channel at the groundwater emergence point. This process is quite efficient because the critical mobilization slope of sediment is significantly smaller than the maximum angle of stability (Lobkovsky et al., 2004;Marra et al., 2014). The channel grows initially downwards and then upwards (headward). The channel profile is typically concave with an increase of slope toward the headcut. As a consequence, destabilization of the sediments surrounding the headcut area occurs and the slumping process starts. Continued headward erosion on the extending channel results in a migration of the groundwater flux toward the emergence point and activation of fluidification process. Therefore, the headward erosion rate increases rapidly Table 1). The formative time scale derived from the calculated flow discharge and measured volume of water (T w , blue lines) in Aram Chaos is drawn for the same calculations. The range of depths for which T w =T s matches the range of observed terrace depths (dashed square) and gives the likely time scale of tens of days (gray area). through the crater rim. Finally, the breaching of rim occurs and the overland flow starts (Fig. 14b). Late outflow event: Aram Valley final carving The initial overland flow conditions were characterized by a higher slope gradient (perhaps 0.02, as a remnant of the groundwater sapping, Fig 14b) and lower water depth (Table 1 -faster scenario). During the outflow event, the channel would have deepened, whereas the lake water level incrementally dropped as it emptied through the channel. The width of that channel was not necessarily equal to the width of the present valley, because the valley would have widened through slab failure and mass wasting while the channel undercut the valley sidewalls. The valley slope rapidly decreased from 0.02 to the present-day value of 0.004 (Fig. 14c, Table 1 -slower scenario). In a waning flow, the sediment transport capacity would have reduced considerably so that the channel floor excavation no longer kept pace with the lake water level lowering. This would have ended the outflow event (Fig. 14c) and may have been abrupt or more gradual depending on the hypsometry of the crater basin immediately after the chaotic terrain collapse event. The formative time scale calculation (Table 1) and its sensitivity to water depth (Fig. 17) suggest that a rapid surface outflow event occurred and was active for tens of days. The entire outflow process from Aram Chaos started shortly before and ended simultaneously with the late outflow event of Ares Vallis, although each single carving event lasted a maximum of tens of days. To have sufficient water volume available to carve the channels so rapidly, a lake at least 1500 m deep would have had to occur within Aram Chaos, which is consistent with the independent estimate for lake depth and subsidence. Discussion In this section, we discuss certain open questions that remain regarding the buried sub-ice lake scenario and its possible applicability to the other chaotic terrains. We also present a possible connection between the proposed scenario and formation of the Interior Layered Deposits, which are frequently associated with chaotic terrains. Open questions concerning the buried sub-ice lake scenario Although the sub-ice lake model is able to explain most characteristic features of chaotic terrains and outflow channels, certain questions remain open, such as those concerning the equatorial ice occurrence and long-term stability of the sub-ice lake. Although thick accumulations of ice on Mars occurred in high latitude areas, the location of Aram Chaos raises certain questions as to the process through which a several kilometers-thick ice deposit was initially formed in the Aram crater. During high-obliquity periods, snow/ice thickness that had accumulated in the equatorial regions could have reached centimeters over several years (Jakosky and Carr, 1985;Mischna et al., 2003) rand resulted in a thick ice cover over millions of years. Many regions in the ancient southern highlands of Mars exhibited extensive fluvial activity, especially during the Noachian period. The sinks of the overland flow were craters and basins where alluvial fans, debris flow fans, deltas and sublacustrine fans are today observable on Mars (Grotzinger and Milliken, 2012). Lakes generated from such craters and basins may have frozen and resulted in thick ice layers. A thick ice deposit could also be accumulated by the freezing of an impact crater lake generated as a result of melting of the cryosphere induced by the increase in temperature related to the impact energy (Newsom et al., 1996;Barnhart et al., 2010). A groundwater recharge from Ares Vallis during the earliest outflow event (S1-S4, Warner et al., 2009) and successive water freezing should also be considered, although the low subsurface gradient (Fig. 14c) makes this hypothesis unlikely. The remaining accessible observations in Aram Chaos do not allow for the choice of a most likely scenario for the formation of a thick ice fill of the initial crater, but the different options indicate that this is a feasible starting-point. The sub-ice lake model implies the stability of a growing subsurface lake for several millions of years to achieve a liquid water layer thick enough to produce an approximately 10 5 km 3 outflow volume in one or possibly two events by using the geothermal heat flow as the sole heat source. This is only possible if the liquid water was contained in the subsurface and could not escape through the pore space and cracks in the surrounding solid rocks and ice. Under geologic timescales and at temperatures close to the melting point, water ice is extremely ductile and behaves like a viscous fluid, analogous to salt units in the subsurface on Earth (Hudec and Jackson, 2007). The ice lid therefore acts as an impermeable lid to the liquid layer, closing any permeability that may form by cracks in the overburden. The liquid water to the sides and bottom of the subsurface lake would be in direct contact with rocky material at a temperature above the melting curve .Under particular conditions of porosity, permeability and hydraulic head, it would be possible for the sub-ice lake to drain and discharge through a subsurface aquifer away from Aram crater. However, such a scenario would produce a collapsed basin without a surface run-off feature, whereas in Aram Chaos, there is evidence of surface water run-off in the form of the Aram Valley. Buried sub-ice lake scenario and other chaotic terrains Other chaotic terrains on Mars share several characteristics with Aram Chaos, such as the fracturing and major subsidence of the fractured terrain, but differ in other respects. This has implications for the applicability of the buried sub-ice lake scenario for other chaotic terrains. The buried sub-ice lake scenario needs the occurrence of a confined basin as initial condition, where an ice layer can accumulate and become buried by sediments afterwards. Some of the chaotic terrains on Mars have a clear circular shape suggesting an original impact crater (Aram Chaos, Marusky Chaos, Chaos SE of Hydaspis and SE of Pyrrhae, Orson Welles Chaos). The western part of Hydaspis Chaos shows circular fractures and round-shape boundaries that can be interpreted as the result of a collapse of two coalescent impact craters. Other chaotic terrains show irregular shapes with a random distribution of fractures and no circular crater shape is detectable. However, the occurrence of pre-existing basins can be deduced from the substantial subsidence characterizing all of the chaotic terrains. The irregular shape of the original basin may have been the result of coalescence of more than two impact craters, as can be observed in several regions of the Highlands. The occurrence of old fault bound basins cannot be excluded; however, it seems unlikely because the original shape would still be visible in the shape of chaotic terrains. If certain basins occur, snow/ice thickness can be accumulated in the equatorial regions during high-obliquity periods or by partial melting of the cryosphere, producing liquid water to fill the basins before the subsequent lake freezing. argue for extensive groundwater upwelling during the Late Noachian that was concentrated in the region around Xanthe Terra and Arabia Terra, where the water table exceeds the topography. The water can be achieved in preexisting basins or craters during the Late Noachian. This can explain the concentration of the chaotic terrains around Xanthe Terra. The climate change occurring in the Early Hesperian would induce the lakes to freeze. Zegers et al. (2010) show that planetary heat loss greater than 25 mW/m 2 is sufficient to melt 2 km thick ice layer if a sediment layer insulates ice from low surface temperatures. This is consistent with estimates of different authors and suggests the Table 2: Comparison between formative models proposed for the chaotic terrains and Aram Chaos -Valley system: the water volume, number of floods, total time scale for carving, amount of subsidence and fractures occurrence suggested by the models are compared with those estimated for Aram Chaos. Model Water volume (km 3 ) Events Carving time scale (days) subsidence (m) Fracturing Volcanic 10 0 -10 3 ≥200 4000 >500 yes Aquifer 10 0 -10 3 ≥200 4000 max 500 pre-existing Salt (gas) hydrated 10 0 -10 3 few 4000 salt thickness yes/pre-existing Sub-ice lake 9.2·10 4 1-2 20-40 1500 yes Aram Chaos 9.3·10 4 1-2 max 30 1500 yes occurrence of planetary heat loss greater than 25 mW/m 2 during the Noachian-Hesperian period (Hauck and Phillips, 2002;Williams and Nimmo, 2004;Ruiz et al., 2011), and it may justify the large-scale melting that occurred in the chaotic terrains region. However, a temporary increase in the surface and subsurface temperatures can be related to impacts (Mangold et al., 2012) or magmatic intrusions (Meresse et al., 2008). For Aram Chaos, the amount of water generated by the melting of buried ice layer was sufficient to carve the Aram Valley outflow channel. Although the calculation of water volume balance for other chaotic terrains and relative outflow channels is difficult because they were highly disturbed and modified by subsequent flows, larger outflow channels (e.g., Ares Vallis, Tiu Valles, Ravi Valles) seem to require larger amount of water than that achieved in the surrounding chaotic terrains. In these cases, perhaps a combination of the ice layer melting and other water supply processes occurred. However, a unique mechanism valid for all chaotic terrains has not been defined yet. Some mechanisms show particular features not shared with the other chaotic terrains. The question is if a unique mechanism is required to explain all of them or if each chaotic terrain can have one particular formative mechanism. Further studies and analysis are required to solve the controversy. Buried sub-ice lake scenario and Interior Layered Deposits The Interior Layered Deposits (ILDs) within Aram Chaos a have planar to gentle mound shape (Glotch and Christensen, 2005;Masse et al., 2008;Lichtenberg et al., 2010). They were deposed after the chaotization event and are characterized, from bottom to top, by mono-hydrated sulfates, poly-hydrated sulfates and hematite layers. Rossi et al. (2008) suggest an Amazonian age for the ILDs of Aram Chaos. For their mineralogy, layering and shape, they are interpreted as the result of deposition in lacustrine or playa ambient (Glotch and Christensen, 2005;Arvidson et al., 2006) or as a result of multiple groundwater upwelling events (Masse et al., 2008;Lichtenberg et al., 2010). Rossi et al. (2008) argue for a possible spring origin. Although the ILDs are clearly deposited after the chaotization process, the formation process of the chaotic terrain may be linked to the deposition of ILDs. In fact, the buried sub-ice scenario accounts for a number of pre-conditions that are favorable to the formation of sulfates: the formation of collapsed basins, where all ILDs occur, and the presence of water to allow for the hydration of originally basaltic rocks. The buried sub-ice lake scenario could also indirectly explain the origin of ILDs. Fractures occurring after the collapse may represent a preferential way for groundwater outflow, resulting in spring deposits as proposed by Rossi et al. (2008). Perhaps multiple mechanisms acted at the same time or in sequence to form the chaotic terrains of Mars, but the evidence presented in this paper for the case of Aram Chaos demonstrates that chaotic terrain can form from catastrophic collapse of ice sheets. Conclusions The geological and hydrological analyses performed on the Aram Chaos-Valley system indicate that the flow volume required to carve the Aram Valley and two small channels (9.3·10 4 km 3 ) is similar to the volume of water that could have been produced in an event of Aram Chaos by the melting of a buried ice lake (9.2·10 4 km 3 , Zegers et al., 2010). The formative time evaluation confirms that a single, rapid (tens of days) and catastrophic event was sufficient to carve the channel rather than many small groundwater events active for a relatively long time. The resulting Aram Chaos morphology implies large amount of subsidence (1500 m) and one (or two) intense fracturing event(s). Such a scenario is consistent with a model of a buried subice lake system . The thermal insulation and relatively low heat flow may have been sufficient to induce ice melting. When the system became unstable, a massive water outflow occurred with the collapse of sediment and carving of the outflow channel. The sub-ice lake scenario explains many features characteristic of chaotic terrains, although more than one mechanism may have been involved in the carving of larger outflow channels. Acknowledgments Netherlands Organization for Scientific Research (NWO) and Netherlands Space Office (NSO grant) are gratefully acknowledged. We thank Wouter Marra for sharing insights in groundwater seepage erosion and catastrophic formation of valleys in experiments, Santiago de Beguera for stimulating discussion about debris-flow rheology and Rob Govers for useful comments. We gratefully acknowledge the anonymous reviewer and Timothy Glotch for their constructive criticism of the text. The authors contributed in the following proportions to conception and design, data collection, modeling, analysis and conclusions, and manuscript preparation (%): MR(30,60,10,30,80); MGK(30,0,90,30,10); TEZ(30,0,0,30,10); JHPO(10,40,0,10,0). Figure 1 : 1Four different scenarios for the origin of chaotic terrains. a) Magma-cryosphere model (redrawn after Fig. 8 8Fig.8 Fig. 4 Fig. 9 49Fig.4 FigFigure 2 2: a) View of Aram Chaos showing the typical fractured and tilted chaos blocks, and associated outflow channel (Aram Valley). The black squares are referred to the locations of detailed figures (HRSC mosaic). b) Geological map of Aram Chaos afterGlotch and Christensen (2005). This material is reproduced with permission of John Wiley & Sons, Inc. Figure 3 3: a) Perspective view, looking towards the north-west of the southwestern part of the rim of Aram Chaos basin. The white arrow indicates where the original boundary between the crater rim highlands and the crater fill is not disrupted by fractures. Figure 4 4: a) Perspective view, looking towards the north-west of the northern part of the rim of Aram Chaos basin. Figure 5 5: a) Map of Aram chaos showing the asymmetric fracture density (lineament length per km 2 , dark red fracture density is 0.387 km −1 ). Semi-radial fractures patterns are indicated with arrows. b) Fracture orientation in the Aram Chaos region. In red the fractures formed in the outer ring, predominantly in the Highland Unit, in blue the fractures formed in the center of the chaotic terrain, predominantly in the fractured Aram unit. The length-weighted rose diagrams in black represent the orientations of fractures in the squares indicated in Aram Chaos. The rose diagram in red (blue) represents all fractures measured in the red (blue) area. Figure 7 : 7Structural map of Aram Chaos afterOosthoek et al. (2007). Figure 8 : 8Small thrusts at the base of the faults rim indicated by white arrows. CTX image P19 008311 1836 XI 03N021W, 6 m/pixel. channels case allows us to calculate the time scale [T s ] of formation directly from the volume of displaced sediment [V = 460 km 3 ] and the sediment transport rate [Q s ] (corrected for porosity) as T s = V/Q s Figure 9 : 9Visible images of north-eastern rim of Aram Chaos. a) Two small outflow channels are traceable across the rim until Ares Vallis. They are likely cut by the late stages of Ares Vallis outflow. HRSC dataset, orbits H0934 0000, H0945 0000 and H0401 0001 b) Detail of the lower part of northern channels: the channel floor is truncated by the ejecta blankets of two recent impact crates but it is still traceable until its termination in the lower part of Ares Vallis (upper right corner). Visible image, credit Google Mars. smallest likely flow velocity and bracket the likely time scales; Figure 10 : 10a) Location of the cross section of small outflow channels on DEM map. b) W-E profiles along the channels. They show a rather constant and similar slope and their outlets are truncated by the late erosional features of Ares Vallis(Warner et al., 2009). c) N-S cross-sections across the outflow channels. The abandoned terrace identified in the section 5 referred to the southern channels (2) is used as an estimate of channel depth. HRSC dataset, orbits H934 0000 and H945 0000. by melting of the ice in the pore space of the cryosphere and Figure 11 :Figure 12 : 1112a) Location of the cross-sections of Aram Valley on DEM map. b) N-S cross sections across Aram Valley. In the sections 1 and 2, obtained near the valley outlet, some abandoned flow terraces are visible and their depth below the surrounding plateau (from 230 to 520 m) is interpreted as an estimate of channel depth. c) W-E cross-sections along the Aram Valley and surrounding rims. The present-day valley slope (section 6) is lower than the pristine slope suggested by the profile along the north and south rims of the outflow valley (sections 7 and 8). HRSC datasets from orbit H0401 0001 and H0923 0000. the water release. This implies that by using an extremely high constant porosity of 20%, at least 7.5 km of cryosphere should Erosive remnant of the flow convergence zone located at the inlet of Aram Valley. It is characterized by a high number of relative small and deep channels and radial grooved terrains overlying the fractured and knobby unit. This structure suggests a converging flow into the valley. HRSC dataset, orbit H945 0000. Figure 13 : 13Elevation map of Aram Chaos -Valley system: five topographic surfaces are highlighted, from -1500 to -4000m. The most part of the Aram chaos terrain stands at higher elevation compare to the upper level of Aram Valley floor (-4000 m). The highest part of the Aram crater rim is located around the Aram Valley. This area is also characterized by a steep gradient between the inner and outer wall of the pristine crater (distance between inner crater rim and greenish surrounding surfaces). HRSC datasets from orbits H0401 0001 and H0923 0000. Figure 14 :Figure 15 : 1415W-E profile along the Aram Chaos (seeFig. 13for the location) and interpretation of the events involved in the buried sub-ice lake scenario. Dashed lines represent the slopes of outflow channels (green and red) and groundwater (blue). Black dotted lines indicate the base levels of Ares Vallis before (-2600 m) and after (-3000 and -3500 m) the first erosive events (S1-S4,Warner et al., 2009). SU and SI are erosive events of Western Valley(Warner et al., 2010b). With dashed black lines the pristine crater and rim profiles are indicated. With transparency the progressive erosion of Aram Valley is shown. a) After the collapse water ponding, first outflows and groundwater flow along the eastern rim occur; b) The water level drops to -2000 m, the small outflows end and water sapping starts the erosion of Aram Valley; c) Once the breaching of the eastern rim occurs, massive water outflow completes the carving of Aram Valley up to the present-day morphology. N-S profile along Ares Vallis: when the main valley floor crosses the Aram Valley the slope increases as well as the valley width (transition from green to cyan lines). This suggests that the water outflow from the Aram Valley was synchronous with final erosive event of Ares Vallis. HRSC dataset, orbits H923 0000 and H934 0000. Figure 16 : 16Details of the eastern debris flow located in the Aram Valley (white arrow). This debris flow goes slope-upward on the opposite valley wall (south wall) indicating a dry granular rheology flow. CTX image P05 003129 1842 XI 04N018W. Figure 17 : 17Sensitivity of formative time scale and water volume to water depth in the channel. The formative time scale for sediment (T s , red lines) plotted versus the most uncertain parameter: water depth. Five lines are plotted for different combinations of gradient (slope) and larger or smaller sediment volume ( Hydrological modeling of outflow channels and chaos regions on Mars. J C Andrews-Hanna, R J Phillips, Journal of Geophysical Research. 1128001Andrews-Hanna, J.C., Phillips, R.J., 2007. Hydrological modeling of outflow channels and chaos regions on Mars. Journal of Geophysical Research 112, E08001. Meridiani Planum and the global hydrology of Mars. J C Andrews-Hanna, R J Phillips, M T Zuber, Nature. 446Andrews-Hanna, J.C., Phillips, R.J., Zuber, M.T., 2007. Meridiani Planum and the global hydrology of Mars. Nature 446, 163-166. Nature and origin of the hematite-bearing plains of Terra Meridiani based on analyses of orbital and Mars Exploration rover data sets. R E Arvidson, F Poulet, R V Morris, J P Bibring, J F Bell, I Squyres, S W Christensen, P R Bellucci, G Gondet, B Ehlmann, B L Farrand, W H Fergason, R L Golombek, M Griffes, J L Grotzinger, J Guinness, E A Herkenhoff, K E Johnson, J R Klingelhfer, G Langevin, Y Ming, D Seelos, K Sullivan, R J Ward, J G Wiseman, S M Wolff, M , Journal of Geophysical Research. 1111211Arvidson, R.E., Poulet, F., Morris, R.V., Bibring, J.P., Bell, J. F., I., Squyres, S.W., Christensen, P.R., Bellucci, G., Gondet, B., Ehlmann, B.L., Farrand, W.H., Fergason, R.L., Golombek, M., Griffes, J.L., Grotzinger, J., Guin- ness, E.A., Herkenhoff, K.E., Johnson, J.R., Klingelhfer, G., Langevin, Y., Ming, D., Seelos, K., Sullivan, R.J., Ward, J.G., Wiseman, S.M., Wolff, M., 2006. Nature and origin of the hematite-bearing plains of Terra Meridiani based on analyses of orbital and Mars Exploration rover data sets. Journal of Geophysical Research 111, E1211. Water and the martian landscape. V Baker, Nature. 412Baker, V., 2001. Water and the martian landscape. Nature 412, 228-236. Martian post-impact hydrothermal systems incorporating freezing. C J Barnhart, F Nimmo, B J Travis, Icarus. 208Barnhart, C.J., Nimmo, F., Travis, B.J., 2010. Martian post-impact hydrother- mal systems incorporating freezing. Icarus 208, 101 -117. Reorganization of a deeply incised drainage: role of deformation, sedimentation and groundwater flow. G Brocard, C Teyssier, W J Dunlap, C Authemayou, T Simon-Labric, E N Cacao-Chiquín, A Gutiérrez-Orrego, S Morán-Ical, Basin Research. 23Brocard, G., Teyssier, C., Dunlap, W.J., Authemayou, C., Simon-Labric, T., Cacao-Chiquín, E.N., Gutiérrez-Orrego, A., Morán-Ical, S., 2011. Reorga- nization of a deeply incised drainage: role of deformation, sedimentation and groundwater flow. Basin Research 23, 631-651. Formation of martian flood features by release of water from confined aquifers. M H Carr, Journal of Geophysical Research. 84Carr, M.H., 1979. Formation of martian flood features by release of water from confined aquifers. Journal of Geophysical Research 84, 2995-3007. Water on Mars. M H Carr, Oxford University PressCarr, M.H., 1996. Water on Mars. Oxford University Press. Related magma-ice interactions: Possible origins of chasmata, chaos, and surface materials in Xanthe, Margaritifer, and Meridiani Terrae. M Chapman, K Tanaka, Mars. Icarus. 155Chapman, M., Tanaka, K., 2002. Related magma-ice interactions: Possible origins of chasmata, chaos, and surface materials in Xanthe, Margaritifer, and Meridiani Terrae, Mars. Icarus 155, 324-339. A model for the hydrologic and climatic behavior of water on Mars. S M Clifford, Journal of Geophysical Research. 98Clifford, S.M., 1993. A model for the hydrologic and climatic behavior of water on Mars. Journal of Geophysical Research 98, 10973-11016. Depth of the Martian cryosphere: Revised estimates and implications for the existence and detection of subpermafrost groundwater. S M Clifford, J Lasue, E Heggy, J Boisson, P Mcgovern, M D Max, Journal of Geophysical Research. 1157001Clifford, S.M., Lasue, J., Heggy, E., Boisson, J., McGovern, P., Max, M.D., 2010. Depth of the Martian cryosphere: Revised estimates and implications for the existence and detection of subpermafrost groundwater. Journal of Geophysical Research 115, E07001. Martian megaflood-triggered chaos formation, revealing groundwater depth, cryosphere thickness, and crustal heat flux. N M Coleman, Journal of Geophysical Research. 110Coleman, N.M., 2005. Martian megaflood-triggered chaos formation, revealing groundwater depth, cryosphere thickness, and crustal heat flux. Journal of Geophysical Research 110, E12S20. Geologic and mineralogic mapping of Aram Chaos: evidence for a water-rich history. T Glotch, P Christensen, Journal of Geophysical Research. 1109006Glotch, T., Christensen, P., 2005. Geologic and mineralogic mapping of Aram Chaos: evidence for a water-rich history. Journal of Geophysical Research 110, E09006. The Sedimentary Rock Record of Mars: Distribution, Origins, and Global Stratigraphy, in: Sedimentary Geology of Mars. J P Grotzinger, R E Milliken, SEPM (Society for Sedimentary Geology102Grotzinger, J.P., Milliken, R.E., 2012. The Sedimentary Rock Record of Mars: Distribution, Origins, and Global Stratigraphy, in: Sedimentary Geology of Mars. SEPM (Society for Sedimentary Geology). volume 102, pp. 1-48. Multiple flooding events in Martian outflow channels. K P Harrison, R E Grimm, Journal of Geophysical Research. 1132002Harrison, K.P., Grimm, R.E., 2008. Multiple flooding events in Martian outflow channels. Journal of Geophysical Research 113, E02002. Regionally compartmented groundwater flow on Mars. K P Harrison, R E Grimm, Journal of Geophysical Research. 1144004Harrison, K.P., Grimm, R.E., 2009. Regionally compartmented groundwater flow on Mars. Journal of Geophysical Research 114, E04004. Thermal and crustal evolution of Mars. Steven A Hauck, I Phillips, R J , J. Geophys. Res. 1075052Hauck, Steven A., I., Phillips, R.J., 2002. Thermal and crustal evolution of Mars. J. Geophys. Res. 107, 5052. Erosion of cohesionless sediment by groundwater seepage. A D Howard, Charles F Mclane, I , Water Resour. Res. 24Howard, A.D., McLane, Charles F., I., 1988. Erosion of cohesionless sediment by groundwater seepage. Water Resour. Res. 24, 1659-1674. Terra infirma: Understanding salt tectonics. M R Hudec, M P Jackson, Earth-Science Reviews. 82Hudec, M.R., Jackson, M.P., 2007. Terra infirma: Understanding salt tectonics. Earth-Science Reviews 82, 1 -28. Evidence for extensive denudation of the Martian highlands. B Hynek, R Phillips, Geology. 29Hynek, B., Phillips, R., 2001. Evidence for extensive denudation of the Martian highlands. Geology 29, 407-410. Possible precipitation of ice at low latitudes of Mars during periods of high obliquity. B M Jakosky, M H Carr, Nature. 315Jakosky, B.M., Carr, M.H., 1985. Possible precipitation of ice at low latitudes of Mars during periods of high obliquity. Nature 315, 559-561. 3D structure of the Gusev Crater region. M Van Kan Parker, T Zegers, T Kneissl, B Ivanov, B Foing, G Neukum, Earth and Planetary Science Letters. 294van Kan Parker, M., Zegers, T., Kneissl, T., Ivanov, B., Foing, B., Neukum, G., 2010. 3D structure of the Gusev Crater region. Earth and Planetary Science Letters 294, 411 -423. Martian hydrogeology sustained by thermally insulating gas and salt hydrates. J S Kargel, R Furfaro, O Prieto-Ballesteros, J A P Rodriguez, D R Montgomery, A R Gillespie, G M Marion, S E Wood, Geology. 35Kargel, J.S., Furfaro, R., Prieto-Ballesteros, O., Rodriguez, J.A.P., Mont- gomery, D.R., Gillespie, A.R., Marion, G.M., Wood, S.E., 2007. Martian hydrogeology sustained by thermally insulating gas and salt hydrates. Ge- ology 35, 975-978. Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars. M G Kleinhans, Journal of Geophysical Research. 11012003Kleinhans, M.G., 2005. Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars. Journal of Geophysical Research 110, E12003. Martian steppeddelta formation by rapid water release. E R Kraal, M Van Dijk, G Postma, M G Kleinhans, Nature. 451Kraal, E.R., van Dijk, M., Postma, G., Kleinhans, M.G., 2008. Martian stepped- delta formation by rapid water release. Nature 451, 973-976. Fate of outflow channel effluents in the northern lowlands of Mars: The Vastitas Borealis Formation as a sublimation residue from frozen ponded bodies of water. M A Kreslavsky, J W Head, J. Geophys. Res. 1075121Kreslavsky, M.A., Head, J.W., 2002. Fate of outflow channel effluents in the northern lowlands of Mars: The Vastitas Borealis Formation as a sublima- tion residue from frozen ponded bodies of water. J. Geophys. Res. 107, 5121. Formation of Aromatum Chaos, Mars: Morphological development as a result of volcano-ice interactions. H J Leask, L Wilson, K L Mitchell, J. Geophys. Res. 1118071Leask, H.J., Wilson, L., Mitchell, K.L., 2006. Formation of Aromatum Chaos, Mars: Morphological development as a result of volcano-ice interactions. J. Geophys. Res. 111, E08071. Stratigraphy of hydrated sulfates in the sedimentary deposits of Aram Chaos, Mars. K A Lichtenberg, R E Arvidson, R V Morris, S L Murchie, J L Bishop, D Fernandez Remolar, T D Glotch, E N Dobrea, J F Mustard, J Andrews-Hanna, L H Roach, Journal of Geophysical Research. 115Lichtenberg, K.A., Arvidson, R.E., Morris, R.V., Murchie, S.L., Bishop, J.L., Fernandez Remolar, D., Glotch, T.D., Dobrea, E.N., Mustard, J.F., Andrews- Hanna, J., Roach, L.H., 2010. Stratigraphy of hydrated sulfates in the sed- imentary deposits of Aram Chaos, Mars. Journal of Geophysical Research 115, E00D17. Threshold phenomena in erosion driven by subsurface flow. A E Lobkovsky, B Jensen, A Kudrolli, D H Rothman, J. Geophys. Res. 1094010Lobkovsky, A.E., Jensen, B., Kudrolli, A., Rothman, D.H., 2004. Threshold phenomena in erosion driven by subsurface flow. J. Geophys. Res. 109, F04010. The origin and timing of fluvial activity at Eberswalde crater, Mars. N Mangold, E Kite, M Kleinhans, H Newsom, V Ansan, E Hauber, E Kraal, C Quantin, K Tanaka, Icarus. 220Mangold, N., Kite, E., Kleinhans, M., Newsom, H., Ansan, V., Hauber, E., Kraal, E., Quantin, C., Tanaka, K., 2012. The origin and timing of fluvial activity at Eberswalde crater, Mars. Icarus 220, 530 -551. Simulation of martian chaotic terrain and outflow channels. J P Manker, A P Johnson, Icarus. 51Manker, J.P., Johnson, A.P., 1982. Simulation of martian chaotic terrain and outflow channels. Icarus 51, 121-132. Valley formation by groundwater seepage, pressurized groundwater outbursts and crater-lake overflow in flume experiments with implications for Mars. W A Marra, L Braat, A W Baar, M G Kleinhans, Icarus. 232Marra, W.A., Braat, L., Baar, A.W., Kleinhans, M.G., 2014. Valley formation by groundwater seepage, pressurized groundwater outbursts and crater-lake overflow in flume experiments with implications for Mars. Icarus 232, 97- 117. Mineralogical composition, structure, morphology, and geological history of Aram Chaos crater fill on Mars derived from OMEGA Mars Express data. M Masse, S Le Mouelic, O Bourgeois, J P Combe, L Le Deit, C Sotin, J P Bibring, B Gondet, Y Langevin, Journal of Geophysical Research. 11312006Masse, M., Le Mouelic, S., Bourgeois, O., Combe, J.P., Le Deit, L., Sotin, C., Bibring, J.P., Gondet, B., Langevin, Y., 2008. Mineralogical composition, structure, morphology, and geological history of Aram Chaos crater fill on Mars derived from OMEGA Mars Express data. Journal of Geophysical Research 113, E12006. Initiation of Martian outflow channels: Related to the dissociation of gas hydrate?. M Max, S Clifford, Geophysical Research Letters. 28Max, M., Clifford, S., 2001. Initiation of Martian outflow channels: Related to the dissociation of gas hydrate? Geophysical Research Letters 28, 1787- 1790. Formation and evolution of the chaotic terrains by subsidence and magmatism: Hydraotes Chaos, Mars. S Meresse, F Costard, N Mangold, P Masson, G Neukum, Icarus. 194Meresse, S., Costard, F., Mangold, N., Masson, P., Neukum, G., 2008. For- mation and evolution of the chaotic terrains by subsidence and magmatism: Hydraotes Chaos, Mars. Icarus 194, 487-500. On the orbital forcing of Martian water and CO2 cycles: A general circulation model study with simplified volatile schemes. M Mischna, M Richardson, R Wilson, D Mccleese, Journal of Geophysical Research. 1085062Mischna, M., Richardson, M., Wilson, R., McCleese, D., 2003. On the or- bital forcing of Martian water and CO2 cycles: A general circulation model study with simplified volatile schemes. Journal of Geophysical Research 108, 5062. Formation of Martian outflow channels by catastrophic dewatering of evaporite deposits. D Montgomery, A Gillespie, Geology. 33Montgomery, D., Gillespie, A., 2005. Formation of Martian outflow channels by catastrophic dewatering of evaporite deposits. Geology 33, 625-628. Geology of Xanthe Terra outflow channels and the Mars Pathfinder landing site. D Nelson, R Greeley, Journal of Geophysical Research-Planets. 104Nelson, D., Greeley, R., 1999. Geology of Xanthe Terra outflow channels and the Mars Pathfinder landing site. Journal of Geophysical Research-Planets 104, 8653-8669. Impact crater lakes on Mars. H Newsom, G Brittelle, C Hibbitts, L Crossey, A Kudo, Journal of Geophysical Research. 101Newsom, H., Brittelle, G., Hibbitts, C., Crossey, L., Kudo, A., 1996. Impact crater lakes on Mars. Journal of Geophysical Research 101, 14951-14955. Evaluation of melting process of the permafrost on Mars: Its implication for surface features. Y Ogawa, Y Yamagishi, K Kurita, Journal of Geophysical Research. 1088046Ogawa, Y., Yamagishi, Y., Kurita, K., 2003. Evaluation of melting process of the permafrost on Mars: Its implication for surface features. Journal of Geophysical Research 108, 8046. the HRSC Co-Investigator Team. J Oosthoek, T Zegers, A Rossi, B Foing, G Neukum, 3D mapping of Aram Chaos: a record of fracturing and fluid activity. Lunar and Planetary Science XXXVIIIOosthoek, J., Zegers, T., Rossi, A., Foing, B., Neukum, G., the HRSC Co- Investigator Team, 2007. 3D mapping of Aram Chaos: a record of fracturing and fluid activity, in: Lunar and Planetary Science XXXVIII, pp. 1-2. Geological evolution of Ares Vallis on Mars: Formation by multiple events of catastrophic flooding, glacial and periglacial processes. A Pacifici, G Komatsu, M Pondrelli, Icarus. 202Pacifici, A., Komatsu, G., Pondrelli, M., 2009. Geological evolution of Ares Vallis on Mars: Formation by multiple events of catastrophic flooding, glacial and periglacial processes. Icarus 202, 60 -77. Outflow channel sources, reactivation, and chaos formation. J Rodriguez, S Sasaki, R Kuzmin, J Dohm, K Tanaka, H Miyamoto, K Kurita, G Komatsu, A Fairen, J Ferris, Icarus. 175Rodriguez, J., Sasaki, S., Kuzmin, R., Dohm, J., Tanaka, K., Miyamoto, H., Ku- rita, K., Komatsu, G., Fairen, A., Ferris, J., 2005. Outflow channel sources, reactivation, and chaos formation, Xanthe Terra, Mars. Icarus 175, 36-57. Large-scale spring deposits on Mars. A P Rossi, G Neukum, M Pondrelli, S Van Gasselt, T Zegers, E Hauber, A Chicarro, B Foing, Journal of Geophysical Research. 1138016Rossi, A.P., Neukum, G., Pondrelli, M., van Gasselt, S., Zegers, T., Hauber, E., Chicarro, A., Foing, B., 2008. Large-scale spring deposits on Mars? Journal of Geophysical Research 113, E08016. The thermal evolution of Mars as constrained by paleo-heat flows. J Ruiz, P J Mcgovern, A Jiménez-Díaz, V López, J P Williams, B C Hahn, R Tejero, Icarus. 215Ruiz, J., McGovern, P.J., Jiménez-Díaz, A., López, V., Williams, J.P., Hahn, B.C., Tejero, R., 2011. The thermal evolution of Mars as constrained by paleo-heat flows. Icarus 215, 508 -517. Spontaneous channelization in permeable ground: theory, experiment, and observation. N Schorghofer, B Jensen, A Kudrolli, D H Rothman, Journal of Fluid Mechanics. 503Schorghofer, N., Jensen, B., Kudrolli, A., Rothman, D.H., 2004. Spontaneous channelization in permeable ground: theory, experiment, and observation. Journal of Fluid Mechanics 503, 357-374. The Structure and Evolution of Ancient Impact Basins on Mars. P H Schultz, R A Schultz, J Rogers, Journal of Geophysical Research. 87Schultz, P.H., Schultz, R.A., Rogers, J., 1982. The Structure and Evolution of Ancient Impact Basins on Mars. Journal of Geophysical Research 87, 9803-9820. Aram Chaos and its constraints on the surface heat flux of Mars. S Schumacher, T E Zegers, Icarus. 211Schumacher, S., Zegers, T.E., 2011. Aram Chaos and its constraints on the surface heat flux of Mars. Icarus 211, 305 -315. Mars -Fretted and chaotic terrains. R P Sharp, Journal of Geophysical Research. 78Sharp, R.P., 1973. Mars -Fretted and chaotic terrains. Journal of Geophysical Research 78, 4073-4083. Martian surface paleotemperatures from thermochronology of meterorites. D Shuster, B Weiss, Science. 309Shuster, D., Weiss, B., 2005. Martian surface paleotemperatures from ther- mochronology of meterorites. Science 309, 594-597. Resurfacing history of the northern plains of Mars based on geologic mapping of Mars Global Surveyor data. K Tanaka, J Skinner, T Hare, T Joyal, A Wenker, Journal of Geophysical Research-Planets. 1088043Tanaka, K., Skinner, J., Hare, T., Joyal, T., Wenker, A., 2003. Resurfacing history of the northern plains of Mars based on geologic mapping of Mars Global Surveyor data. Journal of Geophysical Research-Planets 108, 8043. Late Noachian to Hesperian climate change on Mars: Evidence of episodic warming from transient crater lakes near Ares Vallis. N Warner, S Gupta, S Y Lin, J R Kim, J P Muller, J Morley, Journal of Geophysical Research. 1156013Warner, N., Gupta, S., Lin, S.Y., Kim, J.R., Muller, J.P., Morley, J., 2010a. Late Noachian to Hesperian climate change on Mars: Evidence of episodic warming from transient crater lakes near Ares Vallis. Journal of Geophysical Research 115, E06013. A refined chronology of catastrophic outflow events in Ares Vallis. N Warner, S Gupta, J P Muller, J R Kim, S Y Lin, Mars. Earth and Planetary Science Letters. 288Warner, N., Gupta, S., Muller, J.P., Kim, J.R., Lin, S.Y., 2009. A refined chronology of catastrophic outflow events in Ares Vallis, Mars. Earth and Planetary Science Letters 288, 58-69. Retreat of a giant cataract in a long-lived (3.7-2.6 Ga) martian outflow channel. N H Warner, S Gupta, J R Kim, S Y Lin, J P Muller, Geology. 38Warner, N.H., Gupta, S., Kim, J.R., Lin, S.Y., Muller, J.P., 2010b. Retreat of a giant cataract in a long-lived (3.7-2.6 Ga) martian outflow channel. Geology 38, 791-794. Constraints on the origin and evolution of Iani Chaos. N H Warner, S Gupta, J R Kim, J P Muller, L Le Corre, J Morley, S Y Lin, C Mcgonigle, Mars. Journal of Geophysical Research. 1166003Warner, N.H., Gupta, S., Kim, J.R., Muller, J.P., Le Corre, L., Morley, J., Lin, S.Y., McGonigle, C., 2011. Constraints on the origin and evolution of Iani Chaos, Mars. Journal of Geophysical Research 116, E06003. Thermal evolution of the Martian core: Implications for an early dynamo. J P Williams, F Nimmo, Geology. 32Williams, J.P., Nimmo, F., 2004. Thermal evolution of the Martian core: Im- plications for an early dynamo. Geology 32, 97-100. Melt and collapse of buried water ice: An alternative hypothesis for the formation of chaotic terrains on Mars. T E Zegers, J H Oosthoek, A P Rossi, J K Blom, S Schumacher, Earth and Planetary Science Letters. 297Zegers, T.E., Oosthoek, J.H., Rossi, A.P., Blom, J.K., Schumacher, S., 2010. Melt and collapse of buried water ice: An alternative hypothesis for the formation of chaotic terrains on Mars. Earth and Planetary Science Letters 297, 496 -504.
[]
[ "Wireless Broadcast with short labels", "Wireless Broadcast with short labels" ]
[ "Gewu Bu [email protected] \nUMR 7606\nSorbonne University\nCNRS\nLIP6ParisFrance\n", "Maria Potop-Butucaru [email protected] \nSorbonne University\nCNRS UMR\nLIP6, 7606ParisFrance\n" ]
[ "UMR 7606\nSorbonne University\nCNRS\nLIP6ParisFrance", "Sorbonne University\nCNRS UMR\nLIP6, 7606ParisFrance" ]
[]
In this paper we study the broadcast problem in wireless networks when the broadcast is helped by a labelling scheme. We focus on two variants of broadcast: broadcast without acknowledgement (i.e. the initiator of the broadcast is not notified at the end of broadcast) and broadcast with acknowledgement. Our contribution is twofold. First, we propose label optimal broadcast algorithms in a class of networks issued from recent studies in Wireless Body Area Networks then we extend our solutions to arbitrary networks. We propose an acknowledgement-free broadcast strategy using 1-bit labels and broadcast with acknowledgement using 2-bits labels. In the class of level-separable networks our algorithms finish within 2D rounds for both broadcast with and without acknowledgement, where D is the eccentricity of the broadcast initiator. Second, we improve a recent [11] labelling-based broadcast scheme with acknowledgement designed for arbitrary networks in terms of memory complexity.
null
[ "https://arxiv.org/pdf/1901.08919v3.pdf" ]
59,291,950
1901.08919
906f9843894af41fb569dd47bdcbe23cf0c7f9f4
Wireless Broadcast with short labels Gewu Bu [email protected] UMR 7606 Sorbonne University CNRS LIP6ParisFrance Maria Potop-Butucaru [email protected] Sorbonne University CNRS UMR LIP6, 7606ParisFrance Wireless Broadcast with short labels Index Terms-Labelling SchemeBroadcastWireless Net- works In this paper we study the broadcast problem in wireless networks when the broadcast is helped by a labelling scheme. We focus on two variants of broadcast: broadcast without acknowledgement (i.e. the initiator of the broadcast is not notified at the end of broadcast) and broadcast with acknowledgement. Our contribution is twofold. First, we propose label optimal broadcast algorithms in a class of networks issued from recent studies in Wireless Body Area Networks then we extend our solutions to arbitrary networks. We propose an acknowledgement-free broadcast strategy using 1-bit labels and broadcast with acknowledgement using 2-bits labels. In the class of level-separable networks our algorithms finish within 2D rounds for both broadcast with and without acknowledgement, where D is the eccentricity of the broadcast initiator. Second, we improve a recent [11] labelling-based broadcast scheme with acknowledgement designed for arbitrary networks in terms of memory complexity. I. INTRODUCTION Broadcast is the most studied communication primitive in networks and distributed systems. Broadcast ensures that once a source node (a.k.a. the broadcast initiator) sends a message then all other nodes in the network should receive this message in a finite time. Limited by the transmission range, messages may not be able to be sent directly from one node to some other arbitrary node in the network. Therefore relay nodes need to assist the source node during the message propagation by re-propagating it. Deterministic centralized broadcast, where nodes have complete network knowledge has been studied by Kowalski et al. in [19]. The authors propose an optimal solution that completes within O(Dlog 2 n) rounds, where n is the number of nodes in network and D is the largest distance from the source to any node of the network. The time lower bound for broadcast, Ω(log 2 n), has been proved in [2] for a family of radius-2 networks. For deterministic distributed broadcast, assuming that nodes only know their IDs (i.e. they do not know the IDs of their neighbors nor the network topology), in [8] is proposed the fastest broadcast within O(nlogDloglogD) rounds, where D is the diameter of network. The lower bound in this case, proposed in [9], is Ω(nlogD), where D is the largest distance from the source to any node of the network. In wireless networks, when a message is sent from a node it goes into the wireless channel in the form of a wireless signal which may be received by all the nodes within the transmission range of the sender. However, when a node is located in the range of more than one node that send messages simultaneously the multiple wireless signals may generate collisions at the receiver. The receiver cannot decode any useful information from the superimposed interference signals. At the MAC layer several solutions have been proposed in the last two decades in order to reduce collisions. All of them offer probabilistic guarantees. Our study follows the recent work that addresses this problem at the application layer. More specifically, we are interested in deterministic solutions for broadcasting messages based on the use of extra information or advise (also referred as labelling) precomputed before the broadcast invocation. Labelling schemes have been designed to compute network size, the father-son relationship and the geographic distance between arbitrary nodes in the network (e.g. [1], [14] and [16]). Labelling schemes have been also used in [13] and [15] in order to improve the efficiency of Minimum Spanning Tree or Leader Election algorithms. Furthermore, [10] and [12] exploit labelling in order to improve the existing solutions for network exploration by a robot/agent moving in the network. Very few works ( e.g. [18] and [11]) exploit labelling schemes to design efficient broadcast primitives. When using labelling schemes nodes record less information than in the case of centralized broadcast, where nodes need to know complete network information. Compared with the existing solutions for deterministic distributed broadcast the time complexity is improved. In [18] the authors prove that for an arbitrary network to achieve broadcast within constant number of rounds a O(n) bits of advice is sufficient but not o(n). Very recently, a labelling scheme with 2-bits advice (3 bits for broadcast with acknowledgement) is proposed in [11]. The authors prove that their algorithms need 2n − 3 rounds for the broadcast without acknowledgement and 3n − 4 rounds for broadcast with acknowledgement in arbitrary network. Contribution: Our work is in the line of research described in [11] and [18]. We first introduce a new family of networks, called level-separable networks issued from in Wireless Body Area Networks (e.g. [3], [5], [6], [4] and [7]). We then propose an acknowledgement-free broadcast strategy using 1-bit labels and a broadcast scheme with acknowledgement using 2-bits labels. In the class of level-separable networks our algorithms are memory optimal and terminate within 2D rounds for both types of broadcast primitives, where D is the eccentricity of the broadcast source. Second, we address the arbitrary networks and improve the broadcast scheme with acknowledgement proposed in [11] in terms of memory and time complexity by efficiently exploiting the 3-bits labelling encoding. Differently from the solution proposed in [11], our solution does not use extra local persistent memory except the 3-bits labels. II. MODEL AND PROBLEM DEFINITION A. Communication Model We model the network as a graph G = (V, E) where V , the set of vertices, represents the set of nodes in the network and E, the set of edges, is a set of unordered pairs e = (u, v), u, v ∈ V , that represents the communications links between nodes u and v. In the following d(u) denotes the set of neighbors of node u. We target wireless networks where due to the limitation of the transmission power, a node may not have connections with the other nodes in the network (i.e., |d(u)| ≤ |V | − 1). However, we assume that the network is connected, i.e., there is a path between any two nodes in the network. We assume that nodes execute the same algorithm and are time synchronized. The system execution is decomposed in rounds. When a node u sends a message at round x, all nodes in d(u) receive the message at the end of round x. Collisions occur at node u in round x if a set of nodes, M ⊆ d(u) and |M | > 1, send a message in round x. In that case it is considered that u has not received any message. In the following we are interested in solving the Broadcast problem: when a source node sends a message, this message should be received by all the nodes in the network in finite bounded time. B. Level-Separable Network In this section, we define a family of networks, Level-Separable Network, issued from WBAN area (e.g. [3], [5], [6], [4] and [7]). We say an arbitrary network is a Level-Separable Network if the underlay communication graph G = (V, E) of the network verifies the Level-Separable propriety defined below. To define the Level-Separable propriety, we introduce some preliminary notations. Let G(V, E) be a network and let s ∈ V , a predefined vertex, be the source node of the broadcast. Each vertex u ∈ V has a geometric distance with respect to s denoted d(s, u). The eccentricity of vertex s, ε G (s), is the farthest distance from s to any other vertex. In the rest of the paper we denote ε G (s) by D. i.e., the level of u is its geometric distance to s. Let S i = {u | u ∈ V, l(u) = i}l(v) − l(u) = 1 ∧ {u, v} ∈ E Let S(u) (P (v)) be the set of sons (parents) of u (v). If v ∈ S(u) (u ∈ P (v)), we say that u (v) has v (u) as son (parent). Level-Separable propriety below defines how to filter nodes in the same level i into two disjoint subsets. Definition 3 (Level-Separable Subsets). Given G(V, E) a network and the set S i (the set of all vertices in the same level i of G), the level-separable subsets of S i are S i,1 and S i,2 , such that S i,1 ∩ S i,2 = ∅, S i,1 ∪ S i,2 = S i There may be many possible pairs of S i,1 and S i,2 for a level i. Let T i be the set of all possible pairs of Level-Separable Subsets: T i = {(S (1) i,1 ,(2 x −1) i,1 , S (2 x −1) i,2 )} where (m) on right-top of each pairs represent the index of pairs (the mth pairs) in T i . Definition 4 (Multi Parents Set). Let G(V, E) be a network and let S i contain all vertices at level i. The Multi Parents Set, F i for any i > 1, contains vertices at level i that have more than one parent at level i − 1. We define F i as: F i = {u | u ∈ S i , l(u) = i ∧ |P (u)| > 1} For level i = 1, as all vertices has only one parent, the root, i,1 | = 1, ∀u ∈ F i+1 i.e., for every vertex u at level i + 1 having multi-parents at level i, u has only one parent in S i,1 . F 1 = ∅.Note that if F i+1 = ∅, then S i,1 = ∅. When S i,1 is fixed, S i,2 is S i \ S i,1 . Definition 6 (Level-Separable Network). A network G(V, E) is a Level-Separable Network, if its underlay graph verifies the Level-Separable property. Note that Level-Separable Graph has similar flavor with Bipartite Graph [17]. A graph G = (V, E) is said to be Bipartite if and only if there exists a partition V = A ∪ B and A ∩ B = ∅. So that all edges share a vertex from both sets A and B, and there is no edge containing two vertices in the same set. A bipartite graph separates nodes into two independent sets. In a level-separable network we aim at separating nodes of the same level. Moreover, we are interested in the relation between the two separated sets at level i and nodes in level i + 1, i.e., node's father-son relationship. Note that a level-separable network is not necessary a tree network. However a tree is a level-separable network. A simple Fig. 1. Example of a Level-2 separable network, which is not a tree network example of level-separable network is a tree network, where the root of the tree is the source node s who begins the broadcast. In a tree topology all non-source nodes have only one parent, i.e. ∀u ∈ V − s, |P (u)| = 1, so that in each level, the F i = ∅. So that all S i,1 = ∅ and S i,2 = S i \ S i,1 = S i . The Level-Separable property is therefore verified. Figure 1 shows an example of a level-separable network that is not a tree. In this network, 16 nodes are connected: one source node (i.e. the node that starts the broadcast) and 15 non-source nodes. Note that this network is not a tree: nodes may have more than one parent (e.g., node 12 has two parents: node 5 and node 6). This network is represented by levels for easy the observation. For any level i ∈ [1, D − 1] all nodes at that level can be separated into two level-separable sets: At the level 1, S 1,1 = {2} and S 1,2 = {1, 3}. That is true because the Multi Parents Set F 2 = {6} and the parents set of node 6 is P (6) = {2, 4}. Therefore |P (u) ∩ S 1,1 | = 1, ∀u ∈ F 2 . According to Definition 5, S 1,1 = {2} and S 1,2 = {1, 3} verify the level-separable propriety. From the same reason, at level 2, S 2,1 = {5, 8} and S 2,2 = {4, 6, 7} also verify the level-separable propriety. Studies conducted in wireless body area networks (e.g. [3], [5], [6], [4] and [7]) show that various postural mobilities can be model as graphs that fit our definition of level-separable network. In [5], authors studied the cross-layer broadcast in wireless body area network and model the network as graphs for different human postures. In this case each graph is a levelseparable network, see Figure 2. In the next section we propose a broadcast algorithm without acknowledgement with 1-bit labels in separable networks. Then, we improve in terms of memory complexity the broadcast algorithm proposed in [11] for arbitrary networks. Finally, we propose a solution for broadcast with acknowledgement in level-separable networks using only 2 bits-labels. III. BROADCAST IN LEVEL-SEPARABLE NETWORK In this section we propose a 1-bit constant-length labelling broadcast Algorithm β LS detailed in Algorithm 1. The algorithm needs 2D rounds to terminate, where D is the eccentricity of the broadcast source node. A. Broadcast with 1-bit Labelling Given a level-separable network whose root is the source of the broadcast, we propose Algorithm β LS (shown as Algorithm 1) to achieve the wireless broadcast, when a 1-bit labelling scheme λ LS is used. Each node in the network has a 1-bit label, X 1 . X 1 is set to 1 or 0 following the labelling scheme λ LS described below. The idea of the broadcast algorithm is to separate nodes at each level into two independent sets. Nodes in the first set transmit at round x and nodes in the second set transmit at round x + 1 (the next round), so that they will not generate valid collisions 1 . The broadcast Algorithm β LS using the labelling scheme λ LS is as follows: the source node sends the message, µ, at round 0. Nodes at level 1 receive µ at the end of round 0. When nodes with X 1 = 1 receive message µ at round 2i − 3 (i > 1) or 2i − 2 (i > 0), where i is the level, they send message µ at round 2i − 1. When nodes with X 1 = 0 receive µ at round 2i − 3 (i > 1) or 2i − 2 (i > 0) , then they send µ at rounds 2i. That is, nodes at level i > 0 will receive µ from their parents (nodes at level i − 1) at round 2i − 3 (i > 1) or 2i − 2 (i > 0), and they will send µ at round 2i or 2i − 1 according to the X 1 . In other words, at each level i, nodes take two rounds to propagate µ to all nodes at level i + 1. Figure 3 presents the propagation of the message. The left side shows the level of a level-separable network, from level 0 to 2. It shows three rounds during the execution. The right side shows that at which round nodes at a level receive (denoted Fig. 3. Execution of the broadcast algorithm β LS with λ LS labelling in a level-separable network R) or transmit (denoted T ) a message. At round 0, source s sends message µ to all nodes at level 1. Nodes at level 1 have been already separated into two sets, blue ones and white ones by the labelling scheme λ LS . At round 1, nodes in the white set send µ, and two nodes at level 2 receive the message. At round 2, the nodes in the blue set send µ and the remaining nodes at level 2 receive the message. 1-bit Labelling Scheme λ LS . To achieve collision free transmission, 1-bit Labelling Scheme λ LS X 1 of all nodes in S i,1 for level i > 0 is 1, and X 1 of all nodes in S i,2 for level i > 0 is 0 where S i,1 and S i,2 are the sets identified in Definition 5. B. Correctness and Complexity of Algorithm β LS In the following we prove that Algorithm β LS is correct. First we show that the previously described scheme when used by Algorithm β LS do not generate collisions. Theorem 1. Algorithm β LS with 1-bit constant Labelling Scheme λ LS implements broadcast in a level-separable network. within 2D rounds. The proof of this theorem is a direct consequence of Lemmas 1, 2 and 3 below. Note 1. Note that 1-bit labelling scheme is optimal for broadcast in a level-separable network. That is, with 0-bit labelling (i.e. without using any labelling) it is possible that some node in the network does not receive the broadcasted message due to the collisions since nodes are synchronized and transmit in the same time. First we show that the previously described scheme when used by Algorithm β LS does not generate collisions. Lemma 1. Let G = (V, E) be a level-separable network such that each node has a label according to the labelling scheme λ LS . If nodes with X 1 = 1 at the same level i ∈ [1, D − 1], send a message concurrently they do not generate collisions at nodes at level i + 1. If nodes with X 1 = 0 at the same level i ∈ [1, D − 1], send a message concurrently they do not generate collisions at nodes having only one parent at level i + 1. Proof. At level i, nodes with X 1 = 1 are the nodes in the subset S i,1 . According to Definition 5, each node at level i + 1 has at most one parent in S i,1 . (Note that nodes at level i + 1 may have no parent in S i,1 ). Therefore, when nodes in S i,1 send a message, none of nodes in level i + 1 will receive more than one message. When nodes will X 1 = 1 send, there will be no collisions at level i + 1. Nodes with X 1 = 0 are nodes in S i,2 . S i,2 contains all parents of each node at level i + 1 who has only one parent at level i. It follows that when nodes with X 1 = 0 send, all nodes having only one parent can receive the message without collisions. Lemma 2. Given a level-separable network whose root is the source node by applying β LS and λ LS , all nodes in level i > 0 finish receiving message µ at round 2i − 2. Proof. We begin from the base case where i = 1, nodes at level i = 1 means nodes that are only one hop away from the source node. At round 0, which is round 2 × i − 2 = 2 × 1 − 2 = 0, the source sends the message. All nodes at level 1 will receive the message at the end of round 0. For i = 2, as all nodes at level 1 can receive message at round 0, they will begin to send at round 1 and round 2 for nodes in S i,1 and S i,2 , respectively. According to Lemma 1, no collision occurs at level i = 2. Therefore all the nodes in level i = 2 can receive the message at the end of round 2, which is round 2 × i − 2 = 2 × 2 − 2 = 2 and they begin to send message at round 3 and 4. For the general case, we assume that all nodes at level i, i > 2, finish receiving message at round 2i − 2. So that nodes begin to send the received message at round 2(i + 1) − 3 and 2(i + 1) − 2, and nodes at level i + 1 receive the message at 2(i + 1) − 3 and 2(i + 1) − 2, that is nodes at level i + 1 finish receiving message at round 2(i + 1) − 2. Lemma 3. Given a level-separable network whose root is the source node by applying β LS and λ LS , the broadcast finishes in 2D rounds. Proof. From Lemma 2, nodes having the longest distance to the source will receive the message at round 2D − 2, where D is the source eccentricity. After receiving the message, these nodes will send it according to the broadcast algorithm, even though they are already the ending nodes in the network which takes two more rounds. So the broadcast finishes at round 2D. The idea of the correctness proof is as follow. Consider the execution of the Algorithm β LS in a level-separable network with labelling scheme λ LS , where nodes in level i have been separated into two sets S i,1 and S i,2 verifying level-separable propriety at level i, ∀i > 0. Nodes in S i,1 have X 1 = 1, and nodes in S i,2 have X 1 = 0. The main idea of β LS is that, nodes in each level i separated into two different sets transmit their received messages µ in different execution rounds to reduce the collisions impact at nodes in level i + 1. According to Algorithm β LS , the message µ will be propagated from level to level. Each propagation from a level to the next one takes two execution rounds. In the first round all Algorithm 1 β LS (µ) executed at each node v %Each node has a variable sourcemsg. The source node has this variable initially set to µ, all other nodes have it initially set to null. A variable k initially set to 0 to ensure each node sends µ only once. for each round r from 0 do if v is the source node and r = 0 then transmit sourcemsg if v is not source node and receives µ then if k = 0 then sourcemsg ← µ if r is odd number then if X 1 = 0 then transmit sourcemsg at round r + 3 else if X 1 = 1 then transmit sourcemsg at round r + 2 else if r is even number then if X 1 = 0 then transmit sourcemsg at round r + 2 else if X 1 = 1 then transmit sourcemsg at round r + 1 set k = 1 nodes in S i,1 send the received message µ. At the end of this round all the nodes that are the sons of nodes in S i,1 receive µ, without collision, see Lemma 1. As sons of nodes in S i,1 contain all the nodes at level i + 1 who have multi-parents, that means it remains only nodes at level i + 1 who have only one parent that haven't received message µ yet. In the second round, all nodes in S i,2 send µ, and the remaining part of the nodes at level i + 1 can therefore receive µ from their unique parent. So that after these two rounds of transmission from level i, all the nodes at i + 1 can successfully receive the message µ. It takes therefore 2D rounds to finish the broadcast. Note that nodes will only send once according to β LS . Therefore the algorithm terminates. C. Labeling Preinstall In this section we propose a strategy to select S i,1 for each level i in a level-separable network. Note that this strategy is executed off line before the execution of the broadcast algorithm. Given two arbitrary successive levels, i and i + 1, let F i+1 be the set of all nodes at level i + 1 that have multi parents at level i (see Definition 4). Let SF be the set of all parents of nodes in F i+1 , such that: SF = u∈Fi+1 P (u) The main idea to select S i,1 is to select from the Power Set of SF , i.e., the set of all the subset of SF . The set S i,1 we should chose from the power set of SF should verify: 1) nodes in S i,1 do not have the same son nodes; 2) nodes in S i,1 contain all parents of nodes in F i+1 . Assume that the mean number of nodes in each level is x, then in each level, we need to chose S i,1 from at most IV. BROADCAST WITH ACK FOR ARBITRARY NETWORKS In [11] the authors propose a broadcast with ACK algorithm β ACK for general networks using a 3-bits labelling scheme λ ACK . At the end of the broadcasting, the last informed node generates and sends back to the source node an ACK message. In a 3-bits labelling, there are 8 states: 000, 001, 010, 011, 100, 101, 110 and 111 available. The algorithm in [11] uses only 5 of them: 000, 001, 010, 100 and 110. In this section, we propose a labelling scheme, λ oACK and a broadcast with ACK that uses all the 8 states of the 3-bits labelling in order to improve the memory complexity of the solution proposed in [11]. Our optimization with respect to the λ ACK proposed in [11] is as follows: instated of only using the last bit X 3 (the third bit) as a mark to point who is (one of) the last informed node(s) during the broadcast, we use also this third bit to show a path back to the source node s from the last informed node. Differently from the solution proposed in [11], nodes do not need to keep additional variables in order to send back to the source the ACK during the execution. Our proposition can therefore save node's memory and computational power. In the following we present our λ oACK labelling scheme. A. 3-bit Labelling Scheme λ oACK The first two bits of the labelling scheme X 1 and X 2 have the same functionality as in the λ ACK scheme of [11] (see [11] for more details and proof). The intuitive idea is as follows: X 1 = 1 for nodes who should propagate the message when they receive it; 2) X 2 = 1 for nodes that need to send stay message back to their parent to notice that they need to send the message one more time in the next round; 3) X 3 = 1 for one of the last receiving node to generate ACK and send it back to the source node. In our scheme λ oACK we set additionally X 3 (the third bit) to 1 for all nodes on the path back from the last informed node (who holds 001) to the source node. Note that, nodes on that path could have four kinds of different labels: 101, 011, 111 and 001, where 001 is the label of the last informed node. States 101, 011 and 111 are not used in the original β ACK , therefore nodes can easily recognize if they are on the path to transmit the ACK message back to the source node. B. Broadcast Algorithm β oACK Our broadcast algorithm β oACK that uses the λ oACK is described in Algorithm 2. Given an arbitrary network applying the labelling scheme λ oACK execute β oACK . Nodes with X 1 = 1 receiving a message at round i − 1 send it at round i. Then nodes who sent at round i wait the stay message, at round i + 1, from other nodes with X 2 = 1. If nodes who sent at round i receive stay at round x + 1, then they continue to send one more time µ at round i + 2. Otherwise, they will stay silent. When nodes with label 001 receive the message, they generate the ACK message and send it. Since λ oACK already marked the path back to the source node, in Algorithm β oACK , the ACK message will only be re-propagated by nodes with X 3 = 1. i.e., node with label 101, 111 and 011. Note that our proposed Algorithm β oACK does not need additional variables to reconstruct the path back to the source Algorithm 2 β oACK (µ) executed at each node v %Each node has a variable sourcemsg. The source node has this variable initially set to µ, all other nodes have it initially set to null. for each round r from 0 do if v is source node and r = 0 then transmit sourcemsg if v is not source node then if message m is received AND m = "stay" then sourcemsg ← m else if The node received µ before round r then if v received sourcemsg for first time in round r − 2 then if X 1 = 1 then transmit sourcemsg else if v received sourcemsg for first time in round r − 1 then if X 1 = 0 and X 2 = 0 and X 3 = 1 then transmit "ACK" else if X 2 = 1 then transmit "stay" else if v received "stay" in round r − 1 then if v transmitted sourcemsg in round r − 2 then transmit sourcemsg else if v received "ACK" in round r − 1 then if X 3 = 1 then transmit "ACK" during the broadcast execution. In Algorithm β ACK [11], two additional variables inf ormedRound (type int) and transmitRounds (type table of int) are needed to rebuild the back-way path. inf ormedRound is used to record the round number in which a node received µ; transmitRounds is a table used to record all the round numbers in which one node transmits µ. However, by using β oACK , the ACK message transfer processing can be completed only by checking the third bit, X 3 . Our Algorithm β oACK does not need any extra local storage for directing the ACK message. C. Labeling Preinstall In the following we propose a strategy to decide the backway path in arbitrary network. According to the idea of λ ACK in [11], the last informed node, the 001 node, can be detected easily. If v is the last informed node, let u = P r (v) be the parent node of v from whom v received µ. Since the computation is done offline, the P r (u) of any node u (if it exists) can always be computed offline. The members of the back-way path belong to the set: Bp = {u, P r (u), P r (P r (u)), ..., s} where u is the last informed node and s is the source node. To mark the back-way path, λ oACK sets the X 3 bit of the labels of all nodes in Bp to 1. Note that we do not change the main architecture of β ACK algorithm in [11] therefore the correctness proof of our algorithm is very similar to the one in [11]. V. BROADCAST WITH ACK IN LEVEL-SEPARABLE NETWORK In this section, we combine the Broadcast algorithm β LS and the labelling scheme λ oACK to propose an algorithm of broadcast with ACK, β LS ACK , and the Labelling Scheme, λ LS ACK , for level-separable networks. Our algorithm β LS ACK (Algorithm 3) uses only 2-bits labelling and the broadcast finishes within 2D rounds. In our solution ACK goes back to the source node in at most 2D rounds, where D is the eccentricity of s (the broadcast source node). A. 2-bit Labelling Broadcast with ACK According to Theorem 1 the broadcast finishes in a levelseparable network within 2D rounds where D is the eccentricity of the source node. If the source node has the knowledge of D, then it automatically can decide if the broadcast is finished. However, when an ACK message is necessary to inform the source node to trigger some additional functions then the source waits for the reception of this message. In order to avoid that the ACK message takes addition time after the end of the broadcast, we propose to send in advance the ACK message at the halfway of the transmission during the broadcast execution. Since in a level-separable network, informing nodes from level to level takes exactly 2 rounds, then ACK also takes 2 rounds to goes back one level above. Therefore, when the last node receives µ, the source node receives the ACK message at the same round. Interestingly, compared with non-ACK broadcasting, our solution uses one extra bit for labelling and no additional rounds for forwarding back to the source the ACK message. Figure 4 gives the intuition of how to send in advance the ACK message: the half-way ACK mechanism. In Figure 4, the network is represented in abstract levels to simplify the presentation. Packets flow shown in the figure represent the propagation of messages µ and ACK. B. 2-bit Labelling Scheme λ LS ACK We use λ LS to set X 1 in λ LS ACK in order to verify Lemma 1. Let X 2 be the second bit of the λ LS ACK labelling scheme. X 2 = 1 for a set of nodes if they are on the way back path from a node at level D/2 − 1 to the source node, where D is the eccentricity of s and s is the broadcast source. For the other nodes, X 2 = 0. In Section V-C, we explain why we chose nodes at level D/2 − 1 to begin sending the ACK. Note 2. Note that 2-bit labelling scheme is optimal to achieve broadcast with acknowledgement in a level-separable network. From Note 1 1-bit is necessary for broadcast without acknowledgement. When an acknowledgement has to be sent back to the source node, at least one additional bit is necessary to indicate the node to generate the acknowledgement message and send it back to the source node. Without this additional bit no node can decide (unless it uses extra local memory) if it is the last receiving node, and who should send back the ACK. C. Correctness and complexity of Algorithm β LS ACK Theorem 2 below proves the correctness of Algorithm β LS ACK . Theorem 2. Algorithm β LS ACK with 2-bit labelling scheme λ LS ACK implements broadcast in a level-separable network. The broadcast terminates in 2D rounds. The ACK message is transmitted back to the source at round 2(D − 1), if D is odd or 2D, if D is even. The proof of the theorem is the direct consequence of Lemma 4, 5 and 6 below. Lemma 4. Given a level-separable network whose root is the source node by applying β LS ACK and λ LS ACK , nodes in level i > 0 receive message µ at round 2i−2. The broadcast finishes at round 2D Proof. β LS ACK follows the same idea as β LS . The additional ACK transmission will not have any impact according to Lemma 2 and 3. Hence the proof follows. Lemma 5. Given a level-separable network whose root is the source node by applying β LS ACK and λ LS ACK , the ACK message goes back to source node at round 2(D − 1), if D is odd; or 2D, if D is even. Proof. When D is odd, ACK and the message will begin to be sent to source and to the ending nodes from levels l ACK and l M SG , respectively. The distances from levels l ACK back to source is the same with that from l M SG to the ending nodes. ACK arrives to the source at the same round as the broadcasted message arrives at the ending nodes. According to Lemma 4, this is round 2(D − 1). When D is even ACK needs to go one level farther compared with the broadcasted message. Therefore, it takes two extra rounds when D is even. Therefore, when D is even the ACK message goes back to source node in 2D rounds. Lemma 6. Given a Level-Separable Network whose root is the source node by applying β LS ACK and λ LS ACK , the algorithm finishes within 2D rounds. Proof. The idea of the correctness proof is as follows. Consider a level-separable network with the labelling scheme λ LS ACK , where all nodes in level i have been separated into two sets S i,1 and S i,2 . Nodes in S i,1 have X 1 = 1, and nodes in S i,2 have X 1 = 0. A way back path is marked with X 2 = 1 between source s and an arbitrary node at level D/2 − 1, where D is the eccentricity of s , i.e., we only mark the way back path from the half-way level D/2 − 1 of the network in this case. The idea is that when the message µ propagates to the halfway level of the network, a node at that level will begin the Algorithm 3 β LS ACK (µ) executed at each node v %Each node has a variable sourcemsg. The source node has this variable initially set to µ, all other nodes have it initially set to null. A variable k and k ack initially set to 0 to ensure each node send µ only once. for each round r from 0 do if v is source node and r = 0 then transmit sourcemsg if v is not source node and received µ then sourcemsg ← µ if k = 0 then if r is odd number then if X 1 = 0 then transmit sourcemsg at round r + 3 if X 2 = 1 then transmit "pACK" at round r + 4 if v does not received "pACK" at r + 6 then transmit "ACK" at round r + 6, set k ack = 1 else if X 1 = 1 then transmit sourcemsg at round r + 2 if X 2 = 1 then transmit "pACK" at round r + 4 if v has not received "pACK" at r + 6 then transmit "ACK" at round r + 6, set k ack = 1 else if r is even number then if X 1 = 0 then transmit sourcemsg at round r + 2 if X 2 = 1 then transmit "pACK" at round r + 3 if v has not received "pACK" at r + 5 then transmit "ACK" at round r + 5, set k ack = 1 else if X 1 = 1 then transmit sourcemsg at round r + 1 if X 2 = 1 then transmit "pACK" at round r + 3 if v has not received "pACK" at r + 5 then transmit "ACK" at round r + 5, set k ack = 1 set k = 1 if v is not source node and received ACK then if X 2 = 1 and k ack = 0 then transmit ACK at round r + 2 set k ack = 1 ACK transmission processing, so that when the µ reaches to the ending node(s) at level D, the ACK message reaches the source s at (almost) the same round. As nodes cannot decide if they are the ones at the half-way of network who should generate and send ACK message, we use a Waiting Period and an extra pACK message. According the β LS ACK , when a node with X 2 = 1, receives µ and finishes the µ retransmission, it cannot decide its position in the way back path. Therefore, it sends a pACK message and begins to wait pACK message sent to him in the following rounds. When a node with X 2 = 1 receives a pACK within the W aitingP eriod, that means it is not the ending node, because there is another node with X 2 = 1 that received µ and sent pACK to him. When a node with X 2 = 1 does not receive any pACK within its W aitingP eriod, this means no node in the next level has X 2 = 1, i.e., it is the half-way ending node, so it generates and sends the ACK message. All the nodes with X 2 = 1 will forward ACK message from the ending node to the source s according to the marked way back path. In the β LS ACK , the W aitingP eriod is delayed two rounds after a node sends pACK message to avoid the collision between pACK/ACK and µ. A node with X 2 = 1 that receives µ at round x, transmits µ at round x + 2, then it sends pACK to its parents at round x + 4, then it waits a Waiting Period until round x + 6. If it doesn't receive another pACK, then it sends ACK at round x + 8. That means, for the half-way ending node, it needs to wait 6 rounds to begin sending ACK. What we want for this half-way mechanism is that the source node can receive ACK as fast as possible, after the broadcast finishes. When D (the eccentricity of the broadcast source s) is odd, then if we chose the node at level D/2 − 1 as the half-way ending node, then the ACK can be received by source node at the same round as the end of the broadcast. Because after waiting 6 rounds at level D/2 − 1, message µ has already been transmitted to level D/2 − 1 + 3 = D/2 + 2. The distance from node sending ACK to source node is d(s, D/2 −1) = D/2 −1; the distance from node sending µ to nodes at level D is also d( D/2 + 2, D) = D/2 − 1. When D is even, if we chose the node at level D/2 − 1 as the half-way ending node, then the ACK can be received by the source node only two rounds after the round of the ending of broadcast. Therefore it takes 2D to finish Broadcast and the ACK can be transmitted back to the source node at round 2(D −1) or round 2D. Note that nodes will only send (both for data message and ACK message) once according to β LS ACK . Therefore the algorithm terminates. D. Labeling Preinstall In the following we propose a strategy to decide the backaway path in a level-separable network from the halfway during the broadcast propagation. Similar to Section IV-C, instead of choosing the last informed node as the generator of ACK message, we chose in level D/2 − 1 a node u and build the set Bp = {u, P r (u), P r (P r (u)), ..., s} from u to s. To mark the way back path, one needs only to set X 2 of all nodes in Bp to 1. When using a 3-bits label instead of 2-bits, the last informed node can be marked directly by the labelling scheme using the third bit. That means that during the broadcast execution any Waiting Period or pACK message is unnecessary, so that during the execution of β LS ACK , we can save the unnecessary pACK message transmission. VI. CONCLUSION We proposed solutions for implementing broadcast in wireless networks when the broadcast is helped by a labelling scheme. We studied broadcast without acknowledgement (i.e. the initiator of the broadcast is not notified at the end of broadcast) and broadcast with acknowledgement. We propose an optimal acknowledgement-free broadcast strategy using 1bit labelling and a broadcast with acknowledgement using a 2-bit labelling in level-separable networks. The complexity of both algorithms is 2D where D is the eccentricity of the broadcast initiator. Then, we improved in terms of memory and time complexity the labelling-based broadcast scheme with acknowledgement proposed in [11] for arbitrary networks. Our improvement fully exploits the encoding of the labels in order to not use extra memory to carry back the acknowledgement to the source. Definition 1 ( 1Level). Let G(V, E) be a network and s the source node. For any vertex u in G(V, E), the level of u is l(u) = d(s, u) denote the set containing all the vertices at level i.Definition 2 (Parents and Sons). Let G(V, E) be a network. A vertex u is parent of vertex v (a vertex v is son of vertex u) in graph G with the root source node s: if Definition 5 ( 5Level-Separable Propriety). Given an arbitrary graph G(V, E), for all level i ∈ [1, D − 1], where D is the eccentricity of source node, G verifies the Level-Separable property, if there exists pairs for every T i (the set of all possible pairs of Level-Separable Subsets at level i), Fig. 2 . 2Graphs that model human postures in wireless body area networks. Numbers on the edges represent the edge reliability. 2 x − 1 possible choices. The offline time complexity of choosing S i,1 at each level is O(2 x − 1). Fig. 4 . 4Anticipating the ACK in a level-separable network Note that collisions that occur at a node who has already received the message successfully are not considered valid collisions. Compact labeling schemes for ancestor queries. Serge Abiteboul, Haim Kaplan, Tova Milo, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms. the twelfth annual ACM-SIAM symposium on Discrete algorithmsSerge Abiteboul, Haim Kaplan, and Tova Milo. Compact labeling schemes for ancestor queries. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 547-556. Society for Industrial and Applied Mathematics, 2001. A lower bound for radio broadcast. Amotz Noga Alon, Nathan Bar-Noy, David Linial, Peleg, Journal of Computer and System Sciences. 432Noga Alon, Amotz Bar-Noy, Nathan Linial, and David Peleg. A lower bound for radio broadcast. Journal of Computer and System Sciences, 43(2):290-298, 1991. Broadcast strategies in wireless body area networks. Wafa Badreddine, Claude Chaudet, Federico Petruzzi, Maria Potop-Butucaru, Proceedings of the 18th ACM International Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems. the 18th ACM International Conference on Modeling, Analysis and Simulation of Wireless and Mobile SystemsACMWafa Badreddine, Claude Chaudet, Federico Petruzzi, and Maria Potop- Butucaru. Broadcast strategies in wireless body area networks. In Proceedings of the 18th ACM International Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems, pages 83-90. ACM, 2015. Convergecast in wireless body area networks. Wafa Badreddine, Nesrine Khernane, Maria Potop-Butucaru, Claude Chaudet, Ad Hoc Networks. 66Wafa Badreddine, Nesrine Khernane, Maria Potop-Butucaru, and Claude Chaudet. Convergecast in wireless body area networks. Ad Hoc Networks, 66:40-51, 2017. Peak transmission rate resilient crosslayer broadcast for body area networks. Wafa Badreddine, Maria Potop-Butucaru, arXiv:1702.05031arXiv preprintWafa Badreddine and Maria Potop-Butucaru. Peak transmission rate resilient crosslayer broadcast for body area networks. arXiv preprint arXiv:1702.05031, 2017. Total order reliable convergecast in wban. Gewu Bu, Maria Potop-Butucaru, Proceedings of the 18th International Conference on Distributed Computing and Networking. the 18th International Conference on Distributed Computing and NetworkingACM26Gewu Bu and Maria Potop-Butucaru. Total order reliable convergecast in wban. In Proceedings of the 18th International Conference on Distributed Computing and Networking, page 26. ACM, 2017. Ban-gzkp: Optimal zero knowledge proof based scheme for wireless body area networks. Gewu Bu, Maria Potop-Butucaru, Ad Hoc Networks. 77Gewu Bu and Maria Potop-Butucaru. Ban-gzkp: Optimal zero knowledge proof based scheme for wireless body area networks. Ad Hoc Networks, 77:28-41, 2018. Faster deterministic communication in radio networks. Ferdinando Cicalese, Fredrik Manne, Qin Xin, Algorithmica. 542Ferdinando Cicalese, Fredrik Manne, and Qin Xin. Faster deterministic communication in radio networks. Algorithmica, 54(2):226-242, 2009. Selective families, superimposed codes, and broadcasting on unknown radio networks. E F Andrea, Angelo Clementi, Riccardo Monti, Silvestri, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms. the twelfth annual ACM-SIAM symposium on Discrete algorithmsAndrea EF Clementi, Angelo Monti, and Riccardo Silvestri. Selective families, superimposed codes, and broadcasting on unknown radio networks. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 709-718. Society for Industrial and Applied Mathematics, 2001. Label-guided graph exploration by a finite automaton. Reuven Cohen, Pierre Fraigniaud, David Ilcinkas, Amos Korman, David Peleg, ACM Transactions on Algorithms (TALG). 4442Reuven Cohen, Pierre Fraigniaud, David Ilcinkas, Amos Korman, and David Peleg. Label-guided graph exploration by a finite automaton. ACM Transactions on Algorithms (TALG), 4(4):42, 2008. Constantlength labeling schemes for deterministic radio broadcast. Faith Ellen, Barun Gorain, Avery Miller, Andrzej Pelc, arXiv:1710.03178arXiv preprintFaith Ellen, Barun Gorain, Avery Miller, and Andrzej Pelc. Constant- length labeling schemes for deterministic radio broadcast. arXiv preprint arXiv:1710.03178, 2017. Tree exploration with advice. Information and Computation. Pierre Fraigniaud, David Ilcinkas, Andrzej Pelc, 206Pierre Fraigniaud, David Ilcinkas, and Andrzej Pelc. Tree exploration with advice. Information and Computation, 206(11):1276-1287, 2008. Local mst computation with short advice. Theory of Computing Systems. Pierre Fraigniaud, Amos Korman, Emmanuelle Lebhar, 47Pierre Fraigniaud, Amos Korman, and Emmanuelle Lebhar. Local mst computation with short advice. Theory of Computing Systems, 47(4):920- 933, 2010. Stéphane Pérennes, and Ran Raz. Cyril Gavoille, David Peleg, Journal of Algorithms. 531Distance labeling in graphsCyril Gavoille, David Peleg, Stéphane Pérennes, and Ran Raz. Distance labeling in graphs. Journal of Algorithms, 53(1):85-112, 2004. Time vs. information tradeoffs for leader election in anonymous trees. Christian Glacet, Avery Miller, Andrzej Pelc, ACM Transactions on Algorithms (TALG). 13331Christian Glacet, Avery Miller, and Andrzej Pelc. Time vs. information tradeoffs for leader election in anonymous trees. ACM Transactions on Algorithms (TALG), 13(3):31, 2017. Finding the size of a radio network with short labels. Barun Gorain, Andrzej Pelc, Proceedings of the 19th International Conference on Distributed Computing and Networking. the 19th International Conference on Distributed Computing and NetworkingACM10Barun Gorain and Andrzej Pelc. Finding the size of a radio network with short labels. In Proceedings of the 19th International Conference on Distributed Computing and Networking, page 10. ACM, 2018. Graph theory and its applications. L Jonathan, Jay Gross, Yellen, CRC pressJonathan L Gross and Jay Yellen. Graph theory and its applications. CRC press, 2005. Fast radio broadcasting with advice. David Ilcinkas, R Dariusz, Andrzej Kowalski, Pelc, Theoretical Computer Science. David Ilcinkas, Dariusz R Kowalski, and Andrzej Pelc. Fast radio broadcasting with advice. Theoretical Computer Science, 411(14- 15):1544-1557, 2010. Optimal deterministic broadcasting in known topology radio networks. R Dariusz, Andrzej Kowalski, Pelc, Distributed Computing. 193Dariusz R Kowalski and Andrzej Pelc. Optimal deterministic broadcasting in known topology radio networks. Distributed Computing, 19(3):185- 195, 2007.
[]
[ "On the Robustness of the Approximate Price of Anarchy in Generalized Congestion Games (Full Version) *", "On the Robustness of the Approximate Price of Anarchy in Generalized Congestion Games (Full Version) *" ]
[ "Vittorio Bilò " ]
[]
[]
One of the main results shown through Roughgarden's notions of smooth games and robust price of anarchy is that, for any sum-bounded utilitarian social function, the worst-case price of anarchy of coarse correlated equilibria coincides with that of pure Nash equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions and for a broad generalization of the class of weighted congestion games with non-negative (and possibly decreasing) latency functions, the worst-case price of anarchy of ǫ-approximate coarse correlated equilibria still coincides with that of ǫ-approximate pure Nash equilibria, for any ǫ ≥ 0. As a byproduct of our proof, it also follows that such a value can always be determined by making use of the primal-dual method we introduced in a previous work. It is important to note that our scenario of investigation is beyond the scope of application of the robust price of anarchy (for as it is currently defined), so that our result seems unlikely to be alternatively proved via the smoothness framework. . 1 To this aim, we recall that the set of coarse correlated equilibria contains that of correlated equilibria, which contains that of mixed Nash equilibria, which contains that of pure Nash equilibria.
10.1016/j.tcs.2022.01.008
[ "https://arxiv.org/pdf/1412.0845v1.pdf" ]
9,174,170
1412.0845
ba5aabb82f776c2a2ad1fd24b07b1583aa5718a7
On the Robustness of the Approximate Price of Anarchy in Generalized Congestion Games (Full Version) * 2 Dec 2014 December 3, 2014 Vittorio Bilò On the Robustness of the Approximate Price of Anarchy in Generalized Congestion Games (Full Version) * 2 Dec 2014 December 3, 2014 One of the main results shown through Roughgarden's notions of smooth games and robust price of anarchy is that, for any sum-bounded utilitarian social function, the worst-case price of anarchy of coarse correlated equilibria coincides with that of pure Nash equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions and for a broad generalization of the class of weighted congestion games with non-negative (and possibly decreasing) latency functions, the worst-case price of anarchy of ǫ-approximate coarse correlated equilibria still coincides with that of ǫ-approximate pure Nash equilibria, for any ǫ ≥ 0. As a byproduct of our proof, it also follows that such a value can always be determined by making use of the primal-dual method we introduced in a previous work. It is important to note that our scenario of investigation is beyond the scope of application of the robust price of anarchy (for as it is currently defined), so that our result seems unlikely to be alternatively proved via the smoothness framework. . 1 To this aim, we recall that the set of coarse correlated equilibria contains that of correlated equilibria, which contains that of mixed Nash equilibria, which contains that of pure Nash equilibria. Introduction The celebrated notion of robust price of anarchy introduced by Roughgarden in [19,20] has lately arouse much interest in the determination of inefficiency bounds for pure Nash equilibria which may automatically extend to some of their appealing generalizations, such as mixed Nash equilibria, correlated equilibria and coarse correlated equilibria. These three types of solutions have a particular flavor since, differently from pure Nash equilibria, they are always guaranteed to exist by Nash's Theorem [16] 1 ; moreover, the last two ones can also be efficiently computed and even easily learned when a game is repeatedly played over time. To this aim, Roughgarden [19,20] identifies a class of games, called smooth games, for which a simple three-line proof, called smoothness argument, shows significant upper bounds on the price of anarchy of pure Nash equilibria as long as the social function measuring the quality of any strategy profile in the game is sum-bounded, that is, upper bounded by the sum of the players' costs 2 . He then defines the robust price of anarchy of a smooth game as the best-possible (i.e., the lowest) upper bound which can be derived by making use of this argument and provides an extension theorem which shows that, still for sum-bounded social functions, the price of anarchy of coarse correlated equilibria of any smooth game is upper bounded by its robust price of anarchy. Finally, he shows that several games considered in the literature happen to be smooth and that the class of (unweighted) congestion games with non-negative and non-decreasing latency functions is tight for the utilitarian social function (that is, the social function defined as the sum of the players' costs), in the sense that, in this class of games, the worst-case price of anarchy of pure Nash equilibria exactly matches the robust price of anarchy. This last result has been subsequently extended to the class of weighted congestion games by Bhawalkar, Gairing and Roughgarden in [3]. Our Contribution and Significance In this work, we generalize the tightness result by Bhawalkar, Gairing and Roughgarden along the following four directions (see Section 2 for formal definitions): 1. the class of games we consider is a broad generalization of that of weighted congestion games. In particular, we focus on generalized weighted congestion games, that is, games in which each player's perceived cost is defined as a certain linear combination of all the players' individual costs originally experienced in some underlying weighted congestion game. Thus, it is quite easy to figure out that the class of generalized weighted congestion games widely extends that of weighted congestion games; 2. the families of social functions we consider are generalizations of both the utilitarian and the egalitarian social functions (where the egalitarian social function is defined as the maximum of the players' costs). In particular, a family of utilitarian social functions is obtained by summing up a certain contribution from each player, whereas a family of egalitarian social functions is obtained by taking the maximum contribution among the players, where each player's contribution is given by a conic combination of the players' individual costs. We stress that such a combination may significantly differ from the one used to define the players' perceived costs, so that there exist social functions in both families that may not be sumbounded; 3. the latency functions we consider in the definition of the players' individual costs are selected from a family of allowable non-negative functions with no additional restrictions. This permits us to encompass also latency functions not considered so far in the previous tightness results known in the literature, such as, for instance, the widely used fair cost sharing rule induced by the Shapley value [21]; 4. the solution concepts we consider are the approximate versions of all the four types of equilibria named so far. In particular, for any real value ǫ ≥ 0, we focus on either ǫ-approximate pure Nash equilibria and ǫ-approximate coarse correlated equilibria. For the special case of ǫ = 0, one reobtains the notions of pure Nash equilibria and coarse correlated equilibria, so that results for these solution concepts can be obtained as a special case of the ones holding for their approximate versions. More precisely, but still informally speaking, we prove the following result (Theorem 1 in Section 3): for a variety of utilitarian and egalitarian social functions and for any real value ǫ ≥ 0, the worst-case price of anarchy of ǫ-approximate pure Nash equilibria coincides with that of ǫ-approximate coarse correlated equilibria in the class generalized weighted congestion games with non-negative latency functions. As it can be appreciated, the above tightness result generalizes the previous one by Bhawalkar, Gairing and Roughgarden along all four directions simultaneously. The technique we use to prove the theorem is the primal-dual method that we introduced in [4]. In fact, as a byproduct of our proof, it also follows that, in the above considered scenario of investigation, the worst-case price of anarchy of ǫ-approximate pure Nash equilibria can always be determined through the primal-dual method. We would like to stress that, when adopting the social functions described at point 2, generalized weighted congestion games are not smooth games in general, so that the above tightness result seems unlikely to be reproved via smoothness arguments, at least in the way in which they have been defined and used so far in the literature. This seems to provide an evidence that the primaldual method may be more powerful than the smoothness framework as far as we focus on congestion games and some of their possible generalizations. Related Work The notion of price of anarchy as a measure of the inefficiency caused by selfish behavior in noncooperative games has been introduced in a seminal paper by Koutsoupias and Papadimitriou [14] in 1999. Since then, several classes of games have been studied under this perspective. Among these classes, congestion games introduced by Rosenthal in [18] and their weighted variants [15] occupy a preeminent role. Awerbuch, Azar and Epstein [2] and Christodoulou and Koutsoupias [10] focus on the worstcase price of anarchy of pure Nash equilibria in either weighted and unweighted congestion games under the utilitarian social function. They independently give tight bounds for the case of affine latency functions and almost tight upper and lower bounds for the case of polynomial latency functions with non-negative coefficients. Such a gap has been subsequently closed by Aland et al. in [1]. Moreover, Christodoulou, Koutsoupias and Spirakis [12] obtain tight bounds on the worst-case price of anarchy of approximate pure Nash equilibria in unweighted congestion games for the case of polynomial latency functions with non-negative coefficients, while Christodoulou and Koutsoupias [11] show that the worst-case price of anarchy of correlated equilibria is the same as that for pure Nash equilibria in weighted and unweighted congestion games when considering affine latency functions. As already said, such an equivalence has been further extended to coarse correlated equilibria and to any class of non-negative and non-decreasing latency functions by Roughgarden [19,20] in the unweighted case and by Bhawalkar, Gairing and Roughgarden [3] in the weighted case, by making use of the smoothness argument and the robust price of anarchy. Robust bounds on the worst-case price of anarchy have been lately achieved via extensions of the smoothness argument in some generalizations of (unweighted) congestion games. In particular, de Keijzer et al. [13] and Rahn and Schäfer [17] consider the altruistic extension of congestion games in which, similarly to our model of generalized congestion games, the perceived cost of each player is defined as a linear combination of the individual costs of all the players in the game. Anyway, while we do not impose any kind of restriction on such a combination, they consider the case in which the multiplicative coefficients lie in the interval [0, 1] and, for each player i, the contribution of the individual cost of player i to her perceived cost has to be always multiplied by 1. Moreover, they restrict their analysis to the case in which the social function is the sum of the players' individual costs. Much less attention has been devoted in the literature to the egalitarian social function, for which Christodoulou and Koutsoupias [10] give an asymptotically tight bound on the worst-case price of anarchy in unweighted congestion games with affine latency functions. We introduced the primal-dual method in [4] as a tool for obtaining tight bounds on the inefficiencies caused by selfish behavior in weighted congestion games and their possible generalizations for a variety of solutions concepts. In particular the primal-dual method has been applied by Bilò, Flammini and Gallotti [7] to derive tight bounds on the worst-case price of anarchy of pure Nash equilibria in congestion games with affine latency functions under the assumption that the players' knowledge is restricted by the presence of an underlying social knowledge graph; by Bilò [5] to derive tight bounds on the worst-case price of stability of pure Nash equilibria in congestion games with affine latency functions and altruistic players; by Bilò and Paladini [9] to derive tight bounds on the approximation ratio of the solutions achieved after a one-round walk of ǫ-approximate bestresponses starting from any initial strategy profile in cut games, for any ǫ ≥ 0; by Bilò et al. [8] to derive a surprising matching lower bound on the price of anarchy of subgame perfect equilibria in sequential cut games; and by Bilò, Fanelli and Moscardelli [6] to derive significant upper bounds on the price of anarchy of lookahead equilibria in congestion games with affine latency functions. Paper Organization The paper is organized as follows. In the next section, we give all necessary definitions and notation and provide also some preliminary remarks. Section 3 contains the technical contribution of the paper, with the proof of our main theorem. In the last section, we conclude and discuss open problems. Definitions, Notation and Preliminaries A weighted congestion game is a tuple CG = [n], (w i ) i∈[n] , E, (Σ i ) i∈[n] , (ℓ e ) e∈E such that [n] = {1, 2, . . . , n} is a set of n ≥ 2 players, w i > 0 is the weight of player i, E is a non-empty set of resources, Σ i ⊆ 2 E \ {∅} is a non-empty set of strategies for player i and ℓ e : R ≥0 → R ≥0 is the latency function of resource e ∈ E. Denote as Σ = i∈[n] Σ i the set of all strategy profiles of CG, that is, the set of outcomes which can be realized when each player i ∈ [n] chooses a strategy in Σ i . A strategy profile σ = (σ 1 , . . . , σ n ) is then a vector of strategies, where, for each i ∈ [n], σ i ∈ Σ i denotes the choice of player i in σ. For a strategy profile σ and a resource e ∈ E, the value n e (σ) = i∈[n]:e∈σ i w i denotes the congestion of resource e in σ, that is, the sum of the weights of all the players choosing e in σ. The individual cost of player i in σ is defined as c i (σ) = w i e∈σ i ℓ e (n e (σ)). Given a finite space of functions F ⊆ {f : R ≥0 → R ≥0 }, let B(F) = {f k : R ≥0 → R ≥0 | k ∈ [r] } be a basis for F of cardinality r, whose elements (functions) are numbered from 1 to r. We say that CG is defined over F if, for each e ∈ E, it holds that ℓ e = k∈ [r] v e k f k , where v e k ∈ R is a scalar. Throughout the paper, we will impose only minimal assumptions on F; in particular, we will assume that any f ∈ F is non-negative with f (x) = 0 if and only if x = 0. For any n-dimensional vector of (positive) weights w = (w 1 , . . . , w n ), we denote with C w (F) the class of all the weighted congestion games with players' weights induced by w and defined over F. Moreover, for a fixed quadruple T w = ([n], w, E, (Σ i ) i∈[n] ), called a congestion model, the set C Tw (F) = {CG ∈ C w (F) | CG = (T w , (ℓ e ) e∈E ) } is the set of all the weighted congestion games induced by T w and defined over F. Note that, since for each game CG ∈ C Tw (F) and e ∈ E there exist r numbers v e 1 , . . . , v e r such that ℓ e = k∈[r] v e k f k , it follows that CG can be specified by the pair (T w , (v e k ) e∈E,k∈[r] ). Moreover, it holds that C w (F) = Tw C Tw (F). Finally, we denote with Σ(T w ) the set of strategy profiles induced by the congestion model T w . A generalized weighted congestion game is a pair (CG, α) where CG = ([n], (w i ) i∈[n] , E, (Σ i ) i∈[n] , (ℓ e ) e∈E ) is a weighted congestion game and α ∈ R n×n is an ndimensional square matrix. Game (CG, α) has the same set of players and strategies of CG, but the perceived cost of player i in the strategy profile σ is defined as c i (σ) = j∈[n] α ij c j (σ) = j∈[n] α ij w j e∈E:e∈σ j ℓ e (n e (σ)) = e∈E k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j α ij w j , where c i (σ) is the individual cost that player i experiences in σ in the underlying weighted congestion game CG. Note that, when α is the identity matrix, (CG, α) coincides with CG, while, in all the other cases, (CG, α) may not be isomorphic to any weighted congestion game, so that the set of generalized weighted congestion games expands that of weighted congestion games. Given a strategy profile σ, a player i ∈ [n] and a strategy x ∈ Σ i , we denote with (σ −i , x) the strategy profile obtained from σ when player i changes her strategy from σ i to x, while the strategies of all the other players are kept fixed. In particular, for any ǫ ≥ 0, the perceived cost suffered by player i in σ minus 1 + ǫ times the perceived cost suffered by player i in (σ −i , x) in a generalized weighted congestion game can be expressed as follows: c i (σ) − (1 + ǫ) · c i (σ −i , x) = j∈[n] α ij c j (σ) − (1 + ǫ) j∈[n] α ij c j (σ −i , x) = α ii w i e∈σ i ℓ e (n e (σ)) − (1 + ǫ) e∈x ℓ e (n e (σ −i , x)) + j∈[n]:j =i α ij w j e∈σ i ℓ e (n e (σ)) − (1 + ǫ) e∈x ℓ e (n e (σ −i , x)) = α ii w i   e∈σ i \x ℓ e (n e (σ)) − (1 + ǫ) e∈x\σ i ℓ e (n e (σ) + w i )   + j∈[n]:j =i α ij w j   e∈σ i \x ℓ e (n e (σ)) − (1 + ǫ) e∈x\σ i ℓ e (n e (σ) + w i )   . Hence, we get c i (σ) − (1 + ǫ) · c i (σ −i , x) = e∈σ i \x ℓ e (n e (σ)) j∈[n]:e∈σ j α ij w j − (1 + ǫ) e∈x\σ i ℓ e (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   . (1) Next two definitions formalize the two concepts of approximate equilibria that we will consider throughout the paper. Definition 1 For any ǫ ≥ 0, an ǫ-approximate coarse correlated equilibrium is a probability distribution p defined over Σ such that, for any player i ∈ [n] and strategy x ∈ Σ i , it holds that σ∈Σ p σ · c i (σ) ≤ (1 + ǫ) σ∈Σ p σ · c i (σ −i , x), where, for each σ ∈ Σ, p σ is the probability assigned to σ by p. Definition 2 For any ǫ ≥ 0, an ǫ-approximate pure Nash equilibrium is a strategy profile σ such that, for any player i ∈ [n] and strategy x ∈ Σ i , it holds that c i (σ) ≤ (1 + ǫ) · c i (σ −i , x). Denote as PNE ǫ (CG, α) and CCE ǫ (CG, α), respectively, the set of ǫ-approximate pure Nash equilibria and ǫ-approximate coarse correlated equilibria of the generalized weighted congestion game (CG, α). It is easy to see that, for any ǫ ≥ 0, an ǫ-approximate pure Nash equilibrium σ is an ǫ-approximate coarse correlated equilibrium p such that p σ = 1 and p τ = 0 for any τ ∈ Σ \ {σ}. So, PNE ǫ (CG, α) ⊆ CCE ǫ (CG, α). Moreover, the sets PNE 0 (CG, α) and CCE 0 (CG, α) coincide with the sets of pure Nash equilibria and coarse correlated equilibria of (CG, α), respectively. For an n-dimensional non-null square matrix β ∈ R n×n ≥0 and a player i ∈ [n], let β-cost i : Σ → R >0 be the contribution of player i to the definition of the social function which is defined as follows: β-cost i (σ) = j∈[n] β ij c j (σ) = e∈E k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j β ij w j . Let ∆(Σ) be the set of all the probability distributions defined over Σ. For a p ∈ ∆(Σ), the β-utilitarian social function is a function β-SUM : ∆(Σ) → R >0 such that β-SUM(p) = i∈[n] E σ∼p [β-cost i (σ)] = E σ∼p   i∈[n] β-cost i (σ)   = σ∈Σ p σ   e∈E k∈[r] v e k f k (n e (σ)) i∈[n] j∈[n]:e∈σ j β ij w j   and the β-egalitarian social function is a function β-MAX : ∆(Σ) → R >0 such that β-MAX(p) = max i∈[n] {E σ∼p [β-cost i (σ)]} = max i∈[n]    σ∈Σ p σ e∈E k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j β ij w j    . 3 Consider the case in which p ∈ ∆(Σ) is indeed a strategy profile σ ∈ Σ. When β is the identity matrix, β-SUM (resp. β-MAX) coincides with the sum (resp. the maximum) of the players' individual costs in the underlying weighted congestion game CG, while, when β = α, β-SUM (resp. β-MAX) coincides with the sum (resp. the maximum) of the players' perceived costs in (CG, α). In general, an infinite variety of social functions can be defined by tuning the choice of matrix β 4 . For a function SF ∈ {SUM, MAX}, we denote with o the social optimum, that is, any strategy profile minimizing β-SF. Note that, by the properties of the latency functions and the definition of β 5 , it follows that β-SF(o) > 0. The ǫ-approximate coarse correlated price of anarchy of (CG, α) under the social function β-SF is defined as CCPoA ǫ (β-SF, CG, α) = max p∈CCEǫ(CG,α) β-SF(p) β-SF(o) , while the ǫ-approximate pure price of anarchy of (CG, α) under the social function β-SF is defined as PPoA ǫ (β-SF, CG, α) = max σ∈PNEǫ(CG,α) β-SF(σ) β-SF(o) . For an n-dimensional vector of weights w = (w 1 , . . . , w n ) and a matrix α ∈ R n×n , we denote with C w (F, α) = {(CG, α) : CG ∈ C w (F)} the set of all the generalized weighted congestion games induced by w and α and defined over F. Similarly, for any congestion model T w , one defines the class C Tw (F, α), so as to obtain C w (F, α) = Tw C Tw (F, α). The worst-case ǫ-approximate coarse correlated price of anarchy of the class C w (F, α) under the social function β-SF is defined as CCPoA ǫ (β-SF, C w (F, α)) = sup (CG,α)∈Cw(F ,α) CCPoA ǫ (β-SF, CG, α). Similarly, one defines the worst-case ǫ-approximate pure price of anarchy of the class C w (F, α) under the social function β-SF. By PNE ǫ (CG, α) ⊆ CCE ǫ (CG, α), it follows that PPoA ǫ (β-SF, C w (F, α)) ≤ CCPoA ǫ (β-SF, C w (F, α) ) for any real value ǫ ≥ 0, n-dimensional vector of weights w, finite space of function F, pair of matrices α ∈ R n×n and β ∈ R n×n ≥0 and function SF ∈ {SUM, MAX}. Throughout the paper, we will also refer to the worst-case ǫ-approximate pure price of anarchy and to the worst-case ǫ-approximate coarse correlated price of anarchy of subsets of C w (F, α) which are naturally defined by restriction. We conclude this section with an easy, although crucial result, stating that, independently of which is the adopted social function, both the worst-case ǫ-approximate pure price of anarchy and the worst-case ǫ-approximate coarse correlated price of anarchy of a class of generalized weighted congestion games remain the same even if one restricts to only those games in the given class whose social optimum has social value equal to one 6 . To this aim, for any function SF ∈ {SUM, MAX} 4 One could even relax the constraint β ∈ R n×n ≥0 and allow for negative entries in matrix β as long as i∈[n] βij ≥ 0 for each j ∈ [n] and i∈[n] βij > 0 for some j ∈ [n] which still guarantees either β-SUM(σ) > 0 and β-MAX(σ) > 0 for each σ ∈ Σ. 5 From now on, we will always assume that β is a non-null matrix. 6 Indeed, such a result implicitly holds for the worst-case ǫ-approximate price of anarchy of any kind of equilibrium. and matrix β ∈ R n×n ≥0 , let C w (F, α) ⊂ C w (F, α) be the subset of all the generalized weighted congestion games induced by w and α and defined over F such that the social optimum o satisfies β-SF(o) = 1. Similarly, for any congestion model T w , one defines the class C Tw (F, α), so as to obtain C w (F, α) = Tw C Tw (F, α). Lemma 1 For any real value ǫ ≥ 0, n-dimensional vector of weights w, finite space of functions F, pair of matrices α ∈ R n×n and β ∈ R n×n ≥0 and function SF ∈ {SUM, MAX}, it holds that PPoA ǫ (β-SF, C w (F, α)) = PPoA ǫ (β-SF, C w (F, α)) and CCPoA ǫ (β-SF, C w (F, α)) = CCPoA ǫ (β-SF, C w (F, α)). Proof: Fix a congestion model T w , a pair of matrices α ∈ R n×n and β ∈ R n×n ≥0 and a function SF ∈ {SUM, MAX}. The claim directly follows from the fact that, for any game G := T w , (v e k ) e∈E,k∈[r] , α ∈ C Tw (F, α) such that β-SF(o) := x > 0, there always exists a game G := T w , (v e k ) e∈E,k∈[r] , α ∈ C Tw (F, α), obtained by setting v e k = v e k /x, such that, for any σ ∈ Σ(T w ), it holds that e∈E k∈ [r] v e k f k (n e (σ)) β ij w j e∈σ j k∈[r] v e k f k (n e (σ)) = x · max i∈[n] j∈[n] β ij w j e∈σ j k∈[r] v e k f k (n e (σ)). Moreover, for any σ ∈ Σ(T w ) and i ∈ [n], it holds that e∈E k∈ [r] v e k f k (n e (σ)) j∈[n]:e∈σ j α ij w j = x e∈E k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j α ij w j . That is, for any strategy profile σ ∈ Σ(T w ), the social value of σ in game G is equal to x times the social value of σ in game G, independently of which is the adopted social function. Moreover, for any strategy profile σ ∈ Σ(T w ) and any i ∈ [n], the perceived cost of player i in σ in game G is equal to x times the perceived cost of player i in σ in game G. This implies that G and G have the same set of equilibria (whatever the concept of equilibrium is defined) and that the ratio between any linear combination of the social values of any set of strategy profiles is the same in both games. The Main Result Our main result is the proof of the following general theorem. Theorem 1 For any real value ǫ ≥ 0, n-dimensional vector of weights w, finite space of functions F, pair of matrices α ∈ R n×n and β ∈ R n×n ≥0 and function SF ∈ {SUM, MAX}, it holds that PPoA ǫ (β-SF, C w (F, α)) = CCPoA ǫ (β-SF, C w (F, α)). Moreover, the value PPoA ǫ (β-SF, C w (F, α)) can always be determined via the primal-dual method. Proof: Fix a real value ǫ ≥ 0, an n-dimensional vector of weights w, a finite space of functions F, a pair of matrices α ∈ R n×n and β ∈ R n×n ≥0 and a function SF ∈ {SUM, MAX}. We prove the claim in four steps. Step 1) Definition of the representative congestion model T * w . Let T * w = ([n], w, E * , (Σ * i ) i∈[n] ) be a congestion model such that 1. Σ * i = {σ * i , o * i } for each i ∈ [n] , i.e., each player i ∈ [n] has exactly two strategies denoted as σ * i and o * i ; 2. the set of resources E * and the strategies σ * i and o * i for each i ∈ [n] are properly defined in such a way that, for each P, Q ⊆ [n], there exists exactly one resource e(P, Q) ∈ E * for which it holds that {i ∈ [n] | e(P, Q) ∈ σ * i } = P and {i ∈ [n] | e(P, Q) ∈ o * i } = Q. Hence, |E * | = 2 n · 2 n = 4 n . Intuitively, the representative congestion model T * w is defined in such a way that the pair of strategy profiles σ * = (σ * 1 , . . . , σ * n ) and o * = (o * 1 , . . . , o * n ) is able to encompass all possible configurations of congestions that may arise in any pair of strategy profiles and for any congestion model induced by w. In particular, the following fundamental property holds. Step 2) Definition of a primal-dual formulation for PPoA ǫ (β-SF, C w (F, α)). Property 1 For any congestion model T w = ([n], w, E, (Σ i ) i∈[n] ), resource e ∈ E and pair of profiles σ ′ , σ ′′ ∈ Σ(T w ), there always exists a resource e ∈ E * such that {i ∈ [n] | e ∈ σ ′ i } = {i ∈ [n] | e ∈ σ * i } and {i ∈ [n] | e ∈ σ ′′ i } = {i ∈ [n] | e ∈ o * i }. Fix a function SF ∈ {SUM, MAX}. Our aim is to use the optimal solution of a linear program PP PNE (SF, T * w , σ * , o * ) to achieve an upper bound on the worst-case ǫ-approximate pure price of anarchy of any game in C T * w (F, α) under the restriction that the latency functions are suitably tuned so as to make σ * the worst ǫ-approximate pure Nash equilibrium and o * a social optimum (of social value 1). The linear program PP PNE (SUM, T * w , σ * , o * ) for the β-utilitarian social function is defined as follows. maximize e∈E * k∈[r] v e k f k (n e (σ * )) i∈[n] j∈[n]:e∈σ * j β ij w j subject to e∈σ * i \o * i k∈[r] v e k f k (n e (σ * )) j∈[n]:e∈σ * j α ij w j −(1 + ǫ) e∈o * i \σ * i ∈E k∈[r] v e k f k (n e (σ * ) + w i )   α ii w i + j∈[n]:e∈σ * j α ij w j   ≤ 0, ∀i ∈ [n] e∈E * k∈[r] v e k f k (n e (o * )) i∈[n] j∈[n]:e∈o * j β ij w j ≤ 1, v e k ≥ 0, ∀e ∈ E * , k ∈ [r] The first n constraints guarantee that no player can lower her perceived cost of a factor more than 1 + ǫ by switching to the strategy she uses in the social optimum o * (see Equation (1)), while the last constraint normalizes to at most 1 the value β-SUM(o * ). The dual program DP PNE (SUM, T * w , σ * , o * ) is the following (we associate a variable y i with the ith constraint of the first n ones and a variable γ with the normalizing constraint). v e k f k (n e (σ * )) j∈[n]:e∈σ * j α ij w j −(1 + ǫ) e∈o * i \σ * i ∈E k∈[r] v e k f k (n e (σ * ) + w i )   α ii w i + j∈[n]:e∈σ * j α ij w j   ≤ 0, ∀i ∈ [n] e∈E * k∈[r] v e k f k (n e (σ * )) j∈[n]:e∈σ * j β 1j w j = t, e∈E * k∈[r] v e k f k (n e (σ * )) j∈[n]:e∈σ * j β ij w j ≤ t, ∀i ∈ [n] \ {1} e∈E * k∈[r] v e k f k (n e (o * )) j∈[n]:e∈o * j β ij w j ≤ 1, ∀i ∈ [n] v e k ≥ 0, ∀e ∈ E * , k ∈ [r] t ≥ 0 Here, again the first n constraints guarantee that no player can lower her perceived cost of a factor more than 1 + ǫ by switching to the strategy she uses in the social optimum o * . The next n constraints impose that the maximum value in the social function β-MAX(σ * ) is attained by player The dual program DP PNE (MAX, T * w , σ * , o * ) is the following (we associate variables y i , z i and γ i with the ith constraint of the first, the middle and the last family of n constraints, respectively). minimize i∈[n] γ i subject to i∈[n]:e∈σ * i \o * i y i f k (n e (σ * )) j∈[n]:e∈σ * j α ij w j −(1 + ǫ) i∈[n]:e∈o * i \σ * i y i f k (n e (σ * ) + w i )   α ii w i + j∈[n]:e∈σ * j α ij w j   +f k (n e (σ * )) i∈[n] z i j∈[n]:e∈σ * j β ij w j +f k (n e (o * )) i∈[n] γ i j∈[n]:e∈o * j β ij w j ≥ 0, ∀e ∈ E * , k ∈ [r] i∈N z i ≤ −1 y i , z i , γ i ≥ 0, ∀i ∈ [n] We stress that, being all the values ǫ, (w i ) i∈[n] , (α ij , β ij ) i,j∈[n] , n e (σ * ) and n e (o * ) fixed constants in the proposed formulations, PP PNE (SUM, T * w , σ * , o * ) is a linear program defined over the variables (v e k ) e∈E * ,k∈ [r] and PP PNE (MAX, T * w , σ * , o * ) is a linear program defined over the variables (v e k ) e∈E * ,k∈[r] and t, as needed. Note that, for SF ∈ {SUM, MAX}, PP PNE (SF, T * w , σ * , o * ) is, in general, under-constrained. In fact, in order to assure that σ * and o * are the worst ǫ-approximate pure Nash equilibrium and the social optimum, respectively, one should guarantee β-SF(σ * ) ≥ β-SF(σ) for each other ǫ-approximate pure Nash equilibrium σ ∈ Σ * , if any, and β-SF(o * ) ≤ β-SF(σ) for each σ ∈ Σ * . Moreover, the normalizing constraints have also been relaxed so as to assure β-SF(o * ) ≤ 1 rather than β-SF(o * ) = 1. Anyway, as we will discuss in the proof of Lemma 2, either removing or relaxing these constraints can only worsen the resulting upper bounds. The significance of the previously defined pairs of primal-dual formulations is witnessed by the following lemma which states that the value of an optimal solution to PP PNE (SF, T * w , σ * , o * ) provides an upper bound on PPoA ǫ (β-SF, C w (F, α)). If PP PNE (SF, T * w , σ * , o * ) is unlimited, then x = ∞ and the claim is trivially true. So, we can assume that PP PNE (SF, T * w , σ * , o * ) admits an optimal solution of value x. As we have already observed, PP PNE (SF, T * w , σ * , o * ) may be under-constrained. Nevertheless, recall that we are only interested in an upper bound on the worst-case ǫ-approximate pure price of anarchy of the class C T * w (F, α) attainable when the latency functions are suitably tuned so as to make σ * the worst ǫ-approximate pure Nash equilibrium and o * a social optimum (of social value 1). Let us denote with C T * w (F, α) such a subclass of C T * w (F, α). Hence, since once fixed the profiles σ * and o * any game in C T * w (F, α) can be specified by a particular choice of the values v e k , and because the removal or the relaxation of some constraints in a maximization problem can only increase the value of the optimal solution, we obtain that the optimal solution to PP PNE (SF, T * w , σ * , o * ) yields an upper bound on the worst-case ǫ-approximate pure price of anarchy of the class C T * w (F, α). That is, PPoA ǫ (β-SF, C T * w (F, α)) ≤ x. Moreover, since the optimal solution to PP PNE (SF, T * w , σ * , o * ) has value x, then, by the Strong Duality Theorem, each optimal solution (y * , γ * ) to DP PNE (SF, T * w , σ * , o * ) satisfies x = γ * . By Property 1, the particular combinatorial structure of the pair σ * and o * implies that, for any alternative pair of strategy profiles σ and o in T * w , the set of constraints of the dual program DP PNE (SF, T * w , σ, o) is a subset of that of DP PNE (SF, T * w , σ * , o * ). This implies that any optimal solution (y, γ) to DP PNE (SF, T * w , σ, o) must obey γ ≤ γ * . Thus, one can claim that γ * = x is indeed an upper bound on the worst-case ǫ-approximate pure price of anarchy of the class C T * w (F, α), that is, PPoA ǫ (β-SF, C T * w (F, α)) ≤ x. Note also that, again by Property 1, for any other congestion model T w = ([n], w, E, (Σ i ) i∈[n] ), the pair of primal-dual formulations PP PNE (SF, T w , σ, o) and DP PNE (SF, T w , σ, o) induced by any pair of strategy profiles σ, o ∈ Σ(T w ) are such that the set of constraints of DP PNE (SF, T w , σ, o) is again a subset of that of DP PNE (SF, T * w , σ * , o * ) and this implies that γ * = x is even an upper bound on the worst-case ǫ-approximate pure price of anarchy of the whole class C w (F, α) = Tw C Tw (F, α), that is, PPoA ǫ (β-SF, C w (F, α)) ≤ x. Step 3) Proof of existence of a game (CG, α) ∈ C w (F, α) such that PPoA ǫ (β-SF, CG, α) = x. PP PNE (SF, T * w , σ * , o * ) , it follows that σ * is an ǫ-approximate pure Nash equilibrium for (CG, α) such that β-SF(σ * ) = x. This implies PPoA ǫ (β-SF, C w (F, α)) ≥ x (recall, in fact, that β-SF(o * ) ≤ 1). In the case in which PP PNE (SF, T * w , σ * , o * ) admits an optimal solution SOL SF of value x, by the above argument, it follows that PPoA ǫ (β-SF, C w (F, α)) ≥ x, which, together with Lemma 2, implies the claim. In the case in which PP PNE (SF, T * w , σ * , o * ) is unlimited, then, for any x ∈ R, there exists a feasible solution SOL SF to PP PNE (SF, T * w , σ * , o * ) of value at least x, which implies that, for any x ∈ R, it holds that PPoA ǫ (β-SF, C w (F, α)) ≥ x. Step 4) Definition of a primal-dual formulation for CCPoA ǫ (β-SF, C w (F, α)) and proof of the "Extension Lemma". Fix a congestion model T w = ([n], w, E, (Σ i ) i∈[n] ), a probability distribution p ∈ ∆(Σ(T w )) and a strategy profile o ∈ Σ(T w ). We define the following primal program PP CCE (SUM, T w , p, o) for the β-utilitarian social function. maximize σ∈Σ p σ e∈E k∈[r] v e k f k (n e (σ)) i∈[n] j∈[n]:e∈σ j β ij w j subject to σ∈Σ p σ e∈σ i \o i k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j α ij w j −(1 + ǫ) σ∈Σ p σ e∈o i \σ i k∈[r] v e k f k (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   ≤ 0, ∀i ∈ [n] e∈E k∈ [r] v e k f k (n e (o)) i∈[n] j∈[n]:e∈o j β ij w j ≤ 1, v e k ≥ 0, ∀e ∈ E, k ∈ [r] The dual program DP CCE (SUM, T w , p, o) is the following (again, we associate a variable y i with the ith constraint of the first n ones and a variable γ with the normalizing constraint). minimize γ subject to σ∈Σ p σ i∈[n]:e∈σ i \o i y i f k (n e (σ)) j∈[n]:e∈σ j α ij w j −(1 + ǫ) σ∈Σ p σ i∈[n]:e∈o i \σ i y i f k (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   +γf k (n e (o)) i∈[n] j∈[n]:e∈o j β ij w j ≥ σ∈Σ p σ f k (n e (σ)) i∈[n] j∈[n]:e∈σ j β ij w j , ∀e ∈ E, k ∈ [r] y i ≥ 0, ∀i ∈ [n] γ ≥ 0 Similarly, for the β-egalitarian social function, the primal program PP CCE (MAX, T w , p, o) is defined as follows. maximize t subject to σ∈Σ p σ e∈σ i \o i k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j α ij w j −(1 + ǫ) σ∈Σ p σ e∈o i \σ i k∈[r] v e k f k (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   ≤ 0, ∀i ∈ [n] σ∈Σ p σ e∈E k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j β 1j w j = t, σ∈Σ p σ e∈E k∈[r] v e k f k (n e (σ)) j∈[n]:e∈σ j β ij w j ≤ t, ∀i ∈ [n] \ {1} e∈E k∈[r] v e k f k (n e (o)) j∈[n]:e∈o j β ij w j ≤ 1, ∀i ∈ [n] v e k ≥ 0, ∀e ∈ E, k ∈ [r] t ≥ 0 The dual program DP CCE (MAX, T w , p, o) is the following (again, we associate variables y i , z i and γ i with the ith constraint of the first, the middle and the last family of n constraints, respectively). y i f k (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   + σ∈Σ p σ f k (n e (σ)) i∈[n] z i j∈[n]:e∈σ j β ij w j +f k (n e (o)) i∈[n] γ i j∈[n]:e∈o j β ij w j ≥ 0, ∀e ∈ E, k ∈ [r] i∈N z i ≤ −1 y i , z i , γ i ≥ 0, ∀i ∈ [n] Again, even though both PP CCE (SUM, T w , p, o) and PP CCE (MAX, T w , p, o) may be, in general, under-constrained, by the same arguments used in the discussion of the pairs of primal-dual formulations used for bounding the worst-case ǫ-approximate pure price of anarchy, it follows that, for each function SF ∈ {SUM, MAX}, the optimal solution to PP CCE (SF, T w , p, o) yields an upper bound on the worst-case ǫ-approximate coarse correlated price of anarchy of the class C Tw (F, α) attainable when p is taken for the worst ǫ-approximate coarse correlated equilibrium and o for the social optimum (of social value 1). Let us denote such a class with C Tw (F, α). The following lemma shows that any upper bound on PPoA ǫ (β-SF, C T * w (F, α)) proved via the primal-dual method automatically extends to CCPoA ǫ (β-SF, C Tw (F, α)). Lemma 4 (Extension Lemma) For any function SF ∈ {SUM, MAX}, congestion model T w = ([n], w, E, (Σ i ) i∈[n] ), probability distribution p ∈ ∆(Σ(T w )) and strategy profile o ∈ Σ(T w ), it holds that any feasible solution to DP PNE (SF, T * w , σ, o) is also a feasible solution to DP CCE (SF, T w , p, o). Proof: Let (y * , γ * ) be a feasible solution to DP PNE (SUM, T * w , σ, o). By Property 1 of the representative congestion model T * w , it follows that, for any pair of strategy profiles σ, o ∈ Σ(T w ), it holds that i∈[n]:e∈σ i \o i y * i f k (n e (σ)) j∈[n]:e∈σ j α ij w j − (1 + ǫ) i∈[n]:e∈o i \σ i y * i f k (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   + γ * f k (n e (o)) i∈[n] j∈[n]:e∈o j β ij w j ≥ f k (n e (σ)) i∈[n] j∈[n]:e∈σ j β ij w j (2) for any e ∈ E and k ∈ [r]. Since p σ ≥ 0 for each σ ∈ Σ(T w ), by multiplying inequality (2) for p σ and then summing up the obtained inequalities for each σ ∈ Σ(T w ), we obtain that, for each e ∈ E and k ∈ [r], it holds that σ∈Σ p σ i∈[n]:e∈σ i \o i y * i f k (n e (σ)) j∈[n]:e∈σ j α ij w j − (1 + ǫ) σ∈Σ p σ i∈[n]:e∈o i \σ i y * i f k (n e (σ) + w i )   α ii w i + j∈[n]:e∈σ j α ij w j   + γ * f k (n e (o)) i∈[n] j∈[n]:e∈o j β ij w j σ∈Σ p σ ≥ σ∈Σ p σ f k (n e (σ)) i∈[n] j∈[n]:e∈σ j β ij w j .(3) By σ∈Σ p σ = 1, it follows that, for any e ∈ E and k ∈ [r], inequality (3) coincides with the relative dual constraint of DP CCE (SUM, T w , p, o) and this shows that the solution (y * , γ * ) is also feasible for DP CCE (SUM, T w , p, o). A similar argument shows the claim for the case of the social function β-MAX. We now have all the ingredients needed to conclude the proof of the theorem. Fix a function SF ∈ {SUM, MAX}. Assume, first, that PP PNE (SF, T * w , σ * , o * ) is unlimited. Then, by Lemma 3, it holds that PPoA ǫ (β-SF, C w (F, α)) = ∞ which, together with PPoA ǫ (β-SF, C w (F, α)) ≤ CCPoA ǫ (β-SF, C w (F, α)), immediately implies that PPoA ǫ (β-SF, C w (F, α)) = CCPoA ǫ (β-SF, C w (F, α)). By applying Lemma 1, we obtain PPoA ǫ (β-SF, C w (F, α)) = CCPoA ǫ (β-SF, C w (F, α)). In the case in which PP PNE (SF, T * w , σ * , o * ) admits an optimal solution of value x, by Lemma 3, it holds that PPoA ǫ (β-SF, C w (F, α)) = x. Moreover, by the Strong Duality Theorem, there exists a feasible solution (y * , γ * ) to DP PNE (SF, T * w , σ, o) of value γ * = x. Choose an arbitrary game (CG, α) ∈ C w (F, α) such that CCPoA ǫ (β-SF, C w (F, α)) = CCPoA ǫ (β-SF, CG, α) and let T w be the congestion model defining CG, p be the worst ǫ-approximate coarse correlated equilibrium of (CG, α) and o be the social optimum (of social value 1). By the definition of T w , p and o, it follows that the optimal solution to PP CCE (SF, T w , p, o) has a value of at least CCPoA ǫ (β-SF, C w (F, α)), which, by the Weak Duality Theorem, implies in turn that any feasible solution to DP CCE (SF, T w , p, o) has a value of at least CCPoA ǫ (β-SF, C w (F, α)). By Lemma 4, it follows that (y * , γ * ) is also a feasible solution to DP CCE (SF, T w , p, o). This implies that CCPoA ǫ (β-SF, C w (F, α)) ≤ γ * = x = PPoA ǫ (β-SF, C w (F, α)). Again, by applying Lemma 1, we obtain that PPoA ǫ (β-SF, C w (F, α)) = CCPoA ǫ (β-SF, C w (F, α)). It is clear from our discussion that the value PPoA ǫ (β-SF, C w (F, α)) = PPoA ǫ (β-SF, C w (F, α)) can always be (theoretically) determined via the primal-dual method, that is, by computing the value of the optimal solution of either the primal program PP PNE (SF, T * w , σ, o) or the dual one DP PNE (SF, T * w , σ, o) for each function SF ∈ {SUM, MAX}, and this concludes the proof (solving the dual program, in particular, requires to determine the minimum value γ * for which all the r · 4 n possible constraints induced by the |E * | = 4 n pairs of values yielded by the representative congestion model T * w on each of the r components of the latency functions are satisfied). Conclusions and Open Problems By introducing the notions of smooth games and robust price of anarchy, Roughgarden [19,20] showed that the class of congestion games with non-negative and non-decreasing latency functions is tight under the utilitarian social function (see Section 5 of the Appendix for formal definitions). This result has been extended to the class of weighted congestion games by Bhawalkar, Gairing and Roughgarden [3]. By exploiting the primal-dual method we introduced in [4], we have generalized this result along four directions. In fact, our tightness result holds for the class of generalized weighted congestion games, for generalizations of both the utilitarian and the egalitarian social functions, for any non-negative (and possibly decreasing) latency functions and for the approximated version of the price of anarchy. The fact that two different and seemingly uncorrelated approaches may produce the same type of general results is quite interesting. Understanding whether there is some kind of relationships between them is an intriguing question. Both approaches set up some machinery (smoothness argument vs. primal-dual formulation) allowing for the proof of significant upper bounds on the pure price of anarchy of the games under analysis and then make use of an extension theorem to show that such bounds extend to the coarse correlated price of anarchy as well. In particular, how is this last step achieved? Note that the proof of the smoothness argument (see the proof of Lemma 5) requires the definition of smoothness to hold only for any pair of strategy profiles (σ, σ ′ ) such that σ is a pure Nash equilibrium and σ ′ is a social optimum. The reason why the definition of smoothness is extended to encompass all possible pairs of strategy profiles is due to the fact that it is indeed the proof of the extension theorem (see the proof of Theorem 2) that asks for such a stronger hypothesis. Finally, being a coarse correlated equilibrium a particular probability distribution defined over the set of strategy profiles, the notion of smoothness characterizing each profile in the support of any such an equilibrium can be suitably exploited by the linearity of expectation. In the primal-dual method, instead, the higher degree of generality that is needed to move from pure Nash equilibria up to coarse correlated equilibria is provided by the representative congestion model which imposes that the variables yielding a feasible solution to the dual formulation for the pure price of anarchy have to satisfy any type of "pure dual constraint" that may eventually arise by considering all possible types of configurations of congestions. Then, since it turns out that the dual constraint characterizing the dual formulation for the coarse correlated price of anarchy is indeed a convex combination of a subset of all the possible "pure dual constraints", the extension theorem follows immediately. Anyway, there is an major difference between the two methods when one aims at showing the tightness of a particular class of games. When adopting the smoothness framework, after having proved that a class of games is (λ, µ)-smooth for a certain pair of parameters λ and µ, one has to show that there exists a game in the class for which the pure price of anarchy is indeed λ 1−µ , that is, that the price of anarchy of pure Nash equilibria matches the robust price of anarchy. We stress that this step can be avoided when adopting the primal-dual formulation, since it is directly implied by the Duality Theory (see Lemma 3). In fact, note that the notion of robust price of anarchy, as the best possible upper bound on the pure price of anarchy achievable via the smoothness argument, has no correspondent in the primal-dual method where this bound is implicitly defined by the optimal solution of the pair of primal-dual programs. By summarizing, our findings seem to reveal that the primal-dual method may be superior to the smoothness framework within the realm of weighted congestion games and their possible generalizations, but, at the same time, the primal-dual method has never been exploited so far outside this realm. Hence, a good starting point would be that of trying to export it to other scenarios of investigation in which the smoothness framework has already been fruitfully applied, such as, for instance, the quantification of the price of anarchy in unrelated scheduling games, valid utility games, opinion formation games and auction theory. Proof: Fix a congestion model T w = ([n], w, E, (Σ i ) i∈[n] ), a resource e ∈ E and pair of profiles σ ′ , σ ′′ ∈ Σ(T w ). Let {i ∈ [n] | e ∈ σ ′ i } := P and {i ∈ [n] | e ∈ σ ′′ i } := Q. To prove the claim, it suffices choosing e = e(P, Q). f k (n e (σ * ) + w i ) k (n e (o * )) i∈[n] j∈[n]:e∈o * j β ij w j ≥ f k (n e (σ * )) i∈[n] j∈[n]:e∈σ * j β ij w j , ∀e ∈ E * , k ∈ [r] y i ≥ 0, ∀i ∈ [n] γ ≥ 0Similarly, the linear program PP PNE (MAX, T * w , σ * , o * ) for the β-egalitarian social function is defined as follows. Lemma 2 2For a fixed SF ∈ {SUM, MAX}, let x be the value of an optimal solution to PP PNE (SF, T * w , σ * , o * ) when this linear problem is not unlimited, otherwise let x = ∞. ThenPPoA ǫ (β-SF, C w (F, α)) ≤ x. Proof: We first show that PP PNE (SUM, T * w , σ * , o * ) and PP PNE (MAX, T * w , σ * , o * ) are both feasible. In fact, fixed an index k * ∈ [r], k (w j )w j i∈[n] β ij−1 if k = k * and e ∈ {e({j}, ∅), e(∅, {j})} for some j ∈ [n], 0 otherwiseis feasible for PP PNE (SUM, T * w , σ * , o * ) and yields an objective value equal to 1. Similarly, assuming, for instance, that the players are numbered in such a way that i∈[n] β 1i ≥ i∈[n] β ji for each j ∈ [n] \ {1}, the solution with t = k = k * and e ∈ {e({j}, ∅), e(∅, {j})} for some j ∈ [for PP PNE (MAX, T * w , σ * , o * ) and yields an objective value equal to 1. Hence, for any SF ∈ {SUM, MAX}, exactly one of the two cases included in the claim may occur. Lemma 3 3For a fixed SF ∈ {SUM, MAX}, let x be the value of an optimal solution to PP PNE (SF, T * w , σ * , o * ) when this linear problem is not unlimited, otherwise let x = ∞. ThenPPoA ǫ (β-SF, C w (F, α)) = x. Proof: Assume that PP PNE (SF, T * w , σ * , o * ) admits a feasible solution SOL SUM = ( v e k ) e∈E * ,k∈[r] or SOL MAX = ( v e k ) e∈E * ,k∈[r] , t , both of value x, depending on which is the value of SF. Consider the game (CG, α), where CG = (T * w , ( v e k ) e∈E * ,k∈[r] ) is defined by the representative congestion model T * w coupled with the values ( v e k ) e∈E * ,k∈[r] . Since, for any SF ∈ {SUM, MAX}, SOL SF is feasible for n]:e∈o i \σ i Throughout the paper, we implicitly assume that all games under consideration are cost minimization ones. All the claimed properties and results can be applied mutatis mutandis to the case of payoff maximization games. (this hypothesis is without loss of generality up to a renumbering of the players) and has value t (which is the objective function to be maximized), while the last n constraints normalizes to at most 1 the value β-MAX(o * ). Appendix 5 The Smoothness Argument and the Robust Price of AnarchyLet G = [n], (Σ i ) i∈[n], (c i ) i∈[n] be a cost minimization game defined by the set of players [n], the set of strategies Σ i and the individual cost function c i :The connection between the notion of smoothness and that of pure price of anarchy is captured by the following lemma.Proof: Let σ be any pure Nash equilibrium for G and o be a social optimum for G under SF. It holds thatand the claim follows by rearranging the terms.The robust price of anarchy is then defined as the best possible upper bound on the pure price of anarchy that can be proved via the smoothness argument. For a game G and a social function SF, we denote with A SF (G) the set of parameters (λ, µ) such that G is (λ, µ)-smooth under SF.Definition 4 (Robust Price of Anarchy) Given a sum-bounded social function SF, the robust price of anarchy of G under SF, is the value ρ SF (G) = inf λ 1−µ : (λ, µ) ∈ A SF (G) .The power of the smoothness argument is then stressed by the following extension theorem.Theorem 2 (Extension Theorem) For each cost minimization game G and sum-bounded social function SF for G, it holds that CCPoA(SF, G) ≤ ρ SF (G).Proof: Let p be any coarse correlated equilibrium for G and o be a social optimum for G under SF. It holds thatand the claim follows by rearranging the terms.Let C be a class of cost minimization games and C ⊆ C be the subclass of the games in C which admit at least one pure Nash equilibrium. Given a social function SF, we denote with A SF (C) the set of parameters (λ, µ) such that each game G ∈ C is (λ, µ)-smooth under SF. Exact price of anarchy for polynomial congestion games. S Aland, D Dumrauf, M Gairing, B Monien, F Schoppmann, SIAM Journal on Computing. 405S. Aland, D. Dumrauf, M. Gairing, B. Monien, and F. Schoppmann. Exact price of anarchy for polynomial congestion games. SIAM Journal on Computing, 40(5):1211-1233, 2011. The price of routing unsplittable flow. B Awerbuch, Y Azar, L Epstein, Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC). the 37th Annual ACM Symposium on Theory of Computing (STOC)ACM PressB. Awerbuch, Y. Azar, and L. Epstein. The price of routing unsplittable flow. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), ACM Press, pp. 57-66, 2005. Weighted congestion games: price of anarchy, universal worst-case examples, and tightness. K Bhawalkar, M Gairing, T Roughgarden, Proceedings of the 18th Annual European Symposium on Algorithms (ESA). the 18th Annual European Symposium on Algorithms (ESA)Springer6346K. Bhawalkar, M. Gairing, and T. Roughgarden. Weighted congestion games: price of anarchy, universal worst-case examples, and tightness. In Proceedings of the 18th Annual European Symposium on Algorithms (ESA), LNCS 6346, Springer, pp. 17-28, 2010. A unifying tool for bounding the quality of non-cooperative solutions in weighted congestion games. V Bilò, Proceedings of the 10th Workshop on Approximation and Online Algorithms (WAOA). the 10th Workshop on Approximation and Online Algorithms (WAOA)Springer7846V. Bilò. A unifying tool for bounding the quality of non-cooperative solutions in weighted con- gestion games. In Proceedings of the 10th Workshop on Approximation and Online Algorithms (WAOA), LNCS 7846, Springer, pp. 229-241, 2012. On linear congestion games with altruistic social context. V Bilò, Proceedings of the 20th International Computing and Combinatorics Conference (COCOON). the 20th International Computing and Combinatorics Conference (COCOON)Springer8591V. Bilò. On linear congestion games with altruistic social context. In Proceedings of the 20th International Computing and Combinatorics Conference (COCOON), LNCS 8591, Springer, pp. 547-558, 2014.. On lookahead equilibria in linear congestion games. V Bilò, A Fanelli, L Moscardelli, Proceedings of the 9th International Workshop On Internet And Network Economics (WINE). the 9th International Workshop On Internet And Network Economics (WINE)Springer8289V. Bilò, A. Fanelli, and L. Moscardelli. On lookahead equilibria in linear congestion games. In Proceedings of the 9th International Workshop On Internet And Network Economics (WINE), LNCS 8289, Springer, pp. 54-67, 2013. On bidimensional congestion games. V Bilò, M Flammini, V Gallotti, Proceedings of the 19th International Colloquium on Structural Information and Communication Complexity (SIROCCO). the 19th International Colloquium on Structural Information and Communication Complexity (SIROCCO)Springer7355V. Bilò, M. Flammini, and V. Gallotti. On bidimensional congestion games. In Proceedings of the 19th International Colloquium on Structural Information and Communication Complexity (SIROCCO), LNCS 7355, Springer, pp. 147-158, 2012. Some anomalies of farsighted strategic behavior. Theory of Computing Systems. V Bilò, M Flammini, G Monaco, L Moscardelli, to appearV. Bilò, M. Flammini, G. Monaco, and L. Moscardelli. Some anomalies of farsighted strategic behavior. Theory of Computing Systems, to appear. On the performance of mildly greedy players in cut games. V Bilò, M Paladini, Proceedings of the 20th International Computing and Combinatorics Conference (COCOON). the 20th International Computing and Combinatorics Conference (COCOON)Springer8591V. Bilò and M. Paladini. On the performance of mildly greedy players in cut games. In Proceed- ings of the 20th International Computing and Combinatorics Conference (COCOON), LNCS 8591, Springer, pp. 513-524, 2014. The price of anarchy of finite congestion games. G Christodoulou, E Koutsoupias, Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC). the 37th Annual ACM Symposium on Theory of Computing (STOC)ACM PressG. Christodoulou and E. Koutsoupias. The price of anarchy of finite congestion games. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), ACM Press, pp. 67-73, 2005. On the price of anarchy and stability of correlated equilibria of linear congestion games. G Christodoulou, E Koutsoupias, Proceedings of the 13th Annual European Symposium on Algorithms (ESA). the 13th Annual European Symposium on Algorithms (ESA)Springer3669G. Christodoulou and E. Koutsoupias. On the price of anarchy and stability of correlated equilibria of linear congestion games. In Proceedings of the 13th Annual European Symposium on Algorithms (ESA), LNCS 3669, Springer, pp. 59-70, 2005. On the performance of approximate equilibria in congestion games. G Christodoulou, E Koutsoupias, P G Spirakis, Algorithmica. 611G. Christodoulou, E. Koutsoupias, and P. G. Spirakis. On the performance of approximate equilibria in congestion games. Algorithmica, 61(1):116-140, 2011. Inefficiency of games with social context. B De Keijzer, Guido Schäfer, A Anagnostopoulos, L Becchetti, Proceedings of the 6th International Symposium on Algorithmic Game Theory (SAGT). the 6th International Symposium on Algorithmic Game Theory (SAGT)Springer8146B. de Keijzer, Guido Schäfer, A. Anagnostopoulos, and L. Becchetti. Inefficiency of games with social context. In Proceedings of the 6th International Symposium on Algorithmic Game Theory (SAGT), LNCS 8146, Springer, pp. 219-230, 2013. Worst-case equilibria. E Koutsoupias, C Papadimitriou, Proceedings of the 16th International Symposium on Theoretical Aspects of Computer Science (STACS). the 16th International Symposium on Theoretical Aspects of Computer Science (STACS)Springer1653E. Koutsoupias and C. Papadimitriou. Worst-case equilibria. In Proceedings of the 16th In- ternational Symposium on Theoretical Aspects of Computer Science (STACS), LNCS 1653, Springer, pp. 404-413, 1999. Potential games. D Monderer, L S Shapley, Games and Economic Behavior. 141D. Monderer and L. S. Shapley. Potential games. Games and Economic Behavior, 14(1):124- 143, 1996. Equilibrium points in n-person games. J F Nash, Proceedings of the National Academy of Science. 361J. F. Nash. Equilibrium points in n-person games. Proceedings of the National Academy of Science, 36(1):48-49, 1950. Bounding the inefficiency of altruism through social contribution games. M Rahn, G Schäfer, Proceedings of the 9th International Conference on Web and Internet Economics (WINE). the 9th International Conference on Web and Internet Economics (WINE)Springer8289M. Rahn and G. Schäfer. Bounding the inefficiency of altruism through social contribution games. In Proceedings of the 9th International Conference on Web and Internet Economics (WINE), LNCS 8289, Springer, pp. 391-404, 2013. A class of games possessing pure-strategy Nash equilibria. R W Rosenthal, International Journal of Game Theory. 21R. W. Rosenthal. A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory, 2(1):65-67, 1973. Intrinsic robustness of the price of anarchy. T Roughgarden, Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC). the 41st Annual ACM Symposium on Theory of Computing (STOC)ACM PressT. Roughgarden. Intrinsic robustness of the price of anarchy. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), ACM Press, pp. 513-522, 2009. Intrinsic robustness of the price of anarchy. T Roughgarden, Communications of the ACM. 557T. Roughgarden. Intrinsic robustness of the price of anarchy. Communications of the ACM, 55(7):116-123, 2012. The value of n-person games. L S Shapley, Contributions to the theory of games. Princeton University PressL. S. Shapley. The value of n-person games. Contributions to the theory of games, Princeton University Press, pp. 31-40, 1953.
[]
[ "DEUTSCHES ELEKTRONEN-SYNCHROTRON On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation", "DEUTSCHES ELEKTRONEN-SYNCHROTRON On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation" ]
[ "Ein Forschungszentrum Der Helmholtz-Gemeinschaft ", "Gianluca Geloni ", "Vitali Kocharyan ", "Evgeni Saldin ", "Gianluca Geloni \nEuropean XFEL GmbH\nHamburgGermany\n", "Vitali Kocharyan \nDeutsches Elektronen-Synchrotron (DESY)\nHamburgGermany\n", "Evgeni Saldin \nDeutsches Elektronen-Synchrotron (DESY)\nHamburgGermany\n", "\nDeutsches Elektronen-Synchrotron DESY\nEuropean XFEL GmbH\nHamburg, Hamburg\n" ]
[ "European XFEL GmbH\nHamburgGermany", "Deutsches Elektronen-Synchrotron (DESY)\nHamburgGermany", "Deutsches Elektronen-Synchrotron (DESY)\nHamburgGermany", "Deutsches Elektronen-Synchrotron DESY\nEuropean XFEL GmbH\nHamburg, Hamburg" ]
[]
In this paper we compare experimental observations and theory of radiation emission from a microbunched beam with microbunching wavefront tilt with respect to the direction of motion. The theory refers to the work [1], which predicts, in this case, exponential suppression of coherent radiation along the kicked direction. The observations refer to a recent experiment performed at the LCLS [2, 3], where a microbunched beam was kicked by a bend and sent to a radiator undulator. The experiment resulted in the emission of strong coherent radiation that had its maximum along the kicked direction of motion, when the undulator parameter was detuned to a value larger than the nominal one. We first analyze the theory in detail, and we confirm the correctness of its derivation according to the conventional theory of radiation emission from charged particles. Subsequently, we look for possible peculiarities in the experiment, which may not be modeled by the theory. We show that only spurious effects are not accounted for. We conclude that the experiment defies explanation in terms of the conventional theory of radiation emission.
10.1016/j.optcom.2017.10.010
[ "https://arxiv.org/pdf/1706.10185v1.pdf" ]
119,189,612
1706.10185
6d653b48abdd47e2c432be2f1e8c6d0c293b6666
DEUTSCHES ELEKTRONEN-SYNCHROTRON On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation June 2017 29 Jun 2017 Ein Forschungszentrum Der Helmholtz-Gemeinschaft Gianluca Geloni Vitali Kocharyan Evgeni Saldin Gianluca Geloni European XFEL GmbH HamburgGermany Vitali Kocharyan Deutsches Elektronen-Synchrotron (DESY) HamburgGermany Evgeni Saldin Deutsches Elektronen-Synchrotron (DESY) HamburgGermany Deutsches Elektronen-Synchrotron DESY European XFEL GmbH Hamburg, Hamburg DEUTSCHES ELEKTRONEN-SYNCHROTRON On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation On Radiation Emission from a Microbunched Beam with Wavefront Tilt and Its Experimental Observation June 2017 29 Jun 2017 In this paper we compare experimental observations and theory of radiation emission from a microbunched beam with microbunching wavefront tilt with respect to the direction of motion. The theory refers to the work [1], which predicts, in this case, exponential suppression of coherent radiation along the kicked direction. The observations refer to a recent experiment performed at the LCLS [2, 3], where a microbunched beam was kicked by a bend and sent to a radiator undulator. The experiment resulted in the emission of strong coherent radiation that had its maximum along the kicked direction of motion, when the undulator parameter was detuned to a value larger than the nominal one. We first analyze the theory in detail, and we confirm the correctness of its derivation according to the conventional theory of radiation emission from charged particles. Subsequently, we look for possible peculiarities in the experiment, which may not be modeled by the theory. We show that only spurious effects are not accounted for. We conclude that the experiment defies explanation in terms of the conventional theory of radiation emission. Introduction and motivation The theory of spontaneous emission from a microbunched beam with wavefront tilt with respect to the velocity of propagation has been developed several years ago [1]. When the microbunching wavefront tilt becomes larger than the coherence angle, this theory predicts a dramatic suppression of coherent emission in the beam propagation direction. In relation to this result, a recent experiment at the LCLS [2,3] yielded remarkable outcomes. The LCLS generates linearly polarized X-ray pulses from a planar undulator. A 3.2 m-long Delta undulator, which allows for a full control of the degree of polarization of the emitted radiation, was recently installed in place of the last LCLS undulator segment. Before going through the Delta undulator, the electron beam is microbunched in the preceding planar undulator segments. This enhances the radiation power by several orders of magnitude. Therefore, the Delta undulator is said to be operating in 'afterburner configuration'. Such configuration leads to the presence of linearlypolarized background radiation from the main undulator, which should be suppressed. In fact, when the efficiency of the regular afterburner mode of operation was tested, a maximum contrast ratio of about 2.5 was achieved [2]. It has been recently proposed [4] that the background radiation component can be greatly reduced by a reverse undulator tapering configuration. By inverting the sign of the baseline undulator tapering the radiation emission is reduced, while microbunching can still develop. The efficiency of this mode of operation was tested and a contrast ratio of about 10 was reported in [2]. From a practical viewpoint, under this conditions, at the entrance of the Delta undulator there is only a micro-bunched beam. References [2,3] further report a final improvement of the degree of polarization up to 100% by X-ray beam splitting at the photon energy of 0.7 keV. This was achieved by kicking the electron beam before entering the Delta undulator, in order to let electron beam and background radiation pass through the Delta undulator at different angles. The quadrupole at the end of the last planar undulator section includes a regular vertical corrector, which was used to control the magnitude of the kick. According to [2], the maximal kick angle was about 3 × 10 −5 rad and was limited only by the 4 mm diameter of the beamline aperture at the distance of 80 m. At this maximum angle, the separation between the two radiation spots on the screen in the experimental hall was about 5 rms times the radiation spot size. Moreover, the energy of the output radiation pulse with and without kick is practically the same. In order to explain this observation in relation with the theory in [1], one would conclude that the LCLS experiment apparently shows a readjustment of the microbunching orientation in the kicked direction. In this way, one could produce coherent radiation in the kicked direction. However, classical particle tracking shows that while the electron beam direction changes after the kick, the orientation of the microbunching wavefront stays unvaried. Therefore, the electron motion and the wavefront normal have different directions. Figure 1 illustrates the issue. If one assumes that no readjustment of the microbunching wavefront takes place, according to [1] the FEL process in the downstream undulator is expected to be dramatically suppressed because the kick angle is larger than the divergence of the output coherent radiation, in contrast with the experimental observation. In order to estimate the loss of radiation efficiency we make the assumption that the spatial profile of the bunching factor is close to that of the electron beam and has a Gaussian shape with standard deviation σ b . A bunched Fig. 1. Illustration of the problem, which arises according to classical particle tracking when a microbunched electron beam is deflected by a dipole magnet. After passing the dipole, the microbunching is preserved, but only along its original direction. electron beam in an FEL amplifier can be considered as a sequence of periodically spaced oscillators. The radiation produced by these oscillators always interferes coherently at zero angle with respect to the undulator axis. In the limit for a small size of the electron beam the interference will be constructive within an angle of about ∆θ c c/(ωL g ), where L g is the FEL gain length. In the limit for a large size of the electron beam, the angle of coherence is about ∆θ c c/(ωσ b ) instead. The boundary between these two asymptotes is for sizes of about σ dif cL g /ω. It is worth noting that the condition σ 2 b σ 2 dif is satisfied in our case of study at the LCLS. Thus, we can conclude that the angular distribution of the radiation power in the far zone has a Gaussian shape with standard deviation σ c c/( √ 2ωσ b ). After the electron beam is kicked, as already mentioned, in classical treatments we have a discrepancy between direction of the electron motion and wavefront normal. Then, the radiation intensity along the new direction of the electron beam can be approximated as I I 0 exp[−θ 2 /(2σ 2 c )], where I 0 is the on-axis intensity without kick and θ is the kick angle. The exponential suppression factor is due to the microbunching wavefront tilt with respect to the direction of motion of the electrons. Beam splitting at the LCLS was done by kicking the electron beam of an angle of about 5 standard deviations of the intensity distribution in the far zone. According to the estimations presented above, the intensity of the coherent radiation in the kicked direction should be suppressed by two orders of magnitude. In spite of this, the experiment showed that the radiation intensity in the kicked direction is practically the same as the intensity without kick at zero angle. In addition to exponential suppression of the intensity, one expects negligible detuning effects in the case of radiation emitted along the direction of the kick. In fact, the effective undulator period is now given by λ w / cos(θ) (1 + θ 2 /2)λ w , where λ w is the actual undulator period. This induces a relative red shift in the resonance wavelength of about ∆λ/λ θ 2 /2 which should be compared with the relative bandwidth of the resonance, the ρ parameter, which is much larger. As a result, the red shift in the res-onance wavelength due to the kick can be neglected in all situations of practical relevance. It is clear from the above that if a microbunched beam is at perfect resonance along the direction of motion without kick, then after the kick the same microbunched beam is at perfect resonance along the new direction of the electron beam motion. However, references [2,3] report that the radiation in the kicked direction is red-shifted with respect to the case when no kick is applied. This experimental result is in contradiction with the theory [1]. In this paper we analyze such contradiction in detail. Logically speaking there are three possibilities to explain this contradiction. First, the theory in [1] is, for some reason, incorrect. Second, the theory in [1] does not model the actual experiment, i.e. there are some peculiarities of the experiment that are not accounted for in the theory. In this case, such peculiarity could be, for example, a readjustment of the microbunching wavefront, which is not foreseen according to usual particle tracking. Third, there are no peculiarities in the experiment that are neglected in the theory in [1], and the theory is correctly derived on the basis of the usual laws of electrodynamics and dynamics of charged particles, so there must be some more fundamental reason for the discrepancy observed. In the next Section we critically review the theory in [1] and we show that it is correctly derived according to the usual laws of electrodynamics and dynamics of charged particles. Subsequently we discuss possible reasons why the experiment might not be fully modeled by this theory. Our conclusion is that there are none: in our view, a readjustment in the microbunching wavefront due to neglected dynamical effects is to be excluded. We argue that, in order to explain the reason why the experiment in [2,3] is in disagreement with the result by [1], more fundamental reasons should be invoked. Undulator radiation from a microbunched electron beam Let us critically review the theory in [1]. We consider an electron beam modulated in density at a single frequency ω as source of coherent undulator radiation. We write the longitudinal current density j z as a sum of two terms: a constant unperturbed term, j o , and a term describing the actual modulation at frequency ωj z , to be considered as a perturbation: j z (z, r ⊥ , t) = j o (z, r ⊥ ) +j z (z, r ⊥ , t) ,(1) with r ⊥ a two-dimensional vector fixing the transverse coordinates and t the time. We follow [5] write the unperturbed part j o as as j o (z, r ⊥ ) = j o ( r ⊥ − r (c) ⊥ (z)) ,(2) where r (c) ⊥ (z) describes a "coherent motion" followed by all particles. In the special case of a single electron j o is a δ-Dirac function. Eq. (2) is certainly valid in the case of a monochromatic beam, assuming homogeneous undulator field in the transverse direction. Eq. (2) is also valid in the case of finite energy spread if the transverse size of the electron beam is larger than the transverse excursion of the electrons during their wiggling motion: the validity of Eq. (2) has an accuracy given by the relative deviation of the particles energy form the average value, δγ/γ -with γ the usual Lorentz factor, and must be small if the electron beam is used for Free-Electron Laser light generation. Following the same notation in [5] we writẽ j z (z, t) = j o r ⊥ − r (c) ⊥ (z) ×         ã 1 z, r ⊥ − r (c) ⊥ (z) exp          iω z 0 dz v z (z ) − iωt          + C.C.          ,(3) where "C.C." indicates the complex conjugate of the first term in parenthesis. Hereã 1 is to be considered as a given complex function describing the evolution of the microbunching, while the longitudinal velocity v z (z) can be recovered from the knowledge of r (c) ⊥ (z) and of the average energy of the beam γ = γ(z). We introduce the possibility of beam deflection angles η x (horizontal) and η y (vertical) with respect to the z axis and we indicate the motion in absence of deflection with the subscript "(nd)". In the case of a short undulator with no focusing elements in between, one simply obtains: v z (z, η) = v z(nd) (z)       1 − η 2 x + η 2 y 2       v ⊥ (z, η) = v ⊥(nd) (z) + v z(nd) (z) η ,(4) and r (c) ⊥ (z, η) = r (c) ⊥(nd) (z) + ηz .(5) The orientation of the microbunching wavefront has an impact on the waỹ a 1 depends on η and can be kept fully general at this stage setting a 1 =ã 1 z, r ⊥ − r (c) ⊥ (z, η) .(6) Still following [5], in the limit for γ 2 1, the total current density can be written as j(z, t, η) = v(z, η) c j o r ⊥ − r (c) ⊥ (z, η) 1 + ã 1 z, r ⊥ − r (c) ⊥ (z, η) × exp iω z 0 dz v z (z , η) − iωt + C.C. ,(7) where c is the speed of light in vacuulm while the charge density is ρ = j z v z j z c ,(8) since we work under the paraxial approximation. We look for solutions of the inhomogeneous wave equation for the electric field E ⊥ in the form E ⊥ = E ⊥ exp [iω(z/c − t)] + C.C.(9) in the case of undulator emission. If the electric field does not vary much over an undulator period, E ⊥ has the physical meaning of a slowly varying envelope. The wave equation in paraxial approximation can then be written (see [5]) as ∇ ⊥ 2 + 2iω c ∂ ∂z E ⊥ = 4π c exp i Φ s − ω z c iω c 2 v ⊥ − ∇ ⊥ j oã1 .(10) where Fig. 2. Geometry of the problem. The angles α control the tilt of the microbunching wavefront with respect to the z direction, while the angles η control the direction of the beam with respect to the z direction. When α = 0 the normal to the microbunching wavefront is along z, while when η = 0 the velocity of the beam is along z. One observes radiation at angles θ with respect to the z axis. Φ s (z, η) = ω z 0 dz v z (z , η) ,(11)v ⊥ (z , η) = − cK γ sin (k w z ) + η x v z x + η y v z y ,(12) and follow the constrained motion r (c) ⊥ (z , η) + l = K γk w (cos (k w z ) − 1) + η x z + l x x + η y z + l y y .(13) Here l x and l y model the electron beam offset, while k w = 2π/λ w , λ w being the undulator period. We define the undulator parameter K as K = λ w eH w 2πm e c 2 ,(14) with (−e) is the (negative) electron charge, m e the electron mass, and H w is the maximum of the magnetic field produced by the undulator on the z axis. Eq. (10) can be solved by means of a proper Green's function choice. Still following [5] we have E ⊥ (z o , r ⊥o ) = − 1 c ∞ −∞ dz 1 z o − z d r ⊥ iω c 2 v ⊥ (z , η) − ∇ ⊥ ×j o r ⊥ − r (c) ⊥ (z , η) ã 2 z , r ⊥ − r (c) ⊥ (z , η) exp iω | r ⊥o − r ⊥ | 2 2c(z o − z ) + i Φ s (z , η) − ω z c ,(15) ∇ ⊥ being the gradient operator with respect to the source point. Moreover, (z o , r ⊥o ) is the observation point. Further integration by parts of the gradient terms gives E ⊥ = − iω c 2 ∞ −∞ dz 1 z o − z d r ⊥ v ⊥ (z , η) c − r ⊥o − r ⊥ z o − z ×j o r ⊥ − r (c) ⊥ (z , η) ã 2 z , r ⊥ − r (c) ⊥ (z , η) exp iΦ T (z , r ⊥ , η) ,(16) where the total phase Φ T is given by Φ T = Φ s − ω z c + ω | r ⊥o − r ⊥ | 2 2c(z o − z ) .(17) We will keep only resonant terms and look near the first harmonic in the far zone. Moreover, we will make use of a new integration variable l = r ⊥ − r (c) ⊥ (z , η). We omit detailed calculations: the interested reader may follow a detailed derivation for the second harmonic in [5]). The overall result is E ⊥ = − KωA JJ 2γc 2 z o exp iωz 0 θ 2 2c d l L w /2 −L w /2 dz × exp − iω c θ · l exp iωz 2c θ − η 2 exp [iz C]ρ (1) (z , l, C) ,(18) where we have defined ρ (1) (z , l, C) = j o l ã 1 z , l ,(19) In Eq. (18) θ is the observation angle, A JJ = J 0 [K 2 /(4 + 2K 2 )] − J 1 [K 2 /(4 + 2K 2 )] and the detuning from resonance is C = ω − ω 1 ω 1 k w .(20) where the resonance frequency ω 1 is ω 1 = 2k w cγ 2 z .(21) If we considerρ (1) as a given function we can allow for any particular presentation of the beam modulation. We now introduce a model forρ (1) : We now consider the case when γ(z) =γ = const, and wheñ ρ (1) (z, l) = j o l a 1 exp i ω 1 c α · l ,(22) with a 1 = const and j o l = I o 2πσ 2 exp       − l 2 x + l 2 y 2σ 2       .(23) Here I o and σ are the bunch current and transverse size respectively. Substitution of Eq. (22) into Eq. (18) and integration yields E ⊥ = − KωA JJ I o L w a 1 2γc 2 z o exp iωz 0 θ 2 2c sinc            L w 2            C + ω θ − η 2 2c                       × exp − σ 2 ω 2 2c 2 θ − α 2 .(24) that is a scalar because in the resonant approximation the field is horizonatlly polarized. The angles α control the tilt of the microbunching wavefront with respect to the z direction, while the angles η control the direction of the beam with respect to the z direction. When α = 0 the normal to the microbunching wavefront is along z, while when η = 0 the velocity of the beam is along z. One observes radiation at angles θ with respect to the z axis. The overall geometry is summarized in Fig. 2. The associated power is given by W = c 4π ∞ −∞ dx o ∞ −∞ dy o |E ⊥ (z o , x o , y o , t)| 2 = c 2π ∞ −∞ dx o ∞ −∞ dy o |Ẽ ⊥ (z o , x o , y o )| 2 = c 2π KωA JJ I o L w a 1 2γc 2 2 × d θ sinc 2            L w 2            C + ω θ − η 2 2c                       exp − σ 2 ω 2 c 2 θ − α 2 (25) where (...) denotes averaging over a cycle of oscillation of the carrier wave. Note that introducing the angle ξ = θ − α between the observation direction and the normal to the microbunching, one can cast the power in the form of a convolution between the single-particle emission and the Fourier transform of the transverse electron beam distribution W ∝ d ξ sinc 2            L w 2            C + ω ξ − ( η − α) 2 2c                       exp − σ 2 ω 2 c 2 ξ 2 .(26) One thus recovers Eq. (4) of reference [1], where the power is shown to be a function of the angle between the electron beam direction and the microbunching wavefront normal, that is η − α according to our notations. For further analysis we introduce normalized units for all angular quantities φ asφ = φ/[c/(ωL w )]φ, with the physical meaning of angles normalized to the diffraction angle, for the distanceẑ o = z/L w , for the detuning from reso-nanceĈ = CL w = 2πN w (ω−ω 1 )/ω, for the electron beam size (basically a Fresnel number)N = ωσ 2 /(L w c), for the electric fieldÊ = E ⊥ KωA JJ I o L w a 1 /(2γc 2 z o ), for the intensityÎ = |Ê| 2 , and for the powerŴ = d θÎ . These definitions giveÊ = exp iθ 2ẑ o 2 sinc           Ĉ 2 + θ − η 2 4            exp −N 2 θ − α 2 .(27)I = sinc 2           Ĉ 2 + θ − η 2 4            exp −N θ − α 2 . (28) andŴ = d ξ sinc 2           Ĉ 2 + ξ − ( η − α) 2 4            exp −Nξ 2 .(29) In order to simplify the study case without losing in generality we assume α = (α x , 0) and η = 0. This can be interpreted as the case depicted in Fig. 3, where a microbunched beam is kicked by an angle α and enters an undulator radiator along the direction of the z axis. Then, looking at θ y = 0 we have the following profile forÎ: I = sinc 2 Ĉ 2 +θ 2 x 4 exp −N(θ x −α x ) 2 .(30) while from Eq. (29) we obtain: W = dξ x dξ y sinc 2       Ĉ 2 +ξ 2 x 4 +ξ 2 y 4        exp −N(ξ x −α x ) 2 exp −Nξ 2 y .(31) Note that while we performed our calculations for the case of a planar undulator, the dimensionless result in Eq. (30) remains valid also for the helical case. Inspection of Eq. (30) shows that the radiation maximum is for θ x =α x and red-shifted ofĈ =α 2 /2 or, in dimensional units, θ x = α x and C = −ωα 2 x /(2c) = −k w α 2 x γ 2 /(1 + K 2 /2). Summing up, for α x 0, the conventional theory predicts emission of radiation as in Fig. 4(top) compared to the case for α x = 0 in Fig. 4(bottom). We then fixĈ =α 2 x /2 and we illustrate further some asymptotic behaviors. First, Eq. (30) and Eq. (31) that becomê I = sinc 2 −α 2 x 4 +θ 2 x 4 exp −N(θ x −α x ) 2 .(32)W = dξ x dξ y sinc 2        −α 2 x 4 +ξ 2 x 4 +ξ 2 y 4        exp −N(ξ x −α x ) 2 exp −Nξ 2 y .(33) We show the behavior ofÎ in Eq. (32) for several values ofN and variouŝ α x in Fig. 6. Whenα x 1 andNα x 1 one finds the well-known limiting relations for α x = 0 (see e.g. [6]) I ≡Î 0 (N,θ x ) = sinc 2 θ 2 x 4 exp −Nθ 2 x .(34)W ≡Ŵ 0 (N) = dξ x dξ y sinc 2       ξ 2 x 4 +ξ 2 y 4        exp −N(ξ 2 x +ξ 2 y ) . = 4π arctan 1 2N + N ln 4N 2 4N 2 + 1(35) This expression is valid for any value ofN. An example is shown in Fig. 6. We now turn to consider special asymptotes whenα x 0 andN is small or large. Case forN 1 Ifα x 1 one gets back the limiting case of Eq. (34) and Eq. (35) describing emission from a single particle moving along the z axis: I = lim N→0Î 0 (N,θ x ) = sinc 2 θ 2 x 4 . (36) W = lim N→0Ŵ 0 (N) = 2π 2(37) The behavior ofÎ andŴ is illustrated in Fig. 7. Ifα x ∼ 1 one has I =Î 1 = sinc 2 −α 2 x 4 +θ 2 4 (38) andŴ =Ŵ 1 = 2π ∞ 0 dξξsinc 2 −α 2 x 4 +ξ 2 4 = 2π 2 1 − 4 πα 2 x + 4 πα 2 x cos α 2 x 2 + 1 π Si α 2 x 2 .(39) with Si the Sin Integral function. This limit corresponds to a red-shifted single-particle case, and is valid for increasing values ofα x , untilNα 2 x 1. The behavior ofÎ andŴ is illustrated in Fig. 9. When α x increases such that Nα 2 x 1 or larger the general Eq. (32) and Eq. (33) must be used. Case forN 1 In the opposite limiting case, Eq. (32) and Eq. (33) always simplifies tô I = exp −N(θ x −α x ) 2 .(40)W = π N .(41) Discussion and Conclusions In the previous Section we reviewed with a critical eye the theory in [1]. As discussed, Eq. (26) is nothing but Eq. (4) in reference [1], and the power is shown to be a function of the angle between the electron beam direction and the microbunching wavefront normal, that is η − α according to our notations. We thus conclude that the results presented in [1] are correctly derived in the framework of the conventional theory of radiation from relativistic charged particles. After deriving Eq. (26) we further analyzed the emission properties of a microbunched electron beam with a wavefront tilt. Our main conclusion is that (see Eq. (30)) the radiation emission from a microbunched electron beam with the microbunching wavefront tilted of an angle α x with respect to the direction of motion is maximum is forθ x =α x and red-shifted ofĈ =α 2 /2 or, in dimensional units, θ x = α x and C = −ωα 2 x /(2c) = −k w α 2 x γ 2 /(1 + K 2 /2). In other words, for α x 0, the conventional theory predicts emission of radiation as in Fig. 4(top) compared to the case for α x = 0 in Fig. 4(bottom). The experiment described in [2,3] shows, at variance, a maximum radiation emission in the direction of motion, that is at θ x = 0 and, still, red-shifted. It should be underlined that the experiment was performed with a helical undulator while our calculations were performed for a planar undulator. However, as discussed above, the dimensionless result in Eq. (30) remains valid also for the helical case. To our understanding, the only way of obtaining maximum radiation emission in the direction of motion would be a readjustment of the microbunching wavefront. However, according to conventional particle tracking, the direction of the microbunching wavefront is not influenced by the kick. Concerning this last point we would like to draw the reader's attention to reference [7], which deals with the issue of separation of circular and linear polarization components from a setup similar to that built at the LCLS (linear undulator followed by a kick and a helical radiator, without inverse tapering). The authors of [7], knew that coherent radiation emission is exponentially suppressed, unless the microbunching is directed along the velocity. Therefore, they proposed a design of an isochronous bending system based on the use of conventional particle tracking and XFEL codes. For the European XFEL it requires about 87 m long of total length and consists of 33 magnets, including 8 dipoles, 9 quadrupoles, and 16 sextupoles. Such a system would allow for a rotation of the microbunching wavefront of an angle equal to the bending angle, thus yielding strong coherent emission at resonance in the direction of motion. In reality, no isochronous bending system was actually needed at the LCLS facility to achieve intense emission of coherent, highly circularly polarized radiation. However, at the LCLS, the radiation observed was red-shifted. The LCLS crew tend to ascribe their observations [2,3] to microbunching wavefront readjusting due to the presence of FEL gain in the final radiator. This explanation is not convincing for us: in our view, the FEL gain in the final radiator should only be accounted for as a spurious effect. At the entrance of the radiator, the initial microbunching -calculated according to the usual particle tracking techniques-, is directed at an angle with respect to the velocity. At this initial position, the microbunching is considerable even if not saturated. The readjusting of the microbunching direction could only happen after smearing of the initial microbunching and development of new microbunching along the velocity direction, but this would require many gain lengths, and is impossile in a short radiator. It is also of fundamental importance to note that a simple "rotation" of the microbunching direction would not explain the observation of red-shifted radiation, in contrast to the observations in [2,3]. We therefore conclude that the observations in [2,3] are currently at odds with the present theoretical understanding of radiation emission from relativistic charged particles and a fundamental explanation of this effect needs to be provided 1 . Fig. 3 . 3Geometry of the problem, simplified case for α = (α x , 0) and η = 0. This can be interpreted as the case where a microbunched beam is kicked by an angle α and enters an undulator radiator along the direction of the z axis.We want to solve Eq. (10) for a planar undulator where electrons have velocities Fig. 4 . 4Maximum emission comparison for microbunched beam entering the undulator after a kick or without a kick, according to conventional theory Fig. 5 . 5From left to right and top to bottom: the behavior ofÎ in Eq. (32) for several values ofN = 10 −4 ,N = 10 −2 ,N = 10 −1 ,N = 1,N = 10,N = 100 and variouŝ α x = 0, 1, 2 (blue circles, orange square, green diamonds). Fig. 6 . 6Left:Î 0 (N,θ x ) forN = 1 as a function ofθ x . Right: The behavior ofŴ 0 as a function ofN. The blue solid line refers to direct computation of Eq. (32) and Eq. (33). The orange circles refer to Eq. (34) and Eq. (35). Fig. 7 . 7Lefta function ofθ x . Right: The behavior ofŴ 0 as a function ofα x forN = 10 −4 . The blue solid line refers to direct computation of Eq. (32) and Eq. (33). The orange circles refer to Eq. (36) and Eq. (37). Fig. 8 . 8Left:Î forN 1 andα x = 1 as a function ofθ x . Right: The behavior ofŴ 0 as a function ofα x ∼ 1 forN = 10 −4 . The blue solid line refers to direct computation of Eq. (32) and Eq. (33). The orange circles refer to Eq. (38) and Eq. (39). Fig. 9 . 9Left:Î forN = 100 andα x = 1 as a function ofθ x . Right: The behavior ofŴ 0 as a function ofN forα x = 1 . The blue solid line refers to direct computation of Eq. (32) and Eq. (33). The orange circles refer to Eq. (40) and Eq. (41). In[8] we proposed an explanation which is in contrast with the mainstream understanding of the coupling between dynamics and electrodynamics, but is nevertheless capable of explaining the effects treated in this paper from a fundamental viewpoint. . T Tanaka, H Kitamura, T Shintake, Nuclear Instrum. and Methods in Phys. Res. A. 528T. Tanaka, H. Kitamura and T. Shintake, Nuclear Instrum. and Methods in Phys. Res. A 528 (2004). Commissioning of the Delta polarizing undulator at LCLS. H.-D Nuhn, Proceedings of the 2015 FEL Conference. the 2015 FEL ConferenceDaejeon, South Korea1H.-D. Nuhn et al., "Commissioning of the Delta polarizing undulator at LCLS", in Proceedings of the 2015 FEL Conference, Daejeon, South Korea, WED01 (2015). . A Lutman, Nat.Photon. 10468472A. Lutman et al., Nat.Photon. 10, 468472 (2016). . E A Schneidmiller, M V Yurkov, Phys. Rev. ST AB. 16110702E. A. Schneidmiller and M.V. Yurkov, Phys. Rev. ST AB 16, 110702 (2013). . Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller, Mikhail Yurkov, Optics Communications. 271207Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller and Mikhail Yurkov, Optics Communications 271, 207 (2007). . E Saldin, E Schneidmiller, M Yurkov, Nuclear Instrum. and Methods in Phys. Res. A. 539499E. Saldin, E. Schneidmiller and M. Yurkov, Nuclear Instrum. and Meth- ods in Phys. Res. A, 539, 499 (2005). . Y Li, Phys. Rev. ST AB. 1380705Y. Li et al., Phys. Rev. ST AB 13, 080705 (2010) Radiation from moving charges. G Geloni, V Kocharyan, E Saldin, 17-047G. Geloni, V. Kocharyan and E. Saldin, "Radiation from moving charges", DESY 17-047, https://arxiv.org/abs/1704.01843
[]
[ "Universal Dynamical Steps in the Exact Time-Dependent Exchange-Correlation Potential", "Universal Dynamical Steps in the Exact Time-Dependent Exchange-Correlation Potential" ]
[ "P Elliott \nDepartment of Physics and Astronomy\nHunter College and the Graduate Center of the City University of New York\n695 Park Avenue10065New YorkNew YorkUSA\n", "J I Fuks \nDpto. Física de Materiales\nCentro de Física de Materiales CSIC\nNano-Bio Spectroscopy group\nUniversidad del País Vasco\nUPV/EHU-MPC and DIPC\nAv. Tolosa 72E-20018San SebastiánSpain\n", "A Rubio \nDpto. Física de Materiales\nCentro de Física de Materiales CSIC\nNano-Bio Spectroscopy group\nUniversidad del País Vasco\nUPV/EHU-MPC and DIPC\nAv. Tolosa 72E-20018San SebastiánSpain\n\nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany\n", "N T Maitra \nDepartment of Physics and Astronomy\nHunter College and the Graduate Center of the City University of New York\n695 Park Avenue10065New YorkNew YorkUSA\n" ]
[ "Department of Physics and Astronomy\nHunter College and the Graduate Center of the City University of New York\n695 Park Avenue10065New YorkNew YorkUSA", "Dpto. Física de Materiales\nCentro de Física de Materiales CSIC\nNano-Bio Spectroscopy group\nUniversidad del País Vasco\nUPV/EHU-MPC and DIPC\nAv. Tolosa 72E-20018San SebastiánSpain", "Dpto. Física de Materiales\nCentro de Física de Materiales CSIC\nNano-Bio Spectroscopy group\nUniversidad del País Vasco\nUPV/EHU-MPC and DIPC\nAv. Tolosa 72E-20018San SebastiánSpain", "Fritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-6D-14195BerlinGermany", "Department of Physics and Astronomy\nHunter College and the Graduate Center of the City University of New York\n695 Park Avenue10065New YorkNew YorkUSA" ]
[]
We show that the exact exchange-correlation potential of time-dependent density-functional theory displays dynamical step structures that have a spatially non-local and time non-local dependence on the density. Using one-dimensional two-electron model systems, we illustrate these steps for a range of non-equilibrium dynamical situations relevant for modeling of photo-chemical/physical processes: field-free evolution of a non-stationary state, resonant local excitation, resonant complete charge-transfer, and evolution under an arbitrary field. Lack of these steps in usual approximations yield inaccurate dynamics, for example predicting faster dynamics and incomplete charge transfer.The vast majority of applications of time-dependent density functional theory (TDDFT) today deal with the calculation of the linear electronic spectra and response of molecules and solids, and provide an unprecedented balance between accuracy and efficiency[1,2]. The theorems of TDDFT also apply to any real-time electron dynamics, not necessarily starting in a ground-state, and possibly subject to strong or weak time-dependent fields. Time-resolved dynamics are particularly important and topical for TDDFT for two reasons. First, there is really no alternative practical method for accurately describing correlated electron dynamics, and second, many fascinating new phenomena and technological applications lie in this realm. These include: attosecond control of electron dynamics [3], photo-induced coupled electron-ion dynamics (for example in describing lightharvesting and artificial photosyntheses), and photochemical/physical processes[4,5]in general. TDDFT in theory yields all observables exactly, solely in terms of the time-dependent density, however in practice, approximations must be made both for the observable as a functional of the density, and for the exchangecorrelation (xc) functional. Thus the question arises as to whether the approximate functionals that have been successful for excitations predict equally well the dynamics in the more general time-dependent context. In particular, the exact xc contribution to the Kohn-Sham (KS) potential at time t functionally depends on the history of the density n(r, t ′ < t), the initial interacting many-body state Ψ 0 , and the choice of the initial KS state Φ 0 : v XC [n; Ψ 0 , Φ 0 ](r, t). However, almost all calculations today use an adiabatic approximation, v A XC = v g.s. XC
10.1103/physrevlett.109.266404
[ "https://arxiv.org/pdf/1211.2012v1.pdf" ]
383,856
1211.2012
3b4054e94ce3b6bf0b497420f351b9b07f2a8dc7
Universal Dynamical Steps in the Exact Time-Dependent Exchange-Correlation Potential 8 Nov 2012 P Elliott Department of Physics and Astronomy Hunter College and the Graduate Center of the City University of New York 695 Park Avenue10065New YorkNew YorkUSA J I Fuks Dpto. Física de Materiales Centro de Física de Materiales CSIC Nano-Bio Spectroscopy group Universidad del País Vasco UPV/EHU-MPC and DIPC Av. Tolosa 72E-20018San SebastiánSpain A Rubio Dpto. Física de Materiales Centro de Física de Materiales CSIC Nano-Bio Spectroscopy group Universidad del País Vasco UPV/EHU-MPC and DIPC Av. Tolosa 72E-20018San SebastiánSpain Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4-6D-14195BerlinGermany N T Maitra Department of Physics and Astronomy Hunter College and the Graduate Center of the City University of New York 695 Park Avenue10065New YorkNew YorkUSA Universal Dynamical Steps in the Exact Time-Dependent Exchange-Correlation Potential 8 Nov 2012(Dated: May 5, 2014) We show that the exact exchange-correlation potential of time-dependent density-functional theory displays dynamical step structures that have a spatially non-local and time non-local dependence on the density. Using one-dimensional two-electron model systems, we illustrate these steps for a range of non-equilibrium dynamical situations relevant for modeling of photo-chemical/physical processes: field-free evolution of a non-stationary state, resonant local excitation, resonant complete charge-transfer, and evolution under an arbitrary field. Lack of these steps in usual approximations yield inaccurate dynamics, for example predicting faster dynamics and incomplete charge transfer.The vast majority of applications of time-dependent density functional theory (TDDFT) today deal with the calculation of the linear electronic spectra and response of molecules and solids, and provide an unprecedented balance between accuracy and efficiency[1,2]. The theorems of TDDFT also apply to any real-time electron dynamics, not necessarily starting in a ground-state, and possibly subject to strong or weak time-dependent fields. Time-resolved dynamics are particularly important and topical for TDDFT for two reasons. First, there is really no alternative practical method for accurately describing correlated electron dynamics, and second, many fascinating new phenomena and technological applications lie in this realm. These include: attosecond control of electron dynamics [3], photo-induced coupled electron-ion dynamics (for example in describing lightharvesting and artificial photosyntheses), and photochemical/physical processes[4,5]in general. TDDFT in theory yields all observables exactly, solely in terms of the time-dependent density, however in practice, approximations must be made both for the observable as a functional of the density, and for the exchangecorrelation (xc) functional. Thus the question arises as to whether the approximate functionals that have been successful for excitations predict equally well the dynamics in the more general time-dependent context. In particular, the exact xc contribution to the Kohn-Sham (KS) potential at time t functionally depends on the history of the density n(r, t ′ < t), the initial interacting many-body state Ψ 0 , and the choice of the initial KS state Φ 0 : v XC [n; Ψ 0 , Φ 0 ](r, t). However, almost all calculations today use an adiabatic approximation, v A XC = v g.s. XC We show that the exact exchange-correlation potential of time-dependent density-functional theory displays dynamical step structures that have a spatially non-local and time non-local dependence on the density. Using one-dimensional two-electron model systems, we illustrate these steps for a range of non-equilibrium dynamical situations relevant for modeling of photo-chemical/physical processes: field-free evolution of a non-stationary state, resonant local excitation, resonant complete charge-transfer, and evolution under an arbitrary field. Lack of these steps in usual approximations yield inaccurate dynamics, for example predicting faster dynamics and incomplete charge transfer. The vast majority of applications of time-dependent density functional theory (TDDFT) today deal with the calculation of the linear electronic spectra and response of molecules and solids, and provide an unprecedented balance between accuracy and efficiency [1,2]. The theorems of TDDFT also apply to any real-time electron dynamics, not necessarily starting in a ground-state, and possibly subject to strong or weak time-dependent fields. Time-resolved dynamics are particularly important and topical for TDDFT for two reasons. First, there is really no alternative practical method for accurately describing correlated electron dynamics, and second, many fascinating new phenomena and technological applications lie in this realm. These include: attosecond control of electron dynamics [3], photo-induced coupled electron-ion dynamics (for example in describing lightharvesting and artificial photosyntheses), and photochemical/physical processes [4,5] in general. TDDFT in theory yields all observables exactly, solely in terms of the time-dependent density, however in practice, approximations must be made both for the observable as a functional of the density, and for the exchangecorrelation (xc) functional. Thus the question arises as to whether the approximate functionals that have been successful for excitations predict equally well the dynamics in the more general time-dependent context. In particular, the exact xc contribution to the Kohn-Sham (KS) potential at time t functionally depends on the history of the density n(r, t ′ < t), the initial interacting many-body state Ψ 0 , and the choice of the initial KS state Φ 0 : v XC [n; Ψ 0 , Φ 0 ](r, t). However, almost all calculations today use an adiabatic approximation, v A XC = v g.s. XC [n(t)], that inputs the instantaneous density into a groundstate xc functional [6,7], completely neglecting both the history-and initial-state-dependence. Further, the ground-state functional v g.s. XC must be approximated; hybrid functionals, that mix in a fraction of exact-exchange to an xc functional that otherwise depends locally or semi-locally in space on the density, are most popular for the spectra of molecules, while the spatially local LDA and semi-local GGA's are most popular for solids (see Ref. [2] and references therein). Although understanding when such approximations are expected to work well or fail has advanced significantly in the linear response regime [2], considerably less is known about the performance of approximate TDDFT for general non-linear time-dependent dynamics [8][9][10]. Part of the reason for this is due to the lack of exact, or highly accurate, results to compare with. Moreover, even in the case where an accurate calculation is available, it is very complicated to extract the exact xc potential (although see Refs. [11,12] for significant progress). Thus, it is critical for the reliability of TDDFT for describing fundamental dynamical processes in the applications mentioned earlier, to first test available xc approximations on systems for which the exact xc potential can be extracted. One such case is that of twoelectrons in a spin-singlet, chosen to start in a KS single-Slater determinant. We show that, in this case, the usual adiabatic and semi-local approximations typically fail to capture a critical and fundamental structure in the exact correlation potential: a time-dependent step, that has a spatially ultranonlocal and non-adiabatic dependence on the density. This feature is missing in all available TDDFT approximations today. Even the exact adiabatic functional misses this dynamical step structure. This leads to erroneous dynamics, e.g. faster time scales are observed in the adiabatic approximations for examples where the step opposes the density evolution. For two-electrons in a spin-singlet we choose, as is usually done, the initial KS state as a doubly-occupied spatial orbital, φ(r, t). Then the exact KS potential for a given density evolution can be found easily [13]. In one-dimension (1D), we have v S (x, t) = − (∂ x n(x, t)) 2 8n 2 (x, t) + ∂ 2 x n(x, t) 4n(x, t) − u 2 (x, t) 2 − x ∂u(x ′ , t) ∂t dx ′ (1) where u(x, t) = j(x, t)/n(x, t) is the local "velocity", n(x, t) is the one-body density, and j(x, t) is the currentdensity. We numerically solve the exact time-dependent Schrödinger equation for the two-electron interacting wavefunction, obtain n(x, t) and j(x, t), and insert them into Eq. 1. The exchange-potential in this case is simply minus half the Hartree potential, v X (x, t) = −v H (x, t)/2, with v H (x, t) = w(x ′ , x)n(x ′ , t)dx ′ , in terms of the twoparticle interaction w(x ′ , x). Therefore, we can directly extract the correlation potential using v C (x, t) = v S (x, t) − v ext (x, t) − v H (x, t)/2 ,(2) where v ext (x, t) is the external potential applied to the system. The two electrons in all our 1D examples interact via the soft-Coulomb interaction [14], w(x ′ , x) = 1/ (x ′ − x) 2 + 1. We use atomic units throughout. We start the analysis with some purely (or largely) two-state systems, in which the exact interacting timedependent wavefunction, |Ψ(t) , can be expanded in a basis consisting of the ground-state, |Ψ g (t) , and the first excited singlet state, |Ψ e (t) : |Ψ(t) = a g (t)|Ψ g + a e (t)|Ψ e (3) where a g (t) and a e (t) are coefficients given by: i∂ t a g (t) a e (t) = E g − d gg E(t) −d eg E(t) −d eg E(t) E e − d ee E(t) a g (t) a e (t) (4) where E g , E e are the energy eigenvalues of the two states, d ab = Ψ * a (x 1 , x 2 )(x 1 + x 2 )Ψ b (x 1 , x 2 )dx 1 dx 2 is the transition dipole moment and E(t) = A cos(ωt) is an applied electric field of strength A and frequency ω. In the weak amplitude limit, with ω ≫ |d eg A| and ω close to the resonant frequency, this reduces to the textbook Rabi problem. When on-resonance, the system oscillates from one state to the other over a Rabi cycle of period T R = 2π/(|d eg |A). By solving Eq. 4 we can easily construct the current and density at any time, their time-derivatives, and hence all pieces entering Eq. 1. In our first example, we consider a "1D He atom", where v ext = −2/ √ x 2 + 1, subject to a weak electric field of strength A = 0.00667au and frequency ω = 0.533au, resonant with the first singlet excitation [15,16]. The most salient feature of the correlation potential is the presence of time-dependent steps, that oscillate on the time-scale of the optical field. These steps arise from the fourth term of Eq. 1: whenever there is a net "acceleration", ∂ t u(x, t), through the system, the spatial-integral is finite, resulting in a potential rising from one end of the system to the other. The correlation potential thus has a spatially ultranonlocal dependence on the density, as it changes far from the system. Further, the time-dependence of the steps is nonadiabatic, meaning that the instantaneous density is not enough to determine the correlation potential functional. This is clear from Figure 2 where the small changes of the density between time steps cannot capture the observed large changes of the step. One is tempted to point to the time-derivatives in the fourth term in Eq. 1 as further evidence for the non-adiabatic dependence, however caution would be needed for such an argument as time non-locality in v S is not the same as time non-locality in v C [2]: the fourth term, for example, may be written as v ext plus other terms, and although v ext has typically strongly non-adiabatic dependence, this is irrelevant because it is never approximated as a functional in practice [2,17], rather it is taken from the problem at hand. Only the xc potential must be approximated, and the functional-dependence of this cannot be deduced directly from Eq. (1). Instead, to unambiguously show the non-adiabatic dependence of the step, we plot the "adiabatically-exact" correlation potential in Fig. 1. This is defined by the exact correlation potential for which both the interacting and KS wavefunctions are ground-state wavefunctions with density equal to the instantaneous one i.e. v adia−ex [18], where v adia ext [n] is the external potential for two interacting electrons whose ground-state has density n, and v adia S [n] is the exact ground-state KS potential for this density (given by the first two terms in Eq.(1). We find v adia ext [n] using similar techniques to Ref. 18 (see also Ref. 19 ). Fig. 1 shows that v adia−ex C [n] indeed does not capture the dynamical step structure. C [n] = v adia S [n]−v adia ext [n]−v H [n]−v X [n] Before turning to our next example, we verify that the two-state approximation for the dynamics is accurate enough for our purposes. Actually one aspect of the potentials we find is indeed an artifact of the twostate approximation: the correlation potential asymptotically has a slope so to exactly cancel the externally applied electric field. This is because the two-state approximation cannot correctly describe polarization arising from occupying many excited states in time. The KS potential obtained from inverting the two-state approximation must therefore be flat asymptotically, as it cannot describe states that are polarized asymptotically. The field is so weak in our case that this effect is hardly noticeable on the scale of Figs. 1 and 2, but to check that our conclusions regarding the dynamics step structure are unaffected by the two-level approximation, we computed the KS potential using the density, current, and their time-derivatives from the numerically exact wavefunction, found using octopus [20,21]. Apart from some extra structure in the tail region (small peaks and steps as we move away from the atom), and the linear field-counteracting term, the correlation potential agrees with that from the two-state model. Dynamical step features have arisen in TDDFT in earlier studies; Refs. [18,22] showed they appear in ionization processes, and linked them to a time-dependent derivative discontinuity, related to fractional charges. In time-resolved transport, step structures have been shown to be essential for describing Coulomb-blockade phenomena [23], again related to the discontinuity. In the response regime, field-counteracting steps develop across long-range molecules [24]. In open-systems-TDDFT, Ref. [25] shows steps arise when using a closed KS system to model an open interacting one. In the linear response regime, the xc kernel for charge-transfer excitations displays frequency-dependent steps [26]. However we argue that the dynamical step structures we are seeing are generic, and moreover, unlike most of the above cases [18,[22][23][24], cannot be captured by an adiabatic approximation. They appear with no need for ionization nor subsystems of fractional charge, nor any applied field (see next example), unlike in Refs. [18,[22][23][24]. In this sense our results are more akin to Ref. [11], which studies the physically very different situation when an electron freely propagates through a wire. The range of the examples we present suggests that such non-adiabatic and non-local steps generically arise when dealing with real electron dynamics. Our second example accentuates the fact that dynamical step structures need neither ionization nor an exter- nal field to appear. We begin in an equal linear superposition of the ground and first-excited state of the 1D He and let it evolve freely, so that |Ψ(t) = e −iEg t |Ψ g + e −iEet |Ψ e / √ 2 .(5) It will oscillate back and forth between the two states with frequency ω 0 = E e − E g . The two-state approximation for this is exact at all times. Again, we see large steps in the correlation potential, as shown in Fig. 3. To support the discussion and provide a microscopic insight behind this phenomenon, we also plot in Fig. 3 the acceleration, a(x, t) = ∂ t u(x, t), and its spatial integral with the external potential subtracted out. The position and magnitude of the step at each time is heavily dependent on this term. Peaks in the acceleration, when integrated, become local steps in the potential and the asymptotic value of the step in v C (x, t) is given by the total step in the spatial integral of a(x, t). Although local step-like features may be cancelled out by the other terms in Eq. (1), the net magnitude of the step is determined from the asymptotic values of this integral. Note that we have the freedom to choose the initial state of the KS system as long as it has the same density and first derivative in time of the exact density [27], and the shape of the exact correlation potential depends on this choice [19]. We used a doubly occupied orbital in the previous example, so that we can calculate the exact v C using the method discussed. A different choice, with a configuration more similar to that of the interacting initial state could well yield a more gentle correlation potential [19], with less dramatic step structure. The generality of the step feature in dynamics is further supported by considering different resonant excita- tions. Consider a double-well as a model of a molecule: v ext (x, t) = − 2 (x + 3.5) 2 + 1.0 − 1 cosh 2 (x − 3.5) −E(t)x(6) with E(t) = 0.006 cos(0.112t). Here the ground-state has two electrons in one well, and a charge-transfer excited state Ψ e , with one electron in each well, at a frequency of 0.112au. We use the ground-state and Ψ e in the twostate model of Eq. (3), and solve for the occupations using Eq. (4); we again checked the two-state result against the exact numerical solution using octopus. The system behaves like the Rabi problem with non-zero dipole moment for the ground-state [28,29]. In Fig. 4 we plot the correlation potential for several times within an optical cycle around T R /8. Again dynamical steps oscillating on the optical frequency time scale emerge. The situation is more complicated as a step related to the delocalization of the density during the charge-transfer process slowly develops (on the time-scale of T R /2) [30]. The dynamical step can then increase, decrease, or even reverse this charge-transfer step. Approximations unable to develop steps lead to incomplete charge-transfer. This, along with other details of time resolved chargetransfer, is investigated in more detail in Ref. [30]. For our purposes, it is sufficient to note that dynamical steps are again present to capture the exact dynamics. Finally, we explicitly demonstrate that the non-local non-adiabatic step feature is a generic aspect of the correlation potential in the following way. We subject the 1D He atom to an electric field that is chosen somewhat arbitrarily: it is relatively strong and linearly switched on over 2 optical cycles, with an off-resonant frequency. In Fig. 5, we show the exact correlation potential at four times, along with the density, and the adiabaticallyexact correlation potential. The time-dependent step in the exact v C is once again evident, and again fails to be captured by the adiabatically-exact approximation. In summary, dynamical steps in the correlation potential are a generic feature of electron dynamics. The step features discussed above arise from part of the fourth term of Eq. (1), which suggests that any time there is a net localized acceleration, there will be a step, and that it will have a very non-local spatial dependence on the density, and is non-adiabatic. This represents a type of time-dependent screening, where the electronelectron interaction hinders electron movement to certain regions. Although two-electron systems were studied here, we expect steps are a more general feature of electron dynamics, as supported by the recent Ref. [11]. The lack of the step in approximations leads to incorrect dynamics. For example, faster time scales in adiabatic approximations were found for the field-free dynamics of a linear superposition state, where the direction of the step tended to oppose the density's motion. The exact dipole and adiabatic exact-exchange (AEXX) dipole for this case are shown in Fig. 6. We computed the dynamics of the the local excitation in Figs 1 and 2 using AEXX, adiabatic LDA, and adiabatic selfinteraction-corrected LDA. In all cases, we found the timescale for the dipole oscillations was faster than in the true case, in spite of providing good linear response spectra [16]. How general this finding is will be investigated in future work. We note that the xc electric field, defined as the gradient of the xc potential, has a more local character than the potential. This suggests that considering functional approximations to this field, or, more generally, to an xc vector potential [11,31], may point to an easier path to develop approximations containing step features. As applications of TDDFT continue to expand, it is crucial to further study what the impact of the missing steps in the approximations are on the predictions of these calculations. When starting in the ground-state, the exact adiabatic potential may follow well the exact dynamics at short times, but as soon as there is an appreciable change in the occupation of an excited state, the exact soution develops the dynamical step, entirely missing in the adiabatic one. This result is general and applies to all available functionals. It raises an important issue when applying TDDFT to fundamental photoinduced processes (e.g. photovoltaics, artificial photo-synthesis, photoactivated chemistry, photophysics, etc): all these involve a significant change of state population. Clearly population of many-body states due to the external field is not a linear process and requires functionals able to cope with the generic features of the dynamical step that we have unveiled in the present work. Figure 1 shows snapshots of the correlation potential over one Rabi cycle, while Fig. 2 shows snapshots over one optical period centered around T R /4. (Note that the system is not exactly periodic over the Rabi period as the two time-scales dictated by the optical frequency and the Rabi frequency are not commensurate). FIG. 1 : 1(color online). Snapshots of the exact correlation potential (solid black), density (blue dotted), and exact-adiabatic (red dashed) over one Rabi cycle; at TR/2 the density of the first excited state is essentially exactly reached. In all graphs in this paper, the correlation potentials are plotted up to an arbitrary irrelevant time-dependent constant. FIG . 2: (color online) Snapshots of the correlation potential (left), and corresponding density (right), at times indicated in the right panel. online) The exact correlation potential (black solid) shown at times 0, 2, 4, and 6 au, for the two-state example (Eq. 5). Also shown are the local acceleration (red dotted),x ∂tu − vext (purple dashed), and the adiabatically-exact correlation potential (blue dash-dot). FIG. 4 : 4(color online) The correlation potential (left) and density (right) shown at snapshots of time TR/8± fractions of the optical period, Topt = 2π/0.112au, that are indicated in the key, for the two-well potential model Eq.(6). FIG. 5 : 5(color online). The exact correlation potentials (solid, black) at times 10,15,20,25au during propagation under E((0.3 t). Also shown are the adiabatically-exact correlation potentials (dashed red), and the density (dotted blue, scale on the right). FIG. 6 : 6(color online). The dipole moments in the field-free propagation of the linear-superposition state (see text), where the TDKS calculation starts in the exact singlet doubly occupied orbital initial state. . E Runge, E K U Gross, Phys. Rev. Lett. 52997E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984). Fundamentals of Time-Dependent Density Functional Theory. M.A.L. Marques, N.T. Maitra, F. Nogueira, E.K.U. Gross, and A. RubioBerlin, HeidelbergSpringer-Verlag837and references thereinFundamentals of Time-Dependent Density Functional The- ory, (Lecture Notes in Physics 837), eds. M.A.L. Marques, N.T. Maitra, F. Nogueira, E.K.U. Gross, and A. Rubio, (Springer-Verlag, Berlin, Heidelberg, 2012); and refer- ences therein. . M F Kling, M J J Vrakking, Annu. Rev. Phys. Chem. 59463M. F. Kling and M. J. J. Vrakking, Annu. Rev. Phys. Chem. 59, 463 (2008). . E Tapavicza, I Tavernelli, U Rothlisberger, C Filippi, M E Casida, J. Chem. Phys. 129124108E. Tapavicza, I. Tavernelli, U. Rothlisberger, C. Filippi, and M. E. Casida, J. Chem. Phys. 129, 124108 (2008). . J Gavnholt, A Rubio, T Olsen, K S Thygesen, J Schiøtz, Phys. Rev. B. 79195405J. Gavnholt, A. Rubio, T. Olsen, K. S. Thygesen, and J. Schiøtz, Phys. Rev. B 79, 195405 (2009). . P Hohenberg, W Kohn, Phys. Rev. 136864P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). . J P Perdew, S Kurth, Lecture Notes in Physics. C. Fiolhas, F. Nogueira, M. Marques620Springer-VerlagJ. P. Perdew and S. Kurth, in Lecture Notes in Physics 620, eds. C. Fiolhas, F. Nogueira, M. Marques (Springer- Verlag, Berlin Heidelberg, 2003). Open Problems and new solutions in time dependent density functional Theory. Chemical Physics. R. Baer, L. Kronik, S. Kuemmel.391Special Issue on "Open Problems and new solutions in time dependent density functional Theory", Chemical Physics 391, eds. R. Baer, L. Kronik, S. Kuemmel. (2011). . N Helbig, J I Fuks, I V Tokatly, H Appel, E K U Gross, A. Rubio Chemical Physics. 3911N. Helbig, J.I. Fuks, I.V. Tokatly, H. Appel, E.K.U. Gross, A. Rubio Chemical Physics 391, 1 (2011). . S Raghunathan, M Nest, J. Chem. Theory and Comput. 72492S. Raghunathan and M. Nest, J. Chem. Theory and Com- put. 7, 2492 (2011). . J D Ramsden, R W Godby, Phys. Rev. Lett. 10936402J. D. Ramsden and R. W. Godby, Phys. Rev. Lett. 109, 036402 (2012). . S E B Nielsen, M Ruggenthaler, R Van Leeuwen, arXiv:1208.0226v2S. E. B. Nielsen, M. Ruggenthaler, R. van Leeuwen, arXiv:1208.0226v2 . P Hessler, N T Maitra, K Burke, J. Chem. Phys. 11772P. Hessler, N. T. Maitra, and K. Burke, J. Chem. Phys. 117, 72 (2002). . J Javanainen, J H Eberly, Q Su, Phys. Rev. A. 383430J. Javanainen, J. H. Eberly, Q. Su, Phys. Rev. A. 38, 3430 (1988). . M Ruggenthaler, D Bauer, Phys. Rev. Lett. 102233001M. Ruggenthaler and D. Bauer, Phys. Rev. Lett. 102, 233001 (2009). . J I Fuks, N Helbig, I V Tokatly, A Rubio, Phys. Rev. B. 8475107J. I. Fuks, N. Helbig, I. V. Tokatly, and A. Rubio, Phys. Rev. B. 84, 075107 (2011). . N T Maitra, R Van Leeuwen, K Burke, Phys. Rev. A. 7856501N. T. Maitra, R. van Leeuwen, and K. Burke, Phys. Rev. A 78, 056501 (2008). . M Thiele, E K U Gross, S Kümmel, Phys. Rev. Lett. 100153004M. Thiele, E. K. U. Gross, and S. Kümmel, Phys. Rev. Lett. 100, 153004 (2008). . P Elliott, N T Maitra, Phys. Rev. A. 8552510P. Elliott and N. T. Maitra, Phys. Rev. A. 85, 052510 (2012). . A Castro, Phys. Stat. Sol. (b). 2432465A. Castro et al., Phys. Stat. Sol. (b) 243, 2465 (2006). . M A L Marques, A Castro, G F Bertsch, A Rubio, Comp. Phys. Comm. 15160M. A. L. Marques, A. Castro, G. F. Bertsch, and A. Rubio, Comp. Phys. Comm. 151, 60 (2003). . M Lein, S Kümmel, Phys. Rev. Lett. 94143003M. Lein and S. Kümmel, Phys. Rev. Lett. 94, 143003 (2005). . S Kurth, G Stefanucci, E Khosravi, C Verdozzi, E K U Gross, Phys. Rev. Lett. 104236801S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, and E. K. U. Gross, Phys. Rev. Lett. 104, 236801 (2010). . S J A Van Gisbergen, P R T Schipper, O V Gritsenko, E J Baerends, J G Snijders, B Champagne, B Kirtman, Phys. Rev. Lett. 83694S.J.A. van Gisbergen, P.R.T. Schipper, O.V. Gritsenko, E.J. Baerends, J.G. Snijders, B. Champagne, B. Kirtman, Phys. Rev. Lett. 83, 694 (1999). . D G Tempel, A Aspuru-Guzik, Chem. Phys. 391130D. G. Tempel and A. Aspuru-Guzik, Chem. Phys. 391, 130 (2011). . M Hellgren, E K U Gross, Phys. Rev. A. 8522514M. Hellgren and E. K. U. Gross, Phys. Rev. A. 85, 022514 (2012). . R Van Leeuwen, Phys. Rev. Lett. 823863R. van Leeuwen, Phys. Rev. Lett. 82, 3863 (1999). . M A Kmetic, W J Meath, Phys. Lett. A. 108340M. A. Kmetic and W. J. Meath, Phys. Lett. A 108, 340 (1985). . A Brown, W J Meath, P Tran, Phys REVIEW A. 64A. Brown, W.J. Meath, P. Tran, Phys REVIEW A, VOL- UME 64,(2000) . J I Fuks, P Elliott, A Rubio, N T Maitra, in prepJ. I. Fuks, P. Elliott, A. Rubio, and N. T. Maitra, in prep. . G Vignale, W Kohn, Phys. Rev. Lett. 772037G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996).
[]